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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.
> 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.