Numerical method for determining the quantization error PDF of single-bit ΣΔ Modulators

Existing models for the quantizer of ΣΔ modulators make assumptions on the probability density function (pdf) of the quantization error, or some other convenient signal of the modulator. In this paper, a method for the determination of this pdf for single-bit ΣΔ modulators is presented. First, a numerical method is proposed in order to solve the simplified equation for the quantization error pdf for first-order systems considering noiseless and noisy dc input signals. Then, it is shown how most practical high-order (>2)ΣΔ modulators, resulting from well-established design methods, can be modeled as first-order systems plus an additive noise source at the input. Hence, their quantization error pdf is analyzed using the proposed method. Simulation results are shown to be in considerable agreement with those of the proposed method.