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~ ~ n1 using none touse none with web,windows applicationcode 128 printing c# x 2. Barcodes FAQs ~ n2 ~ y1 ~ x2 x r ~ y2 ~ nr ~ xr ~ yr Figure 10.3: Parallel Decomposition of the MIMO Channel. Example 10.1: Find the equiva none none lent parallel channel model for a MIMO channel with channel gain matrix .1 .

3 .7 H = .5 .

4 .1 (10.3) .

2 .6 .8 Solution: The SVD of H is given by 0.

555 .3764 .7418 1.

3333 0 0 .2811 .7713 .

5710 H = .3338 .9176 .

2158 0 .5129 0 .5679 .

3459 .7469 . .

7619 0.1278 .6349 0 0 .

0965 .7736 .5342 .

3408. (10.4). Thus, since there are 3 nonze none for none ro singular values, R H = 3, leading to three parallel channels, with channel gains 1 = 1.3333, and 2 = .5129, and 3 = .

0965, respectively. Note that the channels have diminishing gain, with a very small gain on the third channel. Hence, this last channel will either have a high error probability or a low capacity.

. 10.3 MIMO Channel Capacity This section focuses on the S none for none hannon capacity of a MIMO channel, which equals the maximum data rate that can be transmitted over the channel with arbitrarily small error probability. Capacity versus outage de nes the maximum rate that can be transmitted over the channel with some nonzero outage probability. Channel capacity depends on what is known about the channel gain matrix or its distribution at the transmitter and/or receiver.

Throughout this section it is assumed that the receiver has knowledge of the channel matrix H, since for static channels a good estimate of H can be obtained fairly easily. First the static channel capacity will be given, which forms the basis for the subsequent section on capacity of fading channels..

10.3.1 Static Channels The capacity of a MIMO channe none none l is an extension of the mutual information formula for a SISO channel given by (4.3) in 4 to a matrix channel. Speci cally, the capacity is given in terms of the mutual information between the channel input vector x and output vector y as C = max I(X; Y) = max [H(Y) H(Y.

X)] ,. p(x) p(x). (10.5). for H(Y) and H(Y X) the entropy in y and y x, as de ned in 4.1 3 . The de nition of entropy yields that H(Y X) = H(N), the entropy in the noise. Since this noise n has xed entropy independent of the channel input, maximizing mutual information is equivalent to maximizing the entropy in y. The mutual information of y depends on its covariance matrix, which for the narrowband MIMO model is given by (10.

6) Ry = E[yyH ] = HRx HH + IMr , where Rx is the covariance of the MIMO channel input. It turns out that for all random vectors with a given covariance matrix Ry , the entropy of y is maximized when y is a zero-mean circularly-symmetric complex Gaussian (ZMCSCG) random vector [5]. But y is only ZMCSCG if the input x is ZMCSCG, and therefore this is the optimal distribution on x.

This yields H(y) = B log 2 det[ eRy ] and H(n) = B log2 det[ eIMr ], resulting in the mutual information I(X; Y) = B log2 det IMr + HRx HH . (10.7) This formula was derived in [3, 5] for the mutual information of a multiantenna system, and also appeared in earlier works on MIMO systems [6, 7] and matrix models for ISI channels [8, 9].

The MIMO capacity is achieved by maximizing the mutual information (10.7) over all input covariance matrices Rx satisfying the power constraint: C=. Rx :Tr(Rx )= . B log2 det IMr + HRx HH ,. (10.8). where det[A] denotes the dete rminant of the matrix A. Clearly the optimization relative to R x will depend on whether or not H is known at the transmitter. We now consider this maximizing under different assumptions about transmitter CSI.

Channel Known at Transmitter: Water lling The MIMO decomposition described in Section 10.2 allows a simple characterization of the MIMO channel capacity for a xed channel matrix H known at the transmitter and receiver. Speci cally, the capacity equals the sum.

Entropy was de ned in 4.1 fo none none r scalar random variables, but the de nition is identical for random vectors. of capacities on each of the independent parallel channels with the transmit power optimally allocated between these channels. This optimization of transmit power across the independent channels results from optimizing the input covariance matrix to maximize the capacity formula (10.8).

Substituting the matrix SVD (10.2) into (10.8) and using properties of unitary matrices we get the MIMO capacity with CSIT and CSIR as C= max P i : i i .

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