Characterizing Decoding Robustness under Parametric Channel Uncertainty

Transcription

1 Characterizing Decoing Robustness uner Parametric Channel Uncertainty Jay D. Wierer, Wahee U. Bajwa, Nigel Boston, an Robert D. Nowak Abstract This paper characterizes the robustness of ecoing uner parametric channel uncertainty. We consier iscrete channel moels escribe by probabilistic graphical moels, such as finite state channels. Using recent avances from the fiel of tropical geometry, we are able to partition the channel parameter space into cells corresponing to all channels that ientically ecoe a given output coewor. The partition, base on a combinatorial object calle the Newton polytope of the graphical moel, can be efficiently compute in polynomial time (in terms of the number of parameters in the channel moel). This partitioning of the channel parameter space provies two key results: 1) Given a nominal (vector) channel parameter, one can easily gauge the robustness of the ecoing as a function of eviations from this nominal parameter; 2) Rather than obtaining one ecoing for a single parametrization, one can obtain a list of ecoings for a family of channel parametrizations at the ecoer. I. INTRODUCTION Success of a communication system lies in how well it can encoe an ecoe ata. While many of toay s commonly employe channel ecoing techniques are built on the premise that the ecoer has knowlege of the true channel parameters, this is almost never the case in reality. In this paper, we attempt to characterize the robustness of ecoing uner parametric uncertainties in iscrete channels escribe by probabilistic graphical moels. The focus of the paper is not on moeling the uncertainties themselves, but rather on examining the relationship between ecoing outputs an variations in the channel parametrization. Recent avances in the fiel of tropical geometry [1] enable us to partition the channel parameter space into cells consisting of equivalent channels. By equivalent we mean that for a given output coewor, all the channels escribe by the parameters in a given cell will prouce the ientical maximum a posteriori (MAP) ecoing. This partitioning of the channel parameter space, which is a function of the family of graphical moels an the output coewor, can be compute in time that is polynomial in the imension of the channel parametrization an provies tremenous insight into the robustness of a given ecoing. For example, suppose we have a nominal (vector) channel parameter an fin that an ientical ecoing results from a large ball of channels about this nominal setting. In such a case, we may This work was partially supporte by the NSF uner grant nos. CNS , ECS , an CCF JDW, WUB, NB, an JDN are with the Department of Electrical an Computer Engineering, University of Wisconsin-Maison, Maison, WI 5376, USA. NB is also with the Department of Mathematics, University of Wisconsin-Maison ( s: jwierer@wisc.eu, bajwa@cae.wisc.eu, boston@math.wisc.eu, nowak@engr.wisc.eu). conclue that the ecoing is robust to possible eviations of the true parameter from the nominal parameter. On the other han, if the nominal parameter is near the bounary of a partition cell then a slight perturbation of the true parameter coul lea to a completely ifferent ecoing, inicating a non-robust conition. To motivate our investigation further, consier the simple example of a binary symmetric channel (BSC) with crossover probability p. Suppose that we employ block coing as a means of transmitting ata over this BSC. In this situation, it is obvious that the MAP ecoing rule is the nearestneighbor ecoing (in Hamming space) for p < 1/2, an the farthest-neighbor ecoing for p > 1/2. Thus, the parameter space of a BSC can be partitione into the intervals [, 1/2) an (1/2, 1]. If our nominal setting for p is close to 1/2, then we see that we are in a very non-robust situation. While this partitioning of the one-imensional parameter space of a BSC is trivial, the etermination of analogous partitions for iscrete channel moels with higher imensional parameter spaces is highly non-trivial. In the sequel, we present a unifie computational approach to solving this problem for iscrete channels escribe by graphical moels (e.g, finite state (Markov) channels [2]). II. CHANNEL PARAMETER SPACE PARTITIONING In this section, we formally efine the channel parameter space partitioning problem uner consieration. Consier a point-to-point igital communication system with finite input coebook X an finite output coebook Y. An input coewor x X is transmitte through a iscrete channel which prouces an output coewor