Power Laws

More generally,

$$x^k(n) \;\longleftrightarrow\; \underbrace{(X\ast X \ast \cdots \ast X)}_{\mbox{$k$\ times}}(\omega)$$

so that the spectral bandwidth of $x^k(n)$ is $k$ times that of $x(n)$, in general.

In summary, the spectrum at the output of the square-law nonlinearity can be written as

$$Y(\omega) = X(\omega) + \alpha (X\ast X)(\omega)$$

This illustrates the great value of series expansions for nonlinearities. In audio signal processing systems, it is essential to keep track of the spectral representation of all output signals, since we hear directly in terms of signal spectra.^S.4