(p y )

exp -(y n - h n x n ) * y (y n - h n x n )

= e [y n y * ] = h0 I n R + e [x 24 ] n 1 n x n3 4 *

N Together with the i.i.d. channel state sequence p(h1 ) = Pnp(hn), the fast at fading channel is a memoryless channel because the channel transition probability can be decomposed into a product form as in Equation (2.1).

Quasistatic Flat Fading MIMO Channels. As the transmission bit rate increases and the frame duration Tf becomes shorter and shorter, we may have a slow fading situation across the entire symbols of a frame; that is, Tf < Tc, < where Tc is the channel coherence time. In this case, the transmit symbols over a frame will share the same channel fading; that is, hn = hm = h. Such slow fading situation is called quasistatic fading. Mathematically, a quasistatic at fading MIMO channel has the transition probability p(y1, . . . , yN|x1, . . . , xN, h) = Pnp(yn|xn, h). However, because of the

The i.i.d. condition is with respect to the time index n; that is, p(hn, hm) = pH(hn)pH(hm), where pH() is the joint distribution of the nR nT complex channel matrix. Note that the elements of hn need not be i.i.d.

......

quasistatic fading h over the entire coding frame, the quasistatic fading channel is not memoryless because the unconditional channel transition probability cannot be decomposed into the required product form. Moreover, within an encoding frame, there is no ergodic realization of the CSI and hence, the channel is a nonergodic memory channel. Block Fading MIMO Channels. Another very important channel model is the block fading channel. This is a hybrid between fast fading and quasistatic fading. Speci cally, an encoding frame spans over N fading blocks. Each fading block has L symbols. Channel fading remains quasistatic for symbols within a fading block and changes only between fading blocks; this is illustrated in Figure 2.1. This is quite an accurate channel model for frequency hopping with slow mobility. The channel fading remains quasistatic within a fading block and becomes i.i.d. between blocks when the frequency is hopped to another channel. Let XL,N = [X1,1, X2,1, . . . , XL,1, X1,2, . . . , Xl,n, . . . , XL,N] be the sequence of 1,1 transmitted symbols over the encoding frame where Xl,n X denotes the lth L,N transmitted symbol of the nth fading block. Let Y1,1 = [Y1,1, . . . , Yl,n, . . . ,YL,N] be the sequence of received symbols over a decoding frame where Yl,n Y. N Similarly, H1 = [H1, . . . , HN] denotes the sequence of channel state for the N fading blocks. The block fading channel is characterized by the conditional L,N L,N N transition probability p(y1,1 |x1,1 , h1 ) and the channel state sequence probaN bility p(h1 ).

The block fading channel is not memoryless, due to the fading memory over a block of L symbols in a fading slot. However, within an encoding frame, there are ergodic realizations of the CSI and therefore, the block fading channel is an ergodic memory channel. Moreover, if the fading between slots are i.i.d., the channel can be regarded as block-memoryless with respect to the supersymbol Xn = [X1,n, . . . , XL,n]. 2.2.2 General Transmission and CSI Feedback Model

N N N N Given a general probabilistic channel model with states {(p(y1 |x1 , h1 ), p(h1 )) : hn H }, Figure 2.2 illustrates a general framework for transmission and state N feedback. A state sequence generator, H1 , provides a state Hn H to the channel once per symbol. At the same time, the receiver is fed with the corresponding CSIR sequence, Vn H, once per symbol n sequentially in time. The transmitter is fed with a CSIT sequence, Un U, once per symbol n sequentially in time. CSIT Un, CSIR Vn, and the channel state Hn are statistically related by p(Un|Vn, Vn-1, . . .) and p(Vn|Hn, Hn-1, . . .), respectively, meaning that Un is derived statistically from {Vn, Vn-1, . . .} and Vn is derived N statistically from {Hn, Hn-1, . . .}. Note that if the channel state H1 is i.i.d., then p(Un|Vn, Vn-1, . . .) = p(Un|Vn). In other words, feedback of past memory of CSIR does not provide any additional information [85] on the current channel state Hn. This model is general enough to cover any partial CSI feedback strategy and the corresponding adaptive transmission strategy. For example, in the rst special case with no CSIT and no CSIR, we have I(Vn; Hn, Hn-1, . . .) = 0 and I(Un; Vn, Vn-1, . . .) = 0, meaning that the CSIT Un is independent of the CSIR {Vn, Vn-1, . . .} and the CSIR Vn is independent of the actual channel state {Hn, Hn-1, . . .}. In the second special case with no CSIT but perfect CSIR, we