What is the relationship between the width of a signal in time and the width of its Fourier transform?

• A. They are directly proportional.
• B. They are independent.
• C. They are inversely proportional.
• D. They are equal.

Answer: (C)

If a signal is very short in time, it changes quickly, which requires a wide range of frequencies to reproduce those sharp features. Conversely, a signal that lasts longer in time changes more slowly and can be represented mainly with low-frequency components, giving a much narrower spectrum. This inverse relationship is a manifestation of the Fourier duality and is often summarized by the time–frequency uncertainty principle: the product of the temporal width and the spectral width is bounded, so making one larger forces the other to be smaller. For a short pulse of width T, the main spectral content spreads roughly over a bandwidth on the order of 1/T; doubling the duration T reduces the spectral width by about a factor of two. Hence, the widths are inversely proportional.