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The ML detection will choose the matrix x that optimizes the ML function in (5.3). Our goal is to design the best choice of the signal set x so as to minimize the average symbol error probability of ML detector. The organization of this chapter is as follows. In Section 5.2, we discuss the constellation design framework for MIMO links with imperfect CSIR (partially coherent designs). In Section 5.3, we address the coded modulation for
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MIMO links with imperfect CSIR (partially coherent systems). Finally, in Section 5.4, we summarize the main points discussed in this chapter. 5.2 CONSTELLATION DESIGN FOR MIMO CHANNELS WITH IMPERFECT CSIR In practical systems, there are usually pilot symbols or pilot channels available to enable channel estimation at the receiver. Because of the limited energy of pilot symbols and the presence of multiple transmit antennas, the channel estimation can never be perfect. The conventional approach is to design the constellation on the basis of perfect CSIR assumption and then evaluate the performance degradation when there is CSIR estimation error. Yet, with this approach, we have no idea as to the best possible design in the presence of imperfect CSIR. In fact, constellations designed using the statistics of the CSIR error are more desirable than the ones designed for perfect CSIR. In this section, we derive a design criteria for spacetime constellations when only imperfect CSIR is available at the receiver. We design single- and multipleantenna constellations based on this criterion and illustrate signi cant performance improvement over conventional constellations in the presence of imperfect CSIR. 5.2.1 System Model
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We consider a MIMO system with nT transmit and nR receive antennas. The nR N received symbols y is given by (5.1), where N is the length of the spacetime code. With imperfect CSIR h given by (5.2), the conditional probability density of the received signal, p(y| h , x), is given by p(y h, x) = e D [ p(y h, D , x)] which is evaluated in (5.3). Let M be the size of the signal set M = {x1, . . . , xM}. De ne pm(y) = p(y| h , xm). Without loss of generality, assuming that xi is the transmitted signal, the conditional pairwise error probability (with ML detection) of detecting xj instead of xi is given by Pr[x i x j h] = Pr[ p j (y) > pi (y)] 5.2.2 Design Criteria Based on Kullback Leibler Distance (5.4)
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The conditional error probability of the ML detector for a signal set of cardinality M is upper-bounded by summing all conditional pairwise error probabilities Pe (h) p(x i ) Pr[x i x j h]
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where p(xi) is the probability of transmitting the ith signal element. However, this error probability is usually dominated by the largest term inside the summation. Hence, the error probability is approximated by Pe (h) max Pr[x i x j h]
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The signal constellation design can be done by selecting the signal set M such that it minimizes the worst-case pairwise error probability as in (5.6). Unfortunately, even with the approximation above, the exact expression or the Chernoff bound for (5.6) is in general intractable. Hence, we shall seek another simple and effective performance criterion. We rst review a very important lemma, namely, Stein s lemma [30], which relates the Kullback Leibler (KL) distance to the pairwise error probabilities of hypothesis testing. Lemma 5.1 (Kullback Leibler Distance) Let Y1, Y2, . . . , Yn Y be n i.i.d. observations generated according to the probability density function Q(y). Consider the hypothesis testing between two alternatives, Q = P1 and Q = P2, where D(P1||P2) < and D(P1||P2) is the relative entropy or Kullback Leibler distance given by D(P1 P2 ) =
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