By Abbas El Gamal, Young-Han Kim
This finished remedy of community details idea and its purposes presents the 1st unified assurance of either classical and up to date effects. With an method that balances the advent of recent versions and new coding strategies, readers are guided via Shannon's point-to-point info thought, single-hop networks, multihop networks, and extensions to allotted computing, secrecy, instant verbal exchange, and networking. easy mathematical instruments and strategies are used all through, requiring in simple terms simple wisdom of chance, while unified proofs of coding theorems are in accordance with a couple of easy lemmas, making the textual content obtainable to novices. Key issues coated comprise successive cancellation and superposition coding, MIMO instant conversation, community coding, and cooperative relaying. additionally coated are suggestions and interactive conversation, capability approximations and scaling legislation, and asynchronous and random entry channels. This booklet is perfect to be used within the school room, for self-study, and as a reference for researchers and engineers in and academia.
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Let X and Z be independent, zero-mean random variables with variances P and N, respectively. (a) Show that h(X | X + Z) ≤ 2πePN 1 log ???? ???? 2 P+N with equality iff both X and Z are Gaussian. ) (b) Let X ∗ and Z ∗ be independent zero-mean Gaussian random variables with variances P and N, respectively. Use part (a) to show that I(X ∗ ; X ∗ + Z ∗ ) ≤ I(X ∗ ; X ∗ + Z) with equality iff Z is Gaussian. Thus, Gaussian noise is the worst noise when the input to an additive channel is Gaussian. . Joint typicality.
Assume a unicast communication scenario in which source node wishes to communicate an R-bit message M to destination node N. What is the network capacity C, that is, the maximum number of bits R that can be communicated reliably? The answer is given by the max-flow min-cut theorem due to Ford and Fulkerson () and Elias, Feinstein, and Shannon (). , C= min S⊂N :1∈S , N∈S ???? C(S), where C(S) = ∑ j∈S , k∈S ???? C jk is the capacity of the cut (S, S c ). They also showed that the capacity is achieved without errors using simple routing at the intermediate (relay) nodes, that is, the incoming bits at each node are forwarded over its outgoing links.
Vector EPI: Let X n ∼ f (x n ) and Z n ∼ f (z n ) be independent random vectors and Y n = X n + Z n . Then ???? ???? ???? 22h(Y )/n ≥ 22h(X )/n + 22h(Z )/n with equality if X n and Z n are Gaussian with KX = aKZ for some scalar a > 0. ∙ Conditional EPI: Let X n and Z n be conditionally independent given an arbitrary random variable U, with conditional pdfs f (x n |u) and f (z n |u), and Y n = X n + Z n . Then ???? ???? ???? 22h(Y |U )/n ≥ 22h(X |U )/n + 22h(Z |U)/n . The scalar EPI can be proved, for example, using a sharp version of Young’s inequality or de Bruijn’s identity; see Bibliographic Notes.