Taking a time derivative after an FFT would be easy on SPICE if you
had some curve that would FFT into a ramp:  Just multiply the function
by the ramp.

The inverse transform of a ramp on Excel is complex, the real part
being a small negative offset with a large positive zero frequency.
The imaginary part may work but it looks like it would be hard to
fashion with a circuit even if the curve was known.

Just builting up 2**n voltage sources where frequency = amplitude = n
is tedious.  The FFT is, of course, just 2**n peaks that increase
linearly in height on the linear - linear graph.  To get a nice ramp
would require an infinite number of voltage sources & frequencies and
amplitudes.

Is there anyway to get a nice smooth envelope -- "envelope" may have
another technical meaning -- over the time domain curve to get a nice
ramp in the FFT?


Bret Cahill

Bret Cahill wrote:
> Taking a time derivative after an FFT would be easy on SPICE if you
> had some curve that would FFT into a ramp:  Just multiply the function
> by the ramp.
>
> The inverse transform of a ramp on Excel is complex, the real part
> being a small negative offset with a large positive zero frequency.
> The imaginary part may work but it looks like it would be hard to
> fashion with a circuit even if the curve was known.
>
> Just builting up 2**n voltage sources where frequency = amplitude = n
> is tedious.  The FFT is, of course, just 2**n peaks that increase
> linearly in height on the linear - linear graph.  To get a nice ramp
> would require an infinite number of voltage sources&  frequencies and
> amplitudes.
>
> Is there anyway to get a nice smooth envelope -- "envelope" may have
> another technical meaning -- over the time domain curve to get a nice
> ramp in the FFT?
>
>
> Bret Cahill

I suggest getting a copy of "The Fourier Transform and Its Applications" 
by Bracewell.  It's a good read and has all of this sort of stuff in it.

Cheers

Phil Hobbs


-- 
Dr Philip C D Hobbs
Principal
ElectroOptical Innovations
55 Orchard Rd
Briarcliff Manor NY 10510
845-480-2058

email: hobbs (atsign) electrooptical (period) net
http://electrooptical.net

> > Taking a time derivative after an FFT would be easy on SPICE if you
> > had some curve that would FFT into a ramp: =A0Just multiply the functio=
n
> > by the ramp.
>
> > The inverse transform of a ramp on Excel is complex, the real part
> > being a small negative offset with a large positive zero frequency.
> > The imaginary part may work but it looks like it would be hard to
> > fashion with a circuit even if the curve was known.
>
> > Just builting up 2**n voltage sources where frequency =3D amplitude =3D=
 n
> > is tedious. =A0The FFT is, of course, just 2**n peaks that increase
> > linearly in height on the linear - linear graph. =A0To get a nice ramp
> > would require an infinite number of voltage sources& =A0frequencies and
> > amplitudes.
>
> > Is there anyway to get a nice smooth envelope -- "envelope" may have
> > another technical meaning -- over the time domain curve to get a nice
> > ramp in the FFT?
>
> > Bret Cahill
>
> I suggest getting a copy of "The Fourier Transform and Its Applications"
> by Bracewell. =A0It's a good read and has all of this sort of stuff in it=
..

At $500 / copy it better be a good read.  That's what university
libraries pay for Russian books on tech esoterica.

Anyway some nuke medicine page claimed the INV FT of a ramp was just 1/
t.

It's somewhat curious that the INV FFT of a function is equal to the
reciprocal of the function but nevertheless 1/t seemed to approach a
ramp on Excel's FFT, at least at higher frequencies.

I couldn't get anything linear on the SPICE FFT but then, I couldn't
get anything to output a 1/t waveform except a crude 10 point plot.


> Cheers
>
> Phil Hobbs
>
> --
> Dr Philip C D Hobbs
> Principal
> ElectroOptical Innovations
> 55 Orchard Rd
> Briarcliff Manor NY 10510
> 845-480-2058
>
> email: hobbs (atsign) electrooptical (period) nethttp://electrooptical.ne=
t- Hide quoted text -
>
> - Show quoted text -

Bret Cahill wrote:
>>> Taking a time derivative after an FFT would be easy on SPICE if you
>>> had some curve that would FFT into a ramp:  Just multiply the function
>>> by the ramp.
>>
>>> The inverse transform of a ramp on Excel is complex, the real part
>>> being a small negative offset with a large positive zero frequency.
>>> The imaginary part may work but it looks like it would be hard to
>>> fashion with a circuit even if the curve was known.
>>
>>> Just builting up 2**n voltage sources where frequency = amplitude = n
>>> is tedious.  The FFT is, of course, just 2**n peaks that increase
>>> linearly in height on the linear - linear graph.  To get a nice ramp
>>> would require an infinite number of voltage sources&    frequencies and
>>> amplitudes.
>>
>>> Is there anyway to get a nice smooth envelope -- "envelope" may have
>>> another technical meaning -- over the time domain curve to get a nice
>>> ramp in the FFT?
>>
>>> Bret Cahill
>>
>> I suggest getting a copy of "The Fourier Transform and Its Applications"
>> by Bracewell.  It's a good read and has all of this sort of stuff in it.
>
> At $500 / copy it better be a good read.  That's what university
> libraries pay for Russian books on tech esoterica.
>
> Anyway some nuke medicine page claimed the INV FT of a ramp was just 1/
> t.
>
> It's somewhat curious that the INV FFT of a function is equal to the
> reciprocal of the function but nevertheless 1/t seemed to approach a
> ramp on Excel's FFT, at least at higher frequencies.
>
> I couldn't get anything linear on the SPICE FFT but then, I couldn't
> get anything to output a 1/t waveform except a crude 10 point plot.
>
>

Five hundred bucks?  Where do you find that?  Try 
http://www.abebooks.com and you'll find a whole pile of the second 
edition (which is better than good enough) for about $22 plus shipping.

Cheers

Phil Hobbs

-- 
Dr Philip C D Hobbs
Principal
ElectroOptical Innovations
55 Orchard Rd
Briarcliff Manor NY 10510
845-480-2058

email: hobbs (atsign) electrooptical (period) net
http://electrooptical.net

On Feb 24, 9:10=A0am, Bret Cahill <Bret_E_Cah...@yahoo.com> wrote:

> Is there anyway to get a nice smooth envelope -- "envelope" may have
> another technical meaning -- over the time domain curve to get a nice
> ramp in the FFT?

No.  The usual FFT for finite sampled sequences is done with circular
boundary conditions, and you must deal with the enormous discontinuity
at that boundary that any frequency-dependent ramp creates.
It isn't a suitable 'test function' in the sense of nicely behaved
functions for the transformation, so it is going to look very nasty in
the time domain (after the inverse transformation of the 'ramp').

> > Is there anyway to get a nice smooth envelope -- "envelope" may have
> > another technical meaning -- over the time domain curve to get a nice
> > ramp in the FFT?
>
> No. =A0The usual FFT for finite sampled sequences is done with circular
> boundary conditions, and you must deal with the enormous discontinuity
> at that boundary that any frequency-dependent ramp creates.

Would this be true if every point used by the FFT -- SPICE has a 256
minimum -- was plotted and used by the FFT?

> It isn't a suitable 'test function' in the sense of nicely behaved
> functions for the transformation, so it is going to look very nasty in
> the time domain (after the inverse transformation of the 'ramp').

Plotting 1/t with about 8 points, (0.125, 8)  (.25, 4) . . . (8,
0.125) on SPICE then taking the FFT and then taking the reciprocal --
not sure why this step works or is necessary -- isn't a bad ramp, at
least at the lower frequencies.

It would be really convenient if a time derivative could be taken
mathematically without a derivative circuit in either domain.

You cannot take a FFT of a wave form after you've taken its time
derivative in the time domain -- it won't appear in the box -- and in
the frequency domain you only get a derivative w/ respect to
frequency, whatever that is.

There doesn't seem to be an easy way to create a time signal on SPICE
that equals 1/t.


Bret Cahill

In article <y8L9p.18$aC6.13@newsfe03.iad>,
Martin Brown  <|||newspam|||@nezumi.demon.co.uk> wrote:

>Bad things happen when this basic assumption of periodic boundary 
>conditions is for whatever reason invalid. A saw tooth has very obvious 
>boundary discontinuity problems.

Indeed.  In that, it is exactly the same as polynomial fitting of
curves, or any such infinite approximation.  Go outside the domain
of validity and things can go badly wrong.

I am often amazed at the things people use Fourier approximations
for - not because they do, but because they seem to work more often
than a naive analysis would expect.  But, as you say, you don't get
that by just rushing in, blindly.


Regards,
Nick Maclaren.

On 24/02/2011 22:18, Bret Cahill wrote:
>>> Is there anyway to get a nice smooth envelope -- "envelope" may have
>>> another technical meaning -- over the time domain curve to get a nice
>>> ramp in the FFT?
>>
>> No.  The usual FFT for finite sampled sequences is done with circular
>> boundary conditions, and you must deal with the enormous discontinuity
>> at that boundary that any frequency-dependent ramp creates.
>
> Would this be true if every point used by the FFT -- SPICE has a 256
> minimum -- was plotted and used by the FFT?

It is always true that an FFT has an implicit assumption of periodic 
boundary conditions at the length of the transform which are most 
commonly a tiled circular wrap around at the edges, but in some 
implementations may be Dirichlet or mirror boundary conditions leading 
to a DCT variant. In addition there is also two plausible choices of 
origin exactly in the centre of each cell or at the edge.

Bad things happen when this basic assumption of periodic boundary 
conditions is for whatever reason invalid. A saw tooth has very obvious 
boundary discontinuity problems.

