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# E  Information Capacity of MIMO Channels

The topic of multiple-input multiple-output (MIMO) links for wireless communications was discussed in Chapter 9 on signaling over fading channels. To get a measure of the transmission efficiency of MIMO links therein, we resorted to the notion of outage capacity, which is naturally of practical interest. However, in light of its mathematical sophistication, we deferred discussion of the information capacity of MIMO links rooted in Shannon’s information theory to this appendix.

To be specific, in this appendix we discuss two different aspects of information capacity:

1. The channel state is known to the receiver but not the transmitter;

2. The channel state is known to both the receiver and the transmitter.

The discussion will proceed in this order.

## E.1   Log-Det Capacity Formula of MIMO Channels

Consider a communication channel with multiple antennas.1 Let the Nt-by-1 vector s denote the transmitted signal vector and the Nr-by-1 vector x denote the received signal vector. These two vectors are related by the input−output relation of the channel:

where H is the channel matrix of the link and w is the additive channel noise vector. The vectors s, w, and x are realizations of the random vectors S, W, and X, respectively.

In what follows in this appendix, the following assumptions are made:

1. The channel is stationary and ergodic. ...

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