Joint Source-Channel Decoding on Factor Trees - Semantic Scholar

International Symposium on Information Theory and its Applications, ISITA2004 Parma, Italy, October 10–13, 2004

Joint Source-Channel Decoding on Factor Trees: A Scalable Solution for Large-Scale Sensor Networks Michael T¨ uchler

Jo˜ao Barros

Christoph Hausl

Institute for Communications Engineering (LNT), Munich University of Technology (TUM), 80290 M¨ unchen, Germany. email: {micha,joao,hausl}@lnt.ei.tum.de

mapper, and a modulator) and a decoder of increased yet manageable complexity. Our goal is then to devise a practical decoding algorithm for this instance of the sensor reachback problem. Unfortunately, the optimal decoder based on minimum mean square error (MMSE) estimation is unfeasible for large-scale sensor networks. We propose to use the sum-product algorithm for decoding based on factor graphs [8] that model the correlation between the sensor signals in a flexible way depending on the targeted decoding complexity and the desired reconstruction fidelity. We show that cycle-free factor graphs, the so-called factor trees, are particularly well suited for large-scale sensor networks with arbitrary topology. Using the Kullback-Leibler distance as a measure of the fidelity of the approximated correlation model, we give a detailed mathematical treatment of multivariate Gaussian sources and a set of optimization algorithms for factor trees with various degree constraints. Finally, we present numerical results that underline the performance and scalability of the proposed approach.

Consider a large-scale sensor network in which hundreds of sensor nodes pick up samples from a physical process in a field, encode their observations, and transmit the data back to a remote location over an array of reachback channels. The task of the decoder at the remote location is then to produce the best possible estimates of the data sent by all the nodes. The correlation between the measurements of different sensors can in general be exploited to improve the decoding result [1] even when the sensor nodes themselves are not capable of eliminating the redundancy in the data prior to transmission. To fulfill this data compression task, each node would have to use complex SlepianWolf source codes, which may become impractical for large-scale sensor networks. In that case, the decoder can still take advantage of the remaining correlation to produce a more accurate estimate of the sent information. We consider a reachback communications model, in which the system complexity is shifted from the sensor nodes to the receiver, i.e., a reachback network with very simple encoders (e.g., a scalar quantizer, a bit

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1. Introduction

Decoder

We consider the problem of jointly decoding the correlated data picked up and transmitted by the nodes of a large-scale sensor network. We focus on decoding algorithms that exploit the correlation structure of the sensor data to produce the best possible estimates under the minimum mean square error (MMSE) criterion. Since the optimal MMSE decoder is unfeasible for large scale sensor networks, we use factor graphs to obtain a scalable alternative decoder based on a simplified model for the correlation structure of the sensor data. We focus on factor graphs in form of trees, for which we specify the exact decoding complexity (using the sum-product algorithm) as well as the optimal edge distribution under the Kullback-Leibler criterion.

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Abstract

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Figure 1: System model of a sensor network. 2. Problem Setup Consider the following notation: The expression 0N is the length-N all-zero column vector, IN is the N × N identity matrix, and |A| is the determinant of A. The covariance is defined by Cov{a, b} = E{abT }− E{a}E{b}T , where E{·} is the expectation operator.

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variance σ 2 , i.e., the channel outputs are given by yk = xk+nk after demodulation, where nk is distributed with N (0Q , σ 2 IQ ). The decoder uses y = (y1 y2 ... yM )T and the available knowledge of the source correlation R to produce estimates u ˆk of the measurements uk . Assuming that the mean square error (MSE) E{(ˆ uk−˜ u(ik ))2 } between u ˆk and u ˜(ik ) is the fidelity criterion to be minimized by the decoder, the conditional mean estimator (CME) [11] should be applied: X u ˆk = E{˜ u(ik )|y} = u ˜(i) · p(ik = i|y). (2)

