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### Negative Frequencies

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Hi,
Can anyone explain the concept of Negative frequencies clearly. Do
they really exist?

 0
Reply itsbhanu (7) 7/15/2003 6:16:53 AM

"Bhanu Prakash Reddy" <itsbhanu@yahoo.com> wrote in message
> Hi,
> Can anyone explain the concept of Negative frequencies clearly. Do
> they really exist?

If you are expressing elements in frequency space as complex exponentials
(which is the kernel of the Fourier Transform) then to have sin(kt) or
cos(kt), you have to have the sum or difference of complex exponentials at
positive and negative frequencies.

So, the time domain cosine is a real sinusoid at a positive frequency.
But, in Fourier Transform spectral space, a real cosine is made up of
complex exponentials of equal amplitude at positive and negative equal
frequencies.  You can find the identity in a trigonometry book of tables.

So, do they really exist?  It depends on which domain you're looking at
them.  In the time domain of real signals, I'd say no.  In the frequency
domain, yes because of the observation above.

I hope this helps.

Fred


 0

Bhanu Prakash Reddy wrore:
> Hi,
> Can anyone explain the concept of Negative frequencies clearly. Do
> they really exist?

Once upon a time there was a  v_e_r_y  long thread,
(the first hit) and you will become enlightened ;-)
ms


 0

"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message news:<F8NQa.2627$Jk5.1914280@feed2.centurytel.net>... > "Bhanu Prakash Reddy" <itsbhanu@yahoo.com> wrote in message > news:28192a4d.0307142216.4c6ee88@posting.google.com... > > Hi, > > Can anyone explain the concept of Negative frequencies clearly. Do > > they really exist? > > Here's the quick answer: > > If you are expressing elements in frequency space as complex exponentials > (which is the kernel of the Fourier Transform) then to have sin(kt) or > cos(kt), you have to have the sum or difference of complex exponentials at > positive and negative frequencies. > > So, the time domain cosine is a real sinusoid at a positive frequency. > But, in Fourier Transform spectral space, a real cosine is made up of > complex exponentials of equal amplitude at positive and negative equal > frequencies. You can find the identity in a trigonometry book of tables. > > So, do they really exist? It depends on which domain you're looking at > them. In the time domain of real signals, I'd say no. In the frequency > domain, yes because of the observation above. > > I hope this helps. > > Fred As a physical quantity negative frequency is difficult to visualize but is pretty clear through mathematical window. I believe time domain or frequency domain is need-basis and complementary to complete the observation for many applications. I always think how to feel one's image through mirror-does it really exist! Regards, Santosh   0 Reply santosh 7/15/2003 10:09:55 AM itsbhanu@yahoo.com (Bhanu Prakash Reddy) wrote in message news:<28192a4d.0307142216.4c6ee88@posting.google.com>... > Hi, > Can anyone explain the concept of Negative frequencies clearly. Do > they really exist? When we encounter frequency for the first time it is usually expressed as 1/T where T is the time-period of one cycle. Because T is non-negative, our first brush with frequency ingrains in us the notion that it is non-negative. OTOH, when exp(j*w0*t) is explained, we are told w0 is the radian frequency and it can take on both +ve and -ve values. This is so because we can think of exp(j*w0*t) as a phasor that is rotating at a constant angular velocity of w0 radians/sec. Clockwise rotation corresponds to negative (radian) frequency and CCW to its positive counterpart. Therefore, if you unlearn the idea that frequency is 1/T, then accepting negative frequency becomes easier.   0 Reply vanamali 7/15/2003 10:20:06 AM Glen Herrmannsfeldt wrote: > > "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message > news:F8NQa.2627$Jk5.1914280@feed2.centurytel.net...
> >
...
> >
> > So, do [negative frequencies] really exist?  It depends on which
> > domain you're looking at them.  In the time domain of real signals,
> > I'd say no. In the frequency domain, yes because of the observation above.
>
> But if you do a Fourier sine transform and a Fourier cosine transform
>
No amount of math will reach a conclusion here, because the issue is not
math, but philosophy. What is clear is that a Fourier transform with
sines and cosines doesn't use negative frequencies in the analysis.

Calculating with complex exponentials entails using negative
frequencies. That doesn't confirm the existence negative frequencies or
of complex exponentials. It simplifies manipulations while extending the
repertoire of necessary concepts.

There was a time when what we call negative numbers were thought of as
positive numbers written in red. "To subtract a number from a smaller
one, reverse the order of subtraction and write the result in red.
If subtraction, add. If you have no pot of red ink, prepend a dash to
the number."

Rules like that are perfectly consistent. Replacing such a rule with
negative numbers is a great simplification, but that does not in itself
make negative numbers real. There is a marvelous puzzle that is readily
solved by positing negative coconuts*; does that simple solution make
negative coconuts real?

We can readily demonstrate that certain mathematical constructs are
useful, but it is usually fruitless to argue about which are real.

Jerry
_____________________________________
* As part of the elegant solution, a monkey is given a positive coconut
from a pile of four negative coconuts, leaving five negative coconuts.
http://www.psc.edu/~burkardt/puzzles/coconut_puzzle.html version 2.
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

 0

itsbhanu@yahoo.com (Bhanu Prakash Reddy) wrote in message news:<28192a4d.0307142216.4c6ee88@posting.google.com>...
> Hi,
> Can anyone explain the concept of Negative frequencies clearly. Do
> they really exist?

Of coarse they don't. But they are handy.
Does phase (negative or positive) really exist ? It doesn't. But as a
mathematical abstraction it is even more handy than a negative
frequency.

For an explaination - the negative frequency is a case of a phase
going backward.

 0

"Jerry Avins" <jya@ieee.org> wrote in message
news:3F1418BA.822C806@ieee.org...
> >
> No amount of math will reach a conclusion here, because the issue is not
> math, but philosophy. What is clear is that a Fourier transform with
> sines and cosines doesn't use negative frequencies in the analysis.
>

Hello Jerry,
I agree with you here in the sense that if one takes the trivial definition
of frequency as to how often something repeats - then the answer is
certainly a simple positive number. But if we expand our viewpoint to two
dimensional things, a natural extension is to say not only how often
something repeats but, we can now include a direction. While this is a
natural philisophical idea, it is motiviated in some by mathematics. The
primary reason complex numbers (two dimensional) were developed, was that
one dimensional numbers proved inadequate for large classes of problems.

The comment about the sines and cosines being purely real and not requiring
negative frequences (explicitly) gets buried in the old "solutions of
differential equations" idea. The functions: exp(iwt), exp(-iwt), sin(wt),
and cos(wt) are all solutions to the same 2nd order equation. So when you do
an analysis with two of these functions, you really have done it with all of
these functions!

Clay


 0

Bhanu Prakash Reddy wrote:
> Hi,
> Can anyone explain the concept of Negative frequencies clearly. Do
> they really exist?

I do not know why, but S(t)=sin(2*pi*f*t) with
"f" negative makes sense to me.

bye,

--

piergiorgio


 0

"Rune Allnor" <allnor@tele.ntnu.no> wrote in message
> itsbhanu@yahoo.com (Bhanu Prakash Reddy) wrote in message
> > Hi,
> > Can anyone explain the concept of Negative frequencies clearly. Do
> > they really exist?
>
> Negative frequencies must be treated as if they exist. I'll give
> two examples:
>
> - In AM modulated systems the signal is represented as two sidebands
>   "mirrored" around the carrier frequency. The upper side band comes
>   from modulating the baseband signal. The lower side band is due to
>   the negative frequencies. To save bandwidth one sideband
>   can be removed, in which case you have a single-sideband (SSB)
>   modulation scheme.

But consider that the carrier is a sine, and so has both positive and
negative frequencies.  So in addition to the upper and lower sidebands of
the positive frequency carrier there should be upper and lower sidebands of
the negative carrier.

> - When sampling real-valued signals, the negative frequencies are
>   repeated in the band between Fs/2 and Fs (Fs is sampling frequency).
>   The existence of these frequency components are the sole cause of
>   Nyquist's sampling theorem, that restricts the bandwidth of the signal
>   to be sampled to f<Fs/2. If you have a complex-valued signal,
>   you don't need to worry about the Fs/2 limit, only Fs.

The displacement of a violin string, the air pressure in an organ pipe, or
the voltage on a coaxial cable are always real.

You can make mathematical transformations that will convert some of the real
numbers to imaginary numbers, such that Fn/2 complex samples can be used.
Each of those complex numbers should represent two real measurements.

If, for example, you said that your complex valued signal was the real and
imaginary displacement of a violin string, with the imaginary value always
zero, I would say that you could not call that a complex valued signal and
sample at Fn/2.

(If Fs is the sampling frequency then sampling at Fs/2 doesn't make any
sense.  Fn/2 (half the Nyquist frequency) does.)

-- glen


 0

Piergiorgio Sartor wrote:
>
> Bhanu Prakash Reddy wrote:
> > Hi,
> > Can anyone explain the concept of Negative frequencies clearly. Do
> > they really exist?
>
> I do not know why, but S(t)=sin(2*pi*f*t) with
> "f" negative makes sense to me.
>
> bye,
>
> --
>
> piergiorgio

It is exactly the same quantity when "f" is positive and "t" is
negative. How can you tell which is is the real way?

Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

 0

"Jerry Avins" <jya@ieee.org> wrote in message
news:3F1473F2.47CF9A7D@ieee.org...

> It is exactly the same quantity when "f" is positive and "t" is
> negative. How can you tell which is is the real way?

Remember, though, that the universe does not conserve time reversal
symmetry.

Close, but not exactly.

-- glen


 0

Jerry Avins <jya@ieee.org> wrote in message news:<3F141D7A.5E25EF1@ieee.org>...
> Rune Allnor wrote:
> >
>  ...
> >
> > Negative frequencies must be treated as if they exist. I'll give
> > two examples:
>
> Rune,
>
> I appreciate your moderation in writing "as if". It follows that
> negative coconuts must be treated _as_if_ they exist. The example is in
> another message of mine in this thread.
>
> > - In AM modulated systems the signal is represented as two sidebands
> >   "mirrored" around the carrier frequency. The upper side band comes
> >   from modulating the baseband signal. The lower side band is due to
> >   the negative frequencies. To save bandwidth one sideband
> >   can be removed, in which case you have a single-sideband (SSB)
> >   modulation scheme.
>
> Just as the answer to the coconut puzzle can be had without recourse to
> negative coconuts, so can AM sidebands be analyzed without recourse to
> negative frequencies. In both cases, the analyses that forgo the use of
> negative quantities are more awkward.

I am sure you are right. My point was merely that the lower side band
appears because of the negative ferquency components of the baseband
representation are shifted by modulation as well. If you do a spectrum
analysis (positive frequencies only) at baseband and then of the modulated
signal, I have been told[*] that you find that the bandwidth of the
modulated signal is twice the bandwidth of the baseband signal.

The negative frequencies may very well be a mathematical abstraction,
but with a measurable manifestation. The only reason for the observable
doubling of bandwidth is the negative frequency components. Thus, it
is tempting to conclude that negative frequencies exist.

> > - When sampling real-valued signals, the negative frequencies are
> >   repeated in the band between Fs/2 and Fs (Fs is sampling frequency).
> >   The existence of these frequency components are the sole cause of
> >   Nyquist's sampling theorem, that restricts the bandwidth of the signal
> >   to be sampled to f<Fs/2. If you have a complex-valued signal,
> >   you don't need to worry about the Fs/2 limit, only Fs.
>
> Every real-valued sample can be written as a complex number x + j0. What
> does that imply about the becessary sample rate for signals so
> expressed? Anyhow, the representation you describe above is an artifact
> of the elegant computation, but not inevitably necessary. Wiechert and
> Sommerfeld's harmonic analyzer used only positive frequencies.

I suspect I may be too close to a mine field for comfort, but it's
important to distinguish between "representation" and "one and only truth".
A representation is one that takes care of some aspect of the data, while
leaving others out. There are several reasons for using ferquencies, either
in sin(wt) or exp(iwt) in mathematical physics. I don't know Wiechert and
Sommerfeldt's work, so I don't want to comment on that. What I do know
is that the complex numbers and the complex exponentials, with their
negative and positive frequencies, are useful. I don't think it's a
problem with negative frequencies as such, but with the properties of time.
When analysisn wave propagation the poitive and negative wavenumbers (that
play the same role in the spatial Fourier transform as ferquency in the
temporal transform) relate to the direction waves travel.

> Jerry
>
> P.S. Does -sin(at) imply -[sin(at)], sin(-a*t), or sin(a*-t)? Maybe,
> instead of being composed of (relatively) negative frequencies, the
> lower sideband runs in (relatively) negative time. Really! :-)

Again, it's a problem with time, not the frequency concept as such.
What you say appear to make no sense, but only because you explicitly
talk about time. Start out with -sin(kx) and you have outlined the basis
for one of many clues that are used to identify various types of waves
in a seismic data record.

Rune

[*] The phrase "I have been told" is included because I haven'y actually
done that exercise myself, but that's what pop out of the maths.
I do put sufficiently trust in maths and physics to predict that's what's
going to happen.

 0


Glen Herrmannsfeldt wrote:
>
> "Jerry Avins" <jya@ieee.org> wrote in message
> news:3F1473F2.47CF9A7D@ieee.org...
>
> > It is exactly the same quantity when "f" is positive and "t" is
> > negative. How can you tell which is is the real way?
>
> Remember, though, that the universe does not conserve time reversal
> symmetry.
>
> Close, but not exactly.

Really?  What demonstration of that exists?

Bob
--

"Things should be described as simply as possible, but no
simpler."

A. Einstein

 0


Bhanu Prakash Reddy wrote:

> Hi,
> Can anyone explain the concept of Negative frequencies clearly. Do
> they really exist?

No....The Fourier Transform theorem says

F(-t) = F(-w)

so time would have to be reversed. If you think of Fourier series
(ordinary) versus complex Fourier series, the latter has negative
frequencies and so it is a mathematical convenience.

