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### Bandwidth of a time-limited pure sinusoidal signal

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```Hi Folks,

I have a very basic question. I am little bit confused about how to know
the bandwidth of a time-limited pure sinusoidal signal. I understand
bandwidth is defined simply as the difference between highest frequency and
lowest frequency, and the bandwidth of a infinitely long pure sinusoid if 0
Hz. But if I have a N sample long 50 Hz sinusoid (sampled at Fs samples per
second), how to find bandwith of this signal?

Thanks,
-- cwoptn

```
 0
Reply gopi.allu (1) 8/5/2010 10:02:13 PM

```On Aug 5, 3:02=A0pm, "cwoptn" <gopi.allu@n_o_s_p_a_m.gmail.com> wrote:
> Hi Folks,
>
> I have a very basic question. I am little bit confused about how to know
> the bandwidth of a time-limited pure sinusoidal signal. I understand
> bandwidth is defined simply as the difference between highest frequency a=
nd
> lowest frequency, and the bandwidth of a infinitely long pure sinusoid if=
0
> Hz. But if I have a N sample long 50 Hz sinusoid (sampled at Fs samples p=
er
> second), how to find bandwith of this signal?
>
> Thanks,
> -- cwoptn

The bandwidth of the truncated pure sinusoid is equal to the
"effective noise bandwidth" (enbw) of the truncating function, often
given in terms of dft bins (Fs/N). For a rectangular truncation
function (window), the enbw is 1.0, so 1.0 x Fs / N.

For other truncating functions, you can look in the usual windows
references like:
On the Use of Windows for Harmonic Analysis
with the Discrete Fourier Transform
fred harris,
from the IEEE proceedings. available at:
http://web.mit.edu/xiphmont/Public/windows.pdf
(beware errors in some Blackman and Blackman-Harris window parameters)

See section IV, A on page 54.

Dale B. Dalrymple

```
 0

```cwoptn wrote:
> Hi Folks,
>
> I have a very basic question. I am little bit confused about how to know
> the bandwidth of a time-limited pure sinusoidal signal. I understand
> bandwidth is defined simply as the difference between highest frequency and
> lowest frequency, and the bandwidth of a infinitely long pure sinusoid if 0
> Hz. But if I have a N sample long 50 Hz sinusoid (sampled at Fs samples per
> second), how to find bandwith of this signal?
>
> Thanks,
> -- cwoptn
>
>

A lot depends on what you need the "bandwidth" measure for.
A truncated sampled sinusoid will have these characteristics in frequency:
- if the number of periods is an integer then there will be a single
sample pair in frequency.
- if the number of periods isn't an integer then there will be samples
with nonzero value throughout frequency that correspond to the Dirichlet
of the window (like a periodic sinc function).  In that case, the
bandwidth is as much as it can possibly be.  But, the energy is
concentrated at the frequency of the sine above and below fs or zero if
you will.
- if the window isn't rectangular then you may be able to limit the
perceived bandwidth to something less for any particular sinusoid.

Fred
```
 0

```dbd  <dbd@ieee.org> wrote:

>On Aug 5, 3:02�pm, "cwoptn" <gopi.allu@n_o_s_p_a_m.gmail.com> wrote:

>> I have a very basic question. I am little bit confused about how to know
>> the bandwidth of a time-limited pure sinusoidal signal. I understand
>> bandwidth is defined simply as the difference between highest frequency and
>> lowest frequency, and the bandwidth of a infinitely long pure sinusoid if 0
>> Hz. But if I have a N sample long 50 Hz sinusoid (sampled at Fs samples per
>> second), how to find bandwith of this signal?

>The bandwidth of the truncated pure sinusoid is equal to the
>"effective noise bandwidth" (enbw) of the truncating function, often
>given in terms of dft bins (Fs/N). For a rectangular truncation
>function (window), the enbw is 1.0, so 1.0 x Fs / N.

>For other truncating functions, you can look in the usual windows
>references like:
>On the Use of Windows for Harmonic Analysis
>with the Discrete Fourier Transform
>fred harris,
>from the IEEE proceedings. available at:
>http://web.mit.edu/xiphmont/Public/windows.pdf
>(beware errors in some Blackman and Blackman-Harris window parameters)

I find it interesting how often a continuous-time question

S.
```
 0

```On Aug 5, 7:23=A0pm, spop...@speedymail.org (Steve Pope) wrote:
> dbd =A0<d...@ieee.org> wrote:
> >On Aug 5, 3:02=A0pm, "cwoptn" <gopi.allu@n_o_s_p_a_m.gmail.com> wrote:
> >> I have a very basic question. I am little bit confused about how to kn=
ow
> >> the bandwidth of a time-limited pure sinusoidal signal. I understand
> >> bandwidth is defined simply as the difference between highest frequenc=
y and
> >> lowest frequency, and the bandwidth of a infinitely long pure sinusoid=
if 0
> >> Hz. But if I have a N sample long 50 Hz sinusoid (sampled at Fs sample=
s per
> >> second), how to find bandwith of this signal?
> >The bandwidth of the truncated pure sinusoid is equal to the
> >"effective noise bandwidth" (enbw) of the truncating function,

this i get...