An N -dimensional random variable with realizations a ∈ RN is Gaussian distributed with mean µ = E{a} and covariance matrix Σ = Cov{a, a}, when its probability density function (PDF) p(a) is given by 1 2

p(a) = exp(− (a − µ )T Σ−1 (a − µ ))/(2π|Σ|)1/2 . (1)

µ, Σ). Such a PDF is denoted as N (µ The basic system model we consider in this paper is shown in Fig. 1. Each sensor k observes at time t continuous real-valued data samples uk (t), with k = 1, 2, . . . , M . We consider only the spatial correlation of the measurements, i.e., we drop the time variable t and consider only one time step. The sample vector u = (u1 u2 ... uM )T is assumed to be one realization of an M -dimensional Gaussian random variable, whose PDF p(u) is given by N (0M , R) with   1 ρ1,2 · · · ρ1,M  ρ2,1 1 · · · ρ2,M    R= . ..  . . . . . .  . . .  . ρM,1 ρM,2 · · · 1

∀i∈L

Notice that for PDF-optimized quantizers this estimator also minimizes the MSE E{(ˆ uk −uk )2 } between u ˆk and uk [7]. The required posterior probabilities p(ik |y) are given by X p(ik = i|y) = γ · p(y|i)p(i), (3) ∀i∈LM :ik =i

where i = (i1 i2 ... iM )T and γ = 1/p(y) is a constant normalizing the sum over the product of probabilities to one. Since the AWGN channels are independent, QM the PDF p(y|i) factors into k=1 p(yk |ik ), where each p(yk |ik ) given by N (xk (ik ), σ 2 IQ ). The probability mass function (PMF) p(i) of the index vector i can be obtained by numerically integrating the source PDF p(u) over the quantization region indexed by i. Alternatively, one can resort to Monte Carlo simulations in order to estimate p(i), a task which needs to be carried out only once and can therefore be performed offline. The computational complexity of the decoding process is determined by the number of additions and multiplications required to compute the estimates u ˆ k for all k. The most demanding decoding operation is the QM marginalization of ik in p(i) · k=1 p(yk |ik ):

Gaussian models for capturing the spatial correlation between sensors at different locations are discussed in [12], whereas reasonable models for the correlation coefficients ρi,j of physical processes unfolding in a field can be found in [5]. We assume for simplicity that the sensors are randomly placed in a unit square according to a uniform distribution and that the correlation ρi,j between sensor i and j decays exponentially with their Euclidean distance di,j , i.e., ρi,j = exp(−β ·di,j ), where β is a positive constant. The sensors are “cheap” devices consisting of a scalar quantizer, a bit mapper, and a modulator1 . Each sensor k quantizes uk to the index ik ∈ L = {1, 2, . . . , 2Q }, representing Q bits, i.e., there are 2Q reconstruction values u ˜(ik ) ∈ R. The modulator maps ik to a tuple xk of channel symbols, which are transmitted to the remote receiver. In our examples we use binary phase shift keying (BPSK), such that in a discrete-time baseband description of our transmission scheme, ik is mapped to Q symbols xk = (xk,1 ... xk,Q ) from the alphabet {+1, −1}. As argued in [1], we assume that the reachback channel is virtually interference-free, Q i.e., the joint PDF M p(y1 , ..., yM |x1 , ..., xM ) factors into k=1 p(yk |xk ). In addition, we model the reachback channel as an array of additive white Gaussian noise channels with noise

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which requires 2Q(M−1) −1 additions and M 2QM multiplications per index k. The calculation of the PDF P p(yk |ik ) and the estimate u ˆk = γ ∀i∈L u ˜(i) · mk (i) for all k requires a number of additions and multiplications which is linear in M . We conclude that the MMSE-optimal decoder is unfeasible for networks with a large number M of sensors with (4) as the major bottleneck, since its computational complexity grows exponentially with M . Our goal is thus to find a scalable decoding algorithm yielding the best possible trade-off between complexity and estimation error.

