Analysis and Design of Analog Integrated Circuits. Solutions by Gray,Hurst

By Gray,Hurst

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28) k=1 This is plotted in Fig. 6. 5, an asymptotic expression of C as t → ∞ can be obtained, yielding C ∼ log(1 + ρ). This approximation to the capacity is also plotted in Fig. 6. s. Hence, the following asymptotic expression holds: C = log det(Ir + (ρ/t)W) → r log(1 + ρ). 2. Ergodic Rayleigh fading channel 31 15 SNR=30 dB 10 C SNR=20 dB SNR=10 dB 5 SNR=0 dB 0 0 2 4 6 8 10 r 12 14 16 18 20 Fig. 5 Capacity of the ergodic Rayleigh MIMO channel with t = 1 (continuous line). The asymptotic approximation C ∼ log(1 + ρr) is also shown (dotted line).

R ρR . . 1 ⎞ ⎟ ⎟ ⎟. ) + (m − 1) log(1 − ρT ) + log(1 − ρT + mρT ) + (m − 1) log(1 − ρR ) + log(1 − ρT + mρR ). When m is large, the asymptotic power loss is about −10 log10 ((1 − ρT )(1 − ρR )) dB. 4 A critique to asymptotic analysis [29] The previous results derived under the assumption r → ∞ should be accepted cum grano salis. Our assumption that the entries of the channelgain matrix H are independent random variables becomes increasingly questionable as r increases. In fact, for this assumption to be justified the antennas should be separated by some multiple of the wavelength, which cannot be obtained when a large number of antennas is packed in a finite volume.

Deterministic channel 23 µ − λi−1 µ λi−1 Fig. ” ˜: we see that, for the corresponding components of the noise vector z i > m, y˜i is independent of the transmitted signal, and x ˜i does not play any role. Maximization of the mutual information requires independent x˜i , i = 1, . . , m, each with independent Gaussian, zero-mean real and imaginary parts. Their variances should be chosen, as indicated in Appendix B, via “water-filling” (Fig. 5) µ − λ−1 ˜i ]2 = E[Re x ˜i ]2 = E[Im x i + 2 where (·)+ max(0, ·).

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