EFFECT OF LIMITED FEEDBACK OPTIMIZING FOR SNR

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between any two lines. Grassmannian line packing forms the mathematical basis for our quantized beamforming codebook design. In the following text, we present a summary of key results of Grassmannian line packing. Consider the space of unit-norm transmit beamforming vectors Wm Cm where Cm is the m-dimensional vector space over the complex eld. De ne an equivalence relation between two unit vectors w1 Wm and w2 Wm by w1 w2 if for some q [0, 2p) w1 = ejq w2. This equivalence relation says that two vectors are equivalent if they are on the same line in Cm. The complex Grassmannian manifold G(m, 1) is the set of all one-dimensional subspaces of the vector space C m. We de ne a distance function on G(m, 1) by letting the distance between the two lines generated from unit vectors w1 and w2 be the sine of the angle q1,2 between the two lines. This distance is expressed as

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H d(w 1 , w 2 ) = sin(q 1,2 ) = 1 - w 1 w 2 2

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(3.13)

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The Grassmannian line packing problem consists in nding the set (or packings) of N lines in C m that has maximum minimum distance between any pairs of lines. Because of the relation to Wm, the problem simpli es down to arranging N unit vectors so that the magnitude correlation between any two unit vectors is as small as possible. We represent a packing of N lines in G(m, 1) by an m N matrix W = [w1 w2 . . . wN], where wi is the vector in Wm whose column space is the ith line in the packing. The packing problem is of interest only in nontrivial cases where N > m because when N m, the packing lines can be simply picked as N orthogonal basis of Wm. The minimum distance of a packing is the sine of the smallest angle between any pair of lines. This is written as d (W) = min

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1 k l N H 1 - wk wl 2

= sin(q min )

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(3.14)

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where qmin is the smallest angle between any pair of lines in the packing. The Rankin bound gives an upper bound on the minimum distance of line packing as a function of m and m N and is given by d ( W)

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(m - 1)N m(N - 1)

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(3.15)

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The problem of nding algorithms to design packings for arbitrary m and N has been studied by many researchers in applied mathematics and information theory [8,28,124]. However, nding the global optimal solution of the minimum distance for arbitrary m and N is not easy either analytically or numerically. For this reason it is often more practical to rely on random computer searches. For example, Figure 3.2 illustrates a tabulation of the optimal search from Reference 123 for the real case. In some cases closed-form solutions are available, such as when N = 2m = pa + 1, where p is prime and a is a positive integer, conference matrices allow explicit constructions of packings [124].

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Figure 3.2. A table of the best packing [123] found for N 50 lines in G(m, 1) where m 9. The values indicate the maximum angle separations.

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EFFECT OF LIMITED FEEDBACK OPTIMIZING FOR SNR

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Figure 3.3. Block diagram of a MIMO system for spatial diversity.

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Grassmannian Precoding for MIMO Systems Spatial Diversity

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We consider the problem of quantized maximum SNR beamforming for an independent and identically distributed (i.i.d.) MIMO system where beamforming and maximum ratio combining (MRC) are employed at the transmitter using nT transmit antennas and at the receiver using nR receive antennas respectively as illustrated in Figure 3.3. Here, the transmitter has access to a low-bandwidth feedback channel from the receiver. In addition, spatial diversity is employed to increase the resilience to fading in the system that we are going to study. Because of the limited capacity of the feedback channel, we assume that the use of a codebook of possible beamforming vectors known to both the transmitter and receiver and the codebook is restricted to having a limited cardinality of Q and is designed of ine. To simplify the problem, the feedback channel is assumed to be error-free and zero-delay. Suppose that the channel is at fading; thus the discrete-time equivalent channel can be modeled by a nR nT channel matrix H. Given a transmitted symbol T, the received signal for this system after diversity combining Y is given by Y = v *HwT + v *z (3.16)

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The vectors w and v are the nT 1 beamforming and nR 1 combining vectors, respectively. The nR 1 noise vector z has i.i.d. entries distributed according to N(0, s 2). We model the channel H as having i.i.d. entries distributed accordz ing to N(0, 1). The channel is assumed to be known perfectly at the receiver. The symbol energy is given by ET[|T|2] = Et. In a beamforming combining system, the key question is how to design w and v to maximize the SNR, which in turn minimizes the average probability of error and maximizes the capacity. For the proposed system, the SNR gr after combining at the receiver is given by gr = Et v H hw

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