C0 C2 (0,4) (2,6) C1 C3 (1,5) (3,7) 6 2

0 4 2 0 1 6

C2 C0 (2,6) (0,4) 3 C1 C3 (3,7) (1,5)

5 4 1 5 (c) 7

Figure 5.8. Uncoded 4PSK and trellis-coded 8PSK modulations: (a) one-state trellis; (b,c) four-state trellis.

Similar techniques have been developed for coded modulation in fast fading scenarios, when the receiver is assumed to have perfect channel state information. The design criteria in this case are maximization of both the symbol Hamming distance and the minimum product Euclidean distance between pairs of codewords. Therefore, the set partitioning and trellis design are performed to maximize the length of the shortest error event path and the product of the Euclidean distances along this path. In the next section, we apply an idea similar to trellis-coded modulation in the AWGN channels, to design a coded modulation scheme for MIMO channels with imperfect CSIR. 5.3.2 Coded Modulation Design for MIMO Channels with Imperfect CSIR In this section, we derive the design criterion for coded modulation over MIMO fading channels with imperfect CSIR. To understand how coding can

help combat fading over MIMO channels, we consider block fading channels where an encoding frame of length N is partitioned into B fading blocks (each of length T). The channel fading is assumed to be quasistatic within a fading block and i.i.d. between the B fading blocks. During the bth fading block, the nR T received signal yb is given by y b = h b xb + zb (5.20)

where hb is the nR nT fading matrix, xb is the nT T transmitted signals, and zb is the nR T channel noise (i.i.d. with unit noise variance) during the bth fading block. Similarly, we de ne the imperfect CSIR hb as hb = hb + Db The equivalent channel model is given by y b = h b xb - D b xb + zb The joint conditional likelihood function is given by p(y 1 , . . . , y B x1 , . . . , x B , h 1 , . . . , h B ) = e D [ p(y 1 , . . . , y B x1 , . . . , x B , h 1 , . . . , h B , D 1 , . . . , D B )] = b=1

(5.21)

(5.22)

exp -tr ( IT + s e2 x b* x b )

(y b - h b x b ) * (y b - h b x b )]} (5.23)

Let pi denote the conditional likelihood p(y1, . . . , yB|x1, . . . , xB, h 1, . . . , i i B). The conditional KL distance between pi and pj is given by h D( pi p j ) = D b (h b )

(5.24)

(5.25)

Similar to Section 5.2.2, we consider the pairwise error proabability. The conditional error probability is approximated by

(5.26)

where xi = [x1, . . . , xB] and h = [ h 1, . . . , h B] are the aggregate transmit signals i i and the aggregate imperfect CSIR over the B fading blocks. Since the CSIR is i.i.d. across the B fading blocks, we have

B p(h) = b=1 p(h b )

The unconditioned error probability is then expressed as Pe (x i x j ) = e h [Pe (x i x j h)] B e h - D b (h b ) b=1

B = b=1 e h b [exp(-D b (h b ))]

= b=1 exp(-D b (x i ; x j ))

B = exp - D b (x i ; x j ) b=1 where D b(xi; xj) is the average KL distance of the bth fading block D b (x i ; x j ) = nR tr IT + s e2 x b*x b IT + s e2 x b*x b i i j j

(5.27)