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Figure 2.2. A general model of transmission and causal partial feedback strategies for block fading channels.
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have I(Un; Vn, Vn-1, . . .) = 0 and p(Vn|Hn = hn) = d(Vn - hn), meaning that the CSIT Un is independent of the channel states {Vn, Vn-1, . . .} while the CSIR Vn equals to the actual channel state Hn. In the third special case with perfect CSIT and no CSIR, we have p(Un|Hn = hn) = d(Un - hn) and I(Vn; Hn, Hn-1, . . .) = 0, meaning that the CSIR is independent of the actual channel state H while the CSIT equals H. In the fourth special case with perfect CSIR and perfect CSIT, we have p(Un|Hn = hn) = d(Un - hn) and p(Vn|Hn = hn) = d(Vn - hn), meaning that both CSIR and CSIT equal to H. In the next chapter, we shall use this generic model to derive the optimal partial state feedback strategy and the optimal transmission strategy. 2.2.3 Adaptive Channel Encoding and Decoding
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Figure 2.3 illustrates the general structure of adaptive channel encoder for probabilistic channels with state feedback. Information message w W (where W is the message set given by W = {1, 2, . . . , 2NR}) is mapped into a frame of N transmitted symbol X1 = [X1, . . . , XN] using an adaptive encoding function n fn : W U1 X. The adaptive encoding function, which is a function of the n message index (w) and causal CSIT sequence U1 , is given by
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n where U1 = {U1, . . . , Un} is the causal CSIT sequence available to the transmitter at the nth symbol and Xn X is the transmitted symbol over the complex eld. Figure 2.4 illustrates a general structure of the channel decoder with CSIR. The receiver decodes the message w based on the entire frame of received
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1st fading block (The encoded frame)
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2nd fading block
nth fading block
X l,n W messa ge index fl,n (w, u1, . . . , un ) lth encoded symbol during the nth fading block
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Figure 2.3. General structure of an adaptive channel encoder.
g(Y, V) Complete frame of received symblols
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Complete frame of CSIR V = (V1, . . . ,VN ) Figure 2.4. General structure of an adaptive channel decoder.
N N symbols, Y1 = [Y1, . . . ,YN], and the entire sequence of CSIR, V1 = [V1, N . . . ,VN]. The channel decoder is given by the function g : Y H W, given by
w = g(Y1N , V1N )
Hence, an adaptive error correction code is characterized by (N, R), where N is the number of symbols in an encoding frame and R is the encoding rate in terms of bits per symbol. Decoding error occurs when w w. A code rate R is N N achievable if the error probability Pe = limN Pr[g(Y1 , V1 |w) w] = 0. The supremum of the achievable code rate is de ned as the channel capacity [30]. 2.2.4 Transmit Power Constraint
The instantaneous transmitted power at the nth transmitted symbol is given by Pn = tr e [X n X * ] n (2.5)
In general, for meaningful design, we must consider the capacity at a given transmitted power constraint because otherwise, we can achieve arbitrarily large capacity at the expense of arbitrarily large transmitted power. There are various possibilities of constraining the transmitted power. For example, we can have Pn P0 for all n where the scope of the expectation is limited to one symbol only. This refers to peak power constraint, which can be important in practical systems when there is nonlinearity in the power ampli er. Obviously, the peak power constraint is very strict because it applies to every transmit symbol. On the other hand, there is a less strict constraint called the short-term average power constraint. The constraint is given by P = tr e [XX *] P0 (2.6)