MIMO CONSTELLATION DESIGN WITH IMPERFECT CHANNEL STATE INFORMATION

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M1 x x x x x x x M2 x M3

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(b) Figure 5.1. Illustration of 2D unitary constellation with nT = 1: (a) unitary 8-point constellation: M = {M1, M2} and M1 = M2 = 8; (b) unitary 16-point constellation: {M1, M2, M3} and M1 = 4, M2 = 8, M3 = 4.

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CONSTELLATION DESIGN FOR MIMO CHANNELS WITH IMPERFECT CSIR

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min D (x i ; x j ) min min Dintra (k), min Dinter (k, k + 1)

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i ,j k [ 1 ,K ] k =1 ,... ,K -1

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(5.13)

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where Dintra(k) is the intrasubset distance de ned as 1 - s e2 Dintra (k) = min log 1 + 4rk2 sin 2 ( x i , x j ) x i , x j Mk 1 + s e2 rk2 and the Dinter(k, k ) is the intersubset distance given by Dinter (k, k ) = 1 + s e2 rk2 1 - s e2 2 1 + s e2 rk2 - 1 - log r - rk + log 1 + 2 2 2 2 k 1 + s e2 rk2 1 + s e rk 1 + s e rk (5.15) (5.14)

Here x is the vector obtained by staggering the real and imaginary parts of x x = [R (x), Z (x)] * (5.16)

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and x i, x j is the angle between two signal vectors x i and x j. Therefore, the optimization problem (Problem 5.1) can be simpli ed as follows. Problem 5.2 (Signal Set Design) Select the unitary constellation parameters, K [1, M], {M1, . . . , MK}, {r1, . . . , rK} so as to maximize the minimum between the intraset distance Dintra(k) and the interset distance Dinter(k, k + 1):

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2 K [ 1 ,M ] , 1 M k rk , M

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max min min Dintra (k), min Dinter (k, k + 1) k [ 1 ,k ] k =1 ,... ,K -1 k M k = M , 0 r1 L rK

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(5.17)

The optimization problem involves discrete variables (K and M1, . . . , MK) as well as continuous variables r1, . . . , rK. For any xed value of K and M1, . . . , MK satisfying the constraints in (5.17), the optimal (r1, . . ., rK) can be obtained numerically. 5.2.4 Single-Transmit Antenna Example

Consider a simple case of constellation design (N = 1) for systems with single transmit antenna nT = 1 in fading channels. Hence, the signal element xi is a complex scalar. The optimization parameter is the constellation set M = {x1, . . . , xM} where the mth constellation point, xm, is a complex scalar. Figure 5.2 illustrates the 8-point and 16-point optimal constellations with average energy P0 of 10 dB for different values of s 2. e The uncoded performance (average symbol error probability) of the 8point and 16-point constellation designs is illustrated in Figure 5.3. The curves with the label coherent refer to the performance of unitary constellation designed for s 2 = 0. Because of the larger minimum average KL distance of e the new constellation, the exponential decay of the symbol error probability

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MIMO CONSTELLATION DESIGN WITH IMPERFECT CHANNEL STATE INFORMATION

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2 sE = 0.0, dmin = 2.2624

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8PSK

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2 sE = 0.2, dmin = 1.3318

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2 sE = 0.5, dmin = 0.8518

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16QAM

2 sE = 0.0, dmin = 1.5841

2 sE = 0.2, dmin = 0.8857

2 sE = 0.5, dmin = 0.5437

(b) Figure 5.2. Illustration of the optimal constellation of point sizes 8 (a) and 16 (b) for nT = nR = 1 and P0 = 10 dB. [20]

CONSTELLATION DESIGN FOR MIMO CHANNELS WITH IMPERFECT CSIR

100 PSK Coherent Optimal

10 1 Symbol error probability

10 2

10 3

100 QAM Coherent Optimal

Symbol error probability

10 1

Figure 5.3. Average symbol error probability versus nR of the optimal constellation of point sizes 2 8 (a) and 16 (b) for nT = 1, P0 = 10, and s e = 0.5. [20]

MIMO CONSTELLATION DESIGN WITH IMPERFECT CHANNEL STATE INFORMATION

versus nR is much higher for the new constellation (comparing the optimal constellation with the regular 8PSK constellation). 5.2.5 Multitransmit Antenna Example

When there is more than one transmit antenna nT > 1, the signal set consists of nT 1 vectors. The constellation design is given by the optimization problem (Problem 5.2). For illustration purposes, we consider nT = nR = 2 uncoded MIMO with throughputs of 4 bits per channel use and 8 bits per channel use. Figure 5.4 illustrates the uncoded performance (average symbol error probability) of the 4-point constellation designed for s 2 = 0.05 and s 2 = 0.1. For come e parison, we consider two baseline references. The rst reference (labeled as QPSK ) is given by regular QPSK constellation per transmit antenna followed by a ML receiver for fair comparison. The second reference (labeled as Alamouti QAM ) is given by regular 16QAM constellation together with Alamouti spacetime coding [9]. These two reference systems have the same throughput of 4 bits per channel use. The performance improvement of the optimal constellations (designed for s 2 = 0.05 and s 2 = 0.10, respectively) is e e substantial compared with the two reference systems. Note that the Alamouti scheme suffers from severe performance degradation due to the CSIR error and its performance becomes worse than the conventional QPSK reference performance without any transmit diversity. Figure 5.5 illustrates the uncoded performance (average symbol error probability) of the 16-point constellation designed for s 2 = 0.00 (perfect CSIR) e and s 2 = 0.01, respectively. Similarly, we consider two baseline references. The e rst reference (labeled as QAM ) refers to the regular 16QAM constellation per transmit antenna followed by ML receiver for fair comparison. The second reference (labeled as Alamouti QAM ) refers to the regular 256QAM constellation together with Alamouti spacetime coding. These two reference systems have the same throughput of 8 bits per channel use for fair comparison. Observe that in the absence of CSIR error (perfect CSIR), the performance of the optimal design (designed for s 2 = 0) and the regular 16QAM e reference remains the same.Also, although Alamouti coding has a larger transmit diversity advantage, the performance is worse than the others because of smaller coding advantage. On the other hand, with s 2 = 0.01, the optimal 16e point constellation (designed for s 2 = 0.01) achieves signi cant performance e relative to the other two references.

5.3 SPACETIME CODING FOR MIMO CHANNELS WITH IMPERFECT CSIR In the previous sections, we focused on the constellation design problem for uncoded MIMO systems (N = 1) with imperfect CSIR. In practice, there is always coding on top of the modulation to improve the reliability of packet