Real applications of FFTs for imaging tend to spend a lot of time and 
effort ameliorating this potential aliasing effect at the boundaries.
>
>> It isn't a suitable 'test function' in the sense of nicely behaved
>> functions for the transformation, so it is going to look very nasty in
>> the time domain (after the inverse transformation of the 'ramp').
>
> Plotting 1/t with about 8 points, (0.125, 8)  (.25, 4) . . . (8,
> 0.125) on SPICE then taking the FFT and then taking the reciprocal --
> not sure why this step works or is necessary -- isn't a bad ramp, at
> least at the lower frequencies.
>
> It would be really convenient if a time derivative could be taken
> mathematically without a derivative circuit in either domain.
>
> You cannot take a FFT of a wave form after you've taken its time
> derivative in the time domain -- it won't appear in the box -- and in
> the frequency domain you only get a derivative w/ respect to
> frequency, whatever that is.
>
> There doesn't seem to be an easy way to create a time signal on SPICE
> that equals 1/t.

What are you trying to do?

Regards,
Martin Brown

On Thu, 24 Feb 2011 09:10:50 -0800 (PST), Bret Cahill
<Bret_E_Cahill@yahoo.com> wrote:

>Taking a time derivative after an FFT would be easy on SPICE if you
>had some curve that would FFT into a ramp:  Just multiply the function
>by the ramp.

Not sure what you are ultimately trying to do, but note that
you can obtain the FFT of the time derivative by taking the
FFT of the raw waveform and applying a +6 dB/octave
"envelope" to that... essentially, you just tilt the
spectrum up at a 6 dB/octave slope.

This turns out to be very handy for measuring frequency
response of a system.  Classically, one can apply an impulse
to the system and take the FFT to get the frequency
response.  But an impulse is pretty narrow (one sample, in a
digital system), so it doesn't have much energy.  A step
response, on the other hand, has a whole lot more.  Since
the derivatve of a step is an impulse, you can get the
frequency response by applying a step, taking the FFT, and
tilting it.  This is so handy that I built this feature into
my Daqarta software.  See "Frequency Response Measurement -
Step Response" at <http://www.daqarta.com/dw_0a0s.htm>.

Best regards,


Bob Masta
 
              DAQARTA  v6.00
   Data AcQuisition And Real-Time Analysis
              www.daqarta.com
Scope, Spectrum, Spectrogram, Sound Level Meter
    Frequency Counter, FREE Signal Generator
           Pitch Track, Pitch-to-MIDI 
          Science with your sound card!

Bob Masta wrote:
> On Thu, 24 Feb 2011 09:10:50 -0800 (PST), Bret Cahill
> <Bret_E_Cahill@yahoo.com>  wrote:
>
>> Taking a time derivative after an FFT would be easy on SPICE if you
>> had some curve that would FFT into a ramp:  Just multiply the function
>> by the ramp.
>
> Not sure what you are ultimately trying to do, but note that
> you can obtain the FFT of the time derivative by taking the
> FFT of the raw waveform and applying a +6 dB/octave
> "envelope" to that... essentially, you just tilt the
> spectrum up at a 6 dB/octave slope.
>
> This turns out to be very handy for measuring frequency
> response of a system.  Classically, one can apply an impulse
> to the system and take the FFT to get the frequency
> response.  But an impulse is pretty narrow (one sample, in a
> digital system), so it doesn't have much energy.  A step
> response, on the other hand, has a whole lot more.  Since
> the derivatve of a step is an impulse, you can get the
> frequency response by applying a step, taking the FFT, and
> tilting it.  This is so handy that I built this feature into
> my Daqarta software.  See "Frequency Response Measurement -
> Step Response" at<http://www.daqarta.com/dw_0a0s.htm>.
>


Are you windowing the data before taking the DFT?  Could get ugly otherwise.

Cheers

Phil Hobbs

-- 
Dr Philip C D Hobbs
Principal
ElectroOptical Innovations
55 Orchard Rd
Briarcliff Manor NY 10510
845-480-2058

email: hobbs (atsign) electrooptical (period) net
http://electrooptical.net

On Feb 24, 11:10=A0am, Bret Cahill <Bret_E_Cah...@yahoo.com> wrote:
> Taking a time derivative after an FFT would be easy on SPICE if you
> had some curve that would FFT into a ramp: =A0Just multiply the function
> by the ramp.
>
> The inverse transform of a ramp on Excel is complex, the real part
> being a small negative offset with a large positive zero frequency.
> The imaginary part may work but it looks like it would be hard to
> fashion with a circuit even if the curve was known.
>
> Just builting up 2**n voltage sources where frequency =3D amplitude =3D n
> is tedious. =A0The FFT is, of course, just 2**n peaks that increase
> linearly in height on the linear - linear graph. =A0To get a nice ramp
> would require an infinite number of voltage sources & frequencies and
> amplitudes.
>
> Is there anyway to get a nice smooth envelope -- "envelope" may have
> another technical meaning -- over the time domain curve to get a nice
> ramp in the FFT?
>
> Bret Cahill

I'm not sure I understand what you're trying to do. Are you trying to
get a FFT whose "amplitude" is a linear (either positive- or negative-
going) ramp?

In article <4d4c208e-fb21-4bfa-9c97-4845cfa6fc85@w9g2000prg.googlegroups.com>,
Bret Cahill  <Bret_E_Cahill@yahoo.com> wrote:
>>
>> What are you trying to do?
>
>Take fractional order derivatives.

Do you know of a reference that explains when that is mathematically
valid?  I have seen plenty of references to it, and even done it on
rare occasions, but never seen anything that analyses the problem
properly.

Of course, it may just be an egregious hack that sometimes works, in
which case there may be no such analysis :-)


Regards,
Nick Maclaren.

> > Taking a time derivative after an FFT would be easy on SPICE if you
> > had some curve that would FFT into a ramp: =A0Just multiply the functio=
n
> > by the ramp.
>
> > The inverse transform of a ramp on Excel is complex, the real part
> > being a small negative offset with a large positive zero frequency.
> > The imaginary part may work but it looks like it would be hard to
> > fashion with a circuit even if the curve was known.
>
> > Just builting up 2**n voltage sources where frequency =3D amplitude =3D=
 n
> > is tedious. =A0The FFT is, of course, just 2**n peaks that increase
> > linearly in height on the linear - linear graph. =A0To get a nice ramp
> > would require an infinite number of voltage sources & frequencies and
> > amplitudes.
>
> > Is there anyway to get a nice smooth envelope -- "envelope" may have
> > another technical meaning -- over the time domain curve to get a nice
> > ramp in the FFT?
>
> > Bret Cahill
>
> I'm not sure I understand what you're trying to do. Are you trying to
> get a FFT whose "amplitude" is a linear (either positive- or negative-
> going) ramp?

Yes.

You can take integer order or fractional order derivatives in the FFT
on Spice if you have something proportional to a ramp.


Bret Cahill

> >>> Is there anyway to get a nice smooth envelope -- "envelope" may have
> >>> another technical meaning -- over the time domain curve to get a nice
> >>> ramp in the FFT?
>
> >> No. =A0The usual FFT for finite sampled sequences is done with circula=
r
> >> boundary conditions, and you must deal with the enormous discontinuity
> >> at that boundary that any frequency-dependent ramp creates.
>
> > Would this be true if every point used by the FFT -- SPICE has a 256
> > minimum -- was plotted and used by the FFT?
>
> It is always true that an FFT has an implicit assumption of periodic
> boundary conditions at the length of the transform which are most
> commonly a tiled circular wrap around at the edges, but in some
> implementations may be Dirichlet or mirror boundary conditions leading
> to a DCT variant. In addition there is also two plausible choices of
> origin exactly in the centre of each cell or at the edge.
>
> Bad things happen when this basic assumption of periodic boundary
> conditions is for whatever reason invalid. A saw tooth has very obvious
> boundary discontinuity problems.

A frequency ramp would need to start at (0,0) to be useful taking
derivatives.

> Real applications of FFTs for imaging tend to spend a lot of time and
> effort ameliorating this potential aliasing effect at the boundaries.

> >> It isn't a suitable 'test function' in the sense of nicely behaved
> >> functions for the transformation, so it is going to look very nasty in
> >> the time domain (after the inverse transformation of the 'ramp').
>
> > Plotting 1/t with about 8 points, (0.125, 8) =A0(.25, 4) . . . (8,
> > 0.125) on SPICE then taking the FFT and then taking the reciprocal --
> > not sure why this step works or is necessary -- isn't a bad ramp, at
> > least at the lower frequencies.
>
> > It would be really convenient if a time derivative could be taken
> > mathematically without a derivative circuit in either domain.
>
> > You cannot take a FFT of a wave form after you've taken its time
> > derivative in the time domain -- it won't appear in the box -- and in
> > the frequency domain you only get a derivative w/ respect to
> > frequency, whatever that is.
>
> > There doesn't seem to be an easy way to create a time signal on SPICE
> > that equals 1/t.
>
> What are you trying to do?

Take fractional order derivatives.


Bret Cahill

In article <01a40481-7894-430a-a37f-53c66638ab59@8g2000prb.googlegroups.com>,
Bret Cahill  <Bret_E_Cahill@yahoo.com> wrote:
>> >> What are you trying to do?
>>
>> >Take fractional order derivatives.
>>
>> Do you know of a reference that explains when that is mathematically
>> valid?  
>
>It seems to work on Excel.  You can check out a lot of different
>fractional order derivatives very quickly because you only have to
>click 3 times on the FT box.

I.e. it doesn't crash and produces a number.  Excel is notorious
for doing that sort of thing.