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model for the encoder may seem too simple, yet it allows us to focus on the essential aspects of the problem and highlight the key features of our decoding algorithm. The latter can be easily extended to include, for example, more sophisticated channel coding.

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lead to a PMF pˆ(i) with the desired properties. As mentioned above, we assume that pˆ(i) factors into N functions gk (·), k = 1, 2, ..., N , where the number of arguments of gk (·) is at most 1, 2, or 3. Fig. 3 depicts possible factor graphs with these constraints on gk (·) for our example network with M = 9 sensors. Running the SP algorithm on the degree-1 factor graph corresponds to scalar decoding, where no information about the correlations between sensors is considered (N = M must hold). Running the SP algorithm on the degree-2 or degree-3 factor graph requires 2·22Q multiplications and 2(2Q−1) addition per degree2 function node and 6·23Q multiplications and 3(22Q−1) per degree-3 function node. In addition, some multiplications are required in the variable nodes. It is shown in [2] that a factor graph consisting of N function nodes with degree dk , k = 1, 2, ..., N , can be a fully connected cycle-free factor graph (a factor tree) PN if M + N − 1 = k=1 dk holds. For example, as shown in Fig. 3 (middle plot), exactly N = M − 1 degree-2 function nodes are required to connect the M variable nodes for uk to a factor tree. There are numerous other factorizations of pˆ(i) yielding different complexity counts, e.g., by increasing the admissible degree of the function nodes, by clustering the variable nodes, or by allowing factor graphs with cycles. In latter case we end up with a very large class of factors graphs, which admit an iterative SP algorithm [3, 9]. However, as shown in Sec. 5, the performance of the scalable decoder based on factor trees with degree-2 or degree-3 functions nodes is already very close to that of the optimal decoder. Moreover, we can construct an optimal set of arguments of the functions gk (·) for these rather simple factor graphs as shown in the next section.

3. Scalable Decoding using Factor Graphs In this section we propose a scalable decoding solution based on factor graphs and the sum-product (SP) algorithm to control the computational complexity of the decoding algorithm. We refer to [8] for an overview about the factor graph concept. The function that needs Q to be factorized in our deM coding problem (4) is p(i) · k=1 p(yk |ik ). The corresponding factor graph, illustrated in Fig. 2 for M = 9 sensors, consists of M variable nodes for each ik , M function nodes for each p(yk |ik ), and one degree-M function node for p(i). 1 8

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Figure 2: Factor graph of the function p(i) · QM k=1 p(yk |ik ) for a sensor network consisting of 9 sensors (the numbered circles). The marginals mk (i) in (4) can be computed by running the SP algorithm [8] on the factor graph in Fig. 2. As long as the factor graph is cycle-free, the SP algorithm yields the correct marginals mk (i) in the M variable nodes. Otherwise, it becomes iterative and the marginals mk (i) cannot be computed exactly. Running the SP algorithm on the factor graph in Fig. 2, which yields the exact marginals mk (i), requires M 2Q multiplications in the M variable nodes for ik and M (2Q(M−1) −1) additions and M (M −1)2QM multiplications in the function node for p(i). Combining these numbers yields the same count as that below equation (4). This complexity count can decrease tremendously if p(i) factors into functions with small numbers of arguments (small function nodes degree). We propose to use factorizations of p(i), which yield a fully connected cycle-free factor graph (a factor tree), whose function nodes have a prescribed degree of at most 1, 2, or 3. Thus, running the SP algorithm on such a graph yields the correct marginals mk (i) and the decoding complexity scales linearly with the number of sensors M . The PMF p(i) derived from p(u) will in most cases not have a structure leading to such a factorization, i.e., we must seek an approximate source distribution pˆ(u) that does

