CHANNEL CAPACITY WITH PERFECT CSIR in .NET

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CHANNEL CAPACITY WITH PERFECT CSIR
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Channel Encoder 1 Channel Encoder 2
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P0 / nT P0 / nT
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P0 / nT
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nT transmit antennas
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Figure 2.7. Optimal transmitter structure of fast fading MIMO channels with perfect CSIR.
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On the other hand, observe that Qx = aInT because every diagonal element is an average of all possible permutations of the diagonal elements. Choosing a to satisfy the transmit power constraint, we have the capacity achieving input P0 In . covariance matrix given by nT T Note that the capacity achieving the input covariance matrix implies uniform power allocation across the nT transmit antennas. Furthermore, since Qx is diagonal, one possible capacity achieving transmitter architecture5 consists of nT parallel and isolated channel encoders as illustrated in Figure 2.7. Let S = (S1 . . . SnT ) be the output symbol vector of the nT-independent channel encoders (with random codebooks). We have e[SS*] = InT. The transP0 S . Hence, the covariance matrix of such mit vector symbol is given by X = nT P0 I n . In other words, no joint transmitter in Figure 2.7 is given by e [XX *] = nT T encoding is needed at the transmitter to realize the MIMO channel capacity if a maximum-likelihood (ML) receiver is used. To evaluate the MIMO capacity, we can write the capacity as P0 P0 Ccsir = e log 2 I nR + HH * = e log 2 I n T + H*H nT h0 nT h0
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(2.34)
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Note that although one possible transmitter architecture is isolated encoding, this does not imply that isolated encoding is the only possible capacity achieving distribution. In fact, a MIMO transmitter with joint encoding can also give a diagonal input covariance matrix as well as a larger error exponent. This will be elaborated in 4.
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De ne HH * nR < nT W= H * H nT < nR This capacity can be expressed in terms of eigenvalues l1, . . . , lm* of W, where m* = min[nT, nR]: P0 m* Ccsir = e log 2 1 + li nT h0 i =1 (2.35)
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where the expectation is over the distribution of the eigenvalues. We consider several important asymptotic cases concerning the MIMO channel capacity in fast fading channels. Transmit diversity nR = 1: When nR = 1, the MIMO capacity becomes
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2 P0 h Ccsir (nR = 1) = e log 2 1 + = e [log 2 (1 + g )] nT h0
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P0 h 2 is the effective SNR. Note that the average SNR nT h0
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P0 is independent of nT but the diversity order of g is nT. Hence, h0 we anticipate capacity increases as nT increases. Figure 2.8a illustrates the ergodic capacity versus number of antennas. Receive diversity nT = 1: When nT = 1, the channel capacity becomes P0 hh * Ccsir (nT = 1) = e log 2 I nR + = e [log 2 (1 + g )] h0 P0 h 2 is the effective SNR. In this case, g enjoys similar h0 nR-order diversity gain in the statistic. In addition, the mean SNR nR P0 e [g ] = increases as nR increases. Hence, we anticipate the ergodic h0 capacity to increase more effectively as nR increases (compare with the previous case of transmit diversity). This is illustrated in Figure 2.8b. Observe that Ccsir(nT = 1) is always 10 log10(nR) dB better than Ccsir(nR = 1). where g = Large number of antennas m* = nT = nR : Since the elements of the 1 hh * I nR, due to the law of large channel matrix h is i.i.d., we have nT numbers. Hence, the MIMO ergodic capacity approaches P0 Ccsir (m * ) = m * log 2 1 + h0
CHANNEL CAPACITY WITH PERFECT CSIR
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C 8 6 4 2 0 0 2 4 6 (a) C 12 10 8 6 4 2 0 0 10 20 (b) Figure 2.8. MIMO ergodic capacity versus number of antennas [126], with SNR ranges of 0 35 dB: (a) Transmit diverity capacity versus nT when nR = 1; (b) receive diversity capacity versus nR when nT = 1. 30 40 50 nR 8 10 12 nT
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Hence, the channel capacity increases linearly with the number of antennas m*. Figure 2.9 illustrates the ergodic capacity versus the number of antennas m*. 2.5.3 Effect of Antenna Correlation on Ergodic MIMO Capacity
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The evaluation of the MIMO capacity in the cases presented above assume i.i.d. channel fading in the nR nT channel matrix. This is achievable when the physical separation of the transmit antennas (as well as the receive antennas) is large. However, in practice, because of the physical constraint or the scattering environment, there might be some correlation between the elements of the fading channel matrix. We shall consider the effect of correlated fading in this section.