Rigorous Analysis of Quantization Error of an A/D Converter Based on β -Map Takaki Makino * , Yukiko Iwata † , Yutaka Jitsumatsu ‡ , Masao Hotta § , Hao San § and Kazuyuki Aihara * * Institute of Industrial Science, The University of Tokyo, Tokyo, 153–8505 Japan Email: mak@sat.t.u-tokyo.ac.jp † FIRST, Aihara Innovative Mathematical Modelling Project, Japan Science and Technology Agency ‡ Kyushu University, Fukuoka, 819–0395 Japan § Tokyo City University, Tokyo, 158–8557 Japan Abstract—A non-binary analog-to-digital converter (ADC) based on β-expansion, called β-encoder, is so far reported to achieve robustness against large process variation and wide environment change. Quantization error of β-encoder is not uniformly distributed, which makes the mean squared error (MSE) evaluation difficult. An analysis method for giving the upper bound of MSE of the quantization error is provided. Signal-to-noise-ratio (SNR) is also evaluated and the result is effective for designing β-encoders ⋆ . Index Terms—Analog-to-Digital and Digital-to-Analog convert- ers, quantization error, β-map, mean squared error. I. I NTRODUCTION Robust analog-to-digital converter (ADC) architecture is desirable for not only nowadays mixed signal LSIs but also next generation CMOS technology, to meet the requirement of ADC performances of high sampling frequency, high reso- lution, small chip area (the same word as low cost), and low power consumption. In the conventional binary architecture, the linearity of ADC is very sensitive to the accuracy of analog components. Therefore, high-gain wideband amplifiers as well as high accuracy matched devices, such as transistors, capacitors, and resistors, are necessary to satisfy the required ADC linearity, leading to large chip area and high power consumption. A recently proposed architecture, called β-encoder [1], is a non-binary ADC based on β-expansion, which has a self- correction property for fluctuations of amplifier factor β and quantizer threshold ν . It is shown [2] that the circuit based on the approach (Fig. 1) has robustness that tolerates the conver- sion errors caused by finite gain of amplifier and mismatches of the devices, and that the proposed β-estimation algorithm eliminates the need for any digital calibration technique. Just by adding a simple conversion sequence with the effective radix-value β, we can realize a reliability-enhanced ADC with greatly relaxed power and area penalties for high-gain amplifier and high-accuracy circuit elements. From a viewpoint of circuit design, it is important to give a theoretical evaluation of mean squared error (MSE) and signal-to-noise ratio (SNR), which guarantee the accuracy and linearity of β-encoders. The only known theoretical results ⋆ This research is supported by the Aihara Innovative Mathematical Model- ling Project, the Japan Society for the Promotion of Science (JSPS) through the “Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program)”, initiated by the Council for Science and Technology Policy (CSTP). Dout Vin β ADC DAC Vres s(β –1) – S1 S2 Fig. 1: Simplified block diagram of the β-encoder (cyclic ADC based on β-expansion). In this paper we use the ordinary β-map, which means that the full-scale width s is set to 1 β-1 . about the quality of β-encoder is its maximum quantization error [1], [3]. The purpose of this paper is to make a theoretical derivation of simple and accurate MSE and SNR evaluation. The fact that the β-expansion map C β,ν (x) is locally eventually onto [ν −1,ν ] implies that we may evaluate MSE by assuming that C L β,ν (X) is uniformly distributed in the interval, where X is a random variable for x. Such an assumption makes the MSE to be 1 12 β −2L , where L is the number of bits 1 . Computer simulation, however, shows that the numerically calculated MSE is deviated from such an evaluation and is not sufficient to guarantee the quality of a β-encoder. The trajectory of C i β,ν (x) for i =0, 1,... with x ∈ [ν − 1, ν ] follows the Parry’s invariant density [4]. Hence, it seems a good way to evaluate MSE based on the Parry’s density. However, this has two difficulties. One is that the main target of L is from 12 to 16 in real applications. Such a size of L is not sufficiently large for the density function of quantization error to converge to the invariant measure. The other is that Parry’s invariant density is expressed as an infinite sum of ±β −n s, which complicates the MSE evaluation based on the direct application of Parry’s density. This situation motivated us to develop a new MSE analysis method. We provide a method for analyzing the MSE of β-encoder by introducing a notion of segments, i.e., linear pieces within the L-nested β-map. Such a method, called level j truncation, enables us to give a tight upper bound of MSE. Using this upper bound, and restricting our attention to the cautious map with ν = β 2(β−1) , we prove that the MSE of such β-encoder is smaller than 1 12 β −2L if 1+ √ 5 2 ≤ β ≤ 2 and 5 ≤ L ≤ 18. Such a sufficient condition is effective for designing β-encoders. We 1 In case of scale-adjusted β-map with a full-scale s, MSE is multiplied by s 2 (β - 1) 2 . For ordinary β-map, s = 1 β-1 . c ⃝2013 IEEE. This paper is an accepted version for 2013 IEEE International Symposium on Circuits and Systems Proceedings. Due to copyright restriction, the final published version is not allowed to be distributed from this server. 1