Tom


 0


Peter Brackett wrote:

> Bhanu:
>
> [snip]
> "Bhanu Prakash Reddy" <itsbhanu@yahoo.com> wrote in message
> > Hi,
> > Can anyone explain the concept of Negative frequencies clearly. Do
> > they really exist?
> [snip]
>
> The short answer is YES they really exist!
>
> And... yes they can be found in both digital and analog signal processing
> systems.  In digital signal processing they can be found in both programmed
> "off-line" systems and in real-time on-line bus oriented systems and in
> analog systems they perforce must be real-time.  Systems support positive
> and negative frequencies must simultaneously support both "real" and
> "imaginary" signals together.  Such systems are rare and are not often seen
> in practice and are more often seen in digital signal processing than in
> analog signal processing.  But they are just as real as any other systems.
>
> I have participated in the design and production of both digital signal
> processors and analog signal processing systems which process complex
> signals having both positive and negative frequencies and I can assure you
> that they are quite practical "real" and often quite useful.
>
> To support negative frequencies simultaneously with positive frequencies and
> to be able to distinguish them from each other one must use physical systems
> that support "complex" signals, i.e. systems which support both "real" and
> "imaginary" signals simultaneously.  Such complex signals will have wires or
> buses that are labeled "real" and "imaginary" for the two separate
> components of the complex signals.
>
> In  the case of analog signal processing there will be two wires or two
> printed circuit board traces each carrying one component of the complex
> signal, one wire or trace is labeled "real" the other is labeled
> "imaginary".  If you find this confusing then, as we often did, simply use
> black insulation for the real signal wires and red insulation for the
> imaginary signal wires :-).  [Complex signals are nothing more than two
> different and separate real signals, one labeled "real" the other labeled
> "imaginary" but which must be handled throughout the processing as a pair
> and operated upon using the rules of complex arithmetic/mathematics.  With
> analog complex signal processing this means that one must be careful about
> the close "matching" of components such as resistors and capacitors and Op
> Amps etc. in the two separate real and imaginary signal paths and in the
> paths where they interact such as in complex filters and complex I/Q
> modulators.  But this is all readily accomplished as long as one does not
> insist upon perfect isolation with zero crosstalk between real and imaginary
> parts of signals.  In complex analog systems that I have implemented I was
> able to maintain up to 30 - 40 dB separation between real and imaginary
> parts through quite complicated complex analog filters.  Some of these
> complex analog filters acutally separated positive frequencies from negative
> frequencies.  Neat and useful. ]
>
> In the case of digital signal processing there are two cases to consider,
> i.e. the off-line programmed case where the complex signals can be easily
> handled programmatically [For instance using Fortran's "COMPLEX" data type
> or by user defined data types say in C++, or other object oriented
> languages.] or in on-line real-time situations where the real and imaginary
> parts are actually run over separate busses, etc.  A few years ago we
> developed a complete complex processor [TASP] for the Navy which had two
> floating point busses, the real bus and the imaginary bus, and separate real
> and imaginary datapaths throughout the processor.  This was an extremely
> fast processor used for SONAR signal processing and all operations were on
> complex hardware signals using complex hardware arithmetic units, busses and
> memory.  The memory arrays even had separate real and imaginary main and
> cache memory banks.
>
> Although there are also incremental/decremental postitive and negative
> frequencies [Units of dB/second or Np/second] as well as oscillitory
> frequencies [Units of cycles/second (Hz) or radians/second] consider for the
> moment only oscillotory signals of fixed maximum extents which oscillate
> [trigonometrically] at a constant frequency f [Hz] between two equal but
> opposite signed amplitude values.
>
> For an angular frequency w = 2*pi*f, where w is in radians/second and f is
> in Hz, w can be either a positive or negative quantity and the physical
> reality of this is easily seen when you explicitly write out the real and
> imaginary parts.  For example:
>
> Positive frequency [rotates counter-clockwise]
>
> exp(jw) = exp(j*2*pi*f) = cos(w) + j sin(w)
>
> Real part [bus or wire] carries the signal "cos(w)" and Imaginary part [bus
> or wire] carries the signal "sin(w)".
>
> With sufficiently low frequency signals, say in the single digit Hz range
> then using an oscilloscope with separate x and y axis inputs, which is what
> we often did with our complex analog signal processor, one can display the
> real part on the horizontal axis and the imaginary part on  the vertical
> axis and actually see the dot tracing out a circle in the counter-clockwise
> direction.  [Positive frequency]
>
> Negative frequency [rotates clockwise]:
>
> exp(-jw) = exp(-j*2*pi*f) = cos(w) -j sin(w)
>
> Again using an oscilloscope with separate x and y inputs displaying the real
> and imaginary parts you will see the dot tracing out the circle in the
> opposite direction [clockwise or negative frequency].
>
> We built complex analog filters using Op Amps, resistors and capacitors to
> filter the complex signals including filters with the transition band about
> zero frequency to suppress the negative frequencies and pass the positive
> frequencies.  As I stated with precision components we were able to achieve
> 30 - 40 dB suppression of negative frequencies in our complex analog
> processor.
>
> cfr:
>
> P. O. Brackett and G. R. Lang, "Complex Analogue Filters", Proceedings 1981
> European Conference on Circuit Theory and Design, The Hague, Netherlands,
> ed. by Boite and DeWilde, Delft University Press, Delft, Netherlands, pp.
> 412 - 419, August 1981.
>
> Just because they are not often used, negative frequencies whether analog or
> digital are just as "real" as positive frequencies.  No mystery just plain
> old complex arithmetic mapped into circuit or algorithm implementations.
>
> Best,
>
> --
> Peter
> Consultant
> Indialantic By-the-Sea, FL.

That is interesting but does this not give some causality problems? For a signal
f(t)
its Fourier transform F(f(t)) is F(w) and F(f(-t)) = F(-w) or are you saying

F(-w) = F(f*(t)) ie the Fourier TF of the complex conjugate as you have a
complex time-domain signal.Therefore the answer would be yes if you have complex
signals in the time-domain and no otherwise?That maybe right...

Tom


 0

"Bob Cain" <arcane@arcanemethods.com> wrote in message
news:3F14CCE5.9F4A5295@arcanemethods.com...
>
>
> Glen Herrmannsfeldt wrote:
> >
> > "Jerry Avins" <jya@ieee.org> wrote in message
> > news:3F1473F2.47CF9A7D@ieee.org...
> >
> > > It is exactly the same quantity when "f" is positive and "t" is
> > > negative. How can you tell which is is the real way?
> >
> > Remember, though, that the universe does not conserve time reversal
> > symmetry.
> >
> > Close, but not exactly.
>
> Really?  What demonstration of that exists?

> "Things should be described as simply as possible, but no
> simpler."
>
>                                              A. Einstein

http://www.phys.washington.edu/~fortson/intro.html

http://www.phys.washington.edu/~wcgriff/romalis/EDM/#Exp

-- glen


 0

Jerry Avins wrote:

>>I do not know why, but S(t)=sin(2*pi*f*t) with
>>"f" negative makes sense to me.
....
> It is exactly the same quantity when "f" is positive and "t" is
> negative. How can you tell which is is the real way?

The fact the I wrote S(t) and not S(f)... ;-)

bye,

--
Piergiorgio Sartor


 0

Jerry Avins wrote:

> Rules like that are perfectly consistent. Replacing such a rule with
> negative numbers is a great simplification, but that does not in itself
> make negative numbers real. There is a marvelous puzzle that is readily
> solved by positing negative coconuts*; does that simple solution make
> negative coconuts real?

You can imagine... Richard P. Feynman was introducing
the concept of negative probability... :-)

bye,

--
Piergiorgio Sartor


 0

"Peter Brackett" <ab4bc@ix.netcom.com> wrote in message
news:bf2kjs$o42$1@slb6.atl.mindspring.net...
> snip<

> Positive frequency [rotates counter-clockwise]
>
>
<snip>
> --
> Peter
> Consultant
> Indialantic By-the-Sea, FL.
>
>
Seems to me we could remove one of life's confusions by
changing the direction clock hands rotate. After all, if
increasing time causes counter clockwise rotation.....

Regards
Ian

;-)


 0

Tom:

[snip]
> > Peter
> > Consultant
> > Indialantic By-the-Sea, FL.
>
> That is interesting but does this not give some causality problems? For a
signal
> f(t)
> its Fourier transform F(f(t)) is F(w) and F(f(-t)) = F(-w) or are you
saying
>
> F(-w) = F(f*(t)) ie the Fourier TF of the complex conjugate as you have a
> complex time-domain signal.Therefore the answer would be yes if you have
complex
> signals in the time-domain and no otherwise?That maybe right...
>
> Tom
[snip]

Negative frequency is NOT negative time.

Consider the argument of exp(phi) with phi = wt = 2*pi*f*t.  phi can be
negative if either of w or t are negative, but not  both.  Just because w is
negative does not mean that t is negative and time is running backwards.

Positive and negative frequencies have nothing whatsoever to do with
causality and the direction of time, rather they simply have  to do with
direction of rotation.  Counterclockwise rotation is a positive angular
frequency and clockwise rotation is a negative angular frequency, nothing
more nothing less.  No mystery, no arm waving, just plain direction of
rotation.  When your car is moving forwards its' tires are rotating with a
positive frequency when the car backs up the tires have a negative
frequency.

Just realize that it takes two co-ordinates [hence complex numbers or
complex  signals] to discern the direction of rotation.

--
Peter
Consultant
Indialantic By-the-Sea, FL.


 0

Rune Allnor wrote:
>
>  ...

>  My point was merely that the lower side band
> appears because of the negative ferquency components of the baseband
> representation are shifted by modulation as well. If you do a spectrum
> analysis (positive frequencies only) at baseband and then of the modulated
> signal, I have been told[*] that you find that the bandwidth of the
> modulated signal is twice the bandwidth of the baseband signal.

That's a possible viewpoint, and a productive one. It's not conclusive
because it isn't the only one. AM modulation if a carrier by a single
baseband cosine is defines by the equation

f(t) = cos(w_c*t)*[1 + m*cos(w_m*t),

where f(t) is the modulated waveform, w_c is the carrier frequency, w_m
is the modulating frequency, and m is the modulation percentage.
Trigonometric identities show that f(t) consists of the original carrier
from the 1 in the bracket term, and two additional frequencies,
w_c + w_m and w_c - w_m, each with amplitude m/2. The math in no way
insists that w_c - w_m be construed as  w_c + (-w_m), although that's
not ruled out.
>
> The negative frequencies may very well be a mathematical abstraction,
> but with a measurable manifestation. The only reason for the observable
> doubling of bandwidth is the negative frequency components. Thus, it
> is tempting to conclude that negative frequencies exist.

No. The bandwidth doubling comes about because for every sideband
frequency some amount above the carrier, there is another frequency
equally far below. It is as reasonable to think of the lower sideband as
positive frequencies subtracted from the carrier as it is to think of it
as negative frequencies added to it.

There is no question that the notions of negative frequencies is
consistent, and that it can greatly simplify thinking about difficult
cases. An example is the carrier frequency descending below the highest
modulating frequency.
>
...

> I don't know Wiechert and
> Sommerfeldt's work, so I don't want to comment on that.

Search on "harmonic" in
http://www.amphilsoc.org/library/guides/ahqp/bios.htm

> What I do know
> is that the complex numbers and the complex exponentials, with their
> negative and positive frequencies, are useful. I don't think it's a
> problem with negative frequencies as such, but with the properties of time.
> When analysisn wave propagation the poitive and negative wavenumbers (that
> play the same role in the spatial Fourier transform as ferquency in the
> temporal transform) relate to the direction waves travel.
>
> > Jerry
> >
> > P.S. Does -sin(at) imply -[sin(at)], sin(-a*t), or sin(a*-t)? Maybe,
> > instead of being composed of (relatively) negative frequencies, the
> > lower sideband runs in (relatively) negative time. Really! :-)
>
> Again, it's a problem with time, not the frequency concept as such.
> What you say appear to make no sense, but only because you explicitly
> talk about time. Start out with -sin(kx) and you have outlined the basis
> for one of many clues that are used to identify various types of waves
> in a seismic data record.
>
> Rune
>
> [*] The phrase "I have been told" is included because I haven'y actually
> done that exercise myself, but that's what pop out of the maths.
> I do put sufficiently trust in maths and physics to predict that's what's
> going to happen.

I make two claims: that negative frequencies can be dispensed with if
one works hard enough (not that it is worth doing), and that showing
that negative frequencies simplify analyses or promote our understanding
of them doesn't serve to establish their reality. Is that heresy?

Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

 0

Glen Herrmannsfeldt wrote:
>
> "Jerry Avins" <jya@ieee.org> wrote in message
> news:3F1473F2.47CF9A7D@ieee.org...
>
> > It is exactly the same quantity when "f" is positive and "t" is
> > negative. How can you tell which is is the real way?
>
> Remember, though, that the universe does not conserve time reversal
> symmetry.
>
> Close, but not exactly.
>
> -- glen

But the math knows nothing about that. a - b is indistinguishable from
a + -b even when a and b have dimensions of time.

Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

 0

Peter Brackett wrote:
>
...

> I have participated in the design and production of both digital signal
> processors and analog signal processing systems which process complex
> signals having both positive and negative frequencies and I can assure you
> that they are quite practical "real" and often quite useful.

Practical, yes. Useful, yes. But any calculation can be completed with
extra effort without invoking them. I don't speak to whether they are
real here, only to your claim that they are. It seems to me you have
demonstrated only utility, not reality.
>
...

Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

 0

Rune Allnor wrote:
>
>  ...

>  My point was merely that the lower side band
> appears because of the negative ferquency components of the baseband
> representation are shifted by modulation as well. If you do a spectrum
> analysis (positive frequencies only) at baseband and then of the modulated
> signal, I have been told[*] that you find that the bandwidth of the
> modulated signal is twice the bandwidth of the baseband signal.

That's a possible viewpoint, and a productive one. It's not conclusive
because it isn't the only one. AM modulation of a carrier by a single
baseband cosine is defined by the equation

f(t) = cos(w_c*t)*[1 + m*cos(w_m*t),

where f(t) is the modulated waveform, w_c is the carrier frequency, w_m
is the modulating frequency, and m is the modulation percentage.
Trigonometric identities show that f(t) consists of the original carrier
from the 1 in the bracket term, and two additional frequencies,
w_c + w_m and w_c - w_m, each with amplitude m/2. The math in no way
insists that w_c - w_m be construed as  w_c + -w_m, although that's
not ruled out.
>
> The negative frequencies may very well be a mathematical abstraction,
> but with a measurable manifestation. The only reason for the observable
> doubling of bandwidth is the negative frequency components. Thus, it
> is tempting to conclude that negative frequencies exist.

No. The bandwidth doubling comes about because for every sideband
frequency some amount above the carrier, there is another frequency
equally far below. It is as reasonable to think of the lower sideband as
positive frequencies subtracted from the carrier as it is to think of it
as negative frequencies added to it.

There is no question that the notions of negative frequencies is
consistent, and that it can greatly simplify thinking about difficult
cases. An example is the carrier frequency descending below the highest
modulating frequency.
>
...

> I don't know Wiechert and
> Sommerfeldt's work, so I don't want to comment on that.

Search on "harmonic" in
http://www.amphilsoc.org/library/guides/ahqp/bios.htm

> What I do know
> is that the complex numbers and the complex exponentials, with their
> negative and positive frequencies, are useful. I don't think it's a
> problem with negative frequencies as such, but with the properties of time.
> When analysisn wave propagation the poitive and negative wavenumbers (that
> play the same role in the spatial Fourier transform as ferquency in the
> temporal transform) relate to the direction waves travel.
>
> > Jerry
> >
> > P.S. Does -sin(at) imply -[sin(at)], sin(-a*t), or sin(a*-t)? Maybe,
> > instead of being composed of (relatively) negative frequencies, the
> > lower sideband runs in (relatively) negative time. Really! :-)
>
> Again, it's a problem with time, not the frequency concept as such.
> What you say appear to make no sense, but only because you explicitly
> talk about time. Start out with -sin(kx) and you have outlined the basis
> for one of many clues that are used to identify various types of waves
> in a seismic data record.
>
> Rune
>
> [*] The phrase "I have been told" is included because I haven'y actually
> done that exercise myself, but that's what pop out of the maths.
> I do put sufficiently trust in maths and physics to predict that's what's
> going to happen.

I make two claims: that negative frequencies can be dispensed with if
one works hard enough (not that it is worth doing), and that showing
that negative frequencies simplify analyses or promote our understanding
of them doesn't serve to establish their reality. Is that heresy?

Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

 0

Jerry:

[snip]
> real here, only to your claim that they are. It seems to me you have
> demonstrated only utility, not reality.
> >
>   ...
>
> Jerry
[snip]

Hmmm... well, we shipped several of the systems to customers [DoD, Lincoln
Labs], and they paid $MM's for the kit. They even looked at the rotating frequency displays with smiles on their faces during acceptance testing. Can't get much more real than that! BTW... in that complex analog system the highest frequencies in use were ~ 20 Hz and so you could actually see the positive and negative frequencies rotating in "real time" on the scope displays and watch the negative frequencies being attenuated into oblivion at the output of the single sided bandpass filters as you decreased the frequency down through zero Hz, neat stuff! -- Peter Consultant Indialantic By-the-Sea, FL.   0 Reply Peter 7/17/2003 5:44:55 AM Jerry Avins <jya@ieee.org> wrote in message news:<3F16006C.4E35B67@ieee.org>... .... > I make two claims: that negative frequencies can be dispensed with if > one works hard enough (not that it is worth doing), and that showing > that negative frequencies simplify analyses or promote our understanding > of them doesn't serve to establish their reality. Is that heresy? > > Jerry No. I think it's realistic reasoning. You choose what tools you want to use. Either a "physically correct" view, that requires hard work but does not induce any controversial issues, or one that is easy to work with but comes at the expence of possible confusion over the "negative frequency" concept. It's a fair choice. Rune   0 Reply allnor 7/17/2003 5:48:27 AM > > And that's my$0.02.
>
> --Randy
Thanks for these two golden cents, worth more than $0.02 Robert   0 Reply yeren 7/17/2003 7:03:00 AM "Peter Brackett" <ab4bc@ix.netcom.com> wrote in message news:bf4mj6$q0u$1@slb9.atl.mindspring.net... > Tom: (snip) > Positive and negative frequencies have nothing whatsoever to do with > causality and the direction of time, rather they simply have to do with > direction of rotation. Counterclockwise rotation is a positive angular > frequency and clockwise rotation is a negative angular frequency, nothing > more nothing less. No mystery, no arm waving, just plain direction of > rotation. When your car is moving forwards its' tires are rotating with a > positive frequency when the car backs up the tires have a negative > frequency. T symmetry specifies whether the universe looks the same if you replace t in all equations with -t. It doesn't mean that time goes backwards, or even violates causality. Parity is a little easier to visualize, though still not so obvious. A good description is in: http://216.239.57.104/search?q=cache:jDM3nbEm9yoJ:www.aps.org/apsnews/1201/1 20107.html+parity+conservation+cobalt+beta+decay+&hl=en&ie=UTF-8 Parity is the symmetry that replaces (x,y,z) with (-x,-y,-z) in all equations. How could the universe be different with that simple transformation? What effect would you expect if you changed the frequency (w) in all equations to (-w). If all frequencies were negative in the whole universe? -- glen   0 Reply Glen 7/17/2003 7:38:36 AM Thank you all. It has been a nice discussion and i got an opportunity to learn different things which i havent heard of and different perspectives i havent visualized..... Thats Gr8. Best Regrads, BP$
" Working with DIGITall Passion......"

yates@ieee.org (Randy Yates) wrote in message news:<567ce618.0307161219.51941182@posting.google.com>...
> itsbhanu@yahoo.com (Bhanu Prakash Reddy) wrote in message news:<28192a4d.0307142216.4c6ee88@posting.google.com>...
> > Hi,
> > Can anyone explain the concept of Negative frequencies clearly. Do
> > they really exist?
>
> Hi Bhanu,
>
> You've gotten lots of responses. Allow me to give you my perspective,
>
> To begin with, let me answer your question with a question: do
> integers really exist? Do real numbers really exist? Do complex
> numbers really exist? All of these questions may potentially be
> answered "no" depending on how philosophical you want to get.
>
> However, in studying math I have found a wonderful field which I think
> is rich in meaning. I think it applies here. That field is called "abstract
> algebra" or "modern algebra" - it deals with the concepts of groups,
> rings, and fields. You will get many many hits if you Google for some
> of these keywords.
>
> A "ring" is basically a set along with two operations, addition (+)
> and multiplication (*), that satisfy certain criteria. For example,
> there must be an identity element for addition and a different
> identity element for multiplication; There must be additive inverses;
> The operations must be associative; etc. A field is a ring in which
> the elements have multiplicative inverses as well.
>
> The common number systems you know about are all rings. For example,
> integers, real numbers, and complex numbers are all rings. There are
> many other not-so-obvious examples as well, such as the set of all
> 8x8 matrices under matrix addition and multiplication. Real numbers
> and complex numbers are fields as well (all fields are rings, but
> not all rings are fields).
>
> Now there's one more concept from abstract algebra that is
> germane - that of "isomorphism." If two rings are isomorphic
> to one another, then they are essentially the same mathematical
> beast. If not, then they're not. For example, the integers
> are not isomorphic to the reals, but the complexes are isomorphic
> to the set of all two-by-two invertible matrices of the form
> [a b; -b a].
>
> OK. So the purpose of stating all this is this: I believe that
> the concept of a ring is so important that any system which
> can be classified as a ring is worthy to be considered "real."
> Further, I consider any two rings that are not isomorphic to
> one another distinct. For example, you cannot equate the real
> numbers to the complex numbers because they are not isomorphic.
>
> Now proceeding on this axiom, we can say that the complex numbers
> are "real." Therefore we can also say that the the quantity
> e^{i*2*pi*f} is "real," where f is any element of the real numbers.
> So a negative value for f is legitimate and "real." Also, this
> uniqueness of negative frequencies only comes out when interpreting
> complex numbers and not real numbers since reals are not isomorphic
> to the complex.
>
> Now we get to the bottom line, which Clay Turner already discussed.
> The basic difference between frequency when thinking in terms of
> real numbers versus complex numbers is the concept of dimension. In
> some sense, real numbers are one-dimensional while complex numbers
> are two-dimensional. So, as Clay illustrated, the sign of the
> frequency can be used to indicate the direction in the plane
> (clockwise or counterclockwise) that a rotating vector is traveling.
> So, in this sense, negative frequency is real because it matters
> in the complex numbers and complex numbers are real if we base
> our definition of "real" on rings. Further, this concept of
> negative frequencies being real is not due to real numbers but
> complex numbers since the real numbers are not isomorphic to
> the complex numbers.
>
> And that's my $0.02. > > --Randy   0 Reply itsbhanu 7/17/2003 12:12:05 PM Peter Brackett wrote: > > Hmmm... well, we shipped several of the systems to customers [DoD, Lincoln > Labs], and they paid$MM's for the kit.  They even looked at the rotating
> frequency displays with smiles on their faces during acceptance testing.
>
> Can't get much more real than that!

Ahh... so anything that can be sold (to competent customers) is real.

I guess it's just a matter of time before the FCC auctions off all that
negative spectrum.

--
Jim Thomas            Principal Applications Engineer  Bittware, Inc
jthomas@bittware.com  http://www.bittware.com          (703) 779-7770
Air conditioning may have destroyed the ozone layer - but it's been
worth it!

 0

Hi Peter,

It sounds like you do most of your work in the mass murder business
(sometimes called defence). When I was a mass murderer we handled almost
everything in complex form. After a while you get to see the entire
universe in complex numbers, and forget this seems alien, not just to
the man in the street, but to most engineers, too.

To a lot of people, having double the aliasing span when you complex
sample means nothing more than "well you've got twice as many numbers,
so you have gathered twice as much information. Obviously that can give
you twice the unambiguous bandwidth". In discussions about negative
frequency it is basically this doubling of the unambiguous span in the
complex world we are really talking about.

To a large extent there *is* nothing special in a complex sampled world.
I can complex sample a radio channel x times per second, and get x Hz of
unambiguous bandwidth. If I real sample at 2x, and apply a Hilbert
transform I get to the same position. There is apparently nothing
special about the complex world. If I can't crank an ADC up to the rate
I need, use 2 at half the speed in quadrature. It causes lots of
engineering troubles - phase matching, temperature tracking and so on -
but it does the job.

The thing that makes the complex world feel different to sonar and radar
people is we don't like anti-alias filters :-) Very often we (or is it
they - I'm a reformed mass murderer now) actually embrace aliases, and
try to resolve correct answers amongst a group of possiblities. When you
do that, the complex sampled world looks a little different. Wrapping
off one end of the band pops you back in the other end - the aliases
rotate, rather than fold. When you've worked in that world for a while
you see it as natural, and the world of folding aliases seems like some
kind of botch-up for the cheapo commercial world. :-)

Regards,
Steve

Peter Brackett wrote:
> Hmmm... well, we shipped several of the systems to customers [DoD, Lincoln
> Labs], and they paid $MM's for the kit. They even looked at the rotating > frequency displays with smiles on their faces during acceptance testing. > > Can't get much more real than that! > > BTW... in that complex analog system the highest frequencies in use were ~ > 20 Hz and so you could actually see the positive and negative frequencies > rotating in "real time" on the scope displays and watch the negative > frequencies being attenuated into oblivion at the output of the single sided > bandpass filters as you decreased the frequency down through zero Hz, neat > stuff!   0 Reply Steve 7/17/2003 2:00:25 PM Steve: Well... you too eh? During my [too long] career I only worked on a couple of defense related projects. One was TASP - Tactical Acoustic Signal Processor, in its' time the world's fastest dsp engine. It had 32 full complex floating point hardware ALU's! The TASP machine was completely complex from input to output, or stem to stern as the Navy types say. Real and imaginary memory, real and imaginary busses, full complex arithmetic Multiply-Add ALU's, etc, etc... Negative frequencies were part and parcel of the whole kit! Most of my career though has been involved with commercial products, many generations of analog dsp modems and xdsl transceivers, all of which make heavy use of negative frequencies. The most interesting negative frequency project I worked on was an analog project which included analog filters and complex analog signals. We actually designed and manufactured analog filters to separate negative frequencies from positive frequencies. That's the one we wrote about in the paper I referenced. Doing negative frequency processing in the analog domain makes it all much more real than doing it in the DSP domain! No? Best Regards, -- Peter Consultant Indialantic By-the-Sea, FL "Steve Underwood" <steveu@dis.org> wrote in message news:bf6a60$knm$1@hfc.pacific.net.hk... > Hi Peter, > > It sounds like you do most of your work in the mass murder business > (sometimes called defence). When I was a mass murderer we handled almost > everything in complex form. After a while you get to see the entire > universe in complex numbers, and forget this seems alien, not just to > the man in the street, but to most engineers, too. > > To a lot of people, having double the aliasing span when you complex > sample means nothing more than "well you've got twice as many numbers, > so you have gathered twice as much information. Obviously that can give > you twice the unambiguous bandwidth". In discussions about negative > frequency it is basically this doubling of the unambiguous span in the > complex world we are really talking about. > > To a large extent there *is* nothing special in a complex sampled world. > I can complex sample a radio channel x times per second, and get x Hz of > unambiguous bandwidth. If I real sample at 2x, and apply a Hilbert > transform I get to the same position. There is apparently nothing > special about the complex world. If I can't crank an ADC up to the rate > I need, use 2 at half the speed in quadrature. It causes lots of > engineering troubles - phase matching, temperature tracking and so on - > but it does the job. > > The thing that makes the complex world feel different to sonar and radar > people is we don't like anti-alias filters :-) Very often we (or is it > they - I'm a reformed mass murderer now) actually embrace aliases, and > try to resolve correct answers amongst a group of possiblities. When you > do that, the complex sampled world looks a little different. Wrapping > off one end of the band pops you back in the other end - the aliases > rotate, rather than fold. When you've worked in that world for a while > you see it as natural, and the world of folding aliases seems like some > kind of botch-up for the cheapo commercial world. :-) > > Regards, > Steve > > > > Peter Brackett wrote: > > Hmmm... well, we shipped several of the systems to customers [DoD, Lincoln > > Labs], and they paid$MM's for the kit.  They even looked at the
rotating
> > frequency displays with smiles on their faces during acceptance testing.
> >
> > Can't get much more real than that!
> >
> > BTW... in that complex analog system the highest frequencies in use were
~
> > 20 Hz and so you could actually see the positive and negative
frequencies
> > rotating in "real time" on the scope displays and watch the negative
> > frequencies being attenuated into oblivion at the output of the single
sided
> > bandpass filters as you decreased the frequency down through zero Hz,
neat
> > stuff!
>


 0


Jim Thomas wrote:

>
> Ahh... so anything that can be sold (to competent customers) is real.
>

I was thinking that maybe he sell me one for negative dollars. Can't get
any more real than that!

-jim

-----= Posted via Newsfeeds.Com, Uncensored Usenet News =-----
http://www.newsfeeds.com - The #1 Newsgroup Service in the World!
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 0

Peter Brackett wrote:
>
> Jerry:
>
> [snip]
> > real here, only to your claim that they are. It seems to me you have
> > demonstrated only utility, not reality.
> > >
> >   ...
> >
> > Jerry
> [snip]
>
> Hmmm... well, we shipped several of the systems to customers [DoD, Lincoln
> Labs], and they paid $MM's for the kit. They even looked at the rotating > frequency displays with smiles on their faces during acceptance testing. > > Can't get much more real than that! I once programmed a titration to appear on the monitor of a Radio Shack Color Computer at the request of the chemistry teacher. The teacher could set the pH of the contents of each of the burettes and the sample beaker. The burette pHs appeared in the display, but the student had to measure the pH of the beaker experimentally. The program exactly duplicated titration in the round by using different keys to permit different drip speeds, and by showing a flash of color that varied in intensity and duration when the neutral endpoint is approached. There was a significant difference which I refused to "correct": as a reminder to everyone that a simulation can be made to do anything, my phenolphthalein turned green. Had my alkaline phenolphthalein been red, would the simulation have been real? > > BTW... in that complex analog system the highest frequencies in use were ~ > 20 Hz and so you could actually see the positive and negative frequencies > rotating in "real time" on the scope displays and watch the negative > frequencies being attenuated into oblivion at the output of the single sided > bandpass filters as you decreased the frequency down through zero Hz, neat > stuff! You saw rotating CRT beams. That's interesting, even educational, but what did it prove? Suppose, as a vehicle travels down the road, the wheels turn clockwise. Is the vehicle going forward, or backward? I think it depends on which side of the vehicle the observer stands on. > > -- > Peter > Consultant > Indialantic By-the-Sea, FL. When designing equipment for three-phase power transmission, I deal with positive- and negative-sequence currents. To reverse the direction of a three-phase motor, exchanging a pair of hot wires reverses the sequence of currents, hence the motor direction. That reversed current sequence isn't related to negative-sequence currents at all: those have no influence on the direction of a motor. Can anyone show which way a motor would turn if it were supplied with negative frequencies? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/17/2003 11:15:45 PM Jim Thomas wrote: > > Peter Brackett wrote: > > > > Hmmm... well, we shipped several of the systems to customers [DoD, Lincoln > > Labs], and they paid$MM's for the kit.  They even looked at the rotating
> > frequency displays with smiles on their faces during acceptance testing.
> >
> > Can't get much more real than that!
>
> Ahh... so anything that can be sold (to competent customers) is real.
>
> I guess it's just a matter of time before the FCC auctions off all that
> negative spectrum.

