Bandwidth of a time-limited pure sinusoidal signal

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

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

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

dbd  <dbd@ieee.org> wrote:

>On Aug 5, 3:02&#4294967295;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
leads to a discrete-time answer on this newsgroup. 

S.

On Aug 5, 7:23�pm, spop...@speedymail.org (Steve Pope) wrote:
> dbd &#4294967295;<d...@ieee.org> wrote:
> >On Aug 5, 3:02&#4294967295;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,

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
> leads to a discrete-time answer on this newsgroup.

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

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
> leads to a discrete-time answer on this newsgroup.
>

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

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
>> leads to a discrete-time answer on this newsgroup.

>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

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

On Aug 6, 9:47&#4294967295;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.

r b-j

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 
adjacent channel" bandwidth.

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 
to change your definition when your immediate purposes change.

* 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