>It might be easier to do electronic circuits on Excel than frequency
>ramp functions on SPICE.

Yeah.  And it's a lot easier to build faster-than-light spaceships
on Excel, too.

>No one will deny it's rigorous elegant civilized orthodox concise
>organized kosher nice philosophical thoughtful lofty and often
>practical, utilitarian and easy to have a nice formal proof.

It also helps to be able to judge when what you are building
isn't going to work, especially when it comes to minor details
like reliability.

>That's why they have witch doctors, aka, "mathematicians" installed at
>universities.

And I am one, looking for a grimoire.

>All an engineer needs to do, however, is be able to say, "well it  w O
>R erks."

That's the sales department.  A good engineer has solid reasons to
believe that what he has built will work according to the actual
requirements and intent.  There are a few of us left :-)

>> Of course, it may just be an egregious hack that sometimes works, in
>> which case there may be no such analysis :-)
>
>There's always an analysis.  Mathematicians just want a pretty one.

Er, like "Well it probably won't break before we have had time to
cash the cheque"?

The history of engineering is littered with projects which failed
because the analysis was not done properly - and where it was known
what the potential problems were.


Regards,
Nick Maclaren.

On 2/24/2011 9:10 AM, Bret Cahill wrote:
> Taking a time derivative after an FFT would be easy on SPICE if you
> had some curve that would FFT into a ramp:  Just multiply the function
> by the ramp.
>
> The inverse transform of a ramp on Excel is complex, the real part
> being a small negative offset with a large positive zero frequency.
> The imaginary part may work but it looks like it would be hard to
> fashion with a circuit even if the curve was known.
>
> Just builting up 2**n voltage sources where frequency = amplitude = n
> is tedious.  The FFT is, of course, just 2**n peaks that increase
> linearly in height on the linear - linear graph.  To get a nice ramp
> would require an infinite number of voltage sources&  frequencies and
> amplitudes.
>
> Is there anyway to get a nice smooth envelope -- "envelope" may have
> another technical meaning -- over the time domain curve to get a nice
> ramp in the FFT?
>
>
> Bret Cahill

It seems to me that the approach might be something like this:

First, a continous, infinite Fourier Transform pair:

df(t)/dt <-> (jw)*F(w)

So, I expect your "ramp" is the jw term above, eh?  Note the "j"

If this is going to be moved into the discrete-periodic world then we 
note that the value switches from jW0 to -jW0 where W0=2*pi*fs/2.
Keeping things in the continuous, infinite but periodic world in 
frequency then we should recognize immediately that the sawtooth in 
frequency will be of infinite extent in time.  So, we have to smooth out 
that discontinuity.

Take a look at a Parks-McClellan time differentiator design.  That has a 
ramp in frequency *but* it's bandlimited so the ramp stops and goes to 
zero at fs/2.

Fred

> >> What are you trying to do?
>
> >Take fractional order derivatives.
>
> Do you know of a reference that explains when that is mathematically
> valid? =A0

It seems to work on Excel.  You can check out a lot of different
fractional order derivatives very quickly because you only have to
click 3 times on the FT box.

It might be easier to do electronic circuits on Excel than frequency
ramp functions on SPICE.

> I have seen plenty of references to it, and even done it on
> rare occasions, but never seen anything that analyses the problem
> properly.

No one will deny it's rigorous elegant civilized orthodox concise
organized kosher nice philosophical thoughtful lofty and often
practical, utilitarian and easy to have a nice formal proof.

That's why they have witch doctors, aka, "mathematicians" installed at
universities.

All an engineer needs to do, however, is be able to say, "well it  w O
R erks."

> Of course, it may just be an egregious hack that sometimes works, in
> which case there may be no such analysis :-)

There's always an analysis.  Mathematicians just want a pretty one.


Bret Cahill


"Picked up a gal, she was ugly too."

-- Commander Cody

nmm1@cam.ac.uk wrote:
> In article<01a40481-7894-430a-a37f-53c66638ab59@8g2000prb.googlegroups.com>,
> Bret Cahill<Bret_E_Cahill@yahoo.com>  wrote:
>>>>> What are you trying to do?
>>>
>>>> Take fractional order derivatives.
>>>
>>> Do you know of a reference that explains when that is mathematically
>>> valid?
>>
>> It seems to work on Excel.  You can check out a lot of different
>> fractional order derivatives very quickly because you only have to
>> click 3 times on the FT box.
>
> I.e. it doesn't crash and produces a number.  Excel is notorious
> for doing that sort of thing.
>
>> It might be easier to do electronic circuits on Excel than frequency
>> ramp functions on SPICE.
>
> Yeah.  And it's a lot easier to build faster-than-light spaceships
> on Excel, too.
>
>> No one will deny it's rigorous elegant civilized orthodox concise
>> organized kosher nice philosophical thoughtful lofty and often
>> practical, utilitarian and easy to have a nice formal proof.
>
> It also helps to be able to judge when what you are building
> isn't going to work, especially when it comes to minor details
> like reliability.
>
>> That's why they have witch doctors, aka, "mathematicians" installed at
>> universities.
>
> And I am one, looking for a grimoire.
>
>> All an engineer needs to do, however, is be able to say, "well it  w O
>> R erks."
>
> That's the sales department.  A good engineer has solid reasons to
> believe that what he has built will work according to the actual
> requirements and intent.  There are a few of us left :-)
>
>>> Of course, it may just be an egregious hack that sometimes works, in
>>> which case there may be no such analysis :-)
>>
>> There's always an analysis.  Mathematicians just want a pretty one.
>
> Er, like "Well it probably won't break before we have had time to
> cash the cheque"?
>
> The history of engineering is littered with projects which failed
> because the analysis was not done properly - and where it was known
> what the potential problems were.
>
>
> Regards,
> Nick Maclaren.


Don't take Cahill too seriously--he's a fairly well known crank who'd 
far rather slag people off than learn anything from them.

Cheers

Phil Hobbs

-- 
Dr Philip C D Hobbs
Principal
ElectroOptical Innovations
55 Orchard Rd
Briarcliff Manor NY 10510
845-480-2058

email: hobbs (atsign) electrooptical (period) net
http://electrooptical.net

> >> >> What are you trying to do?
>
> >> >Take fractional order derivatives.
>
> >> Do you know of a reference that explains when that is mathematically
> >> valid?
>
> >It seems to work on Excel.  You can check out a lot of different
> >fractional order derivatives very quickly because you only have to
> >click 3 times on the FT box.
>
> I.e. it doesn't crash

That seems to be a bigger problem on SPICE.

> and produces a number.

The graphs on Excel check out for a variety of test functions,
certainly for anything as well behaved as the signal to be processed.

.. . .

> >It might be easier to do electronic circuits on Excel than frequency
> >ramp functions on SPICE.

> Yeah.  And it's a lot easier to build faster-than-light spaceships
> on Excel, too.

A series capacitor or inductor is easy on Excel.  Take the FFT, go to
polar, subtract nu*pi/2 from the phase angles, then back to real &
imaginary and then raise each frequency to ^nu and multiply.  Then
take the inverse transform.

Nu => 0+ for the offset block (a large cap) and =>1 for the 1st
derivative (a small cap) circuit.

Nu => -1 to integrate one order with an inductor.

> >No one will deny it's rigorous elegant civilized orthodox concise
> >organized kosher nice philosophical thoughtful lofty and often
> >practical, utilitarian and easy to have a nice formal proof.

> It also helps to be able to judge when what you are building
> isn't going to work,

That's the reason for using a variety of simulators in addition to
whatever theory you may have.

> especially when it comes to minor details
> like reliability.

No one is 100% sure FEA is always reliable but that doesn't stop air
frame engineers from using it as a double check for . . . reliability.

> >That's why they have witch doctors, aka, "mathematicians" installed at
> >universities.

> And I am one, looking for a grimoire.

> >All an engineer needs to do, however, is be able to say, "well it  w O
> >R erks."

> That's the sales department.

The only way to make any money off of a simulator method is to do a
full blown investigation and write a book on it.

> A good engineer has solid reasons to
> believe that what he has built will work according to the actual
> requirements and intent.  There are a few of us left :-)

Even if he "proves" it in a peer reviewed paper with any combination
of first principles and simulators he'll still need to actually build
and test the thing.

But it's not necessary to fully understand a theory to get a patent on
something dependent upon the theory.  Many blamed Newton's defective
lift equation for delaying aviation, but as von Karmen pointed out,
that wouldn't deter inventors.

> >> Of course, it may just be an egregious hack that sometimes works, in
> >> which case there may be no such analysis :-)

> >There's always an analysis.  Mathematicians just want a pretty one.

> Er, like "Well it probably won't break before we have had time to
> cash the cheque"?

RR just tried that with their new wide body turbo fan.  Whether some
engineer should have been fired over it is irrelevant.  There isn't
any way to ever be 100% risk free, no matter the endeavor.

The real issue is a reasonable cost benefit risk analysis.

> The history of engineering is littered with projects which failed
> because the analysis was not done properly - and where it was known
> what the potential problems were.

That's the purpose of getting a ramp into SPICE's FFT.



Bret Cahill


"You can't get there . . . from here."

