Optimal Trade-off Between Sampling Rate and Quantization Precision in Sigma-Delta A/D Conversion Alon Kipnis ∗ , Andrea J. Goldsmith ∗ and Yonina C. Eldar † Abstract—The optimal sampling frequency in a Sigma-Delta analog-to-digital converter with a fixed bitrate at the output is studied. We consider the mean squared error performance metric where the input signal statistics are known. Fixing the output bitrate introduces a trade-off between the sampling rate and the number of bits used to quantize each sample. That is, while increasing the sampling rate reduces the in-band quantization noise, it also reduces the number of bits available to quantize each sample and therefore increases the magnitude of the quantization noise. The optimal sampling rate is the result of the interplay between these two phenomena. In this work we analyze the sampling rate of a Sigma-Delta modulator of arbitrary order under the approximation that the quantization error behaves like additive white noise that is uncorrelated with the signal. We show that for a signal with a spectrum that is constant over its bandwidth, the optimal sampling rate is either the Nyquist rate or the maximal sampling rate corresponding to the output bitrate. The choice between the two is approximately a function of the Sigma-Delta system order and the bitrate per unit bandwidth. I. INTRODUCTION A. Background In analog to digital conversion (ADC) an analog signal is converted into a sequence of bits. Shannon’s distortion-rate function [1] gives the theoretical minimal error as a function of the bitrate of the digital sequence, however it does not provide concrete methods for the A/D conversion. Practical ADC schemes involve operations of sampling and quantization. The overall bitrate in the resulting digital representation is the product of the sampling rate with the average number of bits used to store each sample. In this work we are interested in the trade-off between these two quantities in A/D conversion using Sigma-Delta modulation (ΣΔM). In this ADC scheme, the input process is oversampled (sampled above its Nyquist rate) and quantized using a low-resolution quantizer (usually 1-bit). ΣΔM also employs a negative feedback loop and an integrator so that quantization error of previous samples will be considered in quantizing consecutive samples. While oversampled modulation does not provide any theoretical improvement over sampling at the Nyquist rate or at the minimal rate that achieves the rate-distortion function [2], ΣΔM is commonly used in applications due to its relatively cheap and simple hardware implementation. However, its high sampling rates may be hard to implement in some applications [3]. This makes a performance analysis of ΣΔM relevant for all sampling frequencies, and not only ∗ Department of Electrical Engineering, Stanford University, CA † Department of EE Technion, Israel Institute of Technology, Haifa, Israel in the high over-sampling rate regime. In this work we analyze the ΣΔM as a source coding scheme, that is, we are interested in the minimal error as a function of the bitrate. For that purpose we assume a statistical model on the input process and mean squared error (MSE) as our performance metric. We use the additive white noise assumption [4] for quantization error, where the variance of the quantization noise decreases exponentially with the number of bits per sample q. If the analog source is sampled at frequency f s , the memory rate at the output of the quantizer is R = qf s bits per time unit. Since ΣΔM uses oversampling to reduce the amount of in-band quantization noise, increasing f s decreases the error and effectively improves the resolution of the quantizer. This implies that fixing the memory rate R introduces an interplay between f s and q that induces a trade-off between the amount of in-band quantization noise and the magnitude of this noise. B. Related Work A ΣΔM is based on the principle of oversampling and a negative feedback loop that includes an integrator. The paper [5] provides an extended tutorial of the theoretical and practical aspects in ΣΔM. As in other systems which involve quantization, in ΣΔM analysis it is common to approximate the difference between the quantizer input and output by a white additive noise, see e.g. [6]. While the conditions under which this assumption yields a good approximation are not usually met in ΣΔM, it has been shown in several cases that the white noise assumption does not significantly change the performance results obtained through a rigorous analysis which does not make this assumption. We will discuss this approximation more in Section II. The feedback loop in the ΣΔM is sometimes referred to as a quantization noise shaping system. The white quantization noise assumption implies that in the absence of the feedback loop (zero order ΣΔM), the power of the quantization noise within the signal band decreases linearly with f s . With a simple noise shaping system [6] the quantization noise is attenuated even more and the in-band noise power decreases by a factor of f 2L+1 s , where L is the number of consecutive feedback loops or the modulator order. This implies a mean squared error (MSE) reduction of R 2L+1 in the bitrate. This error reduction is still much slower than the exponential reduction of the optimal distortion-rate trade-off in Shannon’s distortion-rate function [1]. Oversampling schemes which try to bridge this gap and 2015 International Conference on Sampling Theory and Applications (SampTA) 978-1-4673-7353-1/15/$31.00 ©2015 IEEE 627