No, they've already sold it since they double-charged on the positive
spectrum...
--
%% Fuquay-Varina, NC            %       'cause no one knows which side
%%% 919-577-9882                %                   the coin will fall."
%%%% <yates@ieee.org>           %  'Big Wheels', *Out of the Blue*, ELO

 0

Randy Yates <yates@ieee.org> wrote in message news:<3F173FAB.49320045@ieee.org>...
> Jim Thomas wrote:
> >
> > Peter Brackett wrote:
> > >
> > > Hmmm... well, we shipped several of the systems to customers [DoD, Lincoln
> > > Labs], and they paid $MM's for the kit. They even looked at the rotating > > > frequency displays with smiles on their faces during acceptance testing. > > > > > > Can't get much more real than that! > > > > Ahh... so anything that can be sold (to competent customers) is real. > > > > I guess it's just a matter of time before the FCC auctions off all that > > negative spectrum. > > No, they've already sold it since they double-charged on the positive > spectrum... In Oslo there is a building, the "Oslo Spektrum", which is some sort of a parallel to the Radio City Music Hall(?) or the Royal Albert Hall. Some time during my college years I visited Oslo with some friends. Some were fellow engineer students, others were not. The non-engineers didn't see any humour whatsoever when somebody all of a sudden pointed out "Look! A 150 meter wide Spektrum!" Yes, I know. It's time for a vacation... Rune   0 Reply allnor 7/18/2003 6:38:12 AM "Ian Buckner" <Ian_Buckner@agilent.com> asserts: >Seems to me we could remove one of life's confusions by >changing the direction clock hands rotate. After all, if >increasing time causes counter clockwise rotation..... No, no, no. We should all contradict Jerry and insist that *only* negative frequencies exist; it is the positive frequencies that do not exist! That way, increasing time causes clockwise rotation which every one knows is correct.... So, why does the FCC allocate positive frequencies? Well, it is kind of hard to "sell" negative frequencies, and, of course, selling positive frequencies is the same as selling negative frequenciess -- it all comes out correctly through the miracle of modern mathematics. It's like the electric power company charging for positive current delivered to the user even though we all know that it is the negatively-charged electrons that are doing all the work! Come to think of it, could someone explain *why* I have to pay for electricity? After all, I return one electron to the power company to replace each and every electron it sends to me. Not the same electron, of course, but another one that is identical in all respects... -- .-. .-. .-. .-. .-. .-. .-. / D \ I / L \ I / P \ / S \ A / R \ W / A \ T / E \ -' -' -' -' -' -'   0 Reply Dilip 7/18/2003 3:23:56 PM On Fri, 18 Jul 2003 20:43:59 +0000, Eric Jacobsen wrote: > Has anybody kept track of how long it's been since this topic last > came up? I just assume it is one of those things that is always going on, much like the Springfield tire fire... -- Matthew Donadio (m.p.donadio@ieee.org)   0 Reply Matthew 7/18/2003 7:50:23 PM On 16 Jul 2003 13:19:14 -0700, yates@ieee.org (Randy Yates) wrote: >itsbhanu@yahoo.com (Bhanu Prakash Reddy) wrote in message news:<28192a4d.0307142216.4c6ee88@posting.google.com>... >> Hi, >> Can anyone explain the concept of Negative frequencies clearly. Do >> they really exist? > >Now we get to the bottom line, which Clay Turner already discussed. >The basic difference between frequency when thinking in terms of >real numbers versus complex numbers is the concept of dimension. In >some sense, real numbers are one-dimensional while complex numbers >are two-dimensional. So, as Clay illustrated, the sign of the >frequency can be used to indicate the direction in the plane >(clockwise or counterclockwise) that a rotating vector is traveling. >So, in this sense, negative frequency is real because it matters >in the complex numbers and complex numbers are real if we base >our definition of "real" on rings. Further, this concept of >negative frequencies being real is not due to real numbers but >complex numbers since the real numbers are not isomorphic to >the complex numbers. > >And that's my$0.02.
>
>--Randy

I think it's even slightly simpler than that.

All you really need for negative frequency to "really exist" (although
we can discuss what that means for a long time, too) is some arbitrary
reference.   e.g., I construct two pinwheels and put them on sticks
and stick them on a fence across which a nice breeze blows.   I've
twisted the petals of the two pinwheels in opposite directions so that
as the wind blows one pinwheel rotates clockwise and the other
counter-clockwise.

I will argue that in order to have negative numbers all one needs is a
reference across which any counting produces the same magnitude but
different signs, where the signs add information to the quantity that
would otherwise create ambiguity.  For negative numbers the reference
is zero which is a physically recognizable reference.

Consider that in many cases the definition of zero can be arbitrarily
set against a practical physical quantity.  An inventory of shelf
stock can be taken during the day to record net product flow of a
particular item.  In order to see how many units of product move in a
particular day the quantity on the shelf is measured at the beginning
of the day and the end, and flow is then counted from zero at the
beginning of the day.   Flow out of the store at the end of the day
can be negative if a shipment arrives that is greater than the
quantity sold that day.  From this standpoint I argue the negative
numbers are "real" in the sense of being useful to count physical
things in this manner.

The same is true for frequency:   If I count revolutions of the
pinwheels over some equal time and get the same number for each, this
means only that the magnitude (quantity) of rotations is the same, but
there is an ambiguity in direction.  I can resolve this ambiguity in
direction by defining an arbitrary reference across which not only the
quantity of rotations is recognizable but the direction as well.  I
can then distinguish the clockwise and counter-clockwise revolutions
with a mathematical sign, consistent mathematically with the negative
number sign.

I can observe the rotations of the pinwheels and clearly see a
difference that is consistently represented with the negative sign
meaningfully attached to the quantity (magnitude) of the rotations.
To me this makes negative frequency as real as negative quantities
(which is also philosophical), and I happily accept negative
quantities as "real".

From the above reasoning I have a hard time seeing why people who
accept negative numbers as being relevant to physical quantities don't
accept negative frequencies as being equally relevant to physical
quantities.  Whether "relevance" leads to "reality" will be
philosophical as well, but I hope I've made the drift reasonably
clear...

I always like these philosophical threads...