> > often given in terms of dft bins (Fs/N).

.... that i don't.

> I find it interesting how often a continuous-time question

and, i guess i'm not alone.

since a time-limited signal can't also be bandlimited, then the answer
depends on how one defines "bandwidth" for something that stretches
out to infinity on one or both sides.  then for that i think Fred said
it well: "A lot depends on what you need the "bandwidth" measure for."

r b-j

```
 0

```On 08/05/2010 04:23 PM, Steve Pope wrote:
> dbd<dbd@ieee.org>  wrote:
>
>> On Aug 5, 3:02 pm, "cwoptn"<gopi.allu@n_o_s_p_a_m.gmail.com>  wrote:
>
>>> I have a very basic question. I am little bit confused about how to know
>>> the bandwidth of a time-limited pure sinusoidal signal. I understand
>>> bandwidth is defined simply as the difference between highest frequency and
>>> lowest frequency, and the bandwidth of a infinitely long pure sinusoid if 0
>>> Hz.

>>>>>>>  LOOK HERE  >>

But if I have a N sample long 50 Hz sinusoid (sampled at Fs samples per
second),

<<<<<<<<<<

>>> how to find bandwith of this signal?
>
>> The bandwidth of the truncated pure sinusoid is equal to the
>> "effective noise bandwidth" (enbw) of the truncating function, often
>> given in terms of dft bins (Fs/N). For a rectangular truncation
>> function (window), the enbw is 1.0, so 1.0 x Fs / N.
>
>> For other truncating functions, you can look in the usual windows
>> references like:
>> On the Use of Windows for Harmonic Analysis
>> with the Discrete Fourier Transform
>> fred harris,
>>from the IEEE proceedings. available at:
>> http://web.mit.edu/xiphmont/Public/windows.pdf
>> (beware errors in some Blackman and Blackman-Harris window parameters)
>
> I find it interesting how often a continuous-time question
>

Discrete time question -- although the answer is just as valid in
continuous time.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html
```
 0
Reply tim177 (4404) 8/6/2010 4:54:02 AM

```Tim Wescott  <tim@seemywebsite.com> replies to my post,

> >>>>>>>  LOOK HERE  >>
>
>But if I have a N sample long 50 Hz sinusoid (sampled at Fs samples per
>second),

><<<<<<<<<<
>> I find it interesting how often a continuous-time question

>Discrete time question -- although the answer is just as valid in
>continuous time.

Okay you're right.   I should not have jumped on that one.

Steve
```
 0

```Hi Folks,

Thank you again for all your valuable inputs. So if I use rectangular
window of N samples as the truncating function, the bandwidth of the
resulting signal (for all practical purposes) is simply the main lobe width
of the Sinc function (corresponding to N sample long rectangular window in
time domain).

Thanks again,
-- cwoptn
```
 0

```On Aug 6, 9:47=A0am, "cwoptn" <gopi.allu@n_o_s_p_a_m.gmail.com> wrote:
>
>  So if I use rectangular
> window of N samples as the truncating function, the bandwidth of the
> resulting signal (for all practical purposes) is simply the main lobe wid=
th
> of the Sinc function (corresponding to N sample long rectangular window i=
n
> time domain).

if that is how you define the bandwidth of the rectangular pulse
signal to begin with, yes.  some might define such bandwidth
differently (e.g. the difference between the -3 dB points).  there is
no final definitive definition of bandwidth, as far as i can tell from
the lit.  different definitions pop up in different applications.

r b-j

```
 0

```On 08/06/2010 06:47 AM, cwoptn wrote:
> Hi Folks,
>
> Thank you again for all your valuable inputs. So if I use rectangular
> window of N samples as the truncating function, the bandwidth of the
> resulting signal (for all practical purposes) is simply the main lobe width
> of the Sinc function (corresponding to N sample long rectangular window in
> time domain).

That's a good definition of the "useful communications" bandwidth.  But
it's not a good definition at all of the "doesn't interfere with

Any time your spectrum isn't a perfect rectangle* you have to define
what you mean by bandwidth for your immediate purpose -- and be prepared

* And no real-world signal is going to have a perfectly rectangular
spectrum.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html
```
 0
Reply tim177 (4404) 8/6/2010 4:37:54 PM

```robert bristow-johnson  <rbj@audioimagination.com> wrote:

>On Aug 6, 9:47�am, "cwoptn" <gopi.allu@n_o_s_p_a_m.gmail.com> wrote:

>>  So if I use rectangular
>> window of N samples as the truncating function, the bandwidth of the
>> resulting signal (for all practical purposes) is simply the main lobe width
>> of the Sinc function (corresponding to N sample long rectangular window in
>> time domain).

>if that is how you define the bandwidth of the rectangular pulse
>signal to begin with, yes.  some might define such bandwidth
>differently (e.g. the difference between the -3 dB points).  there is
>no final definitive definition of bandwidth, as far as i can tell from
>the lit.  different definitions pop up in different applications.

I concur.  3 dB bandwidth is one common term.  Equivalent noise bandwidth
is another common term and is for most functions somewhat larger.
Yet a third term is "occupied bandwidth", an even larger measure
encompassig almost all of the signal power.

