Nonlinear Modifications

In Fourier terms, the simplest nonlinearity is a square law. Consider an FFT processor that squares each frame spectrum:

$$Y_m(\omega_k) = X_m^2(\omega_k)$$

In the time domain, each frame is convolved with itself:

$$y_m(n) = (x_m*x_m)(n)$$

Since $x_m$ is time limited to $M$ samples, we can avoid time domain aliasing by requiring $N \ge 2M-1$ .

More generally, we can consider

$$Y_m(\omega_k) = X_m^l(\omega_k).$$

This can be thought of as $l$ cascaded convolutions of $x_m$ with itself. The resulting signal will be at most $l(M-l)+1$ samples long. We can avoid time domain aliasing in this case by requiring

$$N \ge l(M-1)+1.$$

We can express a general class of nonlinearities as a polynomial in the spectrum:

$$Y_m(\omega_k) = \sum_l^{K-1} \alpha_l X_m^l(\omega_k)$$

In this case, we require $N \ge K(M-1)+1$ to avoid time aliasing.

For related information, look into Volterra series expansions [20]. The interated-convolution expansion above can be regarded as a special case of a Volterra series expansion.