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If the Fourier transform of an aperiodic continuous time signal has signal components between the minimum frequency w1 and the maximum frequency w2, but not all the frequencies between w1 and w2, is the bandwidth said to be w2-w1? Or does it require that there be signal components at every frequency in the bandwidth?

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The answer to this question could be tricky.

For a simple answer you may take the bandwidth of a bandpass signal as $\omega_2 - \omega_1$ (highest minus the lowest frequencies), and don't care about what happens in between the two. Hence your bandwidth allocation is not fully efficient, and a sampling based on this bandwidth will include redundant information of those empty bands.

Thus, for a more rigorous answer about finding the absolute minimum number of samples to represent a bandpass signal whose spectrum may include empty bands (gaps) between the minimum and maximum fequencies, you would take only the nonzero intervals into count.

Yet when the signal is to be transmitted over a physical channel without frequency allocation control, then you would again consider the minimum and maximum frequencies as the bandwidth, unless cross channel spectrum ovarlapping is allowed.

So the answer depends on your intentions.

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