4. Model optimization The optimal arguments of the functions gk (·) of the QN factorization pˆ(i) = k=1 gk (·) are obtained by the optimal functions fk (·) of the corresponding factorization QN pˆ(u) = k=1 fk (·) of the approximate source distribution pˆ(u), whose arguments are the same as those of gk (·). This is possible because a particular index ik depends only on the source symbol uk (scalar quantization), i.e., any factorization of pˆ(i) implies the same factorization of pˆ(u) and vice-versa. The performance of the decoding algorithm proposed in the previous section depends on how well pˆ(u) approximates p(u). A useful distance measure to determine how close pˆ(u) is to p(u) is the Kullback-Leibler

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QM Figure 3: Factor graph of the function pˆ(i) · k=1 p(yk |ik ) corresponding to a sensor network with QN Q9 M = 9 sensors, where pˆ(i) = k=1 gk (·) is given from left to right by (a) k=1 gk (ik ) (degree-1 function nodes), (b) g(i1 , i6 )g(i2 , i1 )g(i3 , i9 )g(i4 , i7 )g(i5 , i4 )g(i6 , i4 )g(i7 , i8 )g(i9 , i5 ) (degree-2 function nodes), or (c) g(i1 , i2 )g(i3 , i9 )g(i1 , i4 , i6 )g(i4 , i5 , i9 )g(i4 , i7 , i8 ) (degree-2 and -3 function nodes). distance (KLD) measured in bit [4, Sec. 9.5]: Z p(u) du. D(p(u)||ˆ p(u)) = p(u) log2 pˆ(u)

are either empty or contain a single element, yield the factor graphs of interest in this paper, since the factor graph corresponding to pˆ(u) is always a tree [2]. Given our source model, we are particularly interested in CCREs of Gaussian PDFs. The next lemma presents a few very useful properties for this purpose.

(5)

Some motivation for using the KLD as the optimization criterion is included in [2]. To minimize the KLD D(p(u)||ˆ p(u)) subject to the constraints imposed on the functions fk (·) of the facQN torization pˆ(u) = k=1 fk (·), we introduce a few useful mathematical tools: Consider the chain rule expansion p(u1 )p(u2 |u1 )p(u3 , u4 |u1 , u2 )p(u5 |u1 , ..., u4 ) of the source distribution p(u) = p(u1 , u2 , ..., u5 ), which consists of N = 4 factors. A constrained chain rule expansion (CCRE) is obtained by taking the factors of the chain rule expansion and removing some of the conditioning variables, thus yielding an approximate PDF of p(u1 , ..., u5 ). More formally, the PDF pˆ(u) =

YN

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QN Lemma 1 Let pˆ(u) = k=1 p(ak |bk ) be a CCRE of the Gaussian PDF N (0M , R). Let P be an M × M matrix, whose entry in the l-th row and l 0 -th column is 1 if both ul and ul0 are contained in one of the N factors p(ak |bk ) and 0 otherwise. For example, for the CCRE p(u1 )p(u2 |u1 )p(u3 , u4 |u2 )p(u5 |u4 ) we find   1

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The following holds: The PDF pˆ(u) is a Gaussian with ˆ i.e., it is given by N (0M , R). ˆ covariance matrix R, −1 ˆ The entries of R are zero for all zero-positions in P. ˆ −1 equals M , i.e., tr(RR ˆ −1 ) = M . The The trace of RR ˆ entries of R are equal those in R for all one-positions in P if the CCRE is symmetric.

where ak and bk are subsets of the elements in u, is a CCRE of p(u), if the following constraints are met: k−1 [

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For a proof see [2]. Based on Lemma 1, we can prove the following connection between symmetric CCREs and the KLD-optimal functions fk (·) of the factorizaQN tion pˆ(u) = k=1 fk (·):

