第三章-信道容量-习题答案

·17·3.1设信源4.06.0)(21xxXPX通过一干扰信道，接收符号为Y={y1,y2}，信道转移矩阵为43416165，求：(1)信源X中事件x1和事件x2分别包含的自信息量；(2)收到消息yj(j=1,2)后，获得的关于xi(i=1,2)的信息量；(3)信源X和信宿Y的信息熵；(4)信道疑义度H(X/Y)和噪声熵H(Y/X)；(5)接收到信息Y后获得的平均互信息量。解：1)bitxpxIbitxpxI322.14.0log)(log)(737.06.0log)(log)(222221212)bitypxypyxIbitypxypyxIbitypxypyxIbitypxypyxIxypxpxypxpypxypxpxypxpyp907.04.04/3log)()/(log);(263.16.04/1log)()/(log);(263.14.06/1log)()/(log);(474.06.06/5log)()/(log);(4.0434.0616.0)/()()/()()(6.0414.0656.0)/()()/()()(2222222212121222122212111211222121221211113)symbolbitypypYHsymbolbitxpxpXHjjjiii/971.010log)4.0log4.06.0log6.0()(log)()(/971.010log)4.0log4.06.0log6.0()(log)()(224)symbolbitYHXYHXHYXHYXHYHXYHXHsymbolbitxypxypxpXYHijijiji/715.0971.0715.0971.0)()/()()/()/()()/()(/715.010log)43log434.041log414.061log616.065log656.0()/(log)/()()/(2·18·5)symbolbitYXHXHYXI/256.0715.0971.0)/()();(3.2设二元对称信道的传递矩阵为32313132(1)若P(0)=3/4,P(1)=1/4，求H(X),H(X/Y),H(Y/X)和I(X;Y)；(2)求该信道的信道容量及其达到信道容量时的输入概率分布；解：1)symbolbitYXHXHYXIsymbolbitXYHYHXHYXHXYHYHYXHXHYXIsymbolbitypYHxypxpxypxpyxpyxpypxypxpxypxpyxpyxpypsymbolbitxypxypxpXYHsymbolbitxpXHjjijijijiii/062.0749.0811.0)/()();(/749.0918.0980.0811.0)/()()()/()/()()/()();(/980.0)4167.0log4167.05833.0log5833.0()()(4167.032413143)/()()/()()()()(5833.031413243)/()()/()()()()(/918.010log)32lg324131lg314131lg314332lg3243()/(log)/()()/(/811.0)41log4143log43()()(2222212122212212111121112222)21)(/082.010log)32lg3231lg31(2loglog);(max222imixpsymbolbitHmYXIC3.3设有一批电阻，按阻值分70%是2KΩ，30%是5KΩ；按瓦分64%是0.125W，其余是0.25W。现已知2KΩ阻值的电阻中80%是0.125W，问通过测量阻值可以得到的关于瓦数的平均信息量是多少？解：对本题建立数学模型如下：);(求：2.0)/(,8.0)/(36.064.04/18/1)(瓦数3.07.052)(阻值12112121YXIxypxypyyYPYxxXPX以下是求解过程：·19·symbolbitXYHYHXHYXIsymbolbityxpyxpXYHsymbolbitypYHsymbolbitxpXHyxpypyxpyxpyxpypyxpypyxpyxpyxpypxypxpyxpxypxpyxpijjijijjii/186.0638.1943.0881.0)()()();(/638.122.0log22.008.0log08.014.0log14.056.0log56.0)(log)()(/943.036.0log36.064.0log64.0)()(/881.03.0log3.07.0log7.0)()(22.014.036.0)()()()()()(08.056.064.0)()()()()()(14.02.07.0)/()()(56.08.07.0)/()()(222222222122222212111121211112121111113.4若X,Y,Z是三个随机变量，试证明(1)I(X;YZ)=I(X;Y)+I(X;Z/Y)=I(X;Z)+I(X;Y/Z)；证明：)/;();()/()/(log)()()/(log)()/()()