IEEE SIGNAL PROCESSING LETTERS, VOL. 18, NO. 4, APRIL 2011 219 Minimizing the Effect of Sampling Jitters in Wireless Sensor Networks Salman Ahmed and Tongwen Chen Abstract—A wireless sensor network (WSN) consists of low-cost and energy-limited sensors to measure a distributed phenomenon. The finite energy constraint limits the synchronization of sensors at every sampling instant which introduces sampling jitters. In this letter, we model sampling jitters using fractional delay transfer functions. The WSN is modeled using a hybrid multirate filter bank where the objective is to design discrete-time, causal and stable synthesis filters to minimize the effect of sampling jitters. Using a norm-invariant discretization, the hybrid and multirate problem is reduced to a model-matching optimization problem involving linear time-invariant and discrete-time systems. A numerical ex- ample is also presented to show the effectiveness of the proposed approach. Index Terms—Hybrid filter banks, optimization, sampling jitters, wireless sensor networks. I. INTRODUCTION A wireless sensor network (WSN) consists of low-cost and energy-limited sensors which are spatially distributed to accomplish a specific task. Applications of WSNs can be found in many areas such as pipeline monitoring, biomedical research, military applications and traffic surveillance [1]. A fundamental design for a sensor node in a WSN includes sensors, wireless communication hardware and battery sources. The low-cost design and efficient utilization of battery power introduces constraints such as non-synchronization of sensor nodes at every sampling instant. This limitation of non-synchronization introduces sampling jitters. The presence of sampling jitters gives rise to nonuniform sampling. An important research challenge is to efficiently reconstruct the uniformly sampled measurements from the nonuniformly sampled measurements. Most of the standard control and monitoring techniques rely on uniform and synchronized sampling. Therefore, to apply the standard techniques we need to design an algorithm to minimize the effect of unsynchronized sampling. This letter proposes a method to discretize sampling jitters and reconstruct uniformly sampled measurements obtained using a WSN. We assume that the sampling jitter is known and fixed for each sensor, therefore, it can be modeled by a drift (delay/advance) transfer function. Manuscript received September 29, 2010; revised January 09, 2011; accepted January 12, 2011. Date of publication January 31, 2011; current version pub- lished February 10, 2011. This work was supported by NSERC. The associate editor coordinating the review of this manuscript and approving it for publica- tion was Dr. Henry Leung. The authors are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4 Canada (e-mail: salman2@ual- berta.ca; tchen@ece.ualberta.ca). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2011.2109711 As WSNs consist of low-cost sensors, it is not possible to achieve a high sampling rate to sample a signal. In order to overcome this limitation, distributed sampling can be employed [2]. Distributed sampling uses a combination of sensors to sample a common input signal. Each sensor samples at a rate of samples/sec. The outputs of these sensors are multiplexed to produce a fast sampled signal. Therefore, multiple sensors can be used to distribute the sampling load across the sensors. The advantage of distributed sampling is that slow sampling sensors can be added in parallel to act like a fast sampling sensor. A popular approach to model distributed sampling sensors is to use filter banks [3]. Digital multirate filterbanks have been widely studied in signal processing; Vaidyanathan’s book [3] gives a comprehensive historical survey of the literature till 1992. The design of multirate filters using framework was originally proposed in [4]. This design was later extended to discrete-time filter banks in [5] and hybrid filter banks in [6]. However, in both cases the authors considered rational transfer functions whereas in this letter fractional delays are considered which are not rational transfer functions. In [7], the authors considered delays, which are integer multiples of the sampling period, for the design of filter bank. In [8], the authors considered fractional delays for the design of filter bank using optimiza- tion. To our best knowledge, the filter bank design for fractional delays using optimization has not been studied in WSNs. The main contribution of this letter is to propose a technique for dis- cretization of fractional delays and reconstruction of uniformly sampled measurements by designing synthesis filters based on the minimization of the norm of the error system. The norm of a stable system has a physically meaningful interpretation for deterministic as well as stochastic inputs [9]. If we consider the input to be a unit impulse, then the average of the output energy equals the square of the norm of the system. Furthermore, if we consider the input to be white noise having zero mean and unit variance, then the root-mean-square value of the outputs equals the norm of the system. There- fore, the design objective was chosen based on the norm op- timization, which results in getting the optimal and stable syn- thesis filters. The design procedure is implemented in a control room where computing power is abundant. The remainder of this letter is structured as follows: In Section II, the methods of discretization of a continuous-time system and a fractional delay system are presented. The problem formulation and system model using multirate hybrid filterbank are discussed in Section III. The design procedure involving the reduction of the hybrid problem into a single-rate discrete-time model-matching problem is presented in Section IV. An ex- ample is given in Section V. Finally, conclusions and future work are summarized in Section VI. 1070-9908/$26.00 © 2011 IEEE