y Y. It is further assume that the iscrete channel is represente by a irecte acyclic graphical moel an the joint probability of an inputoutput coewor pair can be written as a monomial in a -imensional parameter θ = (a 1,..., a ), that is, P(x, y; θ) = a ν1(x 1 a ν2(x 2 a ν (x, (1) where the powers ν j (x, y), j = 1,...,, are integer-value functions of the pair (x, y), an the parameter θ is an element of the parameter space Θ that efines all possible channels. Finite state channels (FSCs) [2] are one class of channel moels having this form an, for the sake of this exposition, we woul focus exclusively on them. In particular, as a special case of the general FSC, we will carefully investigate a two-state (binary input, quaternary output) intersymbol interference (ISI) channel moel in Section V. Note that in the case of an FSC, one also inclues epenence on the channel state, s, which takes value in the state space

2 a C(x 2 C(x 4 sizeable ball of θ will also ecoe y to x 1. Thus, we can say (loosely) that the ecoing of y base on θ is robust. On the other han, parameter θ 1 is near the bounary of a cell an so the ecoing of y base on θ 1 is not robust. It may be esirable in such cases to provie the ecoer with a list ecoing. In this specific case, the orere list might be (x 1, x 3, x 4, x 2 ) base on the proximity of cell bounaries to θ 1, which is immeiately available from the partition. 1.5 θ θ 1 C(x 1 C(x a 1 Fig. 1. Partitioning of the channel parameter space Θ for a given output coewor y. S [2]. That is, the channel is characterize by the joint probability istribution of the input an output coewors, an the channel state as follows P(x, y, s; θ) = a ν1(x,y,s) 1 a ν2(x,y,s) 2 a ν (x,y,s), (2) where again the powers ν j (x, y, s), j = 1,...,, are integervalue functions. Finally, note that the MAP ecoing rule for a fixe channel parameter θ is given by x(θ ) = argmax x X { max s S P(x, y, s; θ ) or, alternatively, if the state of the channel is known to be s () through some sie information then the MAP ecoing is taken to be } (3) x(θ ) = argmax x X P(x, y, s() ; θ ). (4) Our main interest here concerns the subset Θ Θ such that x(θ) = x(θ ) for every θ Θ. Thus, Θ efines the collection of channel parameters that are equivalent to θ for the output coewor y. More generally, we are intereste in a partition of Θ into subsets/cells corresponing to channel parameters that prouce ientical MAP ecoings for the output coewor y. Partitions of this form allow us to assess the robustness of a given ecoing an simultaneously recover a list ecoing that is orere relative to the proximity of cell bounaries to a nominal parameter setting. To illustrate this iea further, consier the partition of a two-imensional channel parameter space given by Θ = {(a 1, a 2 ) : a 1, a 2 }, as shown in Fig. 1. The partition of Θ is etermine by the output coewor y (an the unerlying graphical moel) an the partition cells can be enumerate in terms of the input coewors. For example, C(x 4, y) Θ is the set of all channel parameters that ecoe y to the input coewor x 4. To illustrate the notion of ecoing robustness, consier the parameter θ. As can be seen from the figure, θ is well containe within the interior of cell C(x 1, y) an hence, all channel parameters within a III. NEWTON POLYTOPES OF FINITE STATE CHANNELS In this section, we introuce concepts from tropical geometry that enable the polynomial-time construction of the partition of the channel parameter space. For channels with joint probability istributions of the form given in (2), we can write the (marginal) probability istribution of y as f y := P(y; θ) = P(x, y, s; θ) = a ν1(x,y,s) 1 a ν2(x,y,s) 2 a ν (x,y,s). (5) Now consier the tropicalization of f y, enote by g y, which is obtaine by replacing the operators (+, ) with the operators (min, +) an a i with log(a i ) [1], [3]. This gives us g y = min min i=1 min = min ( ν i (x, y, s)log(a i )) ν(x, y, s), log(θ), (6) where, is use to enote an inner prouct, ν(x, y, s) = (ν 1 (x, y, s),..., ν (x, y, s)) an log(θ) = ( log(a 1 ),..., log(a )). The interesting thing to observe here is that g y coincies with the negative-log joint probability evaluate at the MAP values of x an s. The Newton polytope of f y, enote by NP(f y ), is simply the convex hull of ν(x, y, s) for all x X an s S, namely, NP(f y ) = conv {ν(x, y, s) : x X, s S}. (7) A nice consequence of tropicalizing f y by g y is that the function g y is piecewise linear on the cones in the normal fan of NP(f y ). Note that the normal cone to a close, convex set K R at the point v K is traitionally efine to be the set NC(v, K) R such that NC(v, K) = {b R : v u, b u K}. (8) However, in this context, we efine the normal cone of a vertex v NP(f y ) to be the set of (log) parameters NC(v, f y ) such that v minimizes v, b for all b NC(v, f y ) an all u NP(f y ), namely, NC(v, f y ) = {b R : v u, b u NP(f y )}. (9) The usefulness of these Newton polytopes lies in the fact that, given the nominal channel parameter θ, they can be use to ecoe a given output coewor y to the optimal

