The relevance of can be appreciated by formulating the Fourier transform of . Let us define

(22.7)

Eqn. (22.7) represents the signal obtained by passing through a filter that admits only frequency around ω. The bandwidth of the filter is 1/T . It can be shown that is related to the spectrum according to

(22.8)

The spectrum thus represents the mean-square amplitude of the filtered signal or the mean-square amplitude of the Fourier coefficient of at ω. It may be thought of as the energy in at that frequency.