-- Maine expression

On Fri, 25 Feb 2011 08:05:00 -0500, Phil Hobbs
<pcdhSpamMeSenseless@electrooptical.net> wrote:

>Bob Masta wrote:
>> On Thu, 24 Feb 2011 09:10:50 -0800 (PST), Bret Cahill
>> <Bret_E_Cahill@yahoo.com>  wrote:
>>
>>> Taking a time derivative after an FFT would be easy on SPICE if you
>>> had some curve that would FFT into a ramp:  Just multiply the function
>>> by the ramp.
>>
>> Not sure what you are ultimately trying to do, but note that
>> you can obtain the FFT of the time derivative by taking the
>> FFT of the raw waveform and applying a +6 dB/octave
>> "envelope" to that... essentially, you just tilt the
>> spectrum up at a 6 dB/octave slope.
>>
>> This turns out to be very handy for measuring frequency
>> response of a system.  Classically, one can apply an impulse
>> to the system and take the FFT to get the frequency
>> response.  But an impulse is pretty narrow (one sample, in a
>> digital system), so it doesn't have much energy.  A step
>> response, on the other hand, has a whole lot more.  Since
>> the derivatve of a step is an impulse, you can get the
>> frequency response by applying a step, taking the FFT, and
>> tilting it.  This is so handy that I built this feature into
>> my Daqarta software.  See "Frequency Response Measurement -
>> Step Response" at<http://www.daqarta.com/dw_0a0s.htm>.
>>
>
>
>Are you windowing the data before taking the DFT?  Could get ugly otherwise.

No, in this case it's important to *not* window the data.
A window function (at least, any of the standard ones) has a
gradual onset and offset, for the specific purpose of
eliminating transients at the start/end of the FFT frame.
But here it is the onset that we are specifically interested
in.  The transient response should be complete (for all
practical purposes) before the end of the frame, or else you
need more samples in the frame.

In general, you never want to window a transient or noise,
only a continous wave.  The FFT analysis presumes a
continuous wave, such that every frame is an identical copy
that can be spliced seamlessly head to tail.  A real-world
continuous wave that does not contain an exact integer
number of cycles in the FFT frame will have a discontinuity
where the next frame is spliced, which results in "spectral
leakage" that appears as "skirts" on what would otherwise be
a single line in the spectrum.  The window function provides
a gradual onset and offset to smooth out this discontinuity,
greatly reducing the spectral leakage.

Interested readers may want to check out my "Gut Level
Fourier Transforms" series at
<http://www.daqarta.com/author.htm>.
In particular, see Part 5 "Dumping Spectral Leakage Out a
Window" <http://www.daqarta.com/eex05.htm>

Best regards,


Bob Masta
 
              DAQARTA  v6.00
   Data AcQuisition And Real-Time Analysis
              www.daqarta.com
Scope, Spectrum, Spectrogram, Sound Level Meter
    Frequency Counter, FREE Signal Generator
           Pitch Track, Pitch-to-MIDI 
          Science with your sound card!

On Fri, 25 Feb 2011 09:34:04 -0800 (PST), Bret Cahill
<Bret_E_Cahill@yahoo.com> wrote:

>> I'm not sure I understand what you're trying to do. Are you trying to
>> get a FFT whose "amplitude" is a linear (either positive- or negative-
>> going) ramp?
>
>Yes.
>
>You can take integer order or fractional order derivatives in the FFT
>on Spice if you have something proportional to a ramp.
>

I'm not a Spice user, but all you need to do is take a
normal FFT and apply a tilt to it, then do an IFFT (if you
want to see the time waveform).  Doesn't Spice allow you to
do this?  

The tilt should be +6 dB/octave for a normal derivative (see
my prior post) or in your case some fraction of that.  Note
that the raw FFT is linear in frequency and the "tilt" I am
talking about is linear in *log* frequency, so you need a
little exponential math here to generate the proper shape.  

Best regards,


Bob Masta
 
              DAQARTA  v6.00
   Data AcQuisition And Real-Time Analysis
              www.daqarta.com
Scope, Spectrum, Spectrogram, Sound Level Meter
    Frequency Counter, FREE Signal Generator
           Pitch Track, Pitch-to-MIDI 
          Science with your sound card!

Bob Masta wrote:
> On Fri, 25 Feb 2011 08:05:00 -0500, Phil Hobbs
> <pcdhSpamMeSenseless@electrooptical.net>  wrote:
>
>> Bob Masta wrote:
>>> On Thu, 24 Feb 2011 09:10:50 -0800 (PST), Bret Cahill
>>> <Bret_E_Cahill@yahoo.com>   wrote:
>>>
>>>> Taking a time derivative after an FFT would be easy on SPICE if you
>>>> had some curve that would FFT into a ramp:  Just multiply the function
>>>> by the ramp.
>>>
>>> Not sure what you are ultimately trying to do, but note that
>>> you can obtain the FFT of the time derivative by taking the
>>> FFT of the raw waveform and applying a +6 dB/octave
>>> "envelope" to that... essentially, you just tilt the
>>> spectrum up at a 6 dB/octave slope.
>>>
>>> This turns out to be very handy for measuring frequency
>>> response of a system.  Classically, one can apply an impulse
>>> to the system and take the FFT to get the frequency
>>> response.  But an impulse is pretty narrow (one sample, in a
>>> digital system), so it doesn't have much energy.  A step
>>> response, on the other hand, has a whole lot more.  Since
>>> the derivatve of a step is an impulse, you can get the
>>> frequency response by applying a step, taking the FFT, and
>>> tilting it.  This is so handy that I built this feature into
>>> my Daqarta software.  See "Frequency Response Measurement -
>>> Step Response" at<http://www.daqarta.com/dw_0a0s.htm>.
>>>
>>
>>
>> Are you windowing the data before taking the DFT?  Could get ugly otherwise.
>
> No, in this case it's important to *not* window the data.
> A window function (at least, any of the standard ones) has a
> gradual onset and offset, for the specific purpose of
> eliminating transients at the start/end of the FFT frame.
> But here it is the onset that we are specifically interested
> in.  The transient response should be complete (for all
> practical purposes) before the end of the frame, or else you
> need more samples in the frame.
>
> In general, you never want to window a transient or noise,
> only a continous wave.  The FFT analysis presumes a
> continuous wave, such that every frame is an identical copy
> that can be spliced seamlessly head to tail.  A real-world
> continuous wave that does not contain an exact integer
> number of cycles in the FFT frame will have a discontinuity
> where the next frame is spliced, which results in "spectral
> leakage" that appears as "skirts" on what would otherwise be
> a single line in the spectrum.  The window function provides
> a gradual onset and offset to smooth out this discontinuity,
> greatly reducing the spectral leakage.
>
> Interested readers may want to check out my "Gut Level
> Fourier Transforms" series at
> <http://www.daqarta.com/author.htm>.
> In particular, see Part 5 "Dumping Spectral Leakage Out a
> Window"<http://www.daqarta.com/eex05.htm>


The usefulness of windowing isn't restricted to the noiseless CW case. 
Unless your signal is actually periodic with the same period as the DFT, 
failure to window will cause every bin to smear out all across the 
interval.  You may not notice it with white noise, because the spectrum 
stays flat in that case, but there are other kinds of noise where you 
certainly will.

If your signal is a pulse, so that it starts from zero somewhere in the 
interval and decays back to zero by the end, then you effectively are in 
the periodic case already.  If not, windowing is good medicine.

Cheers

Phil Hobbs


-- 
Dr Philip C D Hobbs
Principal
ElectroOptical Innovations
55 Orchard Rd
Briarcliff Manor NY 10510
845-480-2058

email: hobbs (atsign) electrooptical (period) net
http://electrooptical.net

Bob Masta wrote:
> On Fri, 25 Feb 2011 09:34:04 -0800 (PST), Bret Cahill
> <Bret_E_Cahill@yahoo.com>  wrote:
>
>>> I'm not sure I understand what you're trying to do. Are you trying to
>>> get a FFT whose "amplitude" is a linear (either positive- or negative-
>>> going) ramp?
>>
>> Yes.
>>
>> You can take integer order or fractional order derivatives in the FFT
>> on Spice if you have something proportional to a ramp.
>>
>
> I'm not a Spice user, but all you need to do is take a
> normal FFT and apply a tilt to it, then do an IFFT (if you
> want to see the time waveform).  Doesn't Spice allow you to
> do this?
>
> The tilt should be +6 dB/octave for a normal derivative (see
> my prior post) or in your case some fraction of that.  Note
> that the raw FFT is linear in frequency and the "tilt" I am
> talking about is linear in *log* frequency, so you need a
> little exponential math here to generate the proper shape.
>

The best you can do for differentiation in a DFT is to take the central 
finite difference, but the DFT of that operation is not a ramp, linear, 
logarithmic, or exponential.

With an appropriately windowed transform, you can make a DFT look like 
samples of the continuous-time Fourier transform of a band-limited 
continuous time function.

Cheers

Phil Hobbs


-- 
Dr Philip C D Hobbs
Principal
ElectroOptical Innovations
55 Orchard Rd
Briarcliff Manor NY 10510
845-480-2058

email: hobbs (atsign) electrooptical (period) net
http://electrooptical.net

On 2/26/2011 10:02 AM, Phil Hobbs wrote:
> The best you can do for differentiation in a DFT is to take the central
> finite difference, but the DFT of that operation is not a ramp, linear,
> logarithmic, or exponential.

Phil,

I don't know what you mean by "best" re: the central finite difference 
in view of the need for using finite discrete transforms and/or windowed 
data.   As far as I know, a minimax FIR filter designed differentiator 
approximates an imaginary ramp in frequency *up to* some cutoff 
frequency - which is necessary to avoid aliasing I do believe.  Another 
approach to getting this is a Taylor approximation to the central finite 
difference of the first derivative which apparently comes in closed form 
(Khan and Ohba, 1999).  I believe these approaches are common practice 
in a case like this.