3 Fig. 2. An example of a Newton polytope. input coewor x an optimal state vector ŝ. To observe this, note that x(θ ), ŝ(θ ) = arg max x X,s S P(x, y, s; θ ) = arg min log(p(x, y, s; θ )) x X,s S = arg min ( ν i (x, y, s)log(a i )) x X,s S i=1 = arg min x X,s S ν(x, y, s), log(θ ) (1) an thus, by a simple convexity argument, the point ν( x(θ ), y, ŝ(θ )) is a vertex of NP(f y ); hence, the point b = log(θ ) lies in NC(ν( x(θ ), y, ŝ(θ )), f y ). If the optimal channel state vector is eeme to be unimportant, we may simply ecoe y to x(θ ) as given in (3). Aitionally, if the channel state vector is known, we may use (4) to ecoe the output coewor y. Finally, note that typically one woul expect the number of possible ecoings to be on the orer of X S, that

4 1 b 2 C(x 3 C(x 1 C(x 4 C(x 2 b Fig. 3. space. Decoing regions for example 1 in the tropical channel parameter Fig. 4. A two-state binary input, quaternary output pure intersymbol interference channel moel. Specifically, let Vx be the vertices of NP(f y ) that o not correspon to x, then v u, b δ = min, (14) u V x v u where v is the vertex of NP(f y ) corresponing to the MAP solution. Example 1 (Continue): We return to the example of Section III. Again, we assume that the channel has only one state. With Newton polytope NP(f y ) as given in Fig. 2, we can also etermine the tropical ecoing regions C(x, y) an the regular ecoing regions C(x, y) for x corresponing to the vertices of the Newton polytope. Let b 1 = log a 1 an b 2 = loga 2. Then, the tropical ecoing regions are given by the normal cones of the vertices of the Newton polytope an are escribe by C(x 1, y) = {(b 1, b 2 ) : b 2 >, b 1 + b 2 > } C(x 2, y) = {(b 1, b 2 ) : b 2 <, b 1 + b 2 > } C(x 3, y) = {(b 1, b 2 ) : b 2 >, b 1 + b 2 < } C(x 4, y) = {(b 1, b 2 ) : b 2 <, b 1 + b 2 < } (15) A plot of these tropical ecoing regions is given in Fig. 3. Similarly, the ecoing regions C(x, y) in the regular channel parameter space can be written as C(x 1, y) = {(a 1, a 2 ) : a 2 < 1, a 1 a 2 < 1} C(x 2, y) = {(a 1, a 2 ) : a 2 > 1, a 1 a 2 < 1} C(x 3, y) = {(a 1, a 2 ) : a 2 < 1, a 1 a 2 > 1} C(x 4, y) = {(a 1, a 2 ) : a 2 > 1, a 1 a 2 > 1} (16) The reaer shoul be remine that a 1, a 2 are not probabilities an hence, they nee not be containe within the unit square. These regular ecoing regions are plotte in Fig. 1. V. APPLICATION TO A SYMMETRIC ISI CHANNEL An intersymbol interference (ISI) channel is a special case of the general FSC in which the channel state vector s epens statistically on the input coewor x [2, pp. 97-1]. A pure ISI channel is the one in which the channel state vector epens eterministically on the input coewor. In this section, we apply the channel partitioning techniques evelope so far to ecoe the output of a two-state binary input, quaternary output pure ISI channel. The channel in Fig. 4 is a simple moel of such an ISI channel. The channel has input coewors x = (x 1,..., x N ) X {, 1} N an output coewors y = (y 1,..., y N ) Y = {a, b, c, } N. Further, as can be seen from the figure, the channel state at any time instant is the same as the channel input at the previous time instant. Without loss of generality, it is also assume that the channel starts from rest in the sense that s = x 1. Thus, the probability assignment on the current output y n conitione on the input coewor x is simply given by p(y n x) = p(y n x n 1 x n ) = p yn x n 1x n (17) for n {2,...,N}, an p(y n x) = p(y n x n ) = p yn x nx n for the specific case of n = 1. Aitionally, it is assume that we have a uniform prior over the input coebook, that is, p(x) = 1/ X, an that the channel outputs are statistically inepenent conitione on the input coewor, that is, N p(y x) = p y1 x 1x 1 p yn x n 1x n. (18) n=2 Finally, we assume that the ISI channel is symmetric in the sense that p a = p 11, p a 1 = p 1, p b = p c 11, p b 1 = p c 1, p c = p b 11, p c 1 = p b 1, p = p a 11, p 1 = p a 1. Note that in this specific case of a symmetric ISI channel, the channel parameter space is 6-imensional in the general case of a non-symmetric ISI channel, however, it can be as large as 14-imensional.

