# Transmit Diversity

Topics: Space–time code, Diversity scheme, MIMO Pages: 32 (7383 words) Published: March 19, 2013
MIMO Systems and Transmit Diversity 1 Introduction

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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

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Figure 1: A single-input-single-output channel

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notes assumes the reader knows the basics of information theory. See [1] 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)

(1)

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...