Here is an example of a length 51 differentiator of the minimax type 
done using the Parks-McClellan program:
[2.83E-05  -7.68E-05  1.34E-04  -2.24E-04  3.53E-04  -5.32E-04 7.73E-04 
  -1.09E-03  1.50E-03  -2.01E-03  2.66E-03  -3.46E-03 4.44E-03 
-5.64E-03  7.10E-03  -8.88E-03  1.11E-02  -1.38E-02  1.72E-02 -2.16E-02 
  2.76E-02  -3.63E-02  5.04E-02  -7.78E-02  1.58E-01
0
-1.58E-01  7.78E-02  -5.04E-02  3.63E-02  -2.76E-02  2.16E-02  -1.72E-02 
  1.38E-02  -1.11E-02  8.88E-03  -7.10E-03  5.64E-03  -4.44E-03 
3.46E-03  -2.66E-03  2.01E-03  -1.50E-03  1.09E-03  -7.73E-04  5.32E-04 
  -3.53E-04  2.24E-04  -1.34E-04  7.68E-05
-2.83E-05]

Fred

Fred Marshall wrote:
> On 2/26/2011 10:02 AM, Phil Hobbs wrote:
>> The best you can do for differentiation in a DFT is to take the central
>> finite difference, but the DFT of that operation is not a ramp, linear,
>> logarithmic, or exponential.
>
> Phil,
>
> I don't know what you mean by "best" re: the central finite difference
> in view of the need for using finite discrete transforms and/or windowed
> data. As far as I know, a minimax FIR filter designed differentiator
> approximates an imaginary ramp in frequency *up to* some cutoff
> frequency - which is necessary to avoid aliasing I do believe. Another
> approach to getting this is a Taylor approximation to the central finite
> difference of the first derivative which apparently comes in closed form
> (Khan and Ohba, 1999). I believe these approaches are common practice in
> a case like this.
>
> Here is an example of a length 51 differentiator of the minimax type
> done using the Parks-McClellan program:
> [2.83E-05 -7.68E-05 1.34E-04 -2.24E-04 3.53E-04 -5.32E-04 7.73E-04
> -1.09E-03 1.50E-03 -2.01E-03 2.66E-03 -3.46E-03 4.44E-03 -5.64E-03
> 7.10E-03 -8.88E-03 1.11E-02 -1.38E-02 1.72E-02 -2.16E-02 2.76E-02
> -3.63E-02 5.04E-02 -7.78E-02 1.58E-01
> 0
> -1.58E-01 7.78E-02 -5.04E-02 3.63E-02 -2.76E-02 2.16E-02 -1.72E-02
> 1.38E-02 -1.11E-02 8.88E-03 -7.10E-03 5.64E-03 -4.44E-03 3.46E-03
> -2.66E-03 2.01E-03 -1.50E-03 1.09E-03 -7.73E-04 5.32E-04 -3.53E-04
> 2.24E-04 -1.34E-04 7.68E-05
> -2.83E-05]
>
> Fred
>

Thanks, that's pretty.  I was perhaps a bit unclear in saying that the 
first finite difference is the best you can do, but the point at issue 
was what happens at frequencies near +-f_sample/2.  The first finite 
difference transforms to a sine wave with a 1/2 sample delay.

Optimal differentiators can be looked at as a variant of windowing a 
ramp, by using linear combinations of the first M finite differences to 
make an M+1 order approximation to the derivative over some frequency 
band, but they always roll off at high frequency, which is a key point.

Bob M. seems to be suggesting using a real ramp with no high-frequency 
cutoff, which will give what the compiler manuals describe as 
"unexpected results".

Cheers

Phil Hobbs


-- 
Dr Philip C D Hobbs
Principal
ElectroOptical Innovations
55 Orchard Rd
Briarcliff Manor NY 10510
845-480-2058

email: hobbs (atsign) electrooptical (period) net
http://electrooptical.net

On 2/26/2011 2:38 PM, Phil Hobbs wrote:

yep!

On 2/26/2011 2:38 PM, Phil Hobbs wrote:
> The first finite difference transforms to a sine wave with a 1/2 sample
> delay.

Yes.  I'm glad you didn't say cosine.  It's an imaginary sine wave at 
the lowest frequency.  So, it has one period.  As such, it goes from 
positive to negative at fs/2.  And, it's "ramp like" near f=0 as sin(x) 
= x for x < whatever small number (0.25 is 1% error)  So, it's a 
reasonable differentiator for frequencies below whatever limit you like 
there.  Not so good at pi/2 or fs/4!!

The "imaginary" part seems to be getting lost in the "ramp" terminology.
Note that the FIR filter I provided is real and purely odd in time - 
thus odd and purely imaginary in frequency (if centered at t=0).  So, 
any of these is an approximation to "jw" - an imaginary ramp.

Fred

On 26/02/2011 14:17, Bob Masta wrote:
> On Fri, 25 Feb 2011 08:05:00 -0500, Phil Hobbs
> <pcdhSpamMeSenseless@electrooptical.net>  wrote:
>
>> Bob Masta wrote:
>>> On Thu, 24 Feb 2011 09:10:50 -0800 (PST), Bret Cahill
>>> <Bret_E_Cahill@yahoo.com>   wrote:
>>>
>>>> Taking a time derivative after an FFT would be easy on SPICE if you
>>>> had some curve that would FFT into a ramp:  Just multiply the function
>>>> by the ramp.
>>>
>>> Not sure what you are ultimately trying to do, but note that
>>> you can obtain the FFT of the time derivative by taking the
>>> FFT of the raw waveform and applying a +6 dB/octave
>>> "envelope" to that... essentially, you just tilt the
>>> spectrum up at a 6 dB/octave slope.
>>>
>>> This turns out to be very handy for measuring frequency
>>> response of a system.  Classically, one can apply an impulse
>>> to the system and take the FFT to get the frequency
>>> response.  But an impulse is pretty narrow (one sample, in a
>>> digital system), so it doesn't have much energy.  A step
>>> response, on the other hand, has a whole lot more.  Since
>>> the derivatve of a step is an impulse, you can get the
>>> frequency response by applying a step, taking the FFT, and
>>> tilting it.  This is so handy that I built this feature into
>>> my Daqarta software.  See "Frequency Response Measurement -
>>> Step Response" at<http://www.daqarta.com/dw_0a0s.htm>.
>>>
>>
>>
>> Are you windowing the data before taking the DFT?  Could get ugly otherwise.
>
> No, in this case it's important to *not* window the data.
> A window function (at least, any of the standard ones) has a
> gradual onset and offset, for the specific purpose of
> eliminating transients at the start/end of the FFT frame.
> But here it is the onset that we are specifically interested
> in.  The transient response should be complete (for all
> practical purposes) before the end of the frame, or else you
> need more samples in the frame.

You will still get a better behaved result (ie one which is mainly due 
to the core transient with more predictable and well controlled 
artefacts if you put that in the middle of the window and tile it with a 
phantom unmeasured vertical axis reflection time shifted (or rather do a 
transform that uses that implied symmetry).

Otherwise you will have the result of your measured transient combined 
with the very sharp step down at the end of the range and associated 
high frequency components that may not be in your measured data.
>
> In general, you never want to window a transient or noise,
> only a continous wave.  The FFT analysis presumes a
> continuous wave, such that every frame is an identical copy
> that can be spliced seamlessly head to tail.  A real-world
> continuous wave that does not contain an exact integer
> number of cycles in the FFT frame will have a discontinuity
> where the next frame is spliced, which results in "spectral
> leakage" that appears as "skirts" on what would otherwise be
> a single line in the spectrum.  The window function provides
> a gradual onset and offset to smooth out this discontinuity,
> greatly reducing the spectral leakage.

Windowing to avoid sharp edge discontinuity artefacts is usually 
worthwhile - and for sampled data that might not be truly bandlimited 
and subject to aliassing it can make sense to preconvolve the raw data 
with a gridding function to get something that behaves much more like 
the ideal DFT. Prolate spheroidal Bessel functions are amongst the 
simplest which give real benefit in this application see for example:

http://www.astron.nl/aips++/docs/glossary/s.html#spheroidal_function

This method requires a multiplicative correction after the FFT but gives 
a final result much closer to the DFT.
>
> Interested readers may want to check out my "Gut Level
> Fourier Transforms" series at
> <http://www.daqarta.com/author.htm>.
> In particular, see Part 5 "Dumping Spectral Leakage Out a
> Window"<http://www.daqarta.com/eex05.htm>

What you say here about power of two transforms is no longer true. FFTW 
and other fast low prime transform kernels are now faster than naieve 
power of two implementations for some favourable sizes and competitive 
for many more. In some cases due to associative cache problems on 
certain processors exact powers of two are slower than they should be!

The pattern of FFT radix for which non power of two beats zero padding 
to the next power of two for my local FFT implementation is:

5, 6, 9, 36, 80, 81, 144, 160, 320, 625, 640, 1152,1250,1280,
1296, 1536, 2187, 2304, 2500, 2460, 2592, 3072, 3125, 3200

Taking only the ones where there is a clear win. Quite a lot of others 
are broadly competitive with the longer highly optimised 2^N transform.

3072 and 3125 are nearly 2x faster than an optimised radix-8 4096 and 3x 
faster than the naive radix-2 implementation for instance. An optimised 
split radix-16 transform would shave roughly another 10% off the 4096 
time but would not change the outcome.

Regards,
Martin Brown

On 2/28/2011 2:54 AM, Martin Brown wrote:

>
> The pattern of FFT radix for which non power of two beats zero padding
> to the next power of two for my local FFT implementation is:
>

I wonder if there isn't some misunderstanding here?  The notion of zero 
padding to achieve a power of two seems an archaic concern - as you well 
point out.  However, zero padding to accommodate circular convolution is 
common - much more common I should think.

