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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.
You can't have a sampled input without having the output of the
transform be periodic. The proof is simple: