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*From*: John Denker <jsd@av8n.com>*Date*: Wed, 07 Jan 2009 16:43:19 -0700

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

**Follow-Ups**:**Re: [Phys-l] some questions related to sampling***From:*Stefan Jeglinski <jeglin@4pi.com>

**References**:**[Phys-l] some questions related to sampling***From:*Stefan Jeglinski <jeglin@4pi.com>

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