Re: [Phys-l] some questions related to sampling

[John Denker <jsd@av8n.com>]

On 01/07/2009 03:46 PM, Stefan Jeglinski wrote:
> I'm ultimately going somewhere with this, but feel I need to get some 
> things clear in my head first. While it may not be the ideal starting 
> place, I want to begin with this article:
>
> http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform
>
> You will have to read/skim the article for context. I'm not sure 
> whether some of my confusion is from being dense, or just reading an 
> article that is not well-written/organized.

I wouldn't worry about feeling confused.  This stuff takes
some getting used to.  When I was in school, on N occasions
people tried really hard to teach me this stuff, but it did
not sink in.  Later, when I really needed it, I figured it
out.  The process involved doing nothing else for a week.  I
had a program that I needed to write.  I wrote the first
version, debugged it for a couple of days, and then decided
it was undebuggable so I threw it out and started over from
scratch.  The second version met the same fate.  The third
version worked fine.

> In particular, I am 
> interested in this quote from the article:
>
> =========
> The DFT and the DTFT can be viewed as the logical result of applying 
> the standard continuous Fourier transform to discrete data. From that 
> perspective, we have the satisfying result that it's not the 
> transform that varies, it's just the form of the input:
>      If it is discrete, the Fourier transform becomes a DTFT.
>      If it is periodic, the Fourier transform becomes a Fourier series.
>      If it is both, the Fourier transform becomes a DFT.
> =========
>
>
> I'm not getting the *significance* of the difference between the DTFT 
> and the DFT. 

There is no significance.  There is no difference for that matter.

The math and the software don't care whether the input is sampled 
in the time direction or the x direction or the y direction or the 
temporal frequency direction or the spatial frequency direction.  
The classification given above is, AFAICT, good for nothing.

There are only two or three things that matter
       Δx          the spacing between input points
        N           the number of samples, and
       N Δx        the interval spanned by the input data

If you are interested in time series you can replace x by t,
but again, the math absolutely does not care.

The spacing between points in _output_ space is given by the
reciprocal of N Δx, so if your input data spans a large 
interval the the output will be finely spaced, and in the
limit begins to resemble a continuous function.

Meanwhile, the output is going to be periodic with period 
equal to the sampling frequency, f_S := 1/Δx.  This gives
rise to _aliasing_.  The output will be defined for all f,
but because it is periodic, there is not much point in
plotting more than one period.  In the limit as Δx becomes
small, the output period gets larger and larger.  

In the limit as Δx gets small and NΔx gets big, the output
becomes nearly continuous and spans a huge frequency range,
in which case the discrete Fourier transform looks a whole
lot like the old fashioned non-discrete Fourier transform.




> I have been applying the DFT to sampled data (specifically a 
> single-pulse, hence non-periodic, waveform) 

There is no such thing.  All DFT methods are tantamount to
*assuming* that your input signal is periodically continued.
In particular, if you are privately assuming that your signal
is zero before and after the interval spanned by your data,
do not assume that the software knows this.  Take your data,
pad it with a bunch of zeros before and after, and take the
transform.  You will get a different result.  You might very
well like the new result better.


For additional detail on all of this, see
  http://www.av8n.com/physics/fourier-refined.htm