課程名稱︰消息理論
課程性質︰選修
課程教師︰林茂昭
開課學院：電資
開課系所︰電信所
考試日期（年月日）︰2013.6.18
考試時限（分鐘）：120
是否需發放獎勵金：yes
（如未明確表示，則不予發放）
試題 :
Final Exam of Information Theory
June 18, 2013
1.(10%)Let the input variable X to a channel be uniformly distributed over
the interval -1/2 <= x <= 1/2. Let the output of the channel be Y=X+Z, where
the noise random variable is uniformly distributed over the interval
-a/2 <= z <= a/2. Find I(X;Y) for a>=1.
2.
(a)(7%)Find the differential entropy h(X)=-∫f lnf for the random variable
with the exponential density, f(x)=λe^(-λx), x>=0
(b)(7%)Please show that the exponential distribution with mean 1/λ is the
maximum entropy distribution among all continuous distributions supported
in [0,∞] that have a mean of 1/λ,
(c)(6%)Let Yi=Xi+Zi, where Zi is i.i.d. exponentially distributed with mean
μi. Assume that we have a mean constraint on the signal (i.e., EXi<=λ).
Show that the capacity of such a channel is C=log(1+λ/μ).
3.(10%)Consider a source X uniformly distributed on the set {1,2,...,m}.
Find the rate distortion function for this source with Hamming distortion;
that is d(x,x')= 0, if x=x'; 1, if x!=x'.
4.(10%)Give the LZ78 (tree-structured Lempel-Ziv algorithm) parsing and
encoding of 00000011010100000110101.
5.(10%)Let (X1,Y1),(X2,Y2),... be a sequence of jointly distributed random
variables i.i.d. ~ p(x,y), where Xi belongs to {0,1}, Yi belongs to {0,1},
p(Xi=a | Yi=b) = 0.1 if a!=b, and p(Xi=a | Yi=b)=0.9 of a=b. Let X^n be
described at rate R1 and Y^n be described at rate R2. What region of rates
allows recovery of X^n and Y^n with probability of error tending to zero?
6.(10%)Please show that there exists a sequence of (2^nR,n) universal source
codes such that P(n) -> 0 for every source Q such that H(Q)<R.
e
7.(10%)Describe the water-filling principle for the parallel Gaussian
channels.
8.(10%)Suppose that X1,...,Xn are drqwn i.i.d.~Q(x). Show that the probability
of x depends only on its type and is Q^n(x)=2^-n(H(Px)+D(Px||Q))
9.(10%)Consider a Gaussian multiple-access channel with two users. Two senders
, X1 and X2 communicate to a single receiver, Y. The received signal at time i
is Yi=X1i+X2i+Zi
where {Zi} is a sequence of i.i.d. zero mean Gaussian random variables with
variance N. Assume that there is a power constraint Pj on sender j; that is,
we have n
1/n(Σ(x_ji)^2 (w_j)) <= Pj, wj belongs to {1,2,...,2^nR}, j=1,2.
i=1
Please describe the achievable regions of R1 and R2.
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