3.1. Solution for white Gaussian noise

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1 Low complexity M-hypotheses detection: M vectors case Mohammed Nae and Ahmed H. Tewk Dept. of Electrical Engineering University of Minnesota, Minneapolis, MN mnae,tewk@ece.umn.edu Abstract Low complexity algorithms are essential in many applications which require low power implementation. In this paper we present a low complexity technique for solving M-hypotheses detection problems, that involve vector observations. Our technique works in these cases where the number of vectors is equal to or smaller than the dimensionality of the vectors. It attempts to optimally trade o complexity with probability of error through solving the problem in a lower dimension. 1. Introduction M-ary hypotheses testing arises in many applications. Optimal detector gas a complexity that is linear in M. However, M can be large, or we may operate under low power constraint. Hence, the need for low power and low complexity detection algorithms. In [1] we presented a progressive renement approach to M-ary detection problems. This approach can lead to a logarithmic reduction in the complexity of the detector. Ideally, we would hope to group the hypotheses in two large groups and subsequently split these two sets in two recursively. Unfortunately, binary partitions cannot always capture the exact boundary between two groups of hypotheses. Therefore, we cannot group the hypotheses in disjoint groupings if we wish to achieve a detection performance close to optimal. On the other hand, we want to minimize overlap between the groupings to minimize the number of comparisons required to make a decision. Hence, our problem then is one of approximating the partitions of the decision plane with the minimal number of binary partitions, or equivalently of designing a tree of minimal depth. In this paper we will discuss a low power detection algorithm applied to M-hypotheses detection when these hypotheses involve vector observations, and the number of hypotheses, M, is smaller than the dimensionality, N, of the vectors. This case arises in many practical applications. Rather than doing the detection in a tree like structure as in [1], the approach we present here projects the received vector onto a lower dimensional space where the detection algorithm can be performed using a relatively smaller number of operations. The projection is done in such a way so as to maximize the minimum distance between the projections of the dierent hypothesis. The rest of this paper is organized as follows: in the next section we will formulate the problem we are trying to solve. In the following section, we explain our proposed projection approach, as applied to orthogonal and non orthogonal vectors. We then end with the conclusion. 2. Problem Formulation Assume that we have M possible hypotheses described by N dimensional vectors, Therefore, under hypothesis i, we receive the vector r = v i + noise, 1 i M. Assume that the noise is white Gaussian noise. We want to select one of the M hypotheses so as to minimize the probability of error. It can be shown [2] that the optimal receiver is to select v i that achieves max 2r T v i? v v i T v i (1) i where r is the received vector. Therefore, we have M matched lter operations and M? 1 comparisons to nally decide on a hypothesis out of the M hypotheses. We have two cases: Case 1: The M-hypotheses lie in N dimensional space, where M N. This will be solved through a projection approach that nds the optimal subspace to project the received vector onto. Case 2: The M-hypotheses lie in N dimensional space, where M > N. In this case, we use the progressive renement approach presented in [1].

2 3. M-ary hypothesis testing problems in larger dimensional spaces In this case, we want to solve the problem using complexity, P, that is lower than M. We rst present a general formulation of the problem, and then we show how to solve this problem in several cases. We want to divide our space into K = 2 P regions bounded by P hyperplanes. Each region is indicated by being above or below each of the P hyperplanes. To minimize the probability of error we solve the following minimization problem min ~x 1; ~x 2;:: x~ P ;th 1;:::th P where MX KX (1?(i?j k ))(P rob:(r 2 R k jh i )) i=1 k=1 (2) j k = index max H i P rob:(r 2 R k jh i ) (3) ~x p 's are unit norm vectors orthogonal on the hyperplanes, th p 's are thresholds determining the exact placement of the hyperplanes, and R k is the kth region bound by these hyperplanes Solution for white Gaussian noise The above formulation holds for any noise density function. The way it is formulated above does not allow any easy solution. In the rest of this paper we will assume we have white Gaussian noise. Since the noise is white, it can be shown that assuming that the hyperplanes are orthogonal does not sacrice any performance. Hence, we have PY P rob:(r k jh i ) = Q((?1) h T (th p? ~x p vi )) (4) p=1 where we dene Q(x) = Z 1 1 p exp? x 2 x2 2 2 (5) and h is either 1 or -1 depending on whether the region R k is above or below the hyperplane designated by ~x p and th p. Now let us assume that the vectors v i are orthogonal vectors, and further assume that M = 2 P. We can simplify the above optimization problem, since we can now pre-assign each of the M regions to any one hypothesis without any loss in the performance. Since the vectors, v i, are orthogonal, we can always transform them to any other set of orthogonal vectors. Therefore, the vectors are \interchangeable". The problem now becomes subject to max x p;th p MX PY Q((?1) h T (th p? ~x p vi )) (6) i=1 p=1 T T ~x p ~xp = 1; x~ m ~xn = ; m 6= n (7) where the h's are chosen according to which region we assign the hypothesis i. We now make the following remark, the way we divide the space is that above each hyperplane we have M 2 vectors and the same number of vectors below it. Therefore, at high signal to noise ratio, we choose that plane such as to maximize the minimum distance between these 2 groups of vectors. Assume, without any loss of generality, that these vectors are represented by the unit vectors, v i = e i, where e i is zeros everywhere except in the ith place where it is a 1. We can write the maxmin problem, and it can be shown to be convex [3]. However, it can easily be seen that to maximize the minimum distance, every vector has to be projected such as its distances from the plane is equal to that of all the other vectors, as, if this is not the case, and we increase the distance of one of the vectors, we have to decrease the distance of some other vector, leading to a decrease in the minimum distance. The vector that we project onto now becomes, y y y : : z z : (8) where a group of vectors is projected onto y and the other group onto z. Therefore we write the problem as max y;z y? z subject to ( M 2 )(y2 + z 2 ) = 1 (9) Therefore, we dierentiate the following equation with respect to y, z, and the Lagrange multiplier and equate to zero We get y? z + ( M 2 )(y2 + z 2 )? 1) (1) y =?z = 1 p M (11) Dividing the space into these optimal regions is equivalent to a projection algorithm, where we map the vectors onto the optimal M? sized constellation in P dimensions, where P is the order of the computations we want to decrease our computations to. If the vectors are orthogonal this constellation is such that we need only to perform P matched lter operations and P comparisons to reach a detection decision.