Cheers,
Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org

 0

"Eric Jacobsen" <eric.jacobsen@ieee.org> wrote in message
> On 16 Jul 2003 13:19:14 -0700, yates@ieee.org (Randy Yates) wrote:
>
> >itsbhanu@yahoo.com (Bhanu Prakash Reddy) wrote in message
> >> Hi,
> >> Can anyone explain the concept of Negative frequencies clearly. Do
> >> they really exist?
> >
> >Now we get to the bottom line, which Clay Turner already discussed.
> >The basic difference between frequency when thinking in terms of
> >real numbers versus complex numbers is the concept of dimension. In
> >some sense, real numbers are one-dimensional while complex numbers
> >are two-dimensional. So, as Clay illustrated, the sign of the
> >frequency can be used to indicate the direction in the plane
> >(clockwise or counterclockwise) that a rotating vector is traveling.
> >So, in this sense, negative frequency is real because it matters
> >in the complex numbers and complex numbers are real if we base
> >our definition of "real" on rings. Further, this concept of
> >negative frequencies being real is not due to real numbers but
> >complex numbers since the real numbers are not isomorphic to
> >the complex numbers.
> >
> >And that's my $0.02. > > > >--Randy > > I think it's even slightly simpler than that. > > All you really need for negative frequency to "really exist" (although > we can discuss what that means for a long time, too) is some arbitrary > reference. e.g., I construct two pinwheels and put them on sticks > and stick them on a fence across which a nice breeze blows. I've > twisted the petals of the two pinwheels in opposite directions so that > as the wind blows one pinwheel rotates clockwise and the other > counter-clockwise. > > I will argue that in order to have negative numbers all one needs is a > reference across which any counting produces the same magnitude but > different signs, where the signs add information to the quantity that > would otherwise create ambiguity. For negative numbers the reference > is zero which is a physically recognizable reference. > > Consider that in many cases the definition of zero can be arbitrarily > set against a practical physical quantity. An inventory of shelf > stock can be taken during the day to record net product flow of a > particular item. In order to see how many units of product move in a > particular day the quantity on the shelf is measured at the beginning > of the day and the end, and flow is then counted from zero at the > beginning of the day. Flow out of the store at the end of the day > can be negative if a shipment arrives that is greater than the > quantity sold that day. From this standpoint I argue the negative > numbers are "real" in the sense of being useful to count physical > things in this manner. > > The same is true for frequency: If I count revolutions of the > pinwheels over some equal time and get the same number for each, this > means only that the magnitude (quantity) of rotations is the same, but > there is an ambiguity in direction. I can resolve this ambiguity in > direction by defining an arbitrary reference across which not only the > quantity of rotations is recognizable but the direction as well. I > can then distinguish the clockwise and counter-clockwise revolutions > with a mathematical sign, consistent mathematically with the negative > number sign. > > I can observe the rotations of the pinwheels and clearly see a > difference that is consistently represented with the negative sign > meaningfully attached to the quantity (magnitude) of the rotations. > To me this makes negative frequency as real as negative quantities > (which is also philosophical), and I happily accept negative > quantities as "real". > > From the above reasoning I have a hard time seeing why people who > accept negative numbers as being relevant to physical quantities don't > accept negative frequencies as being equally relevant to physical > quantities. Whether "relevance" leads to "reality" will be > philosophical as well, but I hope I've made the drift reasonably > clear... > > I always like these philosophical threads... > Cool..   0 Reply Fred 7/18/2003 8:56:02 PM "Jerry Avins" <jya@ieee.org> wrote in message news:3F172675.5FB4F015@ieee.org... > Fred Marshall wrote: > > > > "Jerry Avins" <jya@ieee.org> wrote in message > > news:3F16006C.4E35B67@ieee.org... > > > Rune Allnor wrote: > > > > > > > > ... > > > > > > > My point was merely that the lower side band > > > > appears because of the negative ferquency components of the baseband > > > > representation are shifted by modulation as well. If you do a spectrum > > > > analysis (positive frequencies only) at baseband and then of the > > modulated > > > > signal, I have been told[*] that you find that the bandwidth of the > > > > modulated signal is twice the bandwidth of the baseband signal. > > > > > > That's a possible viewpoint, and a productive one. It's not conclusive > > > because it isn't the only one. AM modulation of a carrier by a single > > > baseband cosine is defined by the equation > > > > > > f(t) = cos(w_c*t)*[1 + m*cos(w_m*t), > > > > > > where f(t) is the modulated waveform, w_c is the carrier frequency, w_m > > > is the modulating frequency, and m is the modulation percentage. > > > Trigonometric identities show that f(t) consists of the original carrier > > > from the 1 in the bracket term, and two additional frequencies, > > > w_c + w_m and w_c - w_m, each with amplitude m/2. The math in no way > > > insists that w_c - w_m be construed as w_c + -w_m, although that's > > > not ruled out. > > > > Jerry, > > > > The point is addressed by: > > > > f(t) = cos(w_c*t)*[1 + m*cos(w_m*t) > > > > expressed using complex exponentials: > > > > f(t) = [exp(w_c*t)/2 + exp(-w_c*t)/2]*{1 + m*[exp(w_m*t)/2 + > > exp(-w_m*t)/2]}, > > > > Multiplying out: > > > > f(t) = [exp(w_c*t)/2 + exp(-w_c*t)/2] + m*{.... > > ...exp(w_c*t)*exp(w_m*t)/2 + exp(w_c*t)*exp(-w_m*t)/2 > > + exp(-w_c*t)*exp(w_m*t)/2 + exp(-w_c*t)*exp(w_m*t)/2} > > > > Collecting exponents in products of exponentials: > > > > f(t) = [exp(w_c*t)/2 + exp(-w_c*t)/2] + m*{.... > > ...exp[(w_c+w_m)*t]/4*exp[(w_c-w_m)*t]/4*+ exp[(-w_c+w_m)*t]/4+ > > exp[(-w_c-w_m)*t]/4} > > > > Collecting terms at or near positive and negative carrier frequencies: > > > > f(t) = exp(w_c*t)/2 + [exp(w_c+w_m)*t]/4 +[exp(w_c-w_m)*t]/4 > > + exp(-w_c*t)/2 + [exp(-w_c+w_m)*t]/4 +[exp(-w_c-w_m)*t]/4 > > > > So, there is a positive and a negative carrier component and there is a > > positive and negative sideband associated with each of those carrier > > components. > > > > Addition is an operation whether the specific addition ends up in > > subtraction - so that's not the point. For example, if we said exp-(+w_c*t) > > or exp(-w_c*t) it wouldn't matter. The function is the same. I really > > think this isn't about the specific operations being performed, it's about > > the function that results. > > > > Fred > > Fred, > > That's one way to do the math. Here's another that yields the same > result, but invites a different interpretation (it's shorter, too): > > f(t) = cos(w_c*t)*[1 + m*cos(w_m*t) > = cos(w_c*t) + m*cos(w_c*t)*cos(w_m*t) > > Given the trig identity 2cos(a)cos(b) = cos(a-b) + cos(a+b), we get > > f(t) = cos(w_c*t) + .5m[cos((w_c - w_m)t) + cos((w_c + w_m)t)] > > directly. I see no negative frequencies there. Do you? We can both > hypothesize them if we want, but neither of us has to. > > Jerry Jerry, OK - that is reasonable as far as it goes. But, the question was about negative frequencies and where they come from, right? As I tried to say before, which agrees with your notation above, if all you care about is the time domain, then you can use the real number notation. That it can be expressed as the sum of complex numbers isn't an issue if that's all you want to do. However, if you're wanting to express the Fourier Transform / spectrum then you're going to get complex quantities - in general. These complex quantities will show up at positive and negative points on the frequency scale with amplitude and phase or, correspondingly, to real and imaginary parts. Examining the Fourier Transform confirms that this is the case. The integral is taken over all time (from minus infinity to plus infinity) and results in a function of frequency. The Fourier Transform looks like a correlation at one frequency at a time over all frequencies. So, sin(wt) correlates at both +w and -w. That makes intuitive sense, no? "Phase" means there is a relationship to a rotating reference vector in that same complex plane, right? The bottom line is: - we *can* express *real time domain signals* without using negative frequencies or complex quantities. Although, as you know, using complex quantities is pretty useful for doing some of the math. - we *can't* express the *spectra of real time domain signals* without using complex quantities and negative frequencies. I think that's correct. So, one would also conclude from this that "negative" frequency is only necessary when examining the spectrum / transform. That's where it shows up. I don't know how to avoid it either! It appears that using sums of complex exponentials in expressing a real time domain signal is overkill. It appears that using sums of diracs at positive and negative frequencies in order to represent the spectrum of real time domain sinusoids is unavoidable. The "argument" seems to overlook which domain is being discussed. There are negative "frequencies" in the *spectrum* for a real signal made up of only positive frequency terms. This is confirmed by using the complex exponential notation in time (which is overkill perhaps but instructive nonetheless). Take the counter example: Put a Dirac in the spectrum at a positive radian frequency w (only). What is the corresponding time domain function? It is a complex sinusoid something like (coswt +/- jsinwt). However, starting with coswt, you get a double Dirac in frequency - one at positive frequency and one at negative frequency. Same with sinwt (n.b.: not jsinwt). It's the complex spectrum that has terms at those negative ordinates...... for a real function. Just transform the real function you provided above and see what you get - you get six Diracs. Fred   0 Reply Fred 7/18/2003 9:24:01 PM Fred Marshall wrote: > ... > > Jerry, > > OK - that is reasonable as far as it goes. But, the question was about > negative frequencies and where they come from, right? I thought the question was, "Are negative frequencies real?" I objected to the answer that they had to be real because we couldn't calculate properly without them. My position was and it that they can be assigned at least arm-waving significance, they are sometimes a convenience but never a necessity for calculating, and they may optionally be considered real by those so inclined. It seems to me from what you wrote below, we agree on most points. > > As I tried to say before, which agrees with your notation above, if all you > care about is the time domain, then you can use the real number notation. > That it can be expressed as the sum of complex numbers isn't an issue > if that's all you want to do. > > However, if you're wanting to express the Fourier Transform / spectrum then > you're going to get complex quantities - in general. Not in general, but specifically when using complex exponential notation. Fourier transforms can calculated using trigonometry, as Fourier himself did. (There is no mention of vector analysis -- curl, divergence, etc. -- In Maxwell's Treatise on Electricity and Magnetism. He used triple integrals instead. It's usually silly to claim that something has to be real because math can't be accomplished without it. > These complex > quantities will show up at positive and negative points on the frequency > scale with amplitude and phase or, correspondingly, to real and imaginary > parts. > Examining the Fourier Transform confirms that this is the case. The > integral is taken over all time (from minus infinity to plus infinity) and > results in a function of frequency. The Fourier Transform looks like a > correlation at one frequency at a time over all frequencies. So, sin(wt) > correlates at both +w and -w. That makes intuitive sense, no? > > "Phase" means there is a relationship to a rotating reference vector in that > same complex plane, right? > > The bottom line is: > > - we *can* express *real time domain signals* without using negative > frequencies or complex quantities. Although, as you know, using complex > quantities is pretty useful for doing some of the math. > - we *can't* express the *spectra of real time domain signals* without using > complex quantities and negative frequencies. > > I think that's correct. So, one would also conclude from this that > "negative" frequency is only necessary when examining the spectrum / > transform. That's where it shows up. I don't know how to avoid it either! > > It appears that using sums of complex exponentials in expressing a real time > domain signal is overkill. > It appears that using sums of diracs at positive and negative frequencies in > order to represent the spectrum of real time domain sinusoids is > unavoidable. > > The "argument" seems to overlook which domain is being discussed. There are > negative "frequencies" in the *spectrum* for a real signal made up of only > positive frequency terms. This is confirmed by using the complex > exponential notation in time (which is overkill perhaps but instructive > nonetheless). > > Take the counter example: > Put a Dirac in the spectrum at a positive radian frequency w (only). > What is the corresponding time domain function? > It is a complex sinusoid something like (coswt +/- jsinwt). > > However, starting with coswt, you get a double Dirac in frequency - one at > positive frequency and one at negative frequency. Same with sinwt (n.b.: > not jsinwt). > > It's the complex spectrum that has terms at those negative ordinates...... > for a real function. Just transform the real function you provided above > and see what you get - you get six Diracs. > > Fred In the early part of the last century, several people, notably Kelvin, I think, designed machines that read out the first N coefficients of the Fourier series of a curve that the stylus traced. (Gibbs showed that the ringing near sharp transitions which appeared in reconstructed waveforms were not defects of workmanship or design, hence "Gibbs' phenomenon.) No negative frequencies appeared in the spectra those harmonic analyzers produced. Just as in the time domain, negative frequencies arise from the way we chose to calculate. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/18/2003 10:42:34 PM Eric Jacobsen wrote: > ... > > From the above reasoning I have a hard time seeing why people who > accept negative numbers as being relevant to physical quantities don't > accept negative frequencies as being equally relevant to physical > quantities. Whether "relevance" leads to "reality" will be > philosophical as well, but I hope I've made the drift reasonably > clear... > > I always like these philosophical threads... Me too, provided they remain civil. > Relevance is clear. But you can't really illustrate frequency with rotation. A better model is vibration, some sort of pendulum. That would address the actual point, not a related one. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/18/2003 10:50:05 PM Jerry, Fred: [snip] > > As I tried to say before, which agrees with your notation above, if all you > > care about is the time domain, then you can use the real number notation. > > That it can be expressed as the sum of complex numbers isn't an issue > > if that's all you want to do. [snip] You can also implement complex time domain waveforms and numbers in physically real systems. And hence make use of [very real] positive and negative frequencies in such signal processing systems. Most common physical systems don't use complex waveforms but that does not mean they don't or can't exist. There are just not many applications for which there is an advantage to using complex waveforms or signals in the signal processing systems and so most system designers never implement complex systems using complex waveforms and signals. But it's easy to do so... since complex waveforms or signals are simply pairs of [ordered/labeled] real waveforms. The signal processing in complex physical systems simply has to be designed to accomodate the complex [pairs of signals] waveforms and to process them according to the rules of complex arithmetic/mathematics. This style of processing pairs of signals can easily be done in both analog and digital networks. It does require a level of "matching" of components on a similar scale to that needed to implement fully differential systems. For complex analog signal processing systems most folks can accept doing this with active RC analog networks using Op Amps, but fewer beleive that it can also be done in completely passive networks, using transformers, to do the sums, etc. it's just that most "modern" engineers don't understand how to use transformers in analog computations :-). [snip] > Not in general, but specifically when using complex exponential > notation. Fourier transforms can calculated using trigonometry, as > Fourier himself did. (There is no mention of vector analysis -- curl, > divergence, etc. -- In Maxwell's Treatise on Electricity and Magnetism. > He used triple integrals instead. It's usually silly to claim that > something has to be real because math can't be accomplished without it. [snip] Maxwell did indeed use some triple integrals, but his original treatise on electromagnetics was all cast in terms of Hamilton's "quaternions". Quaternions were a popular mathematical technique for handling "complex vectors" at the time. Most folks today could not decipher Maxwell's work since like the dead language of Latin the dead language of quaternions is now known only to a few niche area mathematicians. Maxwell's original formulation of his celebrated equations comprised twenty two different quarternion equations! It was left to Oliver Heaviside to recast Maxwell's equations into the modern four equation vector form as we now know them. Heaviside knew quaternions but he hated them and he applied Williard Gibbs modernization of vector calculus to recast Maxwell's 33 quaternion equations down into 4 very concise vector differential equations. Maxwell was a doctoral level graduate of Cambridge, a Fellow of the Royal Society, and a full professor at Edinburgh University, while Oliver Heaviside was a poor disadvantaged youth from the London slums who quit school at age 16 and never studied mathematics or physics formally but taught himself quaternions, vector mathematics and who invented what we now call the LaPlace Transform techniques for solving differential equations. It was known as the Heaviside Operational Calculus, it always gave the correct answers but the Mathematicians said it was just not right because it had no rigorous proofs. Nevertheless he used it to "fix" the first transatlantic cable systems. Which he did for free by publishing the correct analysis and designs in popular magazines of the day. All this to say that... Heaviside knew well the meaning of negative frequencies, from his writings you can tell that he knew that they were real and he explained them often to doubters. Heaviside was eventually appointed a Fellow of the Royal Society and admired on a par with Lord Kelvin and Maxwell himself, but they all had given him so much grief and criticizm over his unorthodox approaches that he never bothered to go down to the Royal Society HQ to receive his Fellow appointment and certificate! [snip] > > The bottom line is: > > > > - we *can* express *real time domain signals* without using negative > > frequencies or complex quantities. Although, as you know, using complex > > quantities is pretty useful for doing some of the math. > > - we *can't* express the *spectra of real time domain signals* without using > > complex quantities and negative frequencies. [snip] What are you gonna do for "complex time domain signals"? [snip] > In the early part of the last century, several people, notably Kelvin, I > think, designed machines that read out the first N coefficients of the > Fourier series of a curve that the stylus traced. (Gibbs showed that the > ringing near sharp transitions which appeared in reconstructed waveforms > were not defects of workmanship or design, hence "Gibbs' phenomenon.) No > negative frequencies appeared in the spectra those harmonic analyzers > produced. Just as in the time domain, negative frequencies arise from > the way we chose to calculate. [snip] Heaviside was a contemporary of Kelvin's and he clearly presented negative and positive frequencies as counter-rotating pairs of signals and outlined how one would need two harmonic analyzers to discern the negative frequencies from the positive. It's interesting how Heaviside back at the turn of the nineteenth century already knew of, accepted and illustrated the meaning of negative frequencies and today folks still doubt and argue about it! What? -- Peter Consultant Indialantic By-the-Sea, FL.   0 Reply Peter 7/19/2003 5:37:36 AM Dilip: [snip] > > So, why does the FCC allocate positive frequencies? Well, > > it is kind of hard to "sell" negative frequencies, and, of > > course, selling positive frequencies is the same as selling > > negative frequenciess -- it all comes out correctly through > > the miracle of modern mathematics. It's like the electric > > power company charging for positive current delivered to > > the user even though we all know that it is the > > negatively-charged electrons that are doing all the work! [snip] The FCC does allocate negative frequencies along with the positives. They just do it in a very compact notation, they allocate frequencies trigonmetrically and not exponentially, but the two are equivalent... cos(wt) = [exp(jwt) + exp(-jwt)]/2 Euler said it all when he explained to us that... exp(j*pi) + 1 = 0 -- Peter Consultant Indialantic By-the-Sea, FL.   0 Reply Peter 7/19/2003 5:43:22 AM Jerry: [snip] > Relevance is clear. But you can't really illustrate frequency with > rotation. A better model is vibration, some sort of pendulum. That would > address the actual point, not a related one. > > Jerry [snip] Wow! That's a very narrow view of frequency! Why can't you illustrate frequency with rotation? Florida Power and Light uses *rotating* not *vibrating* machines to generate the 60Hz frequency electrical current they send to my home, what could be more natural than rotation? -- Peter Consultant Indialantic By-the-Sea, FL.   0 Reply Peter 7/19/2003 5:48:16 AM  Peter Brackett wrote: > > Euler said it all when he explained to us that... > > exp(j*pi) + 1 = 0 > Sorry, Peter, but you have that totally wrong. It was: exp(i*pi) + 1 = 0 :-) Bob (Forever in awe of a universe that holds that little equation to be true.) -- "Things should be described as simply as possible, but no simpler." A. Einstein   0 Reply Bob 7/19/2003 6:17:51 AM  Eric Jacobsen wrote: > I think it's even slightly simpler than that. > > All you really need for negative frequency to "really exist" (although > we can discuss what that means for a long time, too) is some arbitrary > reference. e.g., I construct two pinwheels and put them on sticks > and stick them on a fence across which a nice breeze blows. I've > twisted the petals of the two pinwheels in opposite directions so that > as the wind blows one pinwheel rotates clockwise and the other > counter-clockwise. > > I will argue that in order to have negative numbers all one needs is a > reference across which any counting produces the same magnitude but > different signs, where the signs add information to the quantity that > would otherwise create ambiguity. For negative numbers the reference > is zero which is a physically recognizable reference. > > Consider that in many cases the definition of zero can be arbitrarily > set against a practical physical quantity. An inventory of shelf > stock can be taken during the day to record net product flow of a > particular item. In order to see how many units of product move in a > particular day the quantity on the shelf is measured at the beginning > of the day and the end, and flow is then counted from zero at the > beginning of the day. Flow out of the store at the end of the day > can be negative if a shipment arrives that is greater than the > quantity sold that day. From this standpoint I argue the negative > numbers are "real" in the sense of being useful to count physical > things in this manner. Good analogy. But, you mustn't say this: a) It is impossible to keep track of goods on the shelf without resorting to negative numbers. b) Some days, I have negative number of goods on the shelf. If you do say things analogous to that, its fair to say that you've lost touch with reality. -jim -----= Posted via Newsfeeds.Com, Uncensored Usenet News =----- http://www.newsfeeds.com - The #1 Newsgroup Service in the World! -----== Over 80,000 Newsgroups - 16 Different Servers! =-----   0 Reply jim 7/19/2003 11:33:56 AM Bob: [snip] > Sorry, Peter, but you have that totally wrong. It was: > > exp(i*pi) + 1 = 0 > > :-) > > > Bob (Forever in awe of a universe that holds that little > equation to be true.) > -- > > "Things should be described as simply as possible, but no > simpler." > > A. Einstein [snip] Oh, "I" see! Yes but I thought the 1 and 0 were booleans and not reals and so I used "J". :-) -- Peter Consultant Indialantic By-the-Sea, FL.   0 Reply Peter 7/19/2003 1:25:32 PM Peter Brackett wrote: > > Dilip: > > [snip] > > > So, why does the FCC allocate positive frequencies? ... > > The FCC does allocate negative frequencies along with the positives. > > They just do it in a very compact notation, they allocate frequencies > trigonmetrically and not exponentially, but the two are equivalent... > > cos(wt) = [exp(jwt) + exp(-jwt)]/2 > ... Peter, What if I claim that w is the frequency, always positive, and that I can form the cosine by using it to replace the underscore in either cos(_t) or [exp(j_t) + exp(-j_t)]/2? It's interesting to note that making w a negative number changes nothing at all. Hmmm... :-) Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/19/2003 1:49:10 PM Peter Brackett wrote: > > Jerry: > > [snip] > > Relevance is clear. But you can't really illustrate frequency with > > rotation. A better model is vibration, some sort of pendulum. That would > > address the actual point, not a related one. > > > > Jerry > [snip] > > Wow! That's a very narrow view of frequency! > > Why can't you illustrate frequency with rotation? > > Florida Power and Light uses *rotating* not *vibrating* machines to > generate > the 60Hz frequency electrical current they send to my home, what could be > more > natural than rotation? > > -- > Peter > Consultant > Indialantic By-the-Sea, FL. Would the bulbs suck light out of your room of the generators rotated the other way? The generators rotate, just like the flywheels on the steam engines I built. The voltage reciprocates, just like their pistons. In a portable gasoline generator, the piston reciprocates, rotating the flywheel, causing the generator to rotate, which makes a voltage that alternates. Rotation only in the middle, not the ends. If your 60 Hz rotates, which way does it go? My slide rule has only one set of trig scales, and I haven't yet missed having a set each for clockwise and counterclockwise. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/19/2003 2:00:40 PM On Fri, 18 Jul 2003 18:50:05 -0400, Jerry Avins <jya@ieee.org> wrote: >Eric Jacobsen wrote: >> > ... >> >> From the above reasoning I have a hard time seeing why people who >> accept negative numbers as being relevant to physical quantities don't >> accept negative frequencies as being equally relevant to physical >> quantities. Whether "relevance" leads to "reality" will be >> philosophical as well, but I hope I've made the drift reasonably >> clear... >> >> I always like these philosophical threads... > >Me too, provided they remain civil. >> >Relevance is clear. But you can't really illustrate frequency with >rotation. A better model is vibration, some sort of pendulum. That would >address the actual point, not a related one. > >Jerry That partly comes from my philosophical view that all waves are complex-valued, so it fits my thinking, hence, it influenced the analogy. I'd also opine that negative frequency doesn't mean much for "purely real" measurements unless there is some arbitrary reference selected for phase measurement. As soon as the phase reference is in place the phase vs time of the sinusoid can be determined and then the sign of the frequency is relevant with respect to that reference. This effectively makes the sinusoid representable as a rotating phasor. >But you can't really illustrate frequency with rotation. Rotation can always be represented in terms of frequency and frequency can usually be represented in terms of rotation, and can always be represented as rotation if there is a phase reference. I'm trying to think of an application where selecting an arbitrary phase reference would get in the way of measuring something useful, but I'm not coming up with any examples. The phase reference for complex analysis is always arbitrary, so I don't think there's a danger in doing the same thing for something where only the "real" portion of a quantity is apparent. For the pendulum or mechanical vibration examples it is necessary (but typical) to define a polarity for the magnitude axis of the motion measurement, but once that's done if a phase reference is also defined then the system can be represented as a rotating phasor. Eric Jacobsen Minister of Algorithms, Intel Corp. My opinions may not be Intel's opinions. http://www.ericjacobsen.org   0 Reply eric 7/19/2003 4:29:56 PM Eric Jacobsen wrote: > ... > > ... As soon as the phase reference is in place the > phase vs time of the sinusoid can be determined and then the sign of > the frequency is relevant with respect to that reference. This > effectively makes the sinusoid representable as a rotating phasor. ... That's the nub of it. The uses we put trigonometry to predisposes us to hang more on our analogies than they can support. Trig functions are one-to-one mappings of one scalar to another. That's not enough scaffolding for a solid link to something that rotates. It would be fatuous to pretend that I don't use understand the 'what ifs' and 'sorta likes', but that doesn't entitle me -- or anyone else -- to say, "Sure they're real, and when you grow up, you'll know that". A minus times a minus makes a plus for the convenience of doing math. The strongest one can say about that is that it's consistent. It's not true that it must be that way, just that it's more convenient. Some brilliant people -- my first wife was one -- are put off math by being told that something is a logical necessity when in fact it isn't. "If I can't see this logical necessity, the subject must be beyond me." That can be crippling. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/19/2003 8:20:12 PM On Sat, 19 Jul 2003 16:20:12 -0400, Jerry Avins <jya@ieee.org> wrote: >Eric Jacobsen wrote: >> > ... >> >> ... As soon as the phase reference is in place the >> phase vs time of the sinusoid can be determined and then the sign of >> the frequency is relevant with respect to that reference. This >> effectively makes the sinusoid representable as a rotating phasor. > > ... > >That's the nub of it. The uses we put trigonometry to predisposes us to >hang more on our analogies than they can support. Trig functions are >one-to-one mappings of one scalar to another. That's not enough >scaffolding for a solid link to something that rotates. It would be >fatuous to pretend that I don't use understand the 'what ifs' and 'sorta >likes', but that doesn't entitle me -- or anyone else -- to say, "Sure >they're real, and when you grow up, you'll know that". A minus times a >minus makes a plus for the convenience of doing math. The strongest one >can say about that is that it's consistent. It's not true that it must >be that way, just that it's more convenient. Some brilliant people -- my >first wife was one -- are put off math by being told that something is a >logical necessity when in fact it isn't. "If I can't see this logical >necessity, the subject must be beyond me." That can be crippling. > >Jerry Very much agreed. I figgered we was on essentially the same page... Eric Jacobsen Minister of Algorithms, Intel Corp. My opinions may not be Intel's opinions. http://www.ericjacobsen.org   0 Reply eric 7/19/2003 10:11:10 PM Jerry Avins wrote: > What if I claim that w is the frequency, always positive, and that I can > form the cosine by using it to replace the underscore in either cos(_t) > or [exp(j_t) + exp(-j_t)]/2? It's interesting to note that making w a > negative number changes nothing at all. Hmmm... :-) Well, also the output of abs(x) does not change if I take abs(-x)... Any correlation with the above problem? bye, -- piergiorgio   0 Reply Piergiorgio 7/19/2003 10:25:14 PM Jerry, Eric: Ponderables... Frequency modulation... and associated operations... Consider the case of frequency modulation of a zero frequency carrier. Is this possible? How? Why? Has it ever been done? What exactly are the bandwidth characteristics of a zero frequency carrier modulated by a negative frequency modulation waveform? What exactly is single sideband FM? Can full fidelity modulation waveforms be recovered from a single sideband FM signal? Filter out a conventional FM signal at high frequency, then mix it down to baseband [zero frequency] with a complex IQ demodulator, what do we have supported on the negative frequency domain? Food for thought! Thoughts, comments? -- Peter Consultant Indialantic By-the-Sea, FL. "Jerry Avins" <jya@ieee.org> wrote in message news:3F19A7FC.1640B45@ieee.org... > Eric Jacobsen wrote: > > > ... > > > > ... As soon as the phase reference is in place the > > phase vs time of the sinusoid can be determined and then the sign of > > the frequency is relevant with respect to that reference. This > > effectively makes the sinusoid representable as a rotating phasor. > > ... > > That's the nub of it. The uses we put trigonometry to predisposes us to > hang more on our analogies than they can support. Trig functions are > one-to-one mappings of one scalar to another. That's not enough > scaffolding for a solid link to something that rotates. It would be > fatuous to pretend that I don't use understand the 'what ifs' and 'sorta > likes', but that doesn't entitle me -- or anyone else -- to say, "Sure > they're real, and when you grow up, you'll know that". A minus times a > minus makes a plus for the convenience of doing math. The strongest one > can say about that is that it's consistent. It's not true that it must > be that way, just that it's more convenient. Some brilliant people -- my > first wife was one -- are put off math by being told that something is a > logical necessity when in fact it isn't. "If I can't see this logical > necessity, the subject must be beyond me." That can be crippling. > > Jerry > -- > Engineering is the art of making what you want from things you can get. > �����������������������������������������������������������������������   0 Reply Peter 7/19/2003 11:46:20 PM Jerry Avins <jya@ieee.org> wrote in message news:<3F19A7FC.1640B45@ieee.org>... > Some brilliant people -- my > first wife was one -- are put off math by being told that something is a > logical necessity when in fact it isn't. "If I can't see this logical > necessity, the subject must be beyond me." That can be crippling. Quite true. I think much of the problem is that mathematicians want to condence their results into an as compact formula or proof as possible. That's OK for *doing* maths, but not necessarily for *teaching* maths or engineering. I do some occational teaching on underwater acoustics, and there are a set of formulas regarding Normal Modes that usually are presented in a condensed form. My impression is that students are a bit concerned when they come to this particular lesson, that they don't really see why those formulas pop out that way. So, after presenting the theory, I make a point of writing out the resulting formula in a "physical" way, so that each physical effect gets "its own" easily understood and identified term, in the formula. After having done that, it's a mere reordering of terms to get to the usual formulas. Not a big deal, but you need to see the detailed explanation to see why things pop out the way they do. Mathematicians like to see their craft as some sort of universial truth. Without going into the philosophical aspects of that claim, Man appears to have done his fair share of errors over the ages. For instance, the Romans, while ruling most of Europe by means of a huge army, building structures that lasted for thousands of years, could not do arithmetic tasks we take for granted. Try, for instance, to do this little computation, Subtract MCMXCIX from MIM and you will see two of the main problems with the roman number system. In arabic numbers the task says (provided I haven't messed up the roman numbers) Subtract 1999 from 1999 Of course, having more than one way of representing the same number is a problem: one can't devlop efficient arithmetics. And the romans did not have a way to treat the number 0 (zero). Only after the crusades did the europeans learn (from the arabs) the decimal positional number system, that included the number 0. Arithmetics became much easier after that. Now, even after the crusades there is still some 500 years to go by before Euler establishes most of what we regards as standard basic mathematical notation in the 18th century. In the mean time maths went through various stages of "alchemy and magic". There is, for instance, an hillareous tale about two guys that attend a public "duel" over maths, in Italy some time around the 15th century. At that time equations like x^3 + ax^2 + bx + c = 0 (I think there may be a restriction that a = 0, but I can't find the tale in my books) were the ultimate mathematical challenge. While solutions were known for particular sets of coefficients, no general way of solving for the roots was known. Furthermore, the cases x^3 = ax^2 + bx + c x^3+ax^2 = bx +c etc were considered as *different* equations. They wanted all coefficents a,b,c to be positive. So here comes this duel, where one guy have challenged some other guy (again, I don't remember the details and have lost the book) to solve some fourty equations of the given type. And to disclose his answers in a public event in a city square somewhere (Firenze?). Not totally dissimilar from the present-day public duels between politicians two days before election day. And the challengee has, incidentially, found the method of solution, though kept it secret. Instead of gloating over the chalengee's incapacity to solve the stated problems, the challeneger suffers the disgrace of not being able to state a problem too difficult for the challengee. And there was a huge fuzz some years later, when an apprentice of the discoverer of the solution discloses the method in a book. Not entirely dissimilar to the present day rages about patents and trade secrets. My point is that maths is hard work. The presentation of maths and mathematical concepts as "obvious" is, at best, uneducated. When you look at the time line of discoveries or the introduction of basic concepts, it goes on for decades, centuries and millennia. It appears that some 90% of all mathematical knowledge has been produced during the last two hundred years or so. Before that, people struggeled for centuries or even millennia just to get a working number system in place. Once that was done, they struggled for a few centuries more to get concepts like "negative numbers" in place. The time line alone should indicate that quite a few of those details we take for granted may not be as obvious as we like to think they are. Rune   0 Reply allnor 7/20/2003 6:13:33 AM Rune Allnor wrote: > ... > > My point is that maths is hard work. The presentation of maths and > mathematical concepts as "obvious" is, at best, uneducated. When you > look at the time line of discoveries or the introduction of basic > concepts, it goes on for decades, centuries and millennia. It appears > that some 90% of all mathematical knowledge has been produced during > the last two hundred years or so. Before that, people struggeled for > centuries or even millennia just to get a working number system in place. > Once that was done, they struggled for a few centuries more to get > concepts like "negative numbers" in place. The time line alone should > indicate that quite a few of those details we take for granted may > not be as obvious as we like to think they are. > > Rune Well said! Even further, many of the concepts that some take as necessary and fundamental -- as Euclid's fifth postulate once was -- are in fact merely postulates of convenience. Labeling them "obvious" or "true" is a disservice to those most inclined to ask "why". Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/20/2003 4:48:38 PM Jerry Avins wrote: > [...] > Trig functions are > one-to-one mappings of one scalar to another. y=cos(a) is not a one-to-one mapping from a to y. a1 != a2 does not imply that cos(a1) != cos(a2). -- % Randy Yates % "...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall." %%%% <yates@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO http://home.earthlink.net/~yatescr   0 Reply Randy 7/20/2003 8:36:15 PM Randy Yates wrote: > > Jerry Avins wrote: > > [...] > > Trig functions are > > one-to-one mappings of one scalar to another. > > y=cos(a) is not a one-to-one mapping from a to y. a1 != a2 does > not imply that cos(a1) != cos(a2). > -- > % Randy Yates % "...the answer lies within your soul > %% Fuquay-Varina, NC % 'cause no one knows which side > %%% 919-577-9882 % the coin will fall." > %%%% <yates@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO > http://home.earthlink.net/~yatescr Maybe I said it wrong. What I mean is that for every a, there is one and only one cos(a), and that both a and cos(a) are scalars. