The binary errors-and-erasures channel is given by (1)Find the capacity (2)Specialize to erasures only(ε=0) (3)Specialize to the binary symmetric channel(ρ=0) (4)Would you prefer a binary symmetric channel with crossover probability=0.125 or a simple erasure channel with probability of erasure=0.5?
第1题
iable X,Y and Z should be defined by I(X;Y;Z)=I(X;Y) - I(X;Y|Z) This quantity is symmetric in X,Y and Z,despite the preceding asymmetric define.Unfortunately,I(X;Y;Z)is not nesessarily nonnegative.Find X,Y,Z such that I(X;Y;Z)<0,and Prove the following two identities: I(X;Y;Z)=H(XYZ) - H(X) - H(Y) - H(Z)+I(X;Y)+I(Y;Z)+I(Z;X) I(X;Y;Z)=H(XYZ) - H(XY) - H(YZ) - H(ZX)+H(X)+H(Y)+H(Z) The first identity can be understood using the Venn diargram analogy for entropy and mutual lnlormatlon.The second identity follows easily from the first.
第2题
N.Each sequences an even number of 1’S has probabilitv 2-(N-1) and each sequences with an odd number of 1’s has probability zero.Find the average mutual informations I(X1;X2),I(X3;X2|X1),…,I(XN;XN-1|X1X2…XN-2) Check your result for N=3.
第3题
有一信源输出X∈{0,1,2},其概率为p0=1/4,p1=1/4,p2=1/2。设计两个独立实验去观察它,其结果分别为Y1∈{0,1}和Y2∈{0,1}。已知条件概率为:
求:(1)I(X;Y1)和I(X;Y2),并判断哪个实验好些。 (2)I(X;Y1Y2),并计算做Y1和Y2两个实验比做Y1或Y2中的一个实验各可多得多少关于X的信息。 (3)I(X;Y1|Y2)和I(X;Y2|Y1),并解释它们的含义。
第4题
dent. let ρ=1 - H(X1|X2)/H(X1) (1)Show ρ=I(X1;X2)/H(X1) (2)Show 0≤ρ≤1 (3)When is ρ=0? (4)When is ρ=1?
第5题
(XY) - I(X;Y) =2H(XY) - H(X) - H(Y)
第6题
t ρ(X,Y) is the number of bits needed for X and to communicate their values to each other.
第7题
e variance of I(x;y)=0.Prove that the input assignment achieves channel capacity.
第8题
;0 and VAR[I(x;y)]=0.Calculate I(X;Y).
第10题
(xy)=αP(x)P(y)
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