discrete fourier series and transform

Sirs,

Can you explain what is discrete fourier series vis-a-vis discrete fourier
transform?

Thanks, Manish

"manishp" <58525@dsprelated> writes:

> Sirs,
>
> Can you explain what is discrete fourier series vis-a-vis discrete fourier
> transform?

Hi Manish,

This is actually a very good question. The analysis and synthesis
equations are identical. So what's the difference between the two?

The difference between the two is in the assumptions about the input. In
the DFS, the input is assumed to be periodic. In the N-point DFT, the
input x[n] can be any discrete-time signal (periodic or not), and in
which we construct an intermediate, periodic signal xbar[n] from x[n]
that is the N samples from x[n] we are analyziing periodically extended.

So the DFT provides us with the spectrum of N samples of x[n] as if
those N samples were repeated. That is not the same as the spectrum of
x[n], e.g., as from the Discrete-Time Fourier Transform (see note).

--Randy

Note:

I am using the various Fourier transform terms as in

@BOOK{signalsandsystems,
  title = "{Signals and Systems}",
  author = "{Alan~V.~Oppenheim, Alan~S.~Willsky, with Ian~T.~Young}",
  publisher = "Prentice Hall",
  year = "1983"}

-- 
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com

On Sun, 23 Dec 2012 12:46:16 -0500, Randy Yates
<yates@digitalsignallabs.com> wrote:

>"manishp" <58525@dsprelated> writes:
>
>> Sirs,
>>
>> Can you explain what is discrete fourier series vis-a-vis discrete fourier
>> transform?

Manish, are there specific issues that you're having difficulty with?
This can be a fairly broad topic and the definitions that you may find
may not be universal. 

>Hi Manish,
>
>This is actually a very good question. The analysis and synthesis
>equations are identical. So what's the difference between the two?
>
>The difference between the two is in the assumptions about the input. In
>the DFS, the input is assumed to be periodic. In the N-point DFT, the
>input x[n] can be any discrete-time signal (periodic or not), and in
>which we construct an intermediate, periodic signal xbar[n] from x[n]
>that is the N samples from x[n] we are analyziing periodically extended.

I'd like to note that the construction of an intermediate periodic
signal xbar(n) as Randy has described it and the periodic extension is
merely a conceptual step to help some in understanding the process.
Many do not feel that that is a necessary step, but it can be helpful
for some in understanding the process or interpreting the result.  

>So the DFT provides us with the spectrum of N samples of x[n] as if
>those N samples were repeated. That is not the same as the spectrum of
>x[n], e.g., as from the Discrete-Time Fourier Transform (see note).
>
>--Randy
>
>Note:
>
>I am using the various Fourier transform terms as in
>
>@BOOK{signalsandsystems,
>  title = "{Signals and Systems}",
>  author = "{Alan~V.~Oppenheim, Alan~S.~Willsky, with Ian~T.~Young}",
>  publisher = "Prentice Hall",
>  year = "1983"}
>

I either don't have or can't find my copy of Oppenheim and Willsky,
but in other texts from Oppenheim, e.g., Digitial Signal Processing
and Discrete Time Signal Processing, periodic input sequences are used
for the analysis, but it is not stated (anywhere that I can find)
that this is necessary.  In Discrete Time Signal Processing the
analysis of the DFT is described using an approach that does not
require any assumption of periodicity in the input.   I built on that
approach in this article to amplify the point:

http://www.dsprelated.com/showarticle/175.php

As Randy stated a common distinction between the DFS and DFT is that
the DFS model assumes a periodic input, but the Wikipedia entry (which
I agree with) states otherwise:

http://en.wikipedia.org/wiki/Discrete_Fourier_series

Likewise, a Berkely article starts with an assumption of periodicity,
but then also calls the same treatment the Discrete-Time Fourier
Series, which some people hold as distinct from the DFS:

http://ptolemy.eecs.berkeley.edu/eecs20/week8/dfs.html

I could go on in showing various definitions or descriptions that
differ in significant ways, but I think you get the idea.  So
confusion on the topic is understandable.   If this is for a class,
use whatever definition your Prof. or Department prefers.   Otherwise,
in my experience it is a good idea to clarify with a source what their
definition is to avoid confusion in a particular context.  Depending
on the source or context the DFS may or may not be equivalent to the
DFT.





Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com

On 12/23/12 8:00 AM, manishp wrote:
>
> Can you explain what is discrete fourier series vis-a-vis discrete fourier
> transform?
>

this is a issue that has literally gotten us into a tiff here at 
comp.dsp .  i'm definitely a partisan in that dispute.

Randy's answer is as good as any.

for me, i'm not willing to grant that there is any difference between 
the DFT and the DFS if there is no mathematical difference.  which there 
isn't.

the consequence is, when you yank N adjacent samples from x[n] and pass 
it to the DFT or DFS, you are periodically extending that set of N 
samples.  whether you call it a DFT or DFS, if you perform any action on 
one domain that causes shifting in the other domain, that shifting is 
necessarily circular, as if x[n] is periodic with period N meaning: 
x[n+N] = x[n]  for all integer n.

so periodicity deniers have to use modulo arithmetic in the argument of 
x[n] (i.e. x[ n mod_N ]) when the apply the same theorems regarding 
shifting.  that way they can say that x[n] is undefined for
n < 0 and n > N-1.  i find little point in doing that.  imposing mod_N 
arithmetic to the argument of x[n] is identical to periodically 
extending it.  i don't know why they just can say x[n] = x[n+N] .

so the DFT transforms one periodic discrete sequence of period N to 
another periodic discrete sequence of the same period.  it is also true 
that for a finite set of samples x[n] (for 0 <= n < N), that if you were 
to *zero* extend it out to forever and compute the DTFT of that.  and 
then sample the result of the DTFT at N equally spaced samples, that 
would also be the DFT.  and the act of sampling it exactly corresponds 
to periodic extension.

there's just no way around the property of periodicity.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

manishp <58525@dsprelated> wrote:
 
> Can you explain what is discrete fourier series vis-a-vis discrete fourier
> transform?

Fourier series and Fourier transform are reasonably well defined.

The transform is the integral form, and the series the sum form.

Seems to me that what is commonly called the FFT should instead
be FFS, but it is a little late now.

It might depend on the intent, instead the actual use.

If one intends it as a discrete approximation of a continuous
transform, or as a discrete transform itself.

-- glen

On 12/23/2012 4:33 PM, robert bristow-johnson wrote:

   ...

 > there's just no way around the property of periodicity.

Sure there is. There are people around who avtually believe that the 
world ended when the Mayan calender is purported to have said it would, 
and that what we experience now is merely a periodic extension. 
Periodicity -- or its lack -- seems to be in the eye of the beholder.
Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Jerry, i dunno when, but next time i have to be in Joisey, i wanna look 
you up again.  you are comp.dsp's sage.

On 12/24/12 5:59 PM, Jerry Avins wrote:
> On 12/23/2012 4:33 PM, robert bristow-johnson wrote:
>
> ...
>
>  > there's just no way around the property of periodicity.
>
> Sure there is. There are people around who actually believe that the
> world ended when the Mayan calender is purported to have said it would,
> and that what we experience now is merely a periodic extension.

well, an interesting thought that i first heard from Daniel Dennett to 
support his thesis of physicalism and to deal with the issue of 
consciousness or qualia is that we are essentially automatons who 
*think* they have consciousness.

wouldn't that be a similar existence?

> Periodicity -- or its lack -- seems to be in the eye of the beholder.

sorta like Beauty.

interesting that both are properties of mathematics.  and i understand 
that mathematical aesthetic can be in the eye of the beholder.  i 
usually equate it with elegance and, for me, more elegance is always 
better as long as it remains accurate (well, of course there are times 
to make the elegant approximation and make use of it, sometimes we do 
that with an icky cost function and turn it into something you can turn 
into a closed-form solution).  but i always prefer elegance and, i'll 
admit, i have my own ideas about what elegance means.

but, there is also the concrete in mathematics.  concrete things like 
Boolean algebra.  if something looks, walks, and quacks like a duck, you 
really have to find something clearly incompatible with duck to say it 
ain't.  i don't see anywhere that the periodicity deniers have done 
that.  they *do* show examples where periodicity is not evident, but 
that is different than showing that the DFT is inconsistent with 
periodicity.  where they take issue with me is i am saying that the DFT 
is *never* inconsistent with periodicity and, of course, that there are 
situations where periodicity is required.  not just elegant (it's also 
more elegant because you never have to worry about when it doesn't 
happen) but *required*.  it is required whenever shifting is done and it 
is never incompatible with when shifting is not done.

if you say that x[n] is undefined for n < 0  and  N >= n, then that's 
just a definition and the difference between that and defining it 
outside outside that interval has no consequence.