Nonetheless, you've pointed out some things that are new info for me. 
Thanks.

Fred

Martin Brown wrote:
> On 26/02/2011 14:17, Bob Masta wrote:
>> On Fri, 25 Feb 2011 08:05:00 -0500, Phil Hobbs
>> <pcdhSpamMeSenseless@electrooptical.net> wrote:
>>
>>> Bob Masta wrote:
>>>> On Thu, 24 Feb 2011 09:10:50 -0800 (PST), Bret Cahill
>>>> <Bret_E_Cahill@yahoo.com> wrote:
>>>>
>>>>> Taking a time derivative after an FFT would be easy on SPICE if you
>>>>> had some curve that would FFT into a ramp: Just multiply the function
>>>>> by the ramp.
>>>>
>>>> Not sure what you are ultimately trying to do, but note that
>>>> you can obtain the FFT of the time derivative by taking the
>>>> FFT of the raw waveform and applying a +6 dB/octave
>>>> "envelope" to that... essentially, you just tilt the
>>>> spectrum up at a 6 dB/octave slope.
>>>>
>>>> This turns out to be very handy for measuring frequency
>>>> response of a system. Classically, one can apply an impulse
>>>> to the system and take the FFT to get the frequency
>>>> response. But an impulse is pretty narrow (one sample, in a
>>>> digital system), so it doesn't have much energy. A step
>>>> response, on the other hand, has a whole lot more. Since
>>>> the derivatve of a step is an impulse, you can get the
>>>> frequency response by applying a step, taking the FFT, and
>>>> tilting it. This is so handy that I built this feature into
>>>> my Daqarta software. See "Frequency Response Measurement -
>>>> Step Response" at<http://www.daqarta.com/dw_0a0s.htm>.
>>>>
>>>
>>>
>>> Are you windowing the data before taking the DFT? Could get ugly
>>> otherwise.
>>
>> No, in this case it's important to *not* window the data.
>> A window function (at least, any of the standard ones) has a
>> gradual onset and offset, for the specific purpose of
>> eliminating transients at the start/end of the FFT frame.
>> But here it is the onset that we are specifically interested
>> in. The transient response should be complete (for all
>> practical purposes) before the end of the frame, or else you
>> need more samples in the frame.
>
> You will still get a better behaved result (ie one which is mainly due
> to the core transient with more predictable and well controlled
> artefacts if you put that in the middle of the window and tile it with a
> phantom unmeasured vertical axis reflection time shifted (or rather do a
> transform that uses that implied symmetry).
>
> Otherwise you will have the result of your measured transient combined
> with the very sharp step down at the end of the range and associated
> high frequency components that may not be in your measured data.
>>
>> In general, you never want to window a transient or noise,
>> only a continous wave. The FFT analysis presumes a
>> continuous wave, such that every frame is an identical copy
>> that can be spliced seamlessly head to tail. A real-world
>> continuous wave that does not contain an exact integer
>> number of cycles in the FFT frame will have a discontinuity
>> where the next frame is spliced, which results in "spectral
>> leakage" that appears as "skirts" on what would otherwise be
>> a single line in the spectrum. The window function provides
>> a gradual onset and offset to smooth out this discontinuity,
>> greatly reducing the spectral leakage.
>
> Windowing to avoid sharp edge discontinuity artefacts is usually
> worthwhile - and for sampled data that might not be truly bandlimited
> and subject to aliassing it can make sense to preconvolve the raw data
> with a gridding function to get something that behaves much more like
> the ideal DFT. Prolate spheroidal Bessel functions are amongst the
> simplest which give real benefit in this application see for example:
>
> http://www.astron.nl/aips++/docs/glossary/s.html#spheroidal_function
>
> This method requires a multiplicative correction after the FFT but gives
> a final result much closer to the DFT.
>>
>> Interested readers may want to check out my "Gut Level
>> Fourier Transforms" series at
>> <http://www.daqarta.com/author.htm>.
>> In particular, see Part 5 "Dumping Spectral Leakage Out a
>> Window"<http://www.daqarta.com/eex05.htm>
>
> What you say here about power of two transforms is no longer true. FFTW
> and other fast low prime transform kernels are now faster than naieve
> power of two implementations for some favourable sizes and competitive
> for many more. In some cases due to associative cache problems on
> certain processors exact powers of two are slower than they should be!
>
> The pattern of FFT radix for which non power of two beats zero padding
> to the next power of two for my local FFT implementation is:
>
> 5, 6, 9, 36, 80, 81, 144, 160, 320, 625, 640, 1152,1250,1280,
> 1296, 1536, 2187, 2304, 2500, 2460, 2592, 3072, 3125, 3200
>
> Taking only the ones where there is a clear win. Quite a lot of others
> are broadly competitive with the longer highly optimised 2^N transform.
>
> 3072 and 3125 are nearly 2x faster than an optimised radix-8 4096 and 3x
> faster than the naive radix-2 implementation for instance. An optimised
> split radix-16 transform would shave roughly another 10% off the 4096
> time but would not change the outcome.
>
> Regards,
> Martin Brown

Interesting.  To clarify, you seem to be using 'DFT' to mean a 
sampled-time Fourier transform of infinite length, is that right?  IME 
that's usually used to denote the circular version for which the FFT is 
a fast algorithm.

I'm also less sanguine about resampling.  In the 1-D case at least, the 
orthogonal functions for the actual sampling grid used may be quite 
ill-conditioned, and resampling without taking account of any special 
features of the eigenfunctions can lead to tears.

Cheers

Phil Hobbs

-- 
Dr Philip C D Hobbs
Principal
ElectroOptical Innovations
55 Orchard Rd
Briarcliff Manor NY 10510
845-480-2058

email: hobbs (atsign) electrooptical (period) net
http://electrooptical.net

Fred Marshall wrote:
> On 2/28/2011 2:54 AM, Martin Brown wrote:
>
>>
>> The pattern of FFT radix for which non power of two beats zero padding
>> to the next power of two for my local FFT implementation is:
>>
>
> I wonder if there isn't some misunderstanding here? The notion of zero
> padding to achieve a power of two seems an archaic concern - as you well
> point out. However, zero padding to accommodate circular convolution is
> common - much more common I should think.
>
> Nonetheless, you've pointed out some things that are new info for me.
> Thanks.
>
> Fred
>

Zero padding a transient without windowing does zilch to reduce spectral 
leakage--all it does is to evaluate the exact same transform at 
different sample frequencies.  You can see that from the definition of 
the DFT (circular).

Cheers

Phil Hobbs

-- 
Dr Philip C D Hobbs
Principal
ElectroOptical Innovations
55 Orchard Rd
Briarcliff Manor NY 10510
845-480-2058

email: hobbs (atsign) electrooptical (period) net
http://electrooptical.net

Phil Hobbs wrote:
> Fred Marshall wrote:
>> On 2/28/2011 2:54 AM, Martin Brown wrote:
>>
>>>
>>> The pattern of FFT radix for which non power of two beats zero padding
>>> to the next power of two for my local FFT implementation is:
>>>
>>
>> I wonder if there isn't some misunderstanding here? The notion of zero
>> padding to achieve a power of two seems an archaic concern - as you well
>> point out. However, zero padding to accommodate circular convolution is
>> common - much more common I should think.
>>
>> Nonetheless, you've pointed out some things that are new info for me.
>> Thanks.
>>
>> Fred
>>
>
> Zero padding a transient without windowing does zilch to reduce spectral
> leakage--all it does is to evaluate the exact same transform at
> different sample frequencies. You can see that from the definition of
> the DFT (circular).
>

I should add that leaving space to avoid wraparound in the circular 
convolution (as Fred mentioned) doesn't suffer this problem.

Cheers

Phil Hobbs


-- 
Dr Philip C D Hobbs
Principal
ElectroOptical Innovations
55 Orchard Rd
Briarcliff Manor NY 10510
845-480-2058

email: hobbs (atsign) electrooptical (period) net
http://electrooptical.net

On 28/02/2011 16:21, Phil Hobbs wrote:
> Martin Brown wrote:
>> On 26/02/2011 14:17, Bob Masta wrote:
>>> On Fri, 25 Feb 2011 08:05:00 -0500, Phil Hobbs
>>> <pcdhSpamMeSenseless@electrooptical.net> wrote:
>>>
>>>> Bob Masta wrote:

