ERGODIC AND OUTAGE CHANNEL CAPACITY

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where the expectation is taken for all transmit symbols within an encoding frame. This constraint is less strict because it allows the instantaneous transmit power of some symbols to exceed P0 momentarily as long as the instantaneous transmit power of some other symbols is less than P0 so that on average, the overall transmit power is within P0. A third and nal possibility is called the long-term average power constraint, which is given by Equation (2.6), except that the expectation is taken across multiple encoding frames. This constraint is the least strict one because it allows the short-term average transmit power of one encoding frame to exceed P0 as long as the average power of some other encoding frames is less than P0 to make up the difference and at the end, the overall average power (over multiple frames) is less than P0. In general, the peak power constraint is a stronger constraint than the shortterm average power constraint. Similarly, the short-term average power constraint is a stronger constraint than the long-term average power constraint. These three different constraints usually give different capacity results. For example, for block fading channels with memoryless transition, the transmitted symbols are i.i.d. and hence, the short-term average power constraint and the long-term average power constraint are equivalent while the peak power constraint will give different capacity result. On the other hand, for quasistatic fading channels,4 the peak power constraint and the short-term transmit power constraint are equivalent while the long-term average power constraint will give different capacity results. Unless otherwise speci ed, we shall adopt the short-term average power constraints for mathematical convenience. 2.2.5 Causal Feedback Constraint

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As shown in Figure 2.2, the feedback channel carries the CSIT in a symbolby-symbol manner. Hence, the CSIT has to satisfy the causality constraint. Speci cally, the transmitter at the nth symbol has knowledge of the feedback CSIT {U1, . . . , Un} only. CSIT values {Un+1, Un+2 . . .} correspond to future feedback values and are not yet available during the nth symbol. In other words, adaptation can be performed on the basis of current and past CSIT values only.

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ERGODIC AND OUTAGE CHANNEL CAPACITY

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Before we proceed to discuss the channel capacity in various CSI scenarios, we shall clarify two important concepts of Shannon s capacity for probabilistic channel with states. For ergodic channels, the interpretation of the channel

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Quasistatic fading refers to the slow fading situation where the channel fading realizations of all symbols within an encoding frame are identical.

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MIMO LINK WITH PERFECT CHANNEL STATE INFORMATION

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capacity is pretty straightforward, referring to the capacity in Shannon s sense; speci cally, for any rate R < C, there exists at least one encoder and one decoder that achieves arbitrarily small error probability. However, for nonergodic channels, this is no longer the case because the channel capacity in Shannon s sense can be zero. In general, we may have two important notions of channel capacity, namely, the ergodic capacity and the outage capacity.These concepts are clari ed below. 2.3.1 Ergodic Capacity

Ergodic capacity refers to the channel capacity in Shannon s sense; that is, for any rate R < C, there exists at least one encoder and one decoder that achieves arbitrarily small error probability. Conversely, if R > C, the error probability is always bounded away from zero with any encoder and decoder. For example, if the ergodic capacity of a fading channel is 10 bits per channel use, then we can anticipate zero error probability on the transmitted frames, provided capacity achieving codes are used. Note that not all probabilistic channels with states have nite ergodic capacity. As we shall illustrate, the ergodic capacity can be zero in some cases. That means for any nite transmission rate R, we cannot always guarantee error-free transmission no matter what encoder and decoder we use. For example, consider the transmission of an encoding frame {X1, . . . , XN} with N symbols across a at fading channel with state sequence {H1, . . . , HN}. If the encoding frame is suf ciently long (> coherence time of the fading > channel) so that {H1, . . . , HN} spans across an ergodic realization of the underlying ergodic fading process H(t), the resulting ergodic capacity is nonzero and is given by C = lim 1 N N I (X 1 ; Y1N , H1 ) N N (2.7)

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In the special case of i.i.d. CSI sequence, the ergodic capacity reduces to the following well-known formula: C = I (X; Y, H) = I (X; Y H) (2.8)

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Hence, if the transmitted rate is lower than the ergodic capacity, the error probability is exponentially decaying with the frame length N for capacity achieving codes. More examples will be given in the next section. 2.3.2 Outage Capacity

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The ergodic assumption described above is not necessarily satis ed in practical communication systems operating on fading channels. In fact, if there is a stringent delay requirement as in the case of speech transmission over wire-

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