One possible confusing point: you need to look at the bandwidth
of the sinc function after it has been translated to the center
frequency defined by the sinusoid.  So because of this it is a bandpass,
and not a lowpass function.

Steve
```
 0

```On Aug 5, 7:23=A0pm, spop...@speedymail.org (Steve Pope) wrote:
> dbd =A0<d...@ieee.org> wrote:
> >On Aug 5, 3:02=A0pm, "cwoptn" <gopi.allu@n_o_s_p_a_m.gmail.com> wrote:
> >> I have a very basic question. I am little bit confused about how to kn=
ow
> >> the bandwidth of a time-limited pure sinusoidal signal. I understand
> >> bandwidth is defined simply as the difference between highest frequenc=
y and
> >> lowest frequency, and the bandwidth of a infinitely long pure sinusoid=
if 0
> >> Hz. But if I have a N sample long 50 Hz sinusoid (sampled at Fs sample=
s per
> >> second), how to find bandwith of this signal?
> >The bandwidth of the truncated pure sinusoid is equal to the
> >"effective noise bandwidth" (enbw) of the truncating function, often
> >given in terms of dft bins (Fs/N). For a rectangular truncation
> >function (window), the enbw is 1.0, so 1.0 x Fs / N.
> >For other truncating functions, you can look in the usual windows
> >references like:
> >On the Use of Windows for Harmonic Analysis
> >with the Discrete Fourier Transform
> >fred harris,
> >from the IEEE proceedings. available at:
> >http://web.mit.edu/xiphmont/Public/windows.pdf
> >(beware errors in some Blackman and Blackman-Harris window parameters)
>
> I find it interesting how often a continuous-time question

I find it even more interesting how often a discrete-time

Greg
```
 0
Reply heath (3875) 8/6/2010 5:50:50 PM

```Greg Heath  <heath@alumni.brown.edu> wrote:

>I find it even more interesting how often a discrete-time

Good one!

S.
```
 0

```cwoptn wrote:
> Hi Folks,
>
> Thank you again for all your valuable inputs. So if I use rectangular
> window of N samples as the truncating function, the bandwidth of the
> resulting signal (for all practical purposes) is simply the main lobe width
> of the Sinc function (corresponding to N sample long rectangular window in
> time domain).
>
> Thanks again,
> -- cwoptn

I'd not say that flat out - but you may..  As I mentioned earlier, the
sinc or Dirichlet "tails" show up all over the place - also called
"spectral leakage".  So then it's a matter of what's important in
defining "bandwidth".

Fred
```
 0

```On Aug 6, 6:47=A0am, "cwoptn" <gopi.allu@n_o_s_p_a_m.gmail.com> wrote:
> So if I use rectangular
> window of N samples as the truncating function, the bandwidth of the
> resulting signal (for all practical purposes) is simply the main lobe wid=
th
> of the Sinc function (corresponding to N sample long rectangular window i=
n
> time domain).

The second lobe peaks at over 20% of the main lobe.
Do you care about those outlying "frequencies" in your
bandwidth requirement or definition?

--
rhn A.T nicholson d.0.t C-o-M

```
 0

```On Aug 6, 4:03=A0pm, "Ron N." <rhnlo...@yahoo.com> wrote:
> On Aug 6, 6:47=A0am, "cwoptn" <gopi.allu@n_o_s_p_a_m.gmail.com> wrote:
>
> > So if I use rectangular
> > window of N samples as the truncating function, the bandwidth of the
> > resulting signal (for all practical purposes) is simply the main lobe w=
idth
> > of the Sinc function (corresponding to N sample long rectangular window=
in
> > time domain).
>
> The second lobe peaks at over 20% of the main lobe.
> Do you care about those outlying "frequencies" in your
> bandwidth requirement or definition?
>
> --
> rhn A.T nicholson d.0.t C-o-M

It is merely interesting that the effective bandwidth of the
rectangular truncation function is the same as the (half) width of the
main lobe of the Fourier transform of the truncation function. The
effective bandwidth is calculated from the application of Parseval's
theorem to the samples of the truncation function, as shown in the
cited reference, and includes the power in all sidelobes.

Dale B. Dalrymple
```
 0

```On Aug 6, 7:03=A0pm, "Ron N." <rhnlo...@yahoo.com> wrote:
> On Aug 6, 6:47=A0am, "cwoptn" <gopi.allu@n_o_s_p_a_m.gmail.com> wrote:
>
> > So if I use rectangular
> > window of N samples as the truncating function, the bandwidth of the
> > resulting signal (for all practical purposes) is simply the main lobe w=
idth
> > of the Sinc function (corresponding to N sample long rectangular window=
in
> > time domain).
>
> The second lobe peaks at over 20% of the main lobe.

If the bandwidth is defined w.r.t. power, the 20% amplitude
converts to only 4% in power.

MATLAB:

figure
plot(0:0.1:10, sinc(0:0.1:10))
hold on, plot(0:0.1:10, sinc(0:0.1:10).^2,'r')

Hope this helps.

Greg

```
 0

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