3 In the case of non orthogonal vectors, we require that we only do P matched lter operations, while we allow ourselves more than P comparisons. Therefore, we require that the vectors corresponding to subtraction of the projection of the M hypotheses in the P dimensional space must be in no more than P directions. In that case we need to only project the received vector, r, on these directions, and through some comparisons we can decide on the sent hypothesis Special Cases We now consider some special cases. Assume that M is a power of 2. For example, let us consider the case of M=4 vectors in 4 dimensional space. We have two cases: 1. Orthogonal signal set: Our M vectors are represented by vectors that are orthogonal, i.e. v T i :v j = (12) for all i 6= j. We also assume that the vectors are all equal in norm. We want to project these vectors onto a P = 2 dimensional space. The optimal size four constellation in a 2 dimensional space is a square. Therefore, we project our 4 vectors on the vertices shown in Fig. 1. Then we project the received vector, r, on the x axis and y axis. We then have 2 comparisons to make: compare the x and y components of this projection to zero. performance when the battery is fully charged, but will tend to sacrice some performance for the sake of extending the battery life, when the battery is almost discharged. 2. Nonorthogonal signal set: The M vectors are general. We only assume that they form an independent signal set. We form a matrix such that these M vectors are its columns. If we now multiply this matrix with its inverse from the left side, we have projected the M hypotheses to the same vertices as Fig. 1. The problem is that the noise now is no longer white, and hence the nearest neighbor technique wouldn't be useful here. Therefore, we have to whiten the noise. As a result the vectors now lie on the vertices of Fig. 2. We have to notice that dierent arrangements of the vectors as columns in the matrix will give dierent performance results, and to obtain the best performance, all the possibilities have to be compared. Also, notice that the comparison operations might now be more than P. In the gure shown, notice that if a received vector is closer to hypothesis 1 than 2, then it is closer to 3 than 4, and hence you only need to compare between 3 and 4. While if it is closer to 2, then we still have to compare 2 and 4, and then the winner of these and 3. Therefore the number of comparisons is lower bound by P and upper bound by M? 1. In all cases we only need to do P matched lter operations instead of M. Hyp. 3 Hyp. 1 Hyp. 3 Hyp. 1 Hyp. 4 Hyp. 2 Hyp. 4 Hyp. 2 Figure 1. Orthogonal Signal Set Orthogonal signal sets where the number of vectors is a power of 2, though seemingly restrictive, has many practical applications. For example, in CDMA communications, a user sends one out of 64 orthogonal Walsh functions [4] of length 64. A receiver needs to do matched lter operations with each of these function to be able to decide on the received vector. With the increased usage of antenna arrays, this operation might be repeated several times, making it a large computational burden. Through our technique, we can decrease the computations. As we will show shortly, we can also trade-o complexity and performance. Perhaps, a receiver designer would opt to achieve the maximum Figure 2. Nonorthogonal Signal Set If we have 8 hypotheses in 8 dimensional space, and we wish to solve the problem using 3 operations, we map the hypotheses to the vertices of the cube shown in Fig. 3. If we have a signal set of size smaller than 8, and we want to solve the problem in order 3 operations, we map the hypotheses into a subset of these 8 vertices. This guarantees that, starting from an orthogonal signal set, the projected noise is still white. Therefore our algorithm can be stated in the following: 1. Map the M hypotheses to the vertices of a parallelopoid of dimension P, where log2m P M. 2. Solve the detection problem in the reduced dimensional space, using P matched lter operations. We can now use the nearest neighbor algorithm to solve the problem in the reduced dimensional space. We can

