Polynomial Transfer Functions

• Under certain limited circumstances, the waveshaping operation can be described analytically.

• In particular, when a waveshaping transfer function is expressed in terms of a polynomial, it is relatively simple to determine the resulting output signal spectrum weighting given a sinusoidal input of amplitude 1.

• A generic waveshaping polynomial transfer function can be expressed as:
  $$f(x) = d_0 + d_1 x + d_2 x^2 + \ldots + d_N x^N$$

• Also under these conditions, no spectral energy is produced at frequencies higher than the highest exponent value times the input signal frequency.

• Odd powers of the waveshaping function produce odd harmonic distortion. Even powers of the waveshaping function produce even harmonic distortion.

• Even exponential powers produce even functions having the property $f(-x) = f(x)$, which produced even harmonic distortion.

• Odd exponential powers produce odd functions having the property $f(-x) = -f(x)$, which produced odd harmonic distortion.

• For example, if we process the sinusoid from Fig. 13 with a waveshaping function of the form $f(x) = x + x^2 + x^3$, we get a resulting spectra as shown in Fig. 21.

  Figure 21: Input and output spectra for a waveshaping function of the form $f(x) = x + x^2 + x^3$.

  

• We can also think of waveshaping with polynomials as a form of modulation synthesis. In the above example, when the sinusoidal signal (of frequency $f_1$) is squared, we form sum and different components at 0 and $2f_1$ Hz. When the squared signal is again multiplied by the original sinusoid, we also get components at $f_1$ and $3f_1$.

• The amount of harmonic distortion produced by these memoryless nonlinearities is correlated to the “sharpness” of the characteristic curve.

• Like FM, waveshaping synthesis can be used to vary the bandwidth and spectrum of a sound.

• Unlike FM, waveshaping can be used to create bandlimited spectra with a specific maximum harmonic number (though the FM modulation index will play a similar role).