The intermodulation and distortion due to quantization of sinusoids

The Fourier series representation of the quantization error sawtooth yields exact expressions and convenient approximations for all intermodulation (IM) and distortion components produced by quantization of the sum of two sinusoids whose respective amplitudes are A and a. The mean-squared values of the IM components are also calculated in the case where A and a fluctuate over several quantization steps. When A and a are many times the quantization-step size Q, these mean-squared values turn out to be approximately Q4/(180 π2Aa) except for high-order IM. The quantization is generally assumed to be uniform, but nonuniform quantization is also discussed. The case of A \gg Q and a \ll Q is considered as well as that of a = 0. The inclusion of even a small amount of additive noise in the input, however, is found to reduce the IM and distortion to undetectable levels, thus ensuring that IM cannot be mistaken for an imput signal unless, contrary to assumption, the quantization staircase is curved, i.e., the quantization is nonlinear. Hence, not many quantization bits are needed in order to avoid IM problems.