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In this chapter, we have considered the effect of imperfect CSI on MIMO channel capacity. For instance, there are two roles of CSI in the MIMO wireless link. The CSI at the receiver is used to enhance the signal detection and has a more stringent requirement on the accuracy. The CSI at the transmitter, on the other hand, is used for transmission adaptation and therefore has a more relaxed requirement on the accuracy. In the rst part of the chapter, we consider the effect of imperfect CSIR on the MIMO channel capacity. An asymptotically tight upper bound and lower bound on the capacity are derived. In the second part of the chapter, we consider the effect of limited feedback link capacity on the partial CSI feedback strategy and the transmission adaptation strategy. We found that the optimal feedback and transmission design with the constraint of limited feedback is equivalent to the classical vector quantization problem. When we optimize for the effective SNR, the optimal solution requires Grassmannian partition on the CSIR space. On the other hand, when we optimize for the forward MIMO link capacity, the optimal solution requires a partition de ned over the generalized distortion measure. In either case, the optimal transmission strategy has temporal and spatial water- lling components. At the receiver, there are Q partitions on the CSIR space H. On every received CSIR H, the receiver has to search for the best- t partition region and feedback the partition index. Hence, instead of carrying the quantized CSIR on the feedback link, we found that the optimal structure just needs to carry the partition index on the feedback link. At the transmitter, there is a table of Q entries. Each entry from the table corresponds to a speci c power allocation (across
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Cfb = Cfb = 3 Cfb = 2 10 Average capacity
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Cfb = Cfb = 1 Cfb = 4 Cfb = 3 10 Average capacity
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(b) Figure 3.11. Forward channel capacity versus average SNR with ideal CSIR for 2 1 (a) and 4 1 (b) systems and Cfb = 0,1,2,3,4, .
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nT antennas) as well as a beamforming matrix. Instead of adapting to the instantaneous channel state, the transmitter adapts to the instantaneous partition index. The receiver partition and the transmitter table are obtained from the of ine Lloyd algorithm in much the same way as the classical vector quantization problems.
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EXERCISES 1. Capacity Lower Bound of MIMO Links We assume imperfect CSIR V which is an erroneous measurement of the channel, H. V = H +e We consider a block at fading MIMO channels characterized by Yn = H n X n + Zn As illustrated in the text, the ergodic capacity can be written as, C csir = max I ( X ; Y V ) = max H ( X V ) - H ( X Y , V )
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(a) prove that H(X|V) = log2(|peQ|) where Q = E{xx*|H} where H(.) denotes the entropy and E{.} denotes the expectation. (b) prove that the variance of X - AY is minimized for any A if AY is the linear minimum mean square estimation of X. (c) prove that H(X|Y, V) H(X - AY) for any A. hint: adding a constant does not change the differential entropy and conditioning reduces entropy. (d) prove that H(S - AY) ln(pevar(S - AY)) where var(.) denotes the variance of a random variable. (e) use (c) to (d) to obtain an upper bound of H(X|Y, V) (f) Hence, prove that 1 I lower ( X ; Y V ) = E log 2 I + V* VQ 2 1 + s EP
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2 where sE is the variance of the CSIR error e. (g*) Suppose we have quasi-static fading, is the lower bound on the instantaneous channel capacity in (f) still valid
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Maximum Likelihood Estimator Maximum Likelihood Estimator (MLE) is one popular approach in channel estimation. Consider a block fading channel characterized by Y = HX + Z