From a8b98b01a294b435c67d4d0aa2e7fdf6fc483c86 Mon Sep 17 00:00:00 2001 From: huebli Date: Mon, 4 Dec 2017 12:14:57 +0100 Subject: [PATCH] clarification of CT lemma in practice 6 --- Homework/06-practice-problems.pdf | Bin 196877 -> 197032 bytes Homework/06-practice-problems.tex | 2 +- 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/Homework/06-practice-problems.pdf b/Homework/06-practice-problems.pdf index 5cc851f65cb2723269d887e490fb3869be4fbcd7..1bf59a0669f8d5c942ae0512cbbf1ef5e7630a83 100644 GIT binary patch delta 8572 zcmV-?A%otHf()pG43HxOH8(MrF^~Z$kxmDH`~C`lI~f=2#HXYbMbS$i=%EGrNYNH) z50Gu`t{U5NCA-_)ufLfgB~dbMd1I$(nlH8<)R6O@A%~16cav!H`AszZ4;5a)um9P< zpWpoJGodGGm};&k&u=F&pC~3m!NlbGa`GZ5Hn&p|l+`+$Di-XDaxcJYRZ#; z;MMe(=l=m9u>&NU%oqz(r3{R(rYs7wdN*Y{$Se3>tDv0nD7ZD(Urw0}%DZAQ6)boH z&!h^z$M>6Js)OInXLYfAb@=JW>lSPhEVH%!nnx_Q<4TT?YnNwJCW6I%UOn|-A>vSL z2_I48XZKZ6?}}_=I{pp^vAE9=wVyeEXSW+1f`Z{6b{p$ZNYMb=O_dBXUxz?<#&px} zr>P*&{NAEi^+`3Fiul5qOYfn2L|q92i)Q`DiDTkHI0LEzRK6;zRq`l=0f|0 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z*Ok|zLg~?KN#W^#-ZGrPDV)JMTqt}e|30*yvR^X|BI^oeZkK`q0~`u7H!}()B}Gq0 E3JSYebN~PV diff --git a/Homework/06-practice-problems.tex b/Homework/06-practice-problems.tex index 2f5ff74..b8cace7 100644 --- a/Homework/06-practice-problems.tex +++ b/Homework/06-practice-problems.tex @@ -88,7 +88,7 @@ Prove the lemma below stating that the capacity per transmission is not increased if we use a discrete memoryless channel many times. For inspiration, look again at the proof of the converse of Shannon's noisy-channel coding theorem. \medskip -\noindent\textbf{Lemma 7.9.2 in [CT]}\ Let $Y^n$ be the result of passing $X^n$ through a discrete memoryless channel of capacity $C$. Prove that for all $P_{X^n}$, it holds that $I(X^n; Y^n) \leq nC$. +\noindent\textbf{Lemma 7.9.2 in [CT]}\ Let $X_1, X_2, \ldots X_n = X^n$ be $n$ random variables with arbitrary joint distribution $P_{X^n}$. Let $Y^n$ be the result of passing $X^n$ through a discrete memoryless channel of capacity $C$. Prove that for all $P_{X^n}$, it holds that $I(X^n; Y^n) \leq nC$. \medskip Does your proof also work in case of coding with feedback (i.e.\ $X_{i+1}$ is allowed to depend on $X^i$ and $Y^i$)? If not, point out the steps in your proof where you use that there is no feedback.