Thus, the set b1 is always empty and for the usual Sk−1 chain rule expansion holds bk = l=1 al . A CCRE of p(u1 , ..., u5 ) with at most one conditioning variable is for example given by p(u1 )p(u2 |u1 )p(u3 , u4 |u2 )p(u5 |u4 ). We call a CCRE symmetric, if any bk , k = 2, 3, ..., N , is a subset of (al , bl ) for some l < k, e.g., the CCRE p(u1 )p(u2 |u1 )p(u3 , u4 |u2 )p(u5 |u4 ) of p(u1 , ..., u5 ) is symmetric, but p(u1 , u2 )p(u3 , u4 |u2 )p(u5 |u4 , u1 ) is not. QN It turns out that CCREs pˆ(u) = k=1 p(ak |bk ) of p(u) with at most one conditioning variable, i.e., all bk

Theorem 1 Consider the Gaussian source distribution QN p(u) given by N (0M , R) and the PDF pˆ(u) = k=1 fk (·), which factors into N functions fk (·) with subsets of u as argument. If the latter factorization admits a symmetric CCRE, then the KLD-optimal functions fk (·) minimizing the D(p(u)||ˆ p(u)) are equal to the Gaussian

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PDFs p(ak |bk ) = p(ak , bk )/p(bk ). The corresponding minimal KLD is given by:

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where Rak ,bk and Rbk are the covariance matrices of the Gaussian PDFs p(ak , bk ) and p(bk ), respectively.

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For a proof see [2]. This theorem simplifies our search for KLD-optimal approximate source distribuQN tions pˆ(u) = k=1 fk (·), which yield factor trees with function nodes of degree at most 1, 2, or 3, since we can restrict our attention to the set of symmetric CCREs and determine step by step the factor arguments ak and bk that minimize the KLD. Still, many factor trees exist that connect the M variable nodes in the network and the problem remains to find the factor tree QNfor which the underlying symmetric CCRE pˆ(u) = k=1 p(ak |bk ) yields the smallest KLD D(p(u)||ˆ p(u)). Efficient optimization algorithms to find the KLD-optimal factor tree consisting of only degree-2 or both degree-2 and degree-3 function nodes are outlined in [2]. For example, Fig. 4 shows two KLDoptimal factor trees for a sensor network with M = 100 sensors based on the source model from Sec. 1. Other special cases, including unsymmetric CCREs and factor graphs with cycles, are discussed in [6].

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Figure 4: KLD-optimal factor trees with function node degree of 2 (top) or 2 or 3 (bottom) for M = 100 sensors placed randomly on the unit square according to the source model described in Sec. 1. 1-bit quantization (Q = 1). Clearly, the factor-treebased decoders (degree-2 and degree-3 tree) are nearly as good as the MMSE-optimal decoder. Note also that there is a direct correspondence between the decoding performance and the KLD. The improvement of the degree-3 tree over the degree-2 tree is barely noticeable. The scalar decoder loses a lot of performance, since it does not exploit any information about the source correlations. Fig. 6 depicts the performance results for the network with M = 100 sensors with multiple quantizers. The KLD-optimal degree-2 factor tree for this network is depicted in Fig. 4(b). Again, the KLD of the degree2 tree is nearly as good that of the degree-3 tree, which find a correspondence in their SNR performance.

5. Numerical Examples To evaluate the decoder performance, we measure the output signal-to-noise ratio (SNR) given by ¶ µ kuk2 in dB Output SNR = 10 · log10 ˆ k2 ku − u

versus the channel SNR ES /N0 averaged over a sufficient amount of sample transmissions. We consider two networks with M = 9 or M = 100 sensors, where MMSE-optimal decoding can be simulated for the network with 9 sensors, only. Naturally, the results are highly dependent on the chosen source model. As outlined in Sec. 1, we assume that the correlation between the sensors ui and uj is given by ρi,j = exp(−β ·di,j ). Notice that if we keep increasing the number of sensors in the unit square without altering β, the sensor measurements would become increasingly correlated. Therefore, to obtain a fair result, we set β = 1.05 and β = 4.2 for the simulations with M = 9 sensors and M = 100 sensors, respectively. Each sensor node uses a Lloyd-Max quantizer to map uk to ik , which is then transmitted in accordance with the system setup described in Sec. 1. The decoder performance for the network with M = 9 sensors from Figs. 2 and 3 is illustrated in Fig. 5 for