>>> In general, you never want to window a transient or noise,
>>> only a continous wave. The FFT analysis presumes a
>>> continuous wave, such that every frame is an identical copy
>>> that can be spliced seamlessly head to tail. A real-world
>>> continuous wave that does not contain an exact integer
>>> number of cycles in the FFT frame will have a discontinuity
>>> where the next frame is spliced, which results in "spectral
>>> leakage" that appears as "skirts" on what would otherwise be
>>> a single line in the spectrum. The window function provides
>>> a gradual onset and offset to smooth out this discontinuity,
>>> greatly reducing the spectral leakage.
>>
>> Windowing to avoid sharp edge discontinuity artefacts is usually
>> worthwhile - and for sampled data that might not be truly bandlimited
>> and subject to aliassing it can make sense to preconvolve the raw data
>> with a gridding function to get something that behaves much more like
>> the ideal DFT. Prolate spheroidal Bessel functions are amongst the
>> simplest which give real benefit in this application see for example:
>>
>> http://www.astron.nl/aips++/docs/glossary/s.html#spheroidal_function
>>
>> This method requires a multiplicative correction after the FFT but gives
>> a final result much closer to the DFT.
>>>
>>> Interested readers may want to check out my "Gut Level
>>> Fourier Transforms" series at
>>> <http://www.daqarta.com/author.htm>.
>>> In particular, see Part 5 "Dumping Spectral Leakage Out a
>>> Window"<http://www.daqarta.com/eex05.htm>
>>
>> What you say here about power of two transforms is no longer true. FFTW
>> and other fast low prime transform kernels are now faster than naieve
>> power of two implementations for some favourable sizes and competitive
>> for many more. In some cases due to associative cache problems on
>> certain processors exact powers of two are slower than they should be!
>>
>> The pattern of FFT radix for which non power of two beats zero padding
>> to the next power of two for my local FFT implementation is:
>>
>> 5, 6, 9, 36, 80, 81, 144, 160, 320, 625, 640, 1152,1250,1280,
>> 1296, 1536, 2187, 2304, 2500, 2460, 2592, 3072, 3125, 3200
>>
>> Taking only the ones where there is a clear win. Quite a lot of others
>> are broadly competitive with the longer highly optimised 2^N transform.
>>
>> 3072 and 3125 are nearly 2x faster than an optimised radix-8 4096 and 3x
>> faster than the naive radix-2 implementation for instance. An optimised
>> split radix-16 transform would shave roughly another 10% off the 4096
>> time but would not change the outcome.
>
> Interesting. To clarify, you seem to be using 'DFT' to mean a
> sampled-time Fourier transform of infinite length, is that right? IME

Yes. I am using DFT here in the context of the idealised FT of infinite 
length that is zero outside the region of interest. The one that you 
would get by summing the discrete Fourier equations without the implicit 
FFT assumption of tiled or mirror boundary periodicity.

This becomes important in classical radio astronomy aperture synthesis 
when there are bright point sources in the field of view with grating 
ring artefacts from the telescope baseline configuration that cross the 
reconstructed image boundary!

> that's usually used to denote the circular version for which the FFT is
> a fast algorithm.

Sorry for any confusion.
>
> I'm also less sanguine about resampling. In the 1-D case at least, the
> orthogonal functions for the actual sampling grid used may be quite
> ill-conditioned, and resampling without taking account of any special
> features of the eigenfunctions can lead to tears.

The neat thing about the prolate spheroidal Bessel functions is that 
they are the Eigenfunctions of bandlimiting an FT to some finite width 
and then taking its FT again. This means that when you regrid data using 
one of them all components are accurately represented. The adhoc 
truncated Gaussian, Kaiser-Bessel or truncated Gaussian*Sinc functions 
commonly used have weaker alias rejection and/or in band distortions 
that affect high dynamic range data.

I don't agree with everything in this paper, but it provides an 
introduction and comparison of several popular FT gridding functions. 
The older papers are not online.

http://math.asu.edu/~svetlana/Sampling/Jackson,%20Meyer,%20Nishimura%20(9).pdf

There is another even more sophisticated family of gridding functions 
that sacrifice a guard band around the edge of the transform to obtain 
an extremely high fidelity FT of interferometer and MRI data. Basically 
you can't trust the edges or the corners because some leakage always 
occurs and these methods formalise the error bound in the trusted zone.

Regards,
Martin Brown

On Mon, 28 Feb 2011 10:54:46 +0000, Martin Brown
<|||newspam|||@nezumi.demon.co.uk> wrote:

>On 26/02/2011 14:17, Bob Masta wrote:
>> On Fri, 25 Feb 2011 08:05:00 -0500, Phil Hobbs
>> <pcdhSpamMeSenseless@electrooptical.net>  wrote:
>>
>>> Bob Masta wrote:
>>>> On Thu, 24 Feb 2011 09:10:50 -0800 (PST), Bret Cahill
>>>> <Bret_E_Cahill@yahoo.com>   wrote:
>>>>
>>>>> Taking a time derivative after an FFT would be easy on SPICE if you
>>>>> had some curve that would FFT into a ramp:  Just multiply the function
>>>>> by the ramp.
>>>>
>>>> Not sure what you are ultimately trying to do, but note that
>>>> you can obtain the FFT of the time derivative by taking the
>>>> FFT of the raw waveform and applying a +6 dB/octave
>>>> "envelope" to that... essentially, you just tilt the
>>>> spectrum up at a 6 dB/octave slope.
>>>>
>>>> This turns out to be very handy for measuring frequency
>>>> response of a system.  Classically, one can apply an impulse
>>>> to the system and take the FFT to get the frequency
>>>> response.  But an impulse is pretty narrow (one sample, in a
>>>> digital system), so it doesn't have much energy.  A step
>>>> response, on the other hand, has a whole lot more.  Since
>>>> the derivatve of a step is an impulse, you can get the
>>>> frequency response by applying a step, taking the FFT, and
>>>> tilting it.  This is so handy that I built this feature into
>>>> my Daqarta software.  See "Frequency Response Measurement -
>>>> Step Response" at<http://www.daqarta.com/dw_0a0s.htm>.
>>>>
>>>
>>>
>>> Are you windowing the data before taking the DFT?  Could get ugly otherwise.
>>
>> No, in this case it's important to *not* window the data.
>> A window function (at least, any of the standard ones) has a
>> gradual onset and offset, for the specific purpose of
>> eliminating transients at the start/end of the FFT frame.
>> But here it is the onset that we are specifically interested
>> in.  The transient response should be complete (for all
>> practical purposes) before the end of the frame, or else you
>> need more samples in the frame.
>
>You will still get a better behaved result (ie one which is mainly due 
>to the core transient with more predictable and well controlled 
>artefacts if you put that in the middle of the window and tile it with a 
>phantom unmeasured vertical axis reflection time shifted (or rather do a 
>transform that uses that implied symmetry).
>
>Otherwise you will have the result of your measured transient combined 
>with the very sharp step down at the end of the range and associated 
>high frequency components that may not be in your measured data.

Remember, this is in the context of converting a step
response into an impulse response.  The step response
waveform of a perfect system is a flat line at unity...
there is no step-down.  

In a real system that has an overall low-pass response, the
step response starts below unity and rises up to unity
(maybe with some overshoot and damped oscillation) and then
stays at unity until the end of the FFT period.  We don't
want to taper that down to zero... staying at unity is the
desired result.  

It's true that there is a conceptual step-down that is
off-screen, to prepare for the next rising edge.  But that
has no effect on the desired step response.

In the case of overall high-pass systems, the step response
droops down from the initial peak, and will be somewhere
less than unity by the end of the frame.  Although this
somehow seems more worrisome than the low-pass case, in fact
it is not really a problem for this approach.  The fact that
the step response is not satisfactorily completed means that
the low end of the frequency response is not fully captured.
That's all stuff that is between the 0th (DC) and 1st FFT
lines... a longer FFT is needed to resolve that.  

>> In general, you never want to window a transient or noise,
>> only a continous wave.  The FFT analysis presumes a
>> continuous wave, such that every frame is an identical copy
>> that can be spliced seamlessly head to tail.  A real-world
>> continuous wave that does not contain an exact integer
>> number of cycles in the FFT frame will have a discontinuity
>> where the next frame is spliced, which results in "spectral
>> leakage" that appears as "skirts" on what would otherwise be
>> a single line in the spectrum.  The window function provides
>> a gradual onset and offset to smooth out this discontinuity,
>> greatly reducing the spectral leakage.
>
>Windowing to avoid sharp edge discontinuity artefacts is usually 
>worthwhile - and for sampled data that might not be truly bandlimited 
>and subject to aliassing it can make sense to preconvolve the raw data 
>with a gridding function to get something that behaves much more like 
>the ideal DFT. Prolate spheroidal Bessel functions are amongst the 
>simplest which give real benefit in this application see for example:
>
>http://www.astron.nl/aips++/docs/glossary/s.html#spheroidal_function
>
>This method requires a multiplicative correction after the FFT but gives 
>a final result much closer to the DFT.
>>
>> Interested readers may want to check out my "Gut Level
>> Fourier Transforms" series at
>> <http://www.daqarta.com/author.htm>.
>> In particular, see Part 5 "Dumping Spectral Leakage Out a
>> Window"<http://www.daqarta.com/eex05.htm>
>
>What you say here about power of two transforms is no longer true. FFTW 
>and other fast low prime transform kernels are now faster than naieve 
>power of two implementations for some favourable sizes and competitive 
>for many more. In some cases due to associative cache problems on 
>certain processors exact powers of two are slower than they should be!
>
>The pattern of FFT radix for which non power of two beats zero padding 
>to the next power of two for my local FFT implementation is:
>
>5, 6, 9, 36, 80, 81, 144, 160, 320, 625, 640, 1152,1250,1280,
>1296, 1536, 2187, 2304, 2500, 2460, 2592, 3072, 3125, 3200
>
>Taking only the ones where there is a clear win. Quite a lot of others 
>are broadly competitive with the longer highly optimised 2^N transform.
>
>3072 and 3125 are nearly 2x faster than an optimised radix-8 4096 and 3x 
>faster than the naive radix-2 implementation for instance. An optimised 
>split radix-16 transform would shave roughly another 10% off the 4096 
>time but would not change the outcome.