4 ( 1 8, 1 8, 1 8 ) ( 1 8, 18, 1 8 ) ( 1 8, 1 8, 1 8 ) most increase in the minimum distance between the projected hypotheses, and so on. Fig. 4 shows how the minimum distance changes with increasing the dimensionality at several values of M. Notice here that although the at regions in the graphs correspond to no increase in the minimum distance, we gain some performance by increasing the dimensionality in these regions, as we have a decrease in the number of nearest neighbors as we project on more vectors. 5 Figure 3. Projection of size 8 signal set onto 3 dimensional space vectors 128 vectors 256 vectors 5 prove that the nearest neighbor in the reduced dimensional space is the optimal receiver in that space: We have M vectors, and we receive r = v i + noise, where i is any integer between 1 and M. We now project the received vector, r, by multiplying it by P row vectors, u j ; 1 j P. Therefore we get, r p = v p i + n (13) where r p is the projected received vector, v p i is the projection of v i. If the u j 's are orthogonal, the n vector will be a a P component white Gaussian noise. Therefore the problem is now equivalent to a smaller problem, and the optimal receiver is the nearest neighbor in the new dimensional space. 3. With some additions and comparisons, we can decide on the true hypothesis. Notice that in the case of orthogonal hypotheses, the parallelopoid is a cube Higher Complexity Solutions In some applications, we need to trade-o complexity and performance, so we want to increase the complexity in steps if this achieves a desired gain in performance. Our projection approach allows that by allowing us to progressively project on subspaces of dimensions P log2m. To see how we do this, assume, without loss of generality, that we have an orthogonal signal set composed of M unit vectors in M dimensional space. To project this on a P = log2m dimensional space, we choose a subset of the M length M Walsh functions corresponding to the optimal parallel constellation. The Walsh functions hare orthogonal series of bipolar vectors of the form p 1 p 1 p 1 p 1 : : : 1 M M M M p M When we want to incrementally increase the performance, we project on the Walsh vector that gives the i db loss dimension Figure 4. Minimum distance (in db loss from the optimum ) with dimensions in an orthogonal signal set For nonorthogonal signal sets, we can proceed as with orthogonal unit vectors, by multiplying the matrix of the hypotheses by its inverse. We have an extra step at the end, namely that of whitening the noise. Fig. 5 shows how the performance increases with increased dimensionality for several values of M. The original hypotheses here were chosen to be a random set of unit norm vectors. As for other values of M which are not powers of 2, we can always choose a subset of the projected powerof-2 hypotheses. For example, assume that we have M = 3 unit vectors in a 3 dimensional space. We want to project them onto a 2 dimensional space. We have 2 options, the constellation shown in Fig. 6, and the one shown in Fig. 7. In Fig. 6, we projected the 3 vectors onto 3 corners of the unit square ( the one whose vertices correspond to vectors of norm 1 ), while in Fig. 7, we project the vectors onto the vertices of the unit equilateral triangle. Comparing the minimum distance achieved by these 2 constellations, we found that the latter constellation provides a higher minimum distance. For M=5 projected onto a 3 dimensional space, we can use a pyramid constellation, or use 5 vertices of the

5 2 8 vectors vectors db loss vectors dimension Figure 7. Constellation 2 for 3 vectors projected onto a plane Figure 5. Minimum distance ( in db loss from the optimum )with dimensions in a nonorthogonal signal set Figure 8. Constellation of 13 points in 5 dimensions Conclusion Figure 6. Constellation 1 for 3 vectors projected onto a plane cube shown in Fig.3. For M = 6, we can use 2 parallel constellation of 3. For general M our projection algorithm for projecting onto P = dlog2m e would be: 1. Divide M into factors 4 that are close together as possible (eg. 13=6+7, 7=4+3, 6=3+3 etc.. ) 2. Form the constellation in 2 dimensions using the largest factor rst, then add on parallel constellations as you increase the dimension. 3. Whiten the noise If we want to use more dimensions, we choose an M sized subset of the power-of-2 constellation in that dimension. This subset is chosen according to the factors of M. Fig. 8 shows how we choose the 13 points in 5 dimensions out of the original 16. The solid dots correspond to the 16 points, while the larger circles correspond to the 13 points. In this paper we presented a low power detection approach that leads to an almost logarithmic reduction in complexity in solving hypotheses detection problems when these hypothesis are represented by vectors. This approach relies on projecting the hypothesis onto an optimal constellation in a lower dimensional space. This approach also allows us to trade o complexity and performance, by increasing the dimension of the space where we solve our detection problem. References [1] M. Nae and A. Tewk, \Reduced Complexity M- ary Hypotheses Testing in Wireless Communications," Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, Seattle, Washnigton, [2] H. Van Trees, Detection, estimation and Modulation theory, Wiley, NY, [3] P. Gill, W. Murray, and M. Wright, Practical Optimization, Academic Press, CA, [4] M. Zoltowsky, Talk given in EE Dept. Colloquium Series, University of Minnesota, Winter 1998.