or if you say that x[n] is zero outside of 0 <= n < N the DTFT of that 
is continuous and periodic.  but at this point, of course x[n] is not 
periodic (unless it's zero). then when you define the DFT as the 
sampling of the DTFT at N equally-spaced frequencies between -pi and 
+pi, that action necessarily causes x[n] to be periodically shifted and 
overlap-added.  and since it was zero outside that interval, then 
nothing but zeros get added up except one term, and that is x[n-m*N] for 
some integer m.

or you can just say that the DFT maps a periodic sequence with period N 
to another periodic sequence of the same period and the iDFT back again. 
  this is equivalent to saying that the DFT periodically extends the 
finite data set passed to it, and that is i believe the most concise way 
to put it.

either way, it's unavoidable.  just turning one's face away does not 
change the fact.

Jerry, bestest regrads..

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

On 12/24/2012 10:13 PM, robert bristow-johnson wrote:
>
> Jerry, i dunno when, but next time i have to be in Joisey, i wanna look
> you up again. you are comp.dsp's sage.
>
> On 12/24/12 5:59 PM, Jerry Avins wrote:
>> On 12/23/2012 4:33 PM, robert bristow-johnson wrote:
>>
>> ...
>>
>> > there's just no way around the property of periodicity.
>>
>> Sure there is. There are people around who actually believe that the
>> world ended when the Mayan calender is purported to have said it would,
>> and that what we experience now is merely a periodic extension.
>
> well, an interesting thought that i first heard from Daniel Dennett to
> support his thesis of physicalism and to deal with the issue of
> consciousness or qualia is that we are essentially automatons who
> *think* they have consciousness.

I believe that... for everyone else.  I have no way to distinguish 
whether you are conscious or a very well programmed automaton.  But I 
know I am conscious.  I think, therefore I am conscious.

Rick

On 12/25/12 10:15 PM, rickman wrote:
> On 12/24/2012 10:13 PM, robert bristow-johnson wrote:
>>
>> Jerry, i dunno when, but next time i have to be in Joisey, i wanna look
>> you up again. you are comp.dsp's sage.
>>
>> On 12/24/12 5:59 PM, Jerry Avins wrote:
>>> On 12/23/2012 4:33 PM, robert bristow-johnson wrote:
>>>
>>> ...
>>>
>>> > there's just no way around the property of periodicity.
>>>
>>> Sure there is. There are people around who actually believe that the
>>> world ended when the Mayan calender is purported to have said it would,
>>> and that what we experience now is merely a periodic extension.
>>
>> well, an interesting thought that i first heard from Daniel Dennett to
>> support his thesis of physicalism and to deal with the issue of
>> consciousness or qualia is that we are essentially automatons who
>> *think* they have consciousness.
>
> I believe that... for everyone else. I have no way to distinguish
> whether you are conscious or a very well programmed automaton. But I
> know I am conscious. I think, therefore I am conscious.

so you are a being with consciousness and feelings but everyone else are 
sophisticated automatons?  are you the only such being?

i dunno what this would be called, but could we apply a sorta 
"Copernican principle" but replace "the Earth" with "oneself"?


-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

On Tue, 25 Dec 2012 22:15:45 -0500, rickman <gnuarm@gmail.com> wrote:

>On 12/24/2012 10:13 PM, robert bristow-johnson wrote:
>>
>> Jerry, i dunno when, but next time i have to be in Joisey, i wanna look
>> you up again. you are comp.dsp's sage.
>>
>> On 12/24/12 5:59 PM, Jerry Avins wrote:
>>> On 12/23/2012 4:33 PM, robert bristow-johnson wrote:
>>>
>>> ...
>>>
>>> > there's just no way around the property of periodicity.
>>>
>>> Sure there is. There are people around who actually believe that the
>>> world ended when the Mayan calender is purported to have said it would,
>>> and that what we experience now is merely a periodic extension.
>>
>> well, an interesting thought that i first heard from Daniel Dennett to
>> support his thesis of physicalism and to deal with the issue of
>> consciousness or qualia is that we are essentially automatons who
>> *think* they have consciousness.
>
>I believe that... for everyone else.  I have no way to distinguish 
>whether you are conscious or a very well programmed automaton.  But I 
>know I am conscious.  I think, therefore I am conscious.
>
>Rick

So you think.   You can't prove that your consciousness isn't
deterministic and therefore programmed.


Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com