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*From*: John Denker <jsd@av8n.com>*Date*: Thu, 08 Jan 2009 01:35:46 -0700

On 01/07/2009 11:44 PM, Stefan Jeglinski wrote:

In the limit as É¢x gets small and NÉ¢x 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.

Hmmm. Not sure I would use this wording,

Alas, my wording was not of the best. I used two premises

to draw three conclusions. It would be better to untangle

them. I quote from a new section

http://www.av8n.com/physics/fourier-refined.htm#sec-correspondence

In the spirit of the correspondence principle, let’s examine the

relationship between a discrete Fourier transform and a non-discrete

Fourier transform. We start with a discrete transform and see what

happens:

1. As previously discussed: When dt becomes small, the

frequency-range of the first period of the output becomes large. (The

output is always defined for all frequencies, so we can’t talk about

the range of the output, only the range of one period.)

2. As previously discussed: When Ndt becomes large, the output

becomes nearly continuous.

3. If you do both together – small dt and large Ndt – the discrete

Fourier transform begins to look a whole lot like the old-fashioned

non-discrete Fourier transform.

OK?

I have seen discussions in which the comment is

made: sampling "causes" the periodic extension. I

have felt this is either a very deep statement,

or a very poorly-worded statement. Your use of

the word "tantamount" doesn't help me decide :-)

You can't have a sampled input without having the output of the

transform be periodic. The proof is simple:

Consider the case where the original signal was built up from

sine-wave components to begin with. Draw the signal,

A sin(2 π f t)

for some "signal" frequency f. Now sample it, i.e. circle

some evenly-spaced points on the graph of the signal.

Now draw another curve on top of the first, namely:

A sin(2 π [f + f_S] t)

or more generally

A sin(2 π [f + k f_S] t) for any integer k

where the sampling frequency is f_S := 1/dt. Convince yourself

that the new signal necessarily goes through exactly the same

sample points. It goes nuts between sample points, but it

comes home at the sample points.

That's all there is to it. Evenly sampled input implies

periodically continued output. The spacing of the samples

determines the length of the period of the output, f_S := 1/dt.

**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>

**Re: [Phys-l] some questions related to sampling***From:*John Denker <jsd@av8n.com>

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

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