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(b) Figure 4.11. Comparison of the vector codebooks between random generation (a) and HBLAST (b) architectures.
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where hk is the kth column of h and xk is the transmitted signal from the kth antenna. This is exactly the same channel model as in multiaccess communications, where nT transmitters encode information independently and the nT users have the same transmit power constraint P0/nT. The sum capacity from multiaccess communication theory is given by
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T P0 Ccsir = Ck ,csir = e [I ( X 1 , X 2 , . . . , X nT ; Y h)] = e log 2 I nR + hh* 2 nT s z k =1
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which is the same as the MIMO ergodic capacity. Hence, we can deduce that the H-BLAST architecture is optimal with respect to fast fading channels. At the receiver, we can adopt either the ML receiver or MMSE-SIC receiver because the MMSE is information-lossless.
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Optimal Architecture for Slow Flat Fading Channels
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While H-BLAST architecture is a capacity-achieving design for fast fading channels, we shall illustrate in this section that H-BLAST is strictly suboptimal in slow fading channels. In slow fading channels, there is no temporal variation of the fading channels across the encoding frame. Two factors contribute to packet transmission errors: the nite codelength effect and the channel outage effect. The nite-codelength effect refers to the failure of the error correction code to correct errors due to AWGN noise in the channel. The channel outage effect refers to the fact that the current encoding rate is higher than the actual channel capacity. In fact, even if we employ powerful (capacityachieving) codes at the transmitter, there is still a nite probability of packet error due to the second factor (channel outage effect). In general, following the standard random coding argument, the outage probability of general MIMO channels with CSIR only in slow fading channels is given by 1 Pout (R, Q x ) = Pr log 2 I nR + 2 hQ x h * < R sz (4.27)
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where R is the overall information throughput (bits per channel use) at the transmitter and Qx is the input covariance matrix. Hence, the transmitter design (parameterized by Qx) that minimizes the outage probability is given by the following optimization problem. Problem 4.1 (Minimum Outage Probability) Pout (R) =
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Q x : tr (Q x ) P0
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It has been shown that in MISO channels (nR = 1), the optimizing covariance matrix [which achieves the minimum outage probability P*ut(R)] is given by o P0 Qx = I n . In general, Telatar [126] conjectured that it is also the optimizing nT T input covariance matrix in the outage sense at high SNR. While the optimizing Qx is a scaled identity matrix, we shall illustrate that this does not imply that all different transmitter designs that lead to the same optimizing Qx can also optimize the outage probability. In Theorem 4.1 we shall illustrate that 1 the minimum outage probability Pout(R)* decays like at high SNR. SNR nT nR In other words, the maximum diversity order that can be achieved by slow fading MIMO channels with CSIR only is nTnR. We rst consider the H-BLAST design performance in slow fading channels. Because of the slow fading channels, there is only a single realization of h across the temporal domain of the encoding frame. Even if powerful com-