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/20/2003 9:39:50 PM Jerry Avins <jya@ieee.org> wrote in message news:<3F1AC7E6.9A18EFC7@ieee.org>... > many of the concepts that some take as > necessary and fundamental -- as Euclid's fifth postulate once was -- are > in fact merely postulates of convenience. Labeling them "obvious" or > "true" is a disservice to those most inclined to ask "why". > > Jerry Mathematics is supposed to be based on just a few axioms. I have a vague recollection there are some eleven (or thereabouts) such axioms. Does anyone know where to find those axioms? Rune   0 Reply allnor 7/21/2003 4:01:51 AM Rune Allnor wrote: .... > Mathematics is supposed to be based on just a few axioms. I have a vague > recollection there are some eleven (or thereabouts) such axioms. > > Does anyone know where to find those axioms? I think you mean the Zermelo-Fraenkel (ZF) axioms for set theory, which are usually extended by the axiom of choice (C or AC). The combined axioms are known by the name ZFC. There are seven of them: http://www.jboden.demon.co.uk/SetTheory/axiomsZFC.html Note that the notion of a "set" is not strictly defined. It seems to be impossible without ending in paradox hell. There are some other axioms for set theory, but none have been studied as extensively as ZFC. Also note that these axioms have to be interpreted very carefully (read the passage on Russell's paradox in the above webpage). However, this is not the basis for all maths, just set theory (which is admittedly an important chunk). Regards, Andor   0 Reply an2or 7/21/2003 10:11:10 AM Jerry Avins wrote: > Randy Yates wrote: > > > > Jerry Avins wrote: > > > [...] > > > Trig functions are > > > one-to-one mappings of one scalar to another. > > > > y=cos(a) is not a one-to-one mapping from a to y. a1 != a2 does > > not imply that cos(a1) != cos(a2). .... > Maybe I said it wrong. What I mean is that for every a, there is one and > only one cos(a), and that both a and cos(a) are scalars. When you talk about a map, then the mapping function is only half of the definition. You have to include argument and range (are these the proper words in English?) domain in the definition: Thus cos : [0, Pi] -> [-1,1] is bijective whereas cos : RR -> [-1,1] is surjective but not injective and cos : [0, Pi] -> RR is injective but not surjective and finally cos : RR -> RR is neither injective nor surjective (RR is the set of real numbers)   0 Reply an2or 7/21/2003 11:25:45 AM "Bhanu Prakash Reddy" <itsbhanu@yahoo.com> wrote in message news:28192a4d.0307142216.4c6ee88@posting.google.com... > Hi, > Can anyone explain the concept of Negative frequencies clearly. Do > they really exist? About 30 years ago I wondered the same question (do negative frequencies exist?) Here you have one example from acoustics: Let's assume a sound wave propagating from a source at speed c to certain direction. We sample this sound wave by a microphone moving at speed 2c to the same direction. What we get? The exact time domain wave form but the signal represented in time reversed. We can define the instantaneous frequency as the phase change in time unit (f = d phi/dt). In the microphone signal this change is negative when compared with the original signal. The frequency is negative by definition. In real life we can easily listen and analyze sounds "by negative frequencies" playing recorded signals backwards. Although the signal played forwards and backwards have same spectrum they sounds differently (at least when listened by humans) Kari Pesonen   0 Reply Kari 7/21/2003 1:10:31 PM Hello Rune, A while back Kurt Goedel (1931) proved that any Mathematics based on a finite set of axioms is incomplete. This has sometimes been called the uncertainty theorem of math. So there will always be things in math that are unknowable. For more see: http://www.miskatonic.org/godel.html Euclid had 5 postulates. Clay "Rune Allnor" <allnor@tele.ntnu.no> wrote in message news:f56893ae.0307202001.626f6f27@posting.google.com... > Jerry Avins <jya@ieee.org> wrote in message news:<3F1AC7E6.9A18EFC7@ieee.org>... > > many of the concepts that some take as > > necessary and fundamental -- as Euclid's fifth postulate once was -- are > > in fact merely postulates of convenience. Labeling them "obvious" or > > "true" is a disservice to those most inclined to ask "why". > > > > Jerry > > Mathematics is supposed to be based on just a few axioms. I have a vague > recollection there are some eleven (or thereabouts) such axioms. > > Does anyone know where to find those axioms? > > Rune   0 Reply Clay 7/21/2003 2:38:24 PM Vanamali wrote: > > Jerry Avins <jya@ieee.org> wrote in message news:<3F1418BA.822C806@ieee.org>... > > > No amount of math will reach a conclusion here, because the issue is not > > math, but philosophy. What is clear is that a Fourier transform with > > sines and cosines doesn't use negative frequencies in the analysis. > > > > Calculating with complex exponentials entails using negative > > frequencies. That doesn't confirm the existence negative frequencies or > > of complex exponentials. It simplifies manipulations while extending the > > repertoire of necessary concepts. > > I am in Jerry's camp on this issue. Sometimes I get the feeling that > there are people who think that exp(jwt) is more fundamental than sine > and cosine. I don't think this is correct because one cannot define > exp(jwt) without using sine and cosine. If exp(jwt) were a more basic > building block its definition should not depend on sine and cosine. Believe it or not, exp(jwt) can be defined without any trig. It's just a bit hard to explain it that way to people with intuition as limited as mine. x� x� 3� x^4 x^5 Consider the series 1 + �� + �� + �� + �� + �� + ... !1 !2 !3 !4 !5 f(x) + f(-x) For any function f(x), the even part is ������������ and 2 f(x) - f(-x) the odd part is ������������. 2 Of course, the series is exp(x), and the even and odd parts are cosh(x) and sinh(x). When x is imaginary, ordinary trig appears. Simply from summing the terms, it is clear that exp(i�x) = cos(x) + i�sin(x). So which came first? Historically, trigonometry. Logically, the question means nothing to me. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/21/2003 6:46:41 PM Rune Allnor wrote: > > Jerry Avins <jya@ieee.org> wrote in message news:<3F1AC7E6.9A18EFC7@ieee.org>... > > many of the concepts that some take as > > necessary and fundamental -- as Euclid's fifth postulate once was -- are > > in fact merely postulates of convenience. Labeling them "obvious" or > > "true" is a disservice to those most inclined to ask "why". > > > > Jerry > > Mathematics is supposed to be based on just a few axioms. I have a vague > recollection there are some eleven (or thereabouts) such axioms. > > Does anyone know where to find those axioms? > > Rune Google for "Peano's Axioms"; that may be what you want. "There is a number. For every number, there is a successor. ..." Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/21/2003 6:52:10 PM "Vanamali" <vanamali@netzero.net> wrote in message news:8d4ba7e3.0307210251.6ed3268f@posting.google.com... (snip) > I am in Jerry's camp on this issue. Sometimes I get the feeling that > there are people who think that exp(jwt) is more fundamental than sine > and cosine. I don't think this is correct because one cannot define > exp(jwt) without using sine and cosine. If exp(jwt) were a more basic > building block its definition should not depend on sine and cosine. I would say that they are equally fundamental. Either can be derived from the other. For a related question, is linear or circular polarization more fundamental? It happens that there is a simple way to make linear polarizers, but quantum mechanics seems to like circular better. (Photons are spin one with a missing spin 0 state.) Circular polarization is a linear combination of linear polarization (with the right relative phase), and linear polarization is the appropriate linear combination of circular polarization. Are rectangular or polar coordinates more fundamental? -- glen   0 Reply Glen 7/21/2003 6:56:44 PM Kari Pesonen wrote: > > "Bhanu Prakash Reddy" <itsbhanu@yahoo.com> wrote in message news:28192a4d.0307142216.4c6ee88@posting.google.com... > > Hi, > > Can anyone explain the concept of Negative frequencies clearly. Do > > they really exist? > > About 30 years ago I wondered the same question (do negative frequencies > exist?) > Here you have one example from acoustics: > Let's assume a sound wave propagating from a source at speed c to certain > direction. We sample this sound wave by a microphone moving at speed > 2c to the same direction. What we get? The exact time domain wave form > but the signal represented in time reversed. > We can define the instantaneous frequency as the phase change in time > unit (f = d phi/dt). In the microphone signal this change is negative > when compared with the original signal. The frequency is negative by > definition. > > In real life we can easily listen and analyze sounds "by negative frequencies" > playing recorded signals backwards. Although the signal played forwards > and backwards have same spectrum they sounds differently (at least when > listened by humans) > > Kari Pesonen There is linear or rotary motion involved in playing a tape or record backward, so direction reversal makes sense. What about the loudspeaker cone? Does that vibrate backwards? The examples above are a sort of time reversal. Does that really negate the frequencies? If so, how? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/21/2003 7:04:15 PM Vanamali wrote: > I am in Jerry's camp on this issue. Sometimes I get the feeling that > there are people who think that exp(jwt) is more fundamental than sine > and cosine. I don't think this is correct because one cannot define > exp(jwt) without using sine and cosine. If exp(jwt) were a more basic > building block its definition should not depend on sine and cosine. There is also an other argument: it cannot be more basic, since "j" is more... complex... In fact "j" was introduced in order to solve sqrt(-1), so it extends the basic set of R and thus cannot be a building block. bye, -- piergiorgio   0 Reply Piergiorgio 7/21/2003 7:26:58 PM Jerry Avins wrote: > Kari Pesonen wrote: > >>"Bhanu Prakash Reddy" <itsbhanu@yahoo.com> wrote in message news:28192a4d.0307142216.4c6ee88@posting.google.com... >> >>>Hi, >>>Can anyone explain the concept of Negative frequencies clearly. Do >>>they really exist? >> >>About 30 years ago I wondered the same question (do negative frequencies >>exist?) >>Here you have one example from acoustics: >>Let's assume a sound wave propagating from a source at speed c to certain >>direction. We sample this sound wave by a microphone moving at speed >>2c to the same direction. What we get? The exact time domain wave form >>but the signal represented in time reversed. >>We can define the instantaneous frequency as the phase change in time >>unit (f = d phi/dt). In the microphone signal this change is negative >>when compared with the original signal. The frequency is negative by >>definition. >> >>In real life we can easily listen and analyze sounds "by negative frequencies" >>playing recorded signals backwards. Although the signal played forwards >>and backwards have same spectrum they sounds differently (at least when >>listened by humans) >> >>Kari Pesonen > > > There is linear or rotary motion involved in playing a tape or record > backward, so direction reversal makes sense. What about the loudspeaker > cone? Does that vibrate backwards? In active noise control, a speaker can be used to absorb sound energy. > > The examples above are a sort of time reversal. Does that really negate > the frequencies? If so, how? > > Jerry   0 Reply Stan 7/21/2003 7:29:30 PM Rune Allnor wrote: > Mathematics is supposed to be based on just a few axioms. I have a vague > recollection there are some eleven (or thereabouts) such axioms. > > Does anyone know where to find those axioms? What if I say that there are as many axioms as natural numbers? bye, -- piergiorgio   0 Reply Piergiorgio 7/21/2003 7:35:37 PM Jerry Avins wrote: > Maybe I said it wrong. What I mean is that for every a, there is one and > only one cos(a), and that both a and cos(a) are scalars. For each a there is one cos(a), but not only one. Clearly cos(0) = 1 and cos(2pi) = 1, so there are two a (I should I write? as?), but one cos... bye, -- piergiorgio   0 Reply Piergiorgio 7/21/2003 7:38:39 PM "Piergiorgio Sartor" <piergiorgio.sartor@nexgo.REMOVE.THIS.de> wrote in message news:2drtu-f56.ln1@lazy.lzy... > Vanamali wrote: > > > I am in Jerry's camp on this issue. Sometimes I get the feeling that > > there are people who think that exp(jwt) is more fundamental than sine > > and cosine. I don't think this is correct because one cannot define > > exp(jwt) without using sine and cosine. If exp(jwt) were a more basic > > building block its definition should not depend on sine and cosine. > > There is also an other argument: it cannot be > more basic, since "j" is more... complex... > > In fact "j" was introduced in order to solve > sqrt(-1), so it extends the basic set of R > and thus cannot be a building block. But sin(x)=(exp(j x)-exp(-j x))/(2j) which looks more complex than exp(j x) -- glen   0 Reply Glen 7/21/2003 7:58:10 PM Glen Herrmannsfeldt wrote: > But sin(x)=(exp(j x)-exp(-j x))/(2j) which looks more complex than exp(j x) Let's say you're not allowed to use "j"... :-) bye, -- piergiorgio   0 Reply Piergiorgio 7/21/2003 8:24:04 PM On Sun, 20 Jul 2003 22:01:51 -0700, Rune Allnor wrote: > Mathematics is supposed to be based on just a few axioms. I have a vague > recollection there are some eleven (or thereabouts) such axioms. > > Does anyone know where to find those axioms? This can probably explain things better than I can... http://planetmath.org/encyclopedia/Axiom.html -- Matthew Donadio (m.p.donadio@ieee.org)   0 Reply Matthew 7/21/2003 8:58:52 PM "Piergiorgio Sartor" <piergiorgio.sartor@nexgo.REMOVE.THIS.de> wrote in message news:4outu-676.ln1@lazy.lzy... > Glen Herrmannsfeldt wrote: > > > But sin(x)=(exp(j x)-exp(-j x))/(2j) which looks more complex than exp(j x) > > Let's say you're not allowed to use "j"... :-) OK, use i then. -- glen   0 Reply Glen 7/21/2003 11:08:42 PM Piergiorgio Sartor wrote: > > Jerry Avins wrote: > > > Maybe I said it wrong. What I mean is that for every a, there is one and > > only one cos(a), and that both a and cos(a) are scalars. > > For each a there is one cos(a), but not only one. > > Clearly cos(0) = 1 and cos(2pi) = 1, so there are > two a (I should I write? as?), but one cos... > > bye, > > -- > > piergiorgio The point please? I wrote -- anyway, intended to write, that for any a, there is one and only one cos(a). Do you contradict that? Jerry P.S. "Contradict" means "speak against". When writing, should we use "contrascribe"? -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/21/2003 11:18:39 PM Vanamali wrote: > > Jerry Avins <jya@ieee.org> wrote in message news:<3F1418BA.822C806@ieee.org>... > > > No amount of math will reach a conclusion here, because the issue is not > > math, but philosophy. What is clear is that a Fourier transform with > > sines and cosines doesn't use negative frequencies in the analysis. > > > > Calculating with complex exponentials entails using negative > > frequencies. That doesn't confirm the existence negative frequencies or > > of complex exponentials. It simplifies manipulations while extending the > > repertoire of necessary concepts. > > I am in Jerry's camp on this issue. Sometimes I get the feeling that > there are people who think that exp(jwt) is more fundamental than sine > and cosine. I don't think this is correct because one cannot define > exp(jwt) without using sine and cosine. If exp(jwt) were a more basic > building block its definition should not depend on sine and cosine. I don't think what I'm about to say is concretely related to what you're getting at here, but it is prompted by your ideas, Vanamali. I get the impression that Jerry (and others?) think that there is nothing really gained in using the complex over the reals. I vehemently disagree. Jerry's argument has been, over and over, that whatever you can do with the complex, you can do with the reals - it just may take a few more operations. It's precisely those "few more operations" that make the complex (along with the operations of addition and multiplication) a significantly different mathematical beast. -- % Randy Yates % "...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall." %%%% <yates@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO http://home.earthlink.net/~yatescr   0 Reply Randy 7/22/2003 12:52:19 AM Randy Yates wrote: > ... > > I get the impression that Jerry (and others?) think that there is nothing > really gained in using the complex over the reals. I vehemently disagree. Jerry's > argument has been, over and over, that whatever you can do with the complex, > you can do with the reals - it just may take a few more operations. It's > precisely those "few more operations" that make the complex (along with > the operations of addition and multiplication) a significantly different > mathematical beast. ... Randy, Your impression is unfounded. That whatever you can do with the complex, you can do with the reals refutes the argument that that it must be fundamental because it's necessary. (Anyhow, I thought this discussion was primarily about negative frequencies, even though I did drag complex frequencies into it at one point.) Calculating with only real numbers is almost always harder and more error prone that using complex numbers, and not merely by a few more operations. This discussion (at least my part in it) has been about what is necessary and what is "real". Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/22/2003 1:39:54 AM Jerry Avins wrote: > > Randy Yates wrote: > > > ... > > > > I get the impression that Jerry (and others?) think that there is nothing > > really gained in using the complex over the reals. I vehemently disagree. Jerry's > > argument has been, over and over, that whatever you can do with the complex, > > you can do with the reals - it just may take a few more operations. It's > > precisely those "few more operations" that make the complex (along with > > the operations of addition and multiplication) a significantly different > > mathematical beast. > > ... > > Randy, > > Your impression is unfounded. That whatever you can do with the complex, > you can do with the reals refutes the argument that that it must be > fundamental because it's necessary. My parser is having a hard time extracting the meaning from this sentence. > (Anyhow, I thought this discussion > was primarily about negative frequencies, even though I did drag complex > frequencies into it at one point.) Calculating with only real numbers is > almost always harder and more error prone that using complex numbers, > and not merely by a few more operations. But I'm not talking about differences in "difficulty" and whatnot. They are different things, fundamentally. That is why, e.g., the solution to a polynomial with real coefficients is, in general, not real but rather complex - because there are things you can do with this arithmetic system that you cannot do with the reals. -- % Randy Yates % "...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall." %%%% <yates@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO http://home.earthlink.net/~yatescr   0 Reply Randy 7/22/2003 2:07:48 AM Randy Yates wrote: > > Jerry Avins wrote: > > > > Randy Yates wrote: > > > > > ... > > > > > > I get the impression that Jerry (and others?) think that there is nothing > > > really gained in using the complex over the reals. I vehemently disagree. Jerry's > > > argument has been, over and over, that whatever you can do with the complex, > > > you can do with the reals - it just may take a few more operations. It's > > > precisely those "few more operations" that make the complex (along with > > > the operations of addition and multiplication) a significantly different > > > mathematical beast. > > > > ... > > > > Randy, > > > > Your impression is unfounded. That whatever you can do with the complex, > > you can do with the reals refutes the argument that that it must be > > fundamental because it's necessary. [That is extra! ^^^^] > > My parser is having a hard time extracting the meaning from this sentence. The statement, "whatever you can do with the complex, you can do with the reals" refutes ... I'm sorry for being opaque. "I know what I mean, why don't you?" :-) > > (Anyhow, I thought this discussion > > was primarily about negative frequencies, even though I did drag complex > > frequencies into it at one point.) Calculating with only real numbers is > > almost always harder and more error prone that using complex numbers, > > and not merely by a few more operations. > > But I'm not talking about differences in "difficulty" and whatnot. They > are different things, fundamentally. That is why, e.g., the solution to > a polynomial with real coefficients is, in general, not real but rather > complex - because there are things you can do with this arithmetic system > that you cannot do with the reals. True, but Fourier transforms are not among those things. