By John G. Webster (Editor)
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Extra info for 29.Information Theory
8. R. N. 6–21. 9. R. N. , New York: McGraw-Hill, 1978, pp. 204–215. 10. H. Nyquest, Certain topics in telegraph transmission theory, Trans. AIEE, 47: 617–644, 1928. or equivalently, D(r) = σ 2 e−2R 11. Y. Linde, A. Buzo, and R. M. Gray, An algorithm for vector quantizer design, IEEE Trans. , 28: 84–95, 1980. which provides an optimal trade-off with which real systems can be compared. Shannon and others extended this approach to sources with memory and a variety of coding structures. The Shannon bounds are always useful as lower bounds, but they are often over conservative because they reflect only in the limit of very large dimensions and hence very complicated codes.
Then xs(t) can be drawn as a sequence of arrows of height xs[n] at time t ϭ nT1, as shown with the original signal xa(t) in Fig. 3 for sampling periods of T and 2T. The sampling process usually is not an invertible process. In other words, given a discrete-time sequence, xs[n], it is not always possible to reconstruct the original continuous-time input of the sampler, xa(t). It is very clear that the sampling process is not a one-to-one mapping function. There are many continuous-time signals that may produce the same discretetime sequence output unless they have same bandwidth and sampled at Nyquist rate.
In this case, the estimation problem is said to be regular, and the accuracy of estimation can be bounded from below by the Crame´r-Rao inequality: if ʦ R, then for any estimator ˆ , E|θˆ − θ|2 ≥ 1+ db (θ ) dθ I(θ ) 2 + b2 (θ ) (5) where b() ϭ Eˆ Ϫ is the bias of the estimate ˆ . An analogous inequality holds in the case of multidimensional parameter . Note that if the estimate is unbiased, that is, Eˆ ϭ , then E|θˆ − θ|2 ≥ I −1 (θ ) Moreover, the latter inequality typically holds asymptotically, even for biased estimators when I() ϭ I does not depend on .
29.Information Theory by John G. Webster (Editor)