6. Summary and Conclusions We studied the problem of jointly decoding the correlated measurements picked up by a sensor reachback network. First, we showed that the complexity of the optimal MMSE decoder grows exponentially with the number of nodes in the network, thus motivating the search for scalable solutions offering a trade-off between complexity and end-to-end distortion. Then, we presented a scalable decoding scheme for the sensor reachback problem, which uses a simplified factor graph mo-

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Output SNR against channel SNR for M=9 (random sensor positions). 4.8

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Figure 6: Performance of 3 decoders based on optimized factor graphs for a network with M = 100 sensors using various quantizers (1, 2, or 3-bit quantization): scalar decoder (trivial factor graph) (ˆ p(u) has KLD 45.37), degree-2 factor tree (ˆ p(u) has KLD 6.13), degree-3 factor tree (ˆ p(u) has KLD 5.40).

Figure 5: Performance of the MMSE-optimal decoder and 3 decoders using the SP algorithm on the factor graphs in Fig. 3 for a network with M = 9 sensors: Fig. 3(a): scalar decoder (ˆ p(u) has KLD 3.92), Fig. 3(b): degree-2 factor tree (ˆ p(u) has KLD 0.43), Fig. 3(c): degree-3 factor tree (ˆ p(u) has KLD 0.40).

[5] C. R. Dietrich and G. N. Newsam. Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix. SIAM Journal on Scientific Computing, 18(4):1088–1107, 1997.

del of the dependencies between the sensor measurements such that a belief propagation decoder can produce the required estimates efficiently. Our analysis and simulation results indicate that the proposed approach is well suited for large-scale sensor networks. Natural extensions include (a) extending the factor graph to account for sensor nodes that have more complex features, such as entropy coding, channel coding or higher modulations, and (b) reducing the complexity further by running linear message updates in the nodes of the factor graph based on a Gaussian approximation of the message distributions [10].

[6] Christoph Hausl. Scalable decoding for largescale sensor networks. Diploma Thesis, Lehrstuhl f¨ ur Nachrichtentechnik, Technische Universit¨at M¨ unchen, Munich, Germany, April 2004. [7] N. Jayant and P. Noll. Digital Coding of Waveforms. Prentice Hall, 1984. [8] F. R. Kschischang, B. Frey, and H.-A. Loeliger. Factor graphs and the sum-product algorithm. IEEE Transactions on Information Theory, 47(2):498–519, 2001.

References [1] J. Barros and S. D. Servetto. The sensor reachback problem. Submitted to the IEEE Transactions on Information Theory, November 2003. Available from http://www.ei.tum.de/~barros.

[9] Seong Per Lee. Iterative decoding of correlated sensor data. Diploma Thesis, Lehrstuhl f¨ ur Nachrichtentechnik, Technische Universit¨at M¨ unchen, Munich, Germany, October 2003.

[2] J. Barros and M. T¨ uchler. Scalable decoding on factor trees: A practical solution for wireless sensor networks. In Preparation

[10] H. Loeliger. Least Squares and Kalman Filtering on Forney Graphs. Codes, Graphs, and Systems, R.E. Blahut and R. Koetter, eds., Kluwer, 2002.

[3] J. Barros, M. T¨ uchler, and Seong P. Lee. Scalable source/channel decoding for large-scale sensor networks. In Proc. of the IEEE International Conference in Communications (ICC2004), Paris, June 2004.

[11] H. V. Poor. An Introduction to Signal Detection and Estimation. Springer-Verlag, 1994. [12] A. Scaglione and S. D. Servetto. On the interdependence of routing and data compression in multi-hop sensor networks. In Proc. ACM MobiCom, Atlanta, GA, 2002.

[4] T. M. Cover and J. Thomas. Elements of Information Theory. John Wiley and Sons, Inc., 1991.

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