- When the sinusoidal input amplitude is given by
*a*(instead of 1), the output is then scaled by*a*as:

- 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*) = 2*x*^{2}- 1*T*_{3}(*x*) = 4*x*^{3}- 3*x**T*_{4}(*x*) = 8*x*^{4}- 8*x*^{2}+ 1*T*_{5}(*x*) = 16*x*^{5}- 20*x*^{3}+ 5*x*

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

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