課程名稱︰數位通信系統
課程性質︰選修
課程教師︰蘇炫榮
開課學院：電機資訊學院
開課系所︰電機系
考試日期（年月日）︰2007/1/16
考試時限（分鐘）：150
是否需發放獎勵金：是
試題 :
1.[5 pts each]
(a) What is the envelope delay characteristic of a band-limited channel?
(b) Describe the Nyquist criterion for zero ISI. What is its implication on
the transmission symbol rate?
(c) Sinc waveworm achieves the Nyquist rate. Discuss the disadvantages of
the sinc waveform in practical system implementation.
(d) What is the design criterion of a zero-forcing (ZF) equalizer? What is
the design criterion of a minimum mean square error (MMSE) equalizer?
At high signal-to-noise ratio (SNR), which one performs better?
(e) What is the purpose of the (d,k) modulation coding for magnetic
recording? What do d and k mean?
(f) JPEG uses various source coding techniques to achieve good compression.
List these techniques and describe how they achieve compression.
(g) Describe the properties of cyclic code. Give the mathematical expression
for encoding cyclic code and show that this indeed generates valid code
words.
2. The transmission of a signal pluse with a raise cosine spectrum through a
channel results in the following (noise-free) sampled output from the
demodulator :
Xk= -0.5 (when k=-2)
0.1 (when k=-1)
1 (when k=0)
-0.2 (when k=1)
0.05 (when k=2)
0 (otherwise)
[5 pts each]
(a) Determine the tap coefficients of a three-tap linear equalizer based on
the zero-forcing criterion.
(b) Determine the residual ISI and its span in time for the equalizer
obtained in (a)
3. [10 pts]
Let the running digital sum (RDS) of a binary sequence be defined as the
defference between the total number of accumuated zeros and the total
number of accumulated ones in the sequence. Determine the state transition
matrix and the capacity of a modulation code whose RDS at any time instant
relative to the beginning of the code satisfies -1 =< RDS =< 1
4. Let X be a geometrically distributed random variable with probabilities
P(X=k)=p(1-p)^(k-1), k=1,2,3,.....
[6pts]
(a)Find the entropy of X
[6pts]
(b)Let p>=0.5. Design a binary Huffman code for X. What is the average
codeword length of this Huffman code?
[5pts]
(c)What is the efficiency of the code in (b)? For what value of p is the
efficiency maximized? And what is the maximum efficiency?
5. Let X be a Gaussian random veriable with zero mean and varience σ^2. If
we consider 1-bit scalar quantization with mean square error distortion,
[6 pts]
(a) What are the optimum input and output quantization level?
[6 pts]
(b) What is the mean distortion?
[6 pts]
(c) With the same rate, what is the theoretical lower bound on the
distortion?
6.[10 pts]
Determine the capacity and the capacity achieving input probabilities of
the following channel?
x P(y|x) y
0 ---(0.5)--- 0
1 ---(1)----- 1 P(1|0) = 0.25
2 ---(1)----- 2 P(2|0) = 0.25
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