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...t necessarily arise from the best description of the channel in the mean squared sense. For the description of a Gaussian source (such as the channel vector in our case), the rate distortion function =-=[18]-=- characterizes the difference in performance of a finite bit quantizer and an infinite bit quantizer; where infinite bits lead to zero mean squared distortion. Motivated by the rate distortion functio...

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...n criterion with multiple receive antennas; while SNR maximization always results in unit rank beamforming [5], [11], outage minimization results in generalized beamforming with spatial water-filling =-=[14]-=-. The rest of the paper is organized as follows. The system model is described in Section II. In Section III, the beamforming methodology is described and a universal lower bound on outage probability...

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...ed by the span of the columns of the matrix given by (77) A measure of distance, called the chordal distance, between the subspaces and is given by where and are the principal angles between and (78) =-=[23]-=-. It can be shown that (see Appendix III), so that the design criterion in (71) can be restated as (79)s2576 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 2003 Fig. 7. Performance ...

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...squares of Gaussian random variables with zero mean and variance equal to . Hence, has a central chi-square distribution with degrees of freedom. The probability density function (pdf) of is given by =-=[16]-=- (31) The cumulative distribution function (cdf) of is given by [16] (32) We can now integrate the conditional distribution of successful transmission given in (30) using the distribution of given in ...

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...EDBACK IN MULTIPLE-ANTENNA SYSTEMS 2575 C. Similarity With Unitary Space–Time Constellations Unitary space–time constellations for multiple-antenna systems were introduced by Marzetta and Hochwald=-= in [19]. -=-In particular, it was shown that unitary space–time constellations achieve the capacity of multiple antenna systems when the channel information is not available at either the transmitter or the rec...

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...a multiple transmit antenna system along with a single receive antenna. The channel is assumed to be quasi-static in time and, hence, outage probability can be used as the relevant performance metric =-=[7]-=-. We assume that the channel realization is known perfectly at the receiver. Even though this is not possible in practice, such an assumption allows us to focus on the effects of finite rate feedback ...

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...ity of beamforming with partial channel information at the transmitter is investigated in [4], [8], [9] with mutual information as the performance metric. former codebooks with single receive antenna =-=[11]��-=-�[13]. However, the two approaches lead to a different design criterion with multiple receive antennas; while SNR maximization always results in unit rank beamforming [5], [11], outage minimization re...

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... the channel information is not available at either the transmitter or the receiver. The design criterion for unitary space–time constellations to minimize the pairwise error probability was given i=-=n [20]. -=-A unitary space–time constellation consists of signals , where , where is the block length of the code (less than or equal to the coherence time of the channel), is the number of transmit antennas, ...

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...ncatenated with a scalar codebook (rank one signaling) requires lower decoding complexity than a typical space–time code (full rank codebook). A similar quantized feedback model was first analyzed i=-=n [4]-=- to maximize average mutual information and average received signal-to-noise ratio (SNR), for a multiple transmit and single receive antenna system. The use of Lloyd algorithm for quantization was sug...

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... viz., SNR maximization, also leads to the same design criterion for beam2 The conditions for the optimality of beamforming with partial channel information at the transmitter is investigated in [4], =-=[8], -=-[9] with mutual information as the performance metric. former codebooks with single receive antenna [11]–[13]. However, the two approaches lead to a different design criterion with multiple receive ...

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...t antennas, in the beamforming design problem. Hence, the design of beamformers is equivalent to the problem of the design of unitary space–time constellations under these conditions. Hochwald et al=-=. [21] a-=-ddressed the problem of imposing structure on the unitary space–time constellations for easier encoding/decoding as well as to reduce the dimensionality of the search space during the design process...

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...erm power constraints where the power saved in one frame cannot be used in another. It should be noted that additional gains in outage can be obtained by channel adaptive rate control mechanisms [1], =-=[10]-=-. The contributions of this paper are as follows. 1) Performance bounds on beamforming: We derive a universal lower bound on the outage probability of an arbitrary beamformer with finite feedback bits...

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...ces We conclude the discussion on the construction of beamformer codebooks by showing similarities to the problem of packings in Grassmannian spaces, which was discussed in detail by Conway et al. in =-=[22]-=-. The similarity could potentially lead to new construction methods for beamformer codebooks. Consider the vector . We can write , where and are the real and the imaginary parts of , respectively. Now...

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...erformanceofbeamformer codebooks with finite number of beamforming vectors. 1) Perfect Feedback Bound: Suppose that the channel is known perfectly at the transmitter and the receiver. It was shown in =-=[17]-=- that the optimal scheme for minimizing outage consists of a Gaussian code followed by a beamformer. The optimal beamformer is given by a unit vector in the direction of , where is the channel realiza...

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... the channel, then a noncoherent transmission is optimal. However, with receiver feedback regarding the channel, estimating and feeding back a few bits can appreciably outperform a noncoherent system =-=[2]-=-, especially if the coherence time of the channel is sufficiently large. Second, the proposed formulation directly captures the finite rate nature of the feedback link along with the available feedbac...

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...dback for power control. The problem of designing beamformers to minimize the pairwise error probability of codewords with certain structure imposed on the feedback channel was also addressed in [5], =-=[6]-=-. In this paper, we consider a multiple transmit antenna system along with a single receive antenna. The channel is assumed to be quasi-static in time and, hence, outage probability can be used as the...

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..., SNR maximization, also leads to the same design criterion for beam2 The conditions for the optimality of beamforming with partial channel information at the transmitter is investigated in [4], [8], =-=[9] w-=-ith mutual information as the performance metric. former codebooks with single receive antenna [11]–[13]. However, the two approaches lead to a different design criterion with multiple receive anten...

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...ic is independent of the norm of and depends only on the “direction” of . Hence, if a particular belongs to , then for any , belongs to . Let denote the surface area (sometimes referred to as cont=-=ent [15]-=-) of . We will now characterize the subsets of which can support a given rate of transmission (no-outage regions) and those that cannot support the given rate (outage regions). We will show that the n...

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... for quantization was suggested as a possible solution for the design of 1 If the transmitter also performs rate control, then the outage can be made arbitrarily small at the cost of increasing delay =-=[1].s-=-MUKKAVILLI et al.: ON BEAMFORMING WITH FINITE RATE FEEDBACK IN MULTIPLE-ANTENNA SYSTEMS 2563 Fig. 1. Performance gains of feedback-based schemes compared to space–time codes for four transmit antenn...

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...f beamforming with partial channel information at the transmitter is investigated in [4], [8], [9] with mutual information as the performance metric. former codebooks with single receive antenna [11]�=-=��[13]-=-. However, the two approaches lead to a different design criterion with multiple receive antennas; while SNR maximization always results in unit rank beamforming [5], [11], outage minimization results...