Thanks for the FFTW reference.  I am gratified to see that
there are finally decent non-proprietary FFT routines
available for the public to use.  When I got started in this
adventure, back in the days of the IBM PC and XT, it was
"common knowledge" that you couldn't do real-time spectral
analysis on those systems without a dedicated custom add-in
card, but that "someday soon" we could hope to do it all on
the main CPU.  I realized that I had been hearing this line
for years already, in reference to the Apple, Commodore, and
Sinclair (etc).  It dawned on me that maybe nobody had
really tried to optimize an FFT for this purpose, since all
the references I could find were obviously not well
optimized.  They were written in FORTRAN or BASIC, computed
trig functions on the fly in every iteration, and did a few
other obviously redundant things.  Coding everything in
assembler, avoiding the deathly slow FPU, and doing trig by
table lookup instead of blindly-repeated recalculation gave
a speed improvement of over 100x... plenty fast enough for
real-time on a 4.77 MHz PX/XT when coupled with fast
assembly language display graphics.

Over the years, I have occasionally looked at touted "fast"
FFT routines, but they appeared to all suffer from the same
issues.  (Like assuming that you were only going to perform
one FFT, instead of pre-computing factors and swap tables so
you don't have to repeat it for every FFT.)

Now, however, I suspect that much faster FPUs (and SSE, and
GPUs) have made it actually better to recompute everything,
rather than incur cache misses while trying to access
precomputed tables.  It seems that technology has finally
vindicated the old "don't worry, someday the system will be
fast enough" !  <g>

Best regards,


Bob Masta
 
              DAQARTA  v6.00
   Data AcQuisition And Real-Time Analysis
              www.daqarta.com
Scope, Spectrum, Spectrogram, Sound Level Meter
    Frequency Counter, FREE Signal Generator
           Pitch Track, Pitch-to-MIDI 
          Science with your sound card!

Martin Brown wrote:
> On 28/02/2011 16:21, Phil Hobbs wrote:
>> Martin Brown wrote:
>>> On 26/02/2011 14:17, Bob Masta wrote:
>>>> On Fri, 25 Feb 2011 08:05:00 -0500, Phil Hobbs
>>>> <pcdhSpamMeSenseless@electrooptical.net> wrote:
>>>>
>>>>> Bob Masta wrote:
>
>>>> In general, you never want to window a transient or noise,
>>>> only a continous wave. The FFT analysis presumes a
>>>> continuous wave, such that every frame is an identical copy
>>>> that can be spliced seamlessly head to tail. A real-world
>>>> continuous wave that does not contain an exact integer
>>>> number of cycles in the FFT frame will have a discontinuity
>>>> where the next frame is spliced, which results in "spectral
>>>> leakage" that appears as "skirts" on what would otherwise be
>>>> a single line in the spectrum. The window function provides
>>>> a gradual onset and offset to smooth out this discontinuity,
>>>> greatly reducing the spectral leakage.
>>>
>>> Windowing to avoid sharp edge discontinuity artefacts is usually
>>> worthwhile - and for sampled data that might not be truly bandlimited
>>> and subject to aliassing it can make sense to preconvolve the raw data
>>> with a gridding function to get something that behaves much more like
>>> the ideal DFT. Prolate spheroidal Bessel functions are amongst the
>>> simplest which give real benefit in this application see for example:
>>>
>>> http://www.astron.nl/aips++/docs/glossary/s.html#spheroidal_function
>>>
>>> This method requires a multiplicative correction after the FFT but gives
>>> a final result much closer to the DFT.
>>>>
>>>> Interested readers may want to check out my "Gut Level
>>>> Fourier Transforms" series at
>>>> <http://www.daqarta.com/author.htm>.
>>>> In particular, see Part 5 "Dumping Spectral Leakage Out a
>>>> Window"<http://www.daqarta.com/eex05.htm>
>>>
>>> What you say here about power of two transforms is no longer true. FFTW
>>> and other fast low prime transform kernels are now faster than naieve
>>> power of two implementations for some favourable sizes and competitive
>>> for many more. In some cases due to associative cache problems on
>>> certain processors exact powers of two are slower than they should be!
>>>
>>> The pattern of FFT radix for which non power of two beats zero padding
>>> to the next power of two for my local FFT implementation is:
>>>
>>> 5, 6, 9, 36, 80, 81, 144, 160, 320, 625, 640, 1152,1250,1280,
>>> 1296, 1536, 2187, 2304, 2500, 2460, 2592, 3072, 3125, 3200
>>>
>>> Taking only the ones where there is a clear win. Quite a lot of others
>>> are broadly competitive with the longer highly optimised 2^N transform.
>>>
>>> 3072 and 3125 are nearly 2x faster than an optimised radix-8 4096 and 3x
>>> faster than the naive radix-2 implementation for instance. An optimised
>>> split radix-16 transform would shave roughly another 10% off the 4096
>>> time but would not change the outcome.
>>
>> Interesting. To clarify, you seem to be using 'DFT' to mean a
>> sampled-time Fourier transform of infinite length, is that right? IME
>
> Yes. I am using DFT here in the context of the idealised FT of infinite
> length that is zero outside the region of interest. The one that you
> would get by summing the discrete Fourier equations without the implicit
> FFT assumption of tiled or mirror boundary periodicity.
>
> This becomes important in classical radio astronomy aperture synthesis
> when there are bright point sources in the field of view with grating
> ring artefacts from the telescope baseline configuration that cross the
> reconstructed image boundary!
>
>> that's usually used to denote the circular version for which the FFT is
>> a fast algorithm.
>
> Sorry for any confusion.
>>
>> I'm also less sanguine about resampling. In the 1-D case at least, the
>> orthogonal functions for the actual sampling grid used may be quite
>> ill-conditioned, and resampling without taking account of any special
>> features of the eigenfunctions can lead to tears.
>
> The neat thing about the prolate spheroidal Bessel functions is that
> they are the Eigenfunctions of bandlimiting an FT to some finite width
> and then taking its FT again. This means that when you regrid data using
> one of them all components are accurately represented. The adhoc
> truncated Gaussian, Kaiser-Bessel or truncated Gaussian*Sinc functions
> commonly used have weaker alias rejection and/or in band distortions
> that affect high dynamic range data.
>
> I don't agree with everything in this paper, but it provides an
> introduction and comparison of several popular FT gridding functions.
> The older papers are not online.
>
> http://math.asu.edu/~svetlana/Sampling/Jackson,%20Meyer,%20Nishimura%20(9).pdf
>
>
> There is another even more sophisticated family of gridding functions
> that sacrifice a guard band around the edge of the transform to obtain
> an extremely high fidelity FT of interferometer and MRI data. Basically
> you can't trust the edges or the corners because some leakage always
> occurs and these methods formalise the error bound in the trusted zone.
>
> Regards,
> Martin Brown

Thanks.  I suppose in the design of radiotelescope arrays, folks keep an 
eye on the condition number of the transformation, at least at some 
level, and you know the Jacobian analytically, which is also a big help. 
  Stuff like missing samples in a time series can give rise to all sorts 
of funnies if you try just interpolating.

Cheers

Phil Hobbs

-- 
Dr Philip C D Hobbs
Principal
ElectroOptical Innovations
55 Orchard Rd
Briarcliff Manor NY 10510
845-480-2058

email: hobbs (atsign) electrooptical (period) net
http://electrooptical.net

On 01/03/2011 16:26, Phil Hobbs wrote:

>
> Thanks. I suppose in the design of radiotelescope arrays, folks keep an
> eye on the condition number of the transformation, at least at some
> level, and you know the Jacobian analytically, which is also a big help.

The history is quite interesting in that the interferometer arrays 
became more complex and completely beyond analytical treatment as 
computers became more powerful.

The first Earth rotation aperture synthesis telescope the One Mile 
Telescope at Cambridge had 2 fixed dishes and one movable on a very 
accurately surveyed E-W line. I think 64 baselines at uniform spacing 
(32 days observing) and 32 days moving the scope configuration.

The next generation 5km telescope was about 5 degrees of perfect E-W 
alignment (thanks to Dr Beeching closing the Cambridge-Bedford railway) 
and consisted of 4 fixed and 4 movable. 128 uniformly spaced baselines 
in 8 days observing and 8 days moving the scope around.

Big disadvantage of these E-W arrays is that their beamshape is lousy 
near the celestial equator and you have to observe for 12 hours 
continuous to get a decent image of the target. But on the plus side the 
mathematics for gridding the data is easier.

The VLA moved to a Y shape with 27 antennae I forget how many fixed and 
how many movable and a power law antenna spacing which allows more or 
less instant snapshots. When commissioned at the opening party it did 
some quick give away 5 minute maps of famous objects for VIPs to take 
away. It require massively more computing power to do this. The 
beamshape is pretty hairy and must be deconvolved by either CLEAN or MEM 
to provide an image or map suitable for astrophysics.

If you are interesting in the details a decent talk is online at
http://www.aoc.nrao.edu/epo/powerpoint/interferometry2001.pdf
It loses something by not having a narrator but if you are already 
familiar with the basics it should make reasonable sense. It shows what 
the U-V baseline coverage, beamshape, raw and final maps look like.

International VLBI you are pretty much stuck with where the biggest 
dishes are physically located on the planet and have to make do with 
what is one offer. The beamshape is horrible and it can only 
realistically be used on very bright compact targets. Cunning 
calibration techinques using triples of baselines extract good 
observables despite the unknown phase errors over each telescope. The 
latter method has been applied in the optical at near IR wavelengths.

> Stuff like missing samples in a time series can give rise to all sorts
> of funnies if you try just interpolating.

Indeed. If you have missing data the only reliable way of modelling it 
is to pose the question what is the most probable model of the thing 
being measured that would yield the observed data to within the 
measurement noise. Attempts at direct inverses with missing data can end 
up dominated by the misleading zeroes or guessed missing data.

Astronomers do have some advantaged here. The sky brightness is known to 
be everywhere positive so negative regions have to be artefacts. The 
same cannot be said for electrical signals...

Regards,
Martin Brown