5 TABLE I COMPLEXITY OF MAP DECODINGS BASED ON NEWTON POLYTOPES N X = 2 N 3 # of NP(f y) Vertices A. Experiment 1: Complexity of Newton Polytopes As note earlier, a particularly exciting consequence of using Netwon polytopes for MAP etection is that the number of vertices of these Netwon polytopes grows only polynomially in the number of parameters of the graphical moel [3]. In the context of this symmetric ISI channel, therefore, it means that the number of MAP ecoings can be no larger than O(N 6 ), since the Newton polytope of an output coewor y of this channel can have imensions no larger than six. On the other han, since any reasonable input coebook will have carinality exponential in N, one woul expect the number of possible MAP ecoings to be exponential in the block length N. In this experiment, we numerically emonstrate the polynomial (or subexponential) complexity of MAP ecoings base on Newton polytopes. The input coebook X in this experiment is given by a ranom coebook of carinality 2 N 3, where N is the block length. Note that given an output coewor y, the corresponing Newton polytope NP(f y ) can be calculate by applying (7) to (18). However, these Newton polytopes can also be more easily calculate using the polytope propagation algorithm escribe in [3]. 1 The outcome of this experiment is summarize in Table 1. B. Experiment 2: Perturbation bouns uner parametric variation In this experiment, we consier the effect of varying channel parameters on the perturbation boun δ as escribe in Section IV. The input coebook X in this experiment is also given by a ranom coebook of carinality 2 N 3, where the block length is given by N = 5. In orer to visualize the outcome of this experiment, we reuce the imensionality of the channel parameter space to two by assigning the conitional probabilities as follows p a = p 11 = α A, p a 1 = p 1 = β4 p b = p c 11 = α2 A, p b 1 = p c 1 = β3 p c = p b 11 = α3 A, p c 1 = p b 1 = β2 p = p a 11 = α4 A, p 1 = p a 1 = β where α, β 1 are the two (variable) channel parameters, an A an B are the normalization constants given by A = α + α 2 + α 3 + α 4 an B = β + β 2 + β Ch. 6 an 7 in [4] also contain a etaile escription of this algorithm. log 1 (δ) x* = log 1 (β) Perturbation boun vs. ISI parameters log 1 (α) 4 x* = 111 Fig. 5. Perturbation boun δ in the tropical channel parameter space as a function of the ISI channel parameters α an β. β 4, respectively. Note that these geometrically progressing probability assignments are rather natural in the sense that, for example, if was transmitte then one is more likely to receive an a at the receiver than a b, a c or a. For the sake of this experiment, we fix the output coewor to be y = bcac while the input coebook is given by X = {111, 11, 1111, 1111}. We vary α an β from 1 1 to 1, an calculate the corresponing perturbation boun δ in the tropical channel parameter space as well as the optimal (MAP) input coewor. The results of this experiment are plotte in Fig. 5. The suen rop in the perturbation boun δ in the figure is an inication of the fact that the channel parameter θ = (α, β) is very close to the bounary separating two ecoing regions on one sie of this bounary is the ecoing region for x(θ) = 1111, while on the other sie is the ecoing region for x(θ) = 111. VI. CONCLUSION Base on recent avances in the fiel of tropical geometry, this work provies a new methoology for partitioning the parameter space of finite state channels into separate ecoing regions. The approach is base on a polynomial-time polytope propagation algorithm [3], [4], an the partitioning process is highly efficient an scalable. Furthermore, we have provie a framework for characterizing the robustness of ecoing uner uncertainties in the channel parameters. REFERENCES [1] J. Richter-Gebert, B. Sturmfels, an T. Theobal, First steps in tropical geometry, in Proc. Iempotent Mathematics an Mathematical Physics, ser. Contemporary Mathematics, vol Vienna, Austria: American Mathematical Society, Feb. 23, pp [2] R. G. Gallager, Information Theory an Reliable Communication. New York: John Wiley an Sons, Inc., [3] L. Pachter an B. Sturmfels, Tropical geometry of statistical moels, Proc. Natl. Aca. Sci., vol. 11, pp , 24. [4] L. Pachter an B. Sturmfels, Es., Algebraic Statistics for Computational Biology. Cambrige University Press, 25. [5] F. Kschischang, B. Frey, an H.-A. Loeliger, Factor graphs an the sum-prouct algorithm, IEEE Transactions on Information Theory, vol. 47, pp , 21. 2