Granular quantization noise in the first-order delta-sigma modulator

A unified approach to analyzing the granular quantization error of the first-order ΔΣ modulator is presented. The approach handles many of the previously analyzed input sequences in addition to a large class of new input sequences. By averaging over the arbitrarily small amount of circuit noise assumed to be present at the analog input to the ΔΣ modulator, a simple expression for the autocorrelation of the quantization error is derived. Each term in the expression is formally equal to the quantization error of a nonoverloaded uniform quantizer operating upon a finite partial sum of consecutive input sequence samples. Hence, existing results concerning uniform quantizers are directly applicable in evaluating the autocorrelation expression for specific input sequences. The theory is also applicable to deterministic input sequences, and has been applied to obtain a new closed-form result for sinusoidal input sequences. Ergodic results which assert that, under mild conditions, the autocorrelation equals the time-average autocorrelation in probability are presented