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

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

In the limit as ɢx gets small and Nɢx 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

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

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.


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.