Amplitude Dependence

• When the sinusoidal input amplitude is given by $a$ (instead of 1), the output is then scaled by $a$ as:
  $$f(ax) = d_0 + d_1 a x + d_2 a^2 x^2 + \ldots + d_N a^N x^N$$

• Because higher values of $a$ generally produce richer spectra, $a$ is referred to as the distortion index.

• Because the results of waveshaping with complex input signals are difficult to predict, most waveshaping applications use sinusoidal input signals only.

• When a sinusoid of unity amplitude is applied to a Chebyshev polynomial of order $k$, the output contains energy only at the $k$th harmonic. This property makes Chebyshev polynomials potentially useful for building more complex waveshaping functions in terms of a specific desired harmonic content.

  
  

  Table 1: Chebyshev Polynomials through the 5th Order.

  $T_0(x) = 1$

  $T_1(x) = x$

  $T_2(x) = 2x^2 - 1$

  $T_3(x) = 4x^3 - 3x$

  $T_4(x) = 8x^4 - 8x^2 + 1$

  $T_5(x) = 16x^5 - 20x^3 + 5x$

  
  

• Waveshaping systems will often incorporate output scaling operations to compensate for excessive gains that result from the waveshaping function.