So far we have investigated the use of antenna arrays in interference cancellation and for receive diversity. This ﬁnal chapter takes a broad view of the use of antenna arrays in wireless communications. In particular, we will investigate the capacity of systems using multiple transmit and/or multiple receive antennas. This provides a fundamental limit on the data throughput in multipleinput multiple-output (MIMO) systems. We will also develop the use of transmit diversity, i.e., the use of multiple transmit antennas to achieve reliability (just as earlier we used multiple receive antennas to achieve reliability via receive diversity). The basis for receive diversity is that each element in the receive array receives an independent copy of the same signal. The probability that all signals are in deep fade simultaneously is then signiﬁcantly reduced. In modelling a wireless communication system one can imagine that this capability would be very useful on transmit as well. This is especially true because, at least in the near term, the growth in wireless communications will be asymmetric internet traﬃc. A lot more data would be ﬂowing from the base station to the mobile device that is, say, asking for a webpage, but is receiving all the multimedia in that webpage. Due to space considerations, it is more likely that the base station antenna comprises multiple elements while the mobile device has only one or two. In addition to providing diversity, intuitively having multiple transmit/receive antennas should allow us to transmit data faster, i.e., increase data throughput. The information theoretic analysis in this chapter will formalize this notion. We will also introduce a multiplexing scheme, transmitting multiple data streams to a single user with multiple transmit and receive antennas. This chapter is organized as follows. Section 2 then presents a theoretical analysis of the capacity of MIMO systems. The following two sections, Sections 3 develops transmit diversity techniques for MIMO systems based on space-time coding. Section 4 then addresses the issue of maximizing data throughput while also providing reliability. We will also consider transmitting multiple data streams to a single user. This chapter ends in Section 5 with stating the fundamental tradeoﬀ between data throughput (also called multiplexing) and diversity (reliability).
MIMO Capacity Analysis
Before investigating MIMO capacity, let us take a brief look at the capacity of single-input singleoutput (SISO) fading channels. We start with the original deﬁnition of capacity. This set of
Cha nnel (h)
Figure 1: A single-input-single-output channel
notes assumes the reader knows the basics of information theory. See  for a detailed background. Consider the input-output system in Fig. 1. The capacity of the channel is deﬁned as the maximum possible mutual information between the input (x) and output (y). The maximization is over the probability distribution of the input fX (x), i.e. C = max [I(X; Y )] = max [h(Y ) − h(Y /X)] , fX (x) fX (x)
where h(Y ) is the entropy of the output Y . with limited input energy (E |x|2 ≤ Es ), one can show that the capacity achieving distribution is Gaussian, i.e., x ∼ CN (0, Es ) and y ∼ CN (0, Es + σ 2 ). It is not diﬃcult to show that if n is Gaussian and has variance σ 2 , h(N ) = log2 (πeσ 2 ). Therefore h(Y ) = log2 (πe(Es + σ 2 )). Also, For a SISO additive white gaussian noise (AWGN) channel, y = x + n, with n ∼ CN (0, σ 2 ) and
h(Y /X) is the residual entropy in Y given the channel input X, i.e., it is the entropy in the noise term N . Therefore, h(Y /X) = log2 (πeσ 2 ) and the channel capacity, in bits/s/Hz, is given by C = [h(Y ) − h(Y /X)] = log2 where ρ = Es /σ 2 is the signal-to-noise ratio (SNR). In the case of a fading SISO channel, the received signal at the k-th symbol instant is y[k] = At this point there are...