i ,j

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where

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1 M m =1 x m M

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and ||xm||2 is the matrix norm given by

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It is shown [20] that the KL distance between the probability densities pi and pj for the MIMO channels described in (5.3) can be expressed as D( pi p j ) = nR tr I N + s e2 x*x i I N + s e2 x*x j i j

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- nR log e I N + s e2 x*x i I N + s e2 x*x j i j + tr I N + s e2 x*x j j

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(x i - x j )*h *h(x i - x j )

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(5.10)

From (5.9), the average KL distance D (xi; xj) is given by D (x i ; x j ) = nR tr I N + s e2 x*x i I N + s e2 x*x j i j

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- nR log e I N + s e2 x*x i I N + s e2 x*x j i j

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(x i - x j ) *

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(5.11)

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Considering one extreme when s 2 = 0, we have perfect CSIR, and the e average KL distance reduces to D (x i ; x j ) = nR log e I nT + (x i - x j )(x i - x j ) * Hence, Problem 5.1 is identical to the design criteria in coherent spacetime codes as described in 4, speci cally, the rank and determinant criteria of (xi - xj)(xi - xj)*. On the other hand, when s 2 = 1, the average KL dise tance reduces to D (x i ; x j ) = nR tr I N + x*x i I N + x*x j i j - nR log e I nT

N i i

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)( + (I

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) + x*x )(I

- nR N + x*x j j

which gives the design criteria in MIMO channels without CSIR (noncoherent). For intermediate values of s 2, the performance criterion is a combinae tion of the two extremes, which implies that for an optimal design, contributions from both perfect CSIR and no-CSIR designs have to be

CONSTELLATION DESIGN FOR MIMO CHANNELS WITH IMPERFECT CSIR

exploited to achieve better performance. We shall consider two examples in the following text. Since the term nR does not affect the optimization in Problem 5.1, we assume nR = 1 for simplicity. 5.2.3 Constellation Design Optimization

For a xed spectral ef ciency, the cardinality of the signal set M grows exponentially with respect to the codelength N. For example, to achieve a spectral ef ciency of 4 bits per channel use, the size of the constellation set M reaches 220 with N = 5. This makes the design problem almost impossible to solve. In this section, we consider the constellation design problem for the uncoded system where N = 1. In other words, the signal element xi is a nT 1 complex vector. The average KL distance between xi and xj is given by [20] D (x i ; x j ) = 1 + s e2 x i 1 + s e2 x j

1 + s e2 x i 2 - 1 - log 1 + s e2 x j 2 (5.12)

xi - x j 2 + log 1 + (1 - s e2 ) 2 1 + s e2 x j

The optimization of {xm} in (5.12) is in general not straightforward. To simplify the optimization problem, we shall impose some extra constraints, namely, the unitary constellation constraint, on the signal set. De nition 5.1 (Unitary Constellation) The constellation set M is unitary if there is a partition on M,{M1, . . . , MK}, such that there are Mk constellak tion points in the constellation partition Mk and Sk=1 Mk = M. In addition, 2 2 ||xi|| = ||xj|| for any xi, xj Mk where ||x|| = tr(x*x) is the norm of x. Figure 5.1a illustrates an example of a two-dimensional unitary constellation. Essentially, unitary constellation can be visualized as constellation points over concentric rings of sphere about the origin of the constellation space. Using the de nition of unitary constellation, we can see that if two constellation points xi, xj lie on the same partition Mk for some k, then ||xi||2 = ||xj||2 and the average KL distance D (xi; xj) can be reduced to xi - x j 2 D (x i ; x j ) = log 1 + (1 - s e2 ) 2 1 + s e2 x j

2 if s e < 1. On the other hand, if the two constellation points lie on different partitions Mk and Mk , the minimum average KL distance will be observed when they lie on a line that passes through the origin of the constellation space, which is determined by the radii of these two constellation spheres rk and rk . Hence, the minimum average KL distance is given by