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for n suf ciently large Therefore, by an appropriate choice of and n, we can make Pe as small as we like So, for any choice of > 0, there exists an and n such that over all randomly chosen rate R codes of block length n, the expected distortion is less than D + Hence, there must exist at least one code C with this rate and block length with average distortion less than D + Since was arbitrary, we have shown that (R, D) is achievable if R > R(D) We have proved the existence of a rate distortion code with an expected distortion close to D and a rate close to R(D) The similarities between the random coding proof of the rate distortion theorem and the random coding proof of the channel coding theorem are now evident.



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in a four-mode adaptive PSK scheme did not achieved a better performance across theBPS 1 throughput range Instead, this four-mode adaptive PSK scheme extended the maximum BPS throughput achievableby any adaptive PSK scheme to 2 BPS, while asymptotically achieving a throughput of 3 BPS, as the average SNR increases On the other hand, the relative SNR advantage of adaptive schemes in comparison to fixed-mode schemes increased as the target average BER became lowerand decreased as the fading became less severe More explicitly, less severe fading corresponds to an increased Nakagami fading parameter m, to an increased number of diversity antennas, or to an increased numberof multi-path components encountered wide-band fading channels the in As fading becomes less severe, the average BPS throughput curvesof our adaptive Square QAM schemes exhibit undulations owing to the absence 3-BPS, 5-BPS and 7-BPS square QAM of modes.





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We will explore the parallels further by considering the Gaussian example, which provides some geometric insight into the problem It turns out that channel coding is sphere packing and rate distortion coding is sphere covering Channel coding for the Gaussian channel Consider a Gaussian channel, Yi = Xi + Zi , where the Zi are iid N(0, N) and there is a power constraint P on the power per symbol of the transmitted codeword Consider a sequence of n transmissions The power constraint implies that the transmitted sequence lies within a sphere of radius nP in Rn The coding problem is equivalent to nding a set of 2nR sequences within this sphere such that the probability of any of them being mistaken for any other is small the spheres of radius nN around each them are of almost disjoint.

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This corresponds to lling a sphere of radius n(P + N ) with spheres of radius nN One would expect that the largest number of spheres that could be t would be the ratio of their volumes, or, equivalently, the nth power of the ratio of their radii Thus, if M is the number of codewords that can be transmitted ef ciently, we have ( n(P + N ))n = M ( nN )n P +N N.

(14.8)

The comparisons between fixed-mode MC-CDMA and adaptive OFDM (AOFDM) were made based on different channel models Section 654 itwas found that fixed-mode MCIn CDMA might outperform adaptive OFDM, when the underlying channel providessufficient diversity However, a definite conclusion could not be drawn since in practice MC-CDMA might suffer fromMU1 and AOFDM might suffer from imperfect channel quality estimation and feedback delays Concatenated space-time block coded and turbo convolutional-coded multi-carrier adaptive systems were investigated Section 655 Thecoded schemes reduced therequired average in SNR by about 6dB-7dB at a throughput of 1 BPS, achieving near error-free transmissionIt was also observed in Section 65.

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