Selecting a Tranfer Function

Spectral Matching: Select a transfer function that matches a desired steady-state spectrum for a particular distortion index $a$.

This may be done using Chebyshev polynomials of the first kind, denoted $T_k(x)$, where $k$ is the order of the polynomial.

The zeroth- and first-order Chebyshev polynomials are given by

$$T_0(x) = 1$$

$$T_1(x) = x$$

and higher-order polynomials are given by

$$T_{k+1}(x) = 2xT_k(x)-T_{k-1}(x).$$

These polynomials have the property that when a sinusoid of unit amplitude is applied to the input, the output signal contains only the $k^{\mbox{th}}$ harmonic.