336 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 2, MARCH 2006 Analysis and Design of Single-Bit Sigma-Delta Modulators Using the Theory of Sliding Modes Shiang-Hwua Yu, Member, IEEE Abstract—The existence and stability conditions for a single-bit sigma-delta modulator operated in sliding mode are derived. Based on these conditions, a stable single-bit modulator of arbi- trary order is synthesized. Also, an alternative interpretation on the modulator’s noise-shaping mechanism is presented. From the sliding mode aspect, the modulator is regarded as a device that performs frequency-weighted minimization of the modulation error, and the modulator’s loop filter as the weighting function. The noise-shaping equation derived from this aspect helps pre- cisely predict the modulator performance. A fifth-order 2.2-MHz wide-band modulator with an effective resolution of 17 bits is de- signed as an illustrative example. The simulation result conforms the accuracy of the analysis. In the end, this paper presents a possible extension of single-bit to multibit modulation while still maintaining the sliding-mode operation. Index Terms—Noise shaping, quantization, relay feedback, sigma-delta modulation, sliding mode, stability. I. INTRODUCTION S IGMA-DELTA modulation converts a continuous input signal to a coarsely quantized signal while elimi- nating in-band quantization noise. This modulation technique continues to be an area of active research because of its wide range of applications in data conversion [1], [2] and power con- version [3], [4]. However, the complex behaviors of sigma-delta modulators have not yet all been clarified [5], [6]. A single-bit sigma-delta modulator is actually a relay feed- back system, as shown in Fig. 1. A notable operational mode of a relay feedback system is the sliding mode [7], [8], in which the system rejects the disturbance by zeroing a switching func- tion. Also, in the sliding mode, the relay feedback system be- haves like a reduced-order linear system. This outstanding “lin- earization” feature greatly simplifies the analysis of the complex behavioral patterns of a sigma-delta modulator, thus making an exact analysis possible. In 1988, Sira-Ramirez [9] first ap- plied this theory to analyze the stability of the first-order sigma- delta modulator. Recently, Zourntos and Johns [10] presented a variable-structure sigma-delta modulator based on the vari- able structure control with sliding mode, in which stability is achieved by adapting the loop filter. Plekhanov et al. [11] pro- posed a second-order modulator with a nonlinear sliding sur- face. However, the modulators presented in [10] and [11] are Manuscript received April 18, 2005. Manuscript received in final form November 7, 2005. Recommended by Associate Editor S. Peresada. This work was supported in part by the National Science Council, Taiwan, R.O.C., under Grant NSC94-2215-E-110-022. The author is with the Electrical Engineering Department, National Sun Yat-Sen University, Kaohsiung 804, Taiwan, R.O.C. (e-mail: shaun@mail.ee.nsysu.edu.tw). Digital Object Identifier 10.1109/TCST.2005.863668 Fig. 1. Single-bit continuous-time modulator. not regular sigma-delta modulators. The sliding-mode behav- iors of regular high-order sigma-delta modulators have not yet been fully explored. This paper, following the author’s previous work [12], [13], gives a thorough analysis of the sliding-mode behaviors of a single-bit sigma-delta modulator, and also provides a simple method for designing a single-bit modulator that operates in the sliding mode. From the sliding mode aspect, a new explana- tion on the noise-shaping mechanism is provided. This paper fo- cuses on the continuous-time modulator as shown in Fig. 1 (The analysis of a discrete-time sigma-delta modulator can be found in [14]). The primary reason for using a continuous-time loop filter rather than a switched capacitor filter is that the contin- uous-time sigma-delta can operate at a higher sampling rate with comparable integrators or operational transconductance ampli- fiers [15, p. 11]. Therefore, the continuous-time implementa- tion is especially appropriate for high-frequency and wide-band sigma-delta data conversion. The main contribution of this work is to analyze the stability and performance of a single-bit sigma-delta modulator of ar- bitrary order in the sliding mode, present an alternative inter- pretation on the noise-shaping mechanism of the modulator, and state a number of new results to improve the understanding of sigma-delta modulation. Particular emphasis is on the exact analysis and precise design of the modulator of arbitrary order. The presented design methodology is simple and especially suit- able for computer aided design. Also, a possible extension to multibit modulation in the sliding mode is presented. II. SINGLE-BIT MODULATOR IN SLIDING MODE Consider a single-bit continuous-time sigma-delta modulator shown in Fig. 1, where loop filter has pole excess one and can be written in the following form: (1) (2) (3) 1063-6536/$20.00 © 2006 IEEE