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less fading channels, the ergodicity requirement NTs > Tc, where Ts is the mod> ulation symbol duration and Tc is the channel coherence time, cannot be satis ed. In this case, there is no signi cant channel variation across the encoding frame and there may be no classical Shannon meaning attached to capacity in this typical situation. In the extreme case, when the frame is very short or the coherence time is very long, the entire encoding frame may share a single fading state realization; speci cally, h1 = h2 = = hN = h. In this case, the channel belongs to the type of nonergodic memory channel and the mutual information becomes information unstable [137]. The channel capacity in Shannon s sense is given by
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N N C = lim inf i( x1 ; y 1 , h)
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N N N N 1 where lim inf is in the probability sense and i(x1 ; y1 , h) = N I (x1 ; y1 , h) is the normalized sequence of mutual information. In fact, this mutual information can be considered as a function of the random realization h and therefore is a random variable itself. Hence, there may be a nonnegligible probability that the value of the transmission rate, no matter how small, exceeds the instantaneous mutual information. This situation gives rise to the error probabilities that do not decay with the increase of block length N. In these circumstances, the channel capacity is viewed as a random variable as it depends on the instantaneous channel state realization. Hence, the ergodic capacity of these channels is zero, meaning that no matter how small the transmission rate is, there is no guarantee that the transmitted frame will be error-free. On the other hand, instead of looking at the ergodic capacity in Shannon s sense, it is also possible to look at the capacity from an outage perspective. The outage capacity Cout at a given outage probability Pout is de ned as the maximum data rate R such that
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Pr{R < instantaneous mutual information} Pout
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In other words, the outage probability Pout is the cumulative distribution function (cdf) of the instantaneous mutual information (which is a random variable). 2.4 CHANNEL CAPACITY WITH NO CSIT AND NO CSIR
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We shall rst consider a special case of the model in Figure 2.2, where there is no CSIT and no CSIR. This corresponds to the case when I(Vn; Hn) = 0 and I(Un; Hn) = 0. In this case, no adaptation at the encoding and decoding is possible and therefore, the channel encoding function, fn(w), is a function of the N message index only. The channel decoding function, g(Y1 ), is a function of the received symbols only. N N N Let p(y1 |x1 , h1 ) be the transition probability of the probabilistic channel with states. Together with the joint distribution of the channel state sequence
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N p(h1 ), the channel transition probability of the equivalent channel (without N N states) with input x1 and output y1 is given by N N N N N N p(y 1 x1 ) = p(y 1 x1 , h 1 ) p(h 1 )
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N N Treating X1 and Y1 as supersymbols and assuming ergodic CSI sequence, the channel capacity (bits per symbol) of the equivalent channel with transition probability in Equation (2.11) is given by
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1 N Cno csit, no csir = lim max I ( X 1 ; Y1N ) N p ( X N ) N 1 We shall consider a few examples in the following subsections. 2.4.1 Fast Flat Fading MIMO Channels
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We consider a fast at fading example where the channel output Yn is given by Yn = H n X n + Zn where Hn(i, j) is the i.i.d. complex Gaussian fading with unit variance and Zn 2 is the i.i.d. complex Gaussian noise with covariance s z InR. Since both the state sequence and the noise sequence are i.i.d., the channel transition probability is memoryless. The channel capacity is simpli ed as Cno csit,no csir = max I (X; Y)
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Note that although the channel noise is Gaussian, the capacity achieving distribution p(X) is no longer complex Gaussian. As shown in reference 5, the capacity achieving distribution for a scalar channel with no CSIT and no CSIR at low SNR is binary and is given by 0 x= a with probability 1 - pa with probability pa (2.14)
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such that apa = e[|X|2].This means that the capacity achieving modulator design at low SNR is a binary constellation. In general, at higher SNR, the capacity achieving distribution is peaky, implying discrete constellations. 2.4.2 Block Fading Channels
Consider a MIMO block fading channel with conditional transition probN N N ability p(y1 |x1 , h1 )PnPl p(yl,n|xl,n, hn) and channel state sequence probability N p(h1 ) = Pnp(hn), the channel capacity (bits per symbol) is given by