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/22/2003 4:03:34 AM Randy Yates wrote: > > Jerry Avins wrote: > > > > Randy Yates wrote: > > > > > ... > > > > > > I get the impression that Jerry (and others?) think that there is nothing > > > really gained in using the complex over the reals. I vehemently disagree. Jerry's > > > argument has been, over and over, that whatever you can do with the complex, > > > you can do with the reals - it just may take a few more operations. It's > > > precisely those "few more operations" that make the complex (along with > > > the operations of addition and multiplication) a significantly different > > > mathematical beast. > > > > ... > > > > Randy, > > > > Your impression is unfounded. That whatever you can do with the complex, > > you can do with the reals refutes the argument that that it must be > > fundamental because it's necessary. [That is extra! ^^^^] > > My parser is having a hard time extracting the meaning from this sentence. The statement, "whatever you can do with the complex, you can do with the reals" refutes the argument that that it must be fundamental because it's necessary. The scope of "anything" there is limited to Fourier transforms and the like; complex exponential replacements for trigonometric functions. I'm sorry for being opaque. "I know what I mean, why don't you?" :-) > > (Anyhow, I thought this discussion > > was primarily about negative frequencies, even though I did drag complex > > frequencies into it at one point.) Calculating with only real numbers is > > almost always harder and more error prone that using complex numbers, > > and not merely by a few more operations. > > But I'm not talking about differences in "difficulty" and whatnot. They > are different things, fundamentally. That is why, e.g., the solution to > a polynomial with real coefficients is, in general, not real but rather > complex - because there are things you can do with this arithmetic system > that you cannot do with the reals. True, but Fourier transforms are not among those things. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/22/2003 4:08:12 AM Piergiorgio Sartor <piergiorgio.sartor@nexgo.REMOVE.THIS.de> wrote in message news:<9trtu-f56.ln1@lazy.lzy>... > Rune Allnor wrote: > > Mathematics is supposed to be based on just a few axioms. I have a vague > recollection there are some eleven (or thereabouts) such axioms. > > Does anyone know where to find those axioms? > That is totally wrong..Every theory in mathematics begins from some basic axioms,but no,there are not only eleven axioms for the whole mathematical structure. Of course if you want to find the axioms for a specific theory in mathematics,that would be easier.. i.e for the Euclidean Geometry,or the Riemann Geometry etc etc..   0 Reply xest 7/22/2003 6:58:26 AM "Randy Yates" <yates@ieee.org> wrote in message news:3F1C8ADD.4796E506@ieee.org... (snip) > I get the impression that Jerry (and others?) think that there is nothing > really gained in using the complex over the reals. I vehemently disagree. Jerry's > argument has been, over and over, that whatever you can do with the complex, > you can do with the reals - it just may take a few more operations. It's > precisely those "few more operations" that make the complex (along with > the operations of addition and multiplication) a significantly different > mathematical beast. There are a number of cases where the math is significantly simpler when expressed in complex notation. Cauchy's theorem and the Kramers-Kronig relations are ones I happen to think of first. It seems obvious to me, though, that if you replace each complex number with an ordered pair of real numbers, and supply the appropriate operations to those ordered pairs such that you get the same results, that you have shown that you can do everything with (ordered pairs of) real numbers. It might look ugly, but it should work. Rulers only measure real lengths, clocks measure real time, voltmeters measure real volts. But some physical quantities can be considered complex with useful results. Dielectric constant and index of refraction, where the imaginary part is related to absorption. (Consider the imaginary part of inductance and capacitance as the resistance and conductance, respectively.) But again, it is just convenience. We can always write them separately. -- glen   0 Reply Glen 7/22/2003 7:15:37 AM Xefteris Stefanos wrote: > That is totally wrong..Every theory in mathematics begins from some > basic axioms,but no,there are not only eleven axioms for the whole > mathematical structure. But the natural numbers themself are axioms... :-) bye, -- Piergiorgio Sartor   0 Reply Piergiorgio 7/22/2003 7:21:55 AM Piergiorgio Sartor wrote: > [...] > Usually the form "one and only one" means that the > expression is invertible (one by one), which, of course > it is not true for the cos() operation. I've never heard "one and only one" in a mathematical context, so I don't know how it would be defined, but if you are using this synonymously with "one-to-one" then this is incorrect. A mapping is invertible if and only if it is both one-to-one and onto (also known and "injective" and "surjective," respectively). If a mapping b:S-->T is "invertible" then there exists a mapping b^{-1} such that b * b^{-1}(a) = a for all a \in S, where "*" denotes composition of mappings. -- % Randy Yates % "...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall." %%%% <yates@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO http://home.earthlink.net/~yatescr   0 Reply Randy 7/22/2003 9:06:45 AM Glen Herrmannsfeldt wrote: > [...] > It seems obvious to me, though, that if you replace each complex number with > an ordered pair of real numbers, and supply the appropriate operations to > those ordered pairs such that you get the same results, that you have shown > that you can do everything with (ordered pairs of) real numbers. Then why do we need complex numbers to solve polynomials with real coefficients??? ... -- % Randy Yates % "...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall." %%%% <yates@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO http://home.earthlink.net/~yatescr   0 Reply Randy 7/22/2003 9:22:33 AM Randy Yates wrote: > > Glen Herrmannsfeldt wrote: > > [...] > > It seems obvious to me, though, that if you replace each complex number with > > an ordered pair of real numbers, and supply the appropriate operations to > > those ordered pairs such that you get the same results, that you have shown > > that you can do everything with (ordered pairs of) real numbers. > > Then why do we need complex numbers to solve polynomials with real > coefficients??? ... Glen, let me retract this. You are right, these two mathematical beasts (the complex and "ordered pairs of real numbers with the appropriate operations") are equivalent. However, this brings us to the heart of the problem: in an arithmetic system, there are only two operations: addition and multiplication, and there is only one set (the reals in this case). Therefore, YOU DON'T GET TO DEFINE NEW OPERATIONS AND NEW SETS (e.g., ordered pairs) from real numbers and call it the same mathematical system! The system is defined in terms of its set and the two operations. -- % Randy Yates % "...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall." %%%% <yates@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO http://home.earthlink.net/~yatescr   0 Reply Randy 7/22/2003 9:32:17 AM Jerry Avins wrote: > What I wrote is, I thought, that cos(a) has only one value, whatever 'a' > may be. It is the same as cos(-a) and cos(a + 2npi), but I didn't > address that. OK. Clear. > I apologize for confusing you. I seem to be doing a lot of that lately. It's OK, I like to learn, without being confused, I cannot learn anything... > I like language almost as much as technical things. Read you later. See you later... ;-) bye, -- Piergiorgio Sartor   0 Reply Piergiorgio 7/22/2003 4:21:46 PM Jerry Avins wrote: > ... > > What I wrote is, I thought, that cos(a) has only one value, whatever 'a' > may be. That's not what I had written! It's not what I meant, either. The truth is, that if 'a' changes, cos(a) probably will too. I summed up what I mean later: > > ... For any a, there is one and only one cos(a). > > Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/22/2003 4:57:45 PM Jerry Avins wrote: > Gimme a break! You know that I'm quite "strict" on these issues... ;-) bye, -- Piergiorgio Sartor   0 Reply Piergiorgio 7/23/2003 7:18:41 AM Piergiorgio Sartor wrote: > > Jerry Avins wrote: > > > For random x, the probability is vanishingly small. Intuition leads me > > to believe that it is the same as the ratio of the number of rationals > > to the number of reals. If a grain of black pepper is put on the number > > line at every rational, and a grain of salt at every non-rational real, > > the line would appear immaculately white, despite the infinite amount of > > pepper. > > I had the same idea, I think even worst, > something like N/R (natural/real). > > On the other hand we cannot neglect that > it could happen... > > I miss something here... > > bye, > > -- > Piergiorgio Sartor Probability often defies my intuition too. Consider a process that can vary about some mean position independently in vertical and horizontal directions with the same Gaussian distribution. The probability density at the mean is clearly a maximum for the cardinal directions, and it can be shown that the density function has the same shape along any line through the mean. (Rotating the axes leaves the picture unchanged.) Now consider the distance from the mean: the Euclidian distance accounting for simultaneous variation of H and V. That turns out to be a Rayleigh distribution, with zero probability at the mean. So what? I sometimes shoot at tied-up bundles of newspaper for target practice. The twine crossing is my aiming point. Sometimes, a shot cuts a string. More rarely, a single shot cuts both pieces of twine, collapsing the bundle and showing that even a .22 caliber bullet has finite diameter. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/23/2003 3:13:29 PM Piergiorgio Sartor wrote: > > Jerry Avins wrote: > > > For random x, the probability is vanishingly small. Intuition leads me > > to believe that it is the same as the ratio of the number of rationals > > to the number of reals. If a grain of black pepper is put on the number > > line at every rational, and a grain of salt at every non-rational real, > > the line would appear immaculately white, despite the infinite amount of > > pepper. > > I had the same idea, I think even worst, > something like N/R (natural/real). > Believe it or not, the ratio of naturals to rationals is unity! There as many points along each line segment as along any other, and the same number of points in a square erected on it. Kantor demonstrated those counter-intuitive truths by inventing a method to show a one-to-one naming correspondence called diagonalization. Goedel's proof of unknowability used a modified form of diagonalization. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������   0 Reply Jerry 7/23/2003 3:27:36 PM Jerry Avins wrote: > Believe it or not, the ratio of naturals to rationals is unity! There as I was unsure about that. > many points along each line segment as along any other, and the same > number of points in a square erected on it. Kantor demonstrated those > counter-intuitive truths by inventing a method to show a one-to-one > naming correspondence called diagonalization. Goedel's proof of > unknowability used a modified form of diagonalization. Also many theorems related to computer science use this method. I guess also the halting problem of Turing is a variation of diagonalization. Not sure. bye, -- Piergiorgio Sartor   0 Reply Piergiorgio 7/23/2003 4:06:44 PM On Wed, 23 Jul 2003 19:06:44 +0200, Piergiorgio Sartor wrote: > I guess also the halting problem of Turing is > a variation of diagonalization. Not sure. The standard proof does use diagonalization. It can be found in Hopcraft and Ullman's book, which is the definitive book on the subject. -- Matthew Donadio (m.p.donadio@ieee.org)   0 Reply Matthew 7/23/2003 4:21:22 PM Piergiorgio Sartor wrote: > Jerry Avins wrote: > > > For random x, the probability is vanishingly small. Intuition leads me > > to believe that it is the same as the ratio of the number of rationals > > to the number of reals. If a grain of black pepper is put on the number > > line at every rational, and a grain of salt at every non-rational real, > > the line would appear immaculately white, despite the infinite amount of > > pepper. > > I had the same idea, I think even worst, > something like N/R (natural/real). > > On the other hand we cannot neglect that > it could happen... That's a problem you face when you use the axiomatic (Kolmogorov) measure theory approach to probability. You get events (sets) which have probability (measure) zero, but which are not empty. For example, when you choose a number from the interval [0,1] with uniform distribution, the probability of it being rational is zero. Which doesn't mean that it can't happen! The other problem you get is that you cannot create a uniform distribution on an infinite countable set. For example, you cannot say what the probability is of choosing an even number from natural numbers (even though you feel that it ought to be 1/2). That is because of the additivity of probabilites (measures) on disjoint sets. It's always fun trying to explain that to somebody :) Regards, Andor > > I miss something here... > > bye,   0 Reply an2or 7/25/2003 10:13:36 AM Piergiorgio Sartor wrote: > > For example, when you choose a number from the interval [0,1] with > > uniform distribution, the probability of it being rational is zero. > > Which doesn't mean that it can't happen! > > Actually that's why they prefer P[x <= y] > instead of P[x = y]... If you have a discrete distribution function, a question like P[X = x] makes very good sense. You can also mix absolute continuous and discrete distributions (think of a density function with a Dirac delta included somewhere). > > The other problem you get is that you cannot create a uniform > > distribution on an infinite countable set. For example, you cannot say > > what the probability is of choosing an even number from natural > > numbers (even though you feel that it ought to be 1/2). That is > > because of the additivity of probabilites (measures) on disjoint sets. > > Well, that's also because even, odd and all natural > are _exactly_ the same amount of numbers... Yes. But there are also just as many number which are multiples of 3 as there are non-multiples of 3. However, your intuition tells you that the probability of choosing a multiple of 3 from the natural number should be 1/3. Again, this probability space is not constructable via the axiomatic approach.   0 Reply an2or 7/27/2003 12:07:38 PM an2or@mailcircuit.com (Andor) writes: >Yes. But there are also just as many number which are multiples of 3 >as there are non-multiples of 3. However, your intuition tells you >that the probability of choosing a multiple of 3 from the natural >number should be 1/3. Again, this probability space is not >constructable via the axiomatic approach. Actually, this probability space **is** constructable via the axiomatic approach. The field of events is defined to contain all sets of the form A_n = {all integer multiples of n} and any other sets that can be constructed from the A_n's by the operations of complementation, countable union and countable intersection. The probability assigned to A_n is 1/n, to the complement of A_n is 1 - 1/n. What this assignment "lacks" is that it does not necessarily assign a probability to each individual outcome because a particular outcome may not be expressible in terms of complements, countable unions, and countable intersections of the A_n's, and is thus not a member of the field of events. That is, P({n}) may have no meaning unless we can express the singleton event {n} in terms of complements, countable unions, and countable intersections of the A_n's Note that the intersection of A_n and A_m is A_{LCM(n,m)} and that A_n and A_m are independent events if m and n are relatively prime integers (that is, have no factors in common other than 1; m and n themselves need not be prime, e.g. 15 and 16). -- .-. .-. .-. .-. .-. .-. .-. / D \ I / L \ I / P \ / S \ A / R \ W / A \ T / E \ -' -' -' -' -' -'   0 Reply Dilip 7/27/2003 4:37:19 PM Dilip V. Sarwate wrote: > an2or@mailcircuit.com (Andor) writes: > > >Yes. But there are also just as many number which are multiples of 3 > >as there are non-multiples of 3. However, your intuition tells you > >that the probability of choosing a multiple of 3 from the natural > >number should be 1/3. Again, this probability space is not > >constructable via the axiomatic approach. > > > Actually, this probability space **is** constructable via the > axiomatic approach. The field of events is defined to contain > all sets of the form > > A_n = {all integer multiples of n} > > and any other sets that can be constructed from the A_n's by > the operations of complementation, countable union and countable > intersection. > > The probability assigned to A_n is 1/n, to the complement of > A_n is 1 - 1/n. What this assignment "lacks" is that it does > not necessarily assign a probability to each individual outcome > because a particular outcome may not be expressible in terms of > complements, countable unions, and countable intersections of > the A_n's, and is thus not a member of the field of events. That > is, P({n}) may have no meaning unless we can express the singleton > event {n} in terms of complements, countable unions, and > countable intersections of the A_n's P[{n}] indeed has meaning for all n in NN, because {n} is an element of sigma(A_1, A_2, ...): B_n := {n} = A_n \ U(A_{n*k})_{k=2}^infty, n>=1, (where U denotes "union") (which also shows that sigma(A_1, A_2, ...) is equal to the power set of NN). Now you have a probability space on NN. But it does not give uniform distribution for the sets B_n, which we are interested in. One could think that P[ B_n ] = 0 for all n, but that cannot be true under the distribution that you specify because NN = A_1 = U (B_n)_{n=1}^infty (where the B_n are disjoint). Taking the probability on both sides we get: 1 = P[ A_1 ] = P[U (B_n)_{n=1}^infty] = Sum(P[B_n])_{n=1}^infty, so not all P[B_n] can be zero. OTH, P[B_n] cannot be constant for all n, because of the convergence of the sum on the right. That's what I meant by not constructable uniform distribution for drawing natural numbers. Regards, Andor   0 Reply Andor 7/27/2003 10:59:33 PM robert bristow-johnson wrote: > jim wrote: > > Good analogy. But, you mustn't say this: > > > > a) It is impossible to keep track of goods on the shelf without resorting > > to negative numbers. > > b) Some days, I have negative number of goods on the shelf. > > > > If you do say things analogous to that, its fair to say that you've lost > > touch with reality. > > > even if all tangible physical quantity could be or would normally be > measured as positive values (which i'm not conceding is the case), > there should always be the notion of the differences of like > dimensioned physical quantities. say pressure differences or > difference of rotational position of some shaft or something (somewhat > alluded to by Jerry below). that's a reality that i am in touch with > and it needs negative numbers. so i think that "real" is a very > appropriate adjective for negative numbers. A mathematician, a physicist and a biologist stand before an elevator. Nine people enter the elevator. After some time the elevator returns. Ten people exit. Biologist: "Oh my, people reproduced in there!" Physicist: "A measurement error of ten precent is negligible." Mathematician: "Now if one person goes back inside, none are left in the elevator." Regards, Andor   0 Reply an2or 7/28/2003 9:41:29 AM My apologies to Andor for mis-understanding what he wrote. Since it is not possible to assign equal nonzero probability to each of the natural numbers, it is natural to ask what alternative might be used to model one's desire that each integer is "equally likely" to be chosen. So, let us consider a uniform distribution on the first N integers, say {1, 2, .. , 30} so that each has probability 1/30 of being chosen. Then, what is the probability that the outcome of the experiment is a multiple of n? The answer is exactly 1/n for n = 1, 2, 3, 5, 6, 10, 15 and **approximately** 1/n for other values of n. If we choose N = 120, we get exact results 1/n for more choices of n, and better approximations for the rest. Thus, heuristically speaking, a reasonable model for a "uniform distribution" on the natural numbers is one that assigns probability 1/n to the set A_n, the multiples of n, for each choice of n. As Andor points out, this does not actually assign equal probability to each singleton event {n}. -- .-. .-. .-. .-. .-. .-. .-. / D \ I / L \ I / P \ / S \ A / R \ W / A \ T / E \ -' -' -' -' -' -'   0 Reply Dilip 7/28/2003 2:55:27 PM Dilip V. Sarwate <sarwate@uiuc.edu> wrote in message news:<giURa.2858$o7.37232@vixen.cso.uiuc.edu>...
>
> Come to think of it, could someone explain *why* I have to
> pay for electricity?  After all, I return one electron to
> the power company to replace each and every electron it
> sends to me.  Not the same electron, of course, but another
> one that is identical in all respects...

Because they do all the work of giving you the electron and taking it
back from you. They come to your doorstep to collect that electron you
are returning :)

 0

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