/()/(log)()()/(log)();()/;();()/()/(log)()()/(log)()/()()/()/(log)()()/(log)();(ZYXIZXIzxpzyxpzyxpxpzxpzyxpzxpxpzxpzyxpzyxpxpzyxpzyxpYZXIYZXIYXIyxpzyxpzyxpxpyxpzyxpyxpxpyxpzyxpzyxpxpzyxpzyxpYZXIijkkikjikjiijkikikjiijkkiikikjikjiijkikjikjiijkjikjikjiijkijikjiijkjiijikjikjiijkikjikji(2)I(X;Y/Z)=I(Y;X/Z)=H(X/Z)–H(X/YZ)；证明：ijkkjkikjkjikjiijkkikjikjizypzxpzypzyxpzyxpzxpzyxpzyxpZYXI)()/()()/(log)()/()/(log)()/;(·20·)/()/()/()/(log)()/()/(log)()/(log)()/(log)()/()/(log)()/;()/;()/()/(log)()/()()(log)()/()()(log)()/()()/()(log)(YZXHZXHYZXHzxpzxpYZXHzxpzyxpzyxpzyxpzxpzyxpzxpzyxpzyxpZYXIZXYIzypzxypzyxpzypzxpzyxpzyxpzypzxpzyxpzyxpzypzpzxpzyxpzyxpikkikiikkijkjiijkkjikjiijkkikjiijkkikjikjiijkkjkijkjiijkkjkikjikjiijkkjkikjikjiijkkjkkikjikji(3)I(X;Y/Z)≥0，当且仅当(X,Y,Z)是马氏链时等式成立。证明：0)/;(0log1)/(log1)/()(log)()/()/()(log1)/()/()()/()/(log)()/;()/()/(log)()/;(2222ZYXIezxpezxpzypezyxpzyxpzxpzyxpezyxpzxpzyxpzyxpzxpzyxpZYXIzxpzyxpzyxpZYXIikiikijkkjijkkjiijkkjikikjiijkkjikikjiijkkjikikjiijkkikjikji当01)/()/(kjikizyxpzxp时等式成立·21·)/()/()/()(/)()/()/()()/()/()()()/()/()()/()/(kjikikjkkjikikjkjikikjkkjkjikikjkjikizyxpzxpzypzpzyxpzxpzypzyxpzxpzypzpzypzyxpzxpzypzyxpzxp所以等式成立的条件是X,Y,Z是马氏链3.5若三个随机变量，有如下关系：Z=X+Y，其中X和Y相互独立，试证明：(1)I(X;Z)=H(Z)-H(Y)；(2)I(XY;Z)=H(Z)；(3)I(X;YZ)=H(X)；(4)I(Y;Z/X)=H(Y)；(5)I(X;Y/Z)=H(X/Z)=H(Y/Z)。解：1))()()/()();()()(log)()()/(log)/()()/(log)()/()(0)()()()/(222YHZHXZHZHZXIYHypypxpxzpxzpxpxzpzxpXZHYxzYxzypxzpxzpYXZijjjiikikikiikikkiikikjikik2))(0)()/()();(0)/(log)/()()/(log)()/()(0)(1)/(22ZHZHXYZHZHZXYIyxzpyxzpyxpyxzpzyxpXYZHzyxzyxyxzpYXZijkjikjikjiijkjikkjikjikjijik3)·22·)(0)()/()();(0)/(log)/()()/(log)()/(01)/(22XHXHYZXHXHYZXIzyxpzyxpzypzyxpzyxpYZXHyzxyzxzyxpYXZjkikjikjikjijkkjikjijkijkikji4))(0)()/()/()/;(0)/(log)/()()/(log)()/(01)/(22YHYHXZYHXYHXZYIzxypzxypzxpzxypzyxpXZYHxzyxzyzxypYXZikjkijkijkiijkkijkjiikjikjkij5))/(0)/()/()/()/;(0)/(log)/()()/(log)()/(01)/(22ZXHZXHYZXHZXHZYXIzyxpzyxpzypzyxpzyxpYZXHyzxyzxzyxpYXZjkikjikjikjijkkjikjijkijkikji)/(0)/()/()/()/;(0)/(log)/()()/(log)()/(01)/(22ZYHZYHXZYHZYHZYXIzxypzxypzxpzxypzyxpXZYHxzyxzyzxypYXZikjkijkijkiijkkijkjiikjikjkij·23·3.6有一个二元对称信道，其信道矩阵为98.002.002.098.0。设该信源以1500二元符号/秒的速