application of discrete fourier series and discrete fourier transform

Hello,

I am able to appreciate what exactly are DFS and DFT, but I would like to
know the actual applications of these. I do know that we do
design/analysis of LTI systems using DFT, mostly using response to impulse
inputs. 

But what is the use of DFS? If yes, wouldn't DFT do the same thing?
If not, what would be effect of taking DFT of a discrete signal?

Thanks a lot ...
---------------------------------------
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Sharan123 <99077@dsprelated> wrote:
 
> I am able to appreciate what exactly are DFS and DFT, but I would like to
> know the actual applications of these. I do know that we do
> design/analysis of LTI systems using DFT, mostly using response to impulse
> inputs. 
 
> But what is the use of DFS? If yes, wouldn't DFT do the same thing?
> If not, what would be effect of taking DFT of a discrete signal?

Yes, it seems to me that DFT (and FFT) are misnamed, as the result is
a series of discrete terms, not the way a transform usually works.

The use of DFT (and rarely called DFS) is similar to the Fourier
transform in general, to go between frequency space and time,
or between spatial frequency and position. 

If you have enough points, and the transform is reasonably smooth,
the discrete forms are a good approximation to the continuous ones.

Among the Fourier transform pairs are:

discrete <--> periodic
continuous <--> not periodic

The usual way to show this (not obvious when taught at 8:00 AM on
a Monday morning by a physics TA) is to consider the periodic case
in the limit as the period goes to infinity. 

-- glen

On Fri, 16 Oct 2015 08:57:32 -0500, Sharan123 wrote:

> Hello,
> 
> I am able to appreciate what exactly are DFS and DFT, but I would like
> to know the actual applications of these. I do know that we do
> design/analysis of LTI systems using DFT, mostly using response to
> impulse inputs.
> 
> But what is the use of DFS? If yes, wouldn't DFT do the same thing?
> If not, what would be effect of taking DFT of a discrete signal?

As Glen said, "Discrete Fourier Series" is (probably, in most books) a 
synonym for "Discrete Fourier Transform".

Deep understanding for this needs to go in reverse order from learning 
the math: to really understand you want to start with the Fourier 
Transform, then go from there to the Fourier series (in continuous 
time).  Once you have that down, then use the model of sampling as a 
train-of-impulses and derive the DFT and discrete-time Fourier transform 
(DTFT, if I have my nomenclature right, is the transform of an infinitely 
long sequence of samples into the frequency domain which is continuous in 
time but bounded in extent, or periodic, your choice).

I'm not sure how much of it can be taught, and how much you have to find 
for yourself by reflection on the pertinent mathematics.

-- 

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

On Fri, 16 Oct 2015 17:21:28 +0000, glen herrmannsfeldt wrote:

> Sharan123 <99077@dsprelated> wrote:
>  
>> I am able to appreciate what exactly are DFS and DFT, but I would like
>> to know the actual applications of these. I do know that we do
>> design/analysis of LTI systems using DFT, mostly using response to
>> impulse inputs.
>  
>> But what is the use of DFS? If yes, wouldn't DFT do the same thing?
>> If not, what would be effect of taking DFT of a discrete signal?
> 
> Yes, it seems to me that DFT (and FFT) are misnamed, as the result is a
> series of discrete terms, not the way a transform usually works.
> 
> The use of DFT (and rarely called DFS) is similar to the Fourier
> transform in general, to go between frequency space and time,
> or between spatial frequency and position.
> 
> If you have enough points, and the transform is reasonably smooth, the
> discrete forms are a good approximation to the continuous ones.
> 
> Among the Fourier transform pairs are:
> 
> discrete <--> periodic continuous <--> not periodic
> 
> The usual way to show this (not obvious when taught at 8:00 AM on a
> Monday morning by a physics TA) is to consider the periodic case in the
> limit as the period goes to infinity.

Hopefully I'm not re-igniting a flame war, but:

discrete <--> periodic or bounded

Depending on how you want to define the "meaning" of the Fourier series.

-- 

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

Tim Wescott <seemywebsite@myfooter.really> wrote:

(snip, I wrote)

>> Among the Fourier transform pairs are:
 
>> discrete <--> periodic continuous <--> not periodic
 
>> The usual way to show this (not obvious when taught at 8:00 AM on a
>> Monday morning by a physics TA) is to consider the periodic case in the
>> limit as the period goes to infinity.
 
> Hopefully I'm not re-igniting a flame war, but:
 
> discrete <--> periodic or bounded
 
> Depending on how you want to define the "meaning" of the Fourier series.

Doesn't bother me enough to start a flame war.

I don't tend to say it, since the inverse transforms are not bounded.

But yes, with a transform using an appropriate period, one can use a
periodic transform on a bounded function. 

I think that there are some transforms that only work for bounded
sources, though I can't think of any right now.

-- glen

Tim Wescott <seemywebsite@myfooter.really> wrote:
> On Fri, 16 Oct 2015 08:57:32 -0500, Sharan123 wrote:
 
(snip)

>> But what is the use of DFS? If yes, wouldn't DFT do the same thing?
>> If not, what would be effect of taking DFT of a discrete signal?
 
> As Glen said, "Discrete Fourier Series" is (probably, in most books) a 
> synonym for "Discrete Fourier Transform".
 
> Deep understanding for this needs to go in reverse order from learning 
> the math: to really understand you want to start with the Fourier 
> Transform, then go from there to the Fourier series (in continuous 
> time).  

I learned about Fourier series pretty early, at least enough to learn
that a square wave was a sum of sines, and maybe a few other shapes, too.

But then, as I noted earlier, I had the Fourier transform explained,
at 8:00 AM Monday (or maybe Wednesday) morning from my Physics 1 TA, 
with the only explanation being the limit of the Fourier series.

It took me some time to actually understand what that meant.

You can draw sin(x), sin(x)+sin(3x)/3, etc., and see how it gets closer.

But for other cases, the continuous form is easier.

There are many functions that I can integrate, but not do an infinite
sum over. (Well, without looking them up.)

> Once you have that down, then use the model of sampling as a 
> train-of-impulses and derive the DFT and discrete-time Fourier transform 
> (DTFT, if I have my nomenclature right, is the transform of an infinitely 
> long sequence of samples into the frequency domain which is continuous in 
> time but bounded in extent, or periodic, your choice).
 
> I'm not sure how much of it can be taught, and how much you have to find 
> for yourself by reflection on the pertinent mathematics.

-- glen