IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 10, OCTOBER 2006 4047 Design of Interpolative Sigma Delta Modulators Via Semi-Infinite Programming Charlotte Yuk-Fan Ho, Bingo Wing-Kuen Ling, Joshua D. Reiss, Yan-Qun Liu, and Kok-Lay Teo Abstract—This correspondence considers the optimized design of inter- polative sigma delta modulators (SDMs). The first optimization problem is to determine the denominator coefficients. The objective of the optimiza- tion problem is to minimize the passband energy of the denominator of the loop filter transfer function (excluding the dc poles) subject to the con- tinuous constraint of this function defined in the frequency domain. The second optimization problem is to determine the numerator coefficients in which the cost function is to minimize the stopband ripple energy of the loop filter subject to the stability condition of the noise transfer function (NTF) and signal transfer function (STF). These two optimization prob- lems are actually quadratic semi-infinite programming (SIP) problems. By employing the dual-parameterization method, global optimal solutions that satisfy the corresponding continuous constraints are guaranteed if the filter length is long enough. The advantages of this formulation are the guarantee of the stability of the transfer functions, applicability to design of rational infinite-impulse-response (IIR) filters without imposing specific filter struc- tures, and the avoidance of iterative design of numerator and denominator coefficients. Our simulation results show that this design yields a significant improvement in the signal-to-noise ratio (SNR) and have a larger stability range, compared with the existing designs. Index Terms—Dual parameterization, interpolative sigma delta modula- tors (SDMs), noise shaping, semi-infinite programming (SIP), stability. I. INTRODUCTION Sigma delta modulation is a popular form of analog-to-digital (A/D) and digital-to-analog (D/A) conversion and is applied in most commer- cial A/D and D/A systems [1]–[4]. The popularity of sigma delta mod- ulators (SDMs) is mainly due to their simple, inexpensive, and robust circuit implementation, as well as achieving very high signal-to-noise ratio (SNR) because of their ability to perform noise shaping [5]. The basic operation of SDMs is to sample the input signal at a much higher rate than the Nyquist frequency, filter the signal and perform noise shaping, and then quantize the output [5]. The block diagram of an interpolative (or feedforward) SDM is depicted in Fig. 1. It consists of a loop filter, a low-bit quantizer, and a negative feedback path. Over- sampling of the input signal and noise shaping in the loop filter is used in order to remove the quantization noise out of the passband, typically the lowpass band [5]. Optimal designs have been performed based on optimizing opera- tional transconductance amplifier structures [6], speed, resolution, and A/D complexity [7] and the ratio of peak SNR plus distortion ratio Manuscript received April 25, 2005; revised September 20, 2005. The asso- ciate editor coordinating the review of this paper and approving it for publication was Dr. Kenneth E. Barner. C. Y.-F. Ho, J. D. Reiss are with the Department of Electronic Engineering, Queen Mary, University of London, London E1 4NS, U.K. (e-mail: charlotte. ho@elec.qmul.ac.uk; josh.reiss@elec.qmul.ac.uk). B. W.-K. Ling is with the Department of Electronic Engineering, Division of Engineering, King’s College London, London WC2R 2LS, U.K. (e-mail: wing- kuen.ling@kcl.ac.uk). Y.-Q. Liu is with the Department of Mathematics and Statistics, Royal Melbourne Institute of Technology, Melbourne VIC 3001, Australia (e-mail: yanqun.liu@rmit.edu.au). K.-L. Teo is with the Department of Mathematics and Statistics, Curtin Uni- versity of Technology, Perth, 00301J, Australia (e-mail: K.L.Teo@curtin.edu. au). Color versions of Figs. 2–4 are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2006.880338 Fig. 1. Block diagram of an interpolative SDM as used for A/D conversion . versus power consumption [8], etc. Although these designs have con- sidered many practical issues, the solutions obtained are not global optimal because the optimization problems involved are not convex. SDMs are typically designed using Butterworth filter design rules [2], and optimal SDM designs based on comb filter [9] and Laguerre filter [10] structures were recently proposed. However, since some struc- tures (such as all the poles of the Laguerre filters are constrained to be the same, that of the Butterworth filters are constrained on the same circle, and many zeros are in the impulse response of the comb filter) are assumed on these filters, a better solution may be obtained if these structural assumptions are relaxed. Design based on the finite horizon method [11] was also proposed. However, this method is only an ap- proximation of an infinite horizon method. Although the approxima- tion is improved as the length of window increases, the computational complexity increases. Genetic algorithms have also been applied to per- form the optimization [12]. However, the convergence of the genetic algorithms is not guaranteed and the computational complexity of this method is very high. Other existing optimal designs, such as reported in [13]–[15], are obtained mainly based on the simulation framework and lack of the theoretical support. Since computational complexity of rational infinite-impulse-re- sponse (IIR) filters is usually lower than that of the finite-impulse-re- sponse (FIR) filters, many SDMs employ rational IIR filters [1]–[3]. However, since rational IIR filters consist of both numerator and denominator coefficients, there are some challenges for designing rational IIR filters. One way to design rational IIR filters is to first initialize a set of the denominator coefficients and then design the numerator coefficients based on this set of initialized denominator coefficients by using ripple energy as the cost function and magni- tude specification as the constraints. Then, update the denominator coefficients based on the obtained numerator coefficients and iterate this procedure until the algorithm converges. However, it is not guaranteed that the iterative procedure will converge [16]. Moreover, the obtained solution depends on the initialization of the denominator coefficients, hence only a local optimal solution is obtained. Although the divergence problem can be solved via weighting the filter coef- ficients in each iteration, the frequency characteristics of the filter will depend on the weights and the result obtained may be degraded [17]. Furthermore, these design methods [16], [17] assume that both the desired magnitude and phase responses of the filter are known. However, sometimes it may be difficult to characterize the desired phase response. This applies to Butterworth and Laguerre filter cases because these are nonlinear phase filters. Under this circumstance, the cost function based on the error energy or the absolute error between the desired and designed energy responses will become a fourth-order function or a nonsmooth function ( and denotes the designed and desired frequency response, respectively). Nevertheless, these problems are not convex. The major issues of designing SDMs are to achieve high SNR with the guarantee of the boundedness of state variables [2], [10]. Since the SDMs consist of a quantizer, which is a nonlinear component, there is no simple relationship among the SNR [18], maximum bound of the 1053-587X/$20.00 © 2006 IEEE