DFT of sinc signal

Sirs,

Looking at the DFT of a sinc signal got me thinking about these questions.

If the DFT is about separating the frequency components of a signal then
how come DFT of sinc signal is a rectangle in the frequency domain, meaning
multiple frequency components in frequency domain.

A related question, are the lower frequencies corresponding to upper values
of the sinc function (that is higher values of n in sinc(n)). And, if I
were to convert the sinc functional to a sinusoidal signal, the lower
frequencie would disappear?

Again, the question is mostly from theoretical perspective. To some extent,
I understand the maths part and I can see the conversion is happening.

Sorry for the long post though ...

thanks, manish

PS: one thing I have been wondering is, how to attach/inline pictures in
dsprelated forum posts. That would be very helpful, especially when dealing
with special chars or signal waveforms ...	 

_____________________________		
Posted through www.DSPRelated.com

"manishp" <58525@dsprelated> writes:

> Sirs,
>
> Looking at the DFT of a sinc signal got me thinking about these
> questions.

How can you take the DFT of a sinc signal? The sinc signal (function) is
infinite.
-- 
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com

Randy Yates <yates@digitalsignallabs.com> wrote:
> "manishp" <58525@dsprelated> writes:
 
>> Looking at the DFT of a sinc signal got me thinking about these
>> questions.
 
> How can you take the DFT of a sinc signal? The sinc signal 
> (function) is infinite.

The same way you do anything in DSP, use a large N and hope it
is large enough. Note, for example, that Nyquist sampling only
works (theoretically) for an infinite number of points. It is
a good approcximation for large N, not so good for small N.

I presume, then, the OP is trying to understand how multiplying
sin(x) by (1/x) changes its spectrum. That is, what modulaion
does to a signal. There should be some good references for
the spectrum of an AM radio signal. That is, an amplitude
modulated sine wave. 

-- glen

glen herrmannsfeldt <gah@ugcs.caltech.edu> writes:

> Randy Yates <yates@digitalsignallabs.com> wrote:
>> "manishp" <58525@dsprelated> writes:
>  
>>> Looking at the DFT of a sinc signal got me thinking about these
>>> questions.
>  
>> How can you take the DFT of a sinc signal? The sinc signal 
>> (function) is infinite.
>
> The same way you do anything in DSP, use a large N and hope it
> is large enough. 

Then that's not "the sinc signal." That was my point. And if you're not
accounting for that fact, you're bound to run into problems.
-- 
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com

On 8/5/13 3:47 AM, manishp wrote:
>
> Looking at the DFT of a sinc signal got me thinking about these questions.
>
> If the DFT is about separating the frequency components of a signal then
> how come DFT of sinc signal is a rectangle in the frequency domain, meaning
> multiple frequency components in frequency domain.
>
> A related question, are the lower frequencies corresponding to upper values
> of the sinc function (that is higher values of n in sinc(n)). And, if I
> were to convert the sinc functional to a sinusoidal signal, the lower
> frequencie would disappear?

not sure what the meaning of the question is.

>
> Again, the question is mostly from theoretical perspective. To some extent,
> I understand the maths part and I can see the conversion is happening.
>
>

the Fourier Transform of the sinc() function is the rectangular pulse 
(often called rect()) and since the DFT is a sorta approximation to the 
F.T., then the DFT of a sinc() (that would have to be windowed or at 
least truncated) is going to be an approximation to the rectangular 
pulse function.

>
> PS: one thing I have been wondering is, how to attach/inline pictures in
> dsprelated forum posts. That would be very helpful, especially when dealing
> with special chars or signal waveforms ...	

comp.dsp is not dsprelated.com (which provides access to comp.dsp).  i 
wouldn't think it's such a good idea to attach docs to USENET posts, but 
if you can find a place that you can park your graphic to (and is in 
some common format like png or something), then put a link to it in your 
comp.dsp post.


-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

Randy Yates <yates@digitalsignallabs.com> wrote:

(snip, I wrote)
>> The same way you do anything in DSP, use a large N and hope it
>> is large enough. 
 
> Then that's not "the sinc signal." That was my point. And if you're 
> not accounting for that fact, you're bound to run into problems.

