Discussion
What does the time-bandwidth product tell a system designer?
Answer: C
It states a floor, not an equality. The product of a signal's effective duration and its effective bandwidth cannot fall below a bound set by the Fourier transform itself, so any attempt to localise a signal sharply in one domain spreads it in the other. That is why radar trades pulse length against bandwidth, why a short observation cannot resolve two close tones, and why the Gaussian, which attains the bound, is of special interest. The relation is an inequality rather than the exact reciprocal, and the product varies with waveform shape while never going below the limit. Longer signals in fact tend to occupy less bandwidth, which is the opposite of the last option.
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