Because we assume a real valued model the units ofthis capacity formula might

Because we assume a real valued model the units

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Because we assume a real-valued model, the units of this capacity formula might also be bits/channel sym- bol/dimension. For a complex-valued model, which requires two di- mensions, the formula would increase by a factor of two, much as the throughput of quaternary phase-shift key- ing (QPSK) is double that of binary phase-shift keying (BPSK) for the same channel symbol rate. In general, for d dimensions, the formula in (13) would increase by a factor of d . Frequently, one is interested in a channel capacity in units of bits per second rather than bits per channel sym- bol. Such a channel capacity is easily obtainable via multi- plication of C Shannon in (13) by the channel symbol rate R s (symbols per second): C 0 Shannon = R s C Shannon = R s 2 log 2 μ 1 + P σ 2 bits/sec. 39
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Rate R ( E b /N 0 ) Shannon (dB) 0.05 -1.440 0.10 -1.284 0.15 -1.133 0.20 -0.976 1/4 -0.817 0.30 -0.657 1/3 -0.550 0.35 -0.495 0.40 -0.333 0.45 -0.166 1/2 0 0.55 0.169 0.60 0.339 0.65 0.511 2/3 0.569 0.70 0.686 3/4 0.860 4/5 1.037 0.85 1.215 9/10 1.396 0.95 1.577 Table 1: E b /N 0 limits for unconstrainted AWGN channel. 41
Rate R ( E b /N 0 ) soft (dB) ( E b /N 0 ) hard (dB) 0.05 -1.440 0.480 0.10 -1.285 0.596 0.15 -1.126 0.713 0.20 -0.963 0.839 1/4 -0.793 0.972 0.30 -0.616 1.112 1/3 -0.497 1.211 0.35 -0.432 1.261 0.40 -0.236 1.420 0.45 -0.030 1.590 1/2 0.187 1.772 0.55 0.423 1.971 0.60 0.682 2.188 0.65 0.960 2.428 2/3 1.059 2.514 0.70 1.275 2.698 3/4 1.626 3.007 4/5 2.039 3.370 0.85 2.545 3.815 9/10 3.199 4.399 0.95 4.190 5.295 Table 2: E b /N 0 limits for binary-input AWGN channel (soft and hard decisions). 42

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