Make N large enough, and you won't see the difference. For 1D cases,
computers are pretty much always fast enough. For 2D, though, it 
isn't so hard to run into problems that take longer than you want
to sit and wait. In that case, run it, eat dinner, get results.

With the built-in graphics, this should take minutes to write
and view in Matlab or Octave. 

(I have Octave on a different computer. Maybe I will try later.)

-- glen

glen herrmannsfeldt <gah@ugcs.caltech.edu> writes:

> Randy Yates <yates@digitalsignallabs.com> wrote:
>
> (snip, I wrote)
>>> The same way you do anything in DSP, use a large N and hope it
>>> is large enough. 
>  
>> Then that's not "the sinc signal." That was my point. And if you're 
>> not accounting for that fact, you're bound to run into problems.
>
> Make N large enough, and you won't see the difference. 

You're in the practical domain. I'm in the analytical domain. In
the analytical domain, they will never be the same. I presumed
the poster was asking a theoretical/analytical question.
-- 
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com

Randy Yates <yates@digitalsignallabs.com> wrote:

(snip, I wrote)
>>>> The same way you do anything in DSP, use a large N and hope it
>>>> is large enough. 
  
>>> Then that's not "the sinc signal." That was my point. And if you're 
>>> not accounting for that fact, you're bound to run into problems.

>> Make N large enough, and you won't see the difference. 
 
> You're in the practical domain. I'm in the analytical domain. In
> the analytical domain, they will never be the same. I presumed
> the poster was asking a theoretical/analytical question.

Even in the analytical domain, you can still do limits.

In the limit as N approaches infinity, it will be a sinc, and
the transform will get closer and closer as N gets larger.

Well, some time ago there was a post about the FT of Gaussian not
being Gaussian. Again, you have to have a large enough N, though
also the origin in the right place. 

Seems like Matlab and Octave have the fftshift() function for
people who get that wrong.

-- glen

On Mon, 05 Aug 2013 21:35:30 +0000, glen herrmannsfeldt wrote:

> Randy Yates <yates@digitalsignallabs.com> wrote:
> 
> (snip, I wrote)
>>> The same way you do anything in DSP, use a large N and hope it is
>>> large enough.
>  
>> Then that's not "the sinc signal." That was my point. And if you're not
>> accounting for that fact, you're bound to run into problems.
> 
> Make N large enough, and you won't see the difference. For 1D cases,
> computers are pretty much always fast enough. For 2D, though, it isn't
> so hard to run into problems that take longer than you want to sit and
> wait. In that case, run it, eat dinner, get results.
> 
> With the built-in graphics, this should take minutes to write and view
> in Matlab or Octave.

Why not just take the Fourier transform of the sinc signal, or look it up?

Or take the discrete-time Fourier transform of the sinc signal (now that 
should be interesting -- but it should work).

Math by experimentation is sometimes absolutely necessary, but doing real 
math (i.e., symbolically) will often get you a deeper understanding.

-- 

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

Tim Wescott <tim@seemywebsite.really> wrote:

(snip, I wrote)
>> Make N large enough, and you won't see the difference. For 1D cases,
>> computers are pretty much always fast enough. For 2D, though, it isn't
>> so hard to run into problems that take longer than you want to sit and
>> wait. In that case, run it, eat dinner, get results.
 
>> With the built-in graphics, this should take minutes to write and view
>> in Matlab or Octave.
 
> Why not just take the Fourier transform of the sinc signal, 
> or look it up?

Well, yes, but sometimes the visual result is more convincing.

High school math classes now tend to require a graphing calculator.
The ability to type a function in and quickly see its shape allows
one to learn faster. 

Similarly, they are tending to do more numerical results, such
as integrals, than we did not so many years ago. (Or many years ago.)

Being able to look at the transform, change the function a little,
and look at the new transform is a different way of learning than
writing down expressions.
 
> Or take the discrete-time Fourier transform of the sinc signal 
> (now that should be interesting -- but it should work).
 
> Math by experimentation is sometimes absolutely necessary, 
> but doing real math (i.e., symbolically) will often get 
> you a deeper understanding.

Deeper, but different. 

-- glen