Distribution Laws of Quantization Noise for Sigma-Delta Modulator

Abstract – The very fast development of sigma-delta analog-to-digital converters is evidently [1, 2]. Therefore analytical methods of their description become more and more important. If the found equations are precise enough, they can be used for verification and validation of modelling and simulation. Analytical equations are also important from the viewpoint of education. As is known, the sigma-delta ADC consists of sigma-delta modulator (SDM) and digital filter. In this paper the quantization noise of modulator is investigated for different forms of input signal. The world known (classical) theory [3-5] of quantization noise is not accurate enough [6]. A new approach was supposed by the authors [7]. The most important point of the approach is to use the discrete two-value distribution law at the modulator output. According to classical theory, for any forms of input signal the quantization noise at the output of the modulator has uniform distribution law. In accordance with new approach, the distribution law of quantization noise depends significantly on the form of input signal. The simulation results are found in agreement with new theory. I. Distribution laws of quantization noise in analytical form for SDM with different input signals A. General approach to definition of the distribution laws The nominal (ideal) transfer function of SDM with input signal X and output signal Y is Y = X. The modulator error is ∆ = Y–X. Mean value M(∆) = 0 for the ideal elements of SDM. Output Y for one bit quantizer (comparator