Principles of Communication System (II)
Final Exam
Spring, 2005
1. (16%) A binary sequence {an} is to be transmitted by a coding scheme described as

0 t T
1
p (t )   An g (t  nT ) , where g(t) is defined as g (t )  
.
otherwise
n  
0
Assume p(an = 1) = p(an = 0 ) = 0.5 for all n, and p(an, an+m) = p(an)p(an+m) for m  0.
 A an 1
(a) Assume An  
.
an  0
 A
Find the power spectral density of p(t).
 A
(b) Assume An  
 A
a n  a n -1
a n  a n -1
(8%)
.
Find the power spectral density of p(t).
(8%)
2. (15%) Assume the generator matrix of a (7, 3) code is
T
1 0 0 1 1 1 0
0 1 0 0 1 1 1 

 .
0 0 1 1 1 0 1
(a)
(b)
(c)
(d)
(e)
Find all possible codewords.
Is this a systematic code?
Is this a cyclic code?
Find the minimum Hamming distance of this (7, 3) code.
Find the corresponding parity check matrix.
(3%)
(3%)
(3%)
(3%)
(3%)
3. (9%) Assume data from a source of nine messages X = {a, b, c, d, e, f, g, h, i} with
respective probability [0.02, 0.35, 0.20, 0.15, 0.10, 0.05, 0.08, 0.04, 0.01] are to
be transmitted through a discrete channel. Do Huffman encoding for these source
symbols.
4. (20%) The block diagram of a binary convolutional code is shown below.
Input
S1
S2
1
1
S3
V1
V2
V3
That is, v1 = s1s2s3, v2 = s1s3, and v3 = s1s2, with  denoting the XOR
operation.
(a) Draw the state-transition diagram for this encoder.
(5%)
(b) Draw the trellis diagram for this code.
(5%)
(c) If the input sequence is 11001010, find the corresponding output bitstream.
(5%)
(d) If the output bitstream in 101010010001, find the decoded bitstream if the
Viterbi algorithm is used.
(5%)
5. (20%) Assume a sequence of binary data is to be transmitted via an AWGN channel
at a rate of 1Mbps. Assume the power loss during the transmission is 40dB and the
double-sided power spectral density of the Gaussian noise is N0/2 = 0.5×10-10 V2/Hz.
For each of the following cases, find the corresponding channel capacity.
(a) s1 (t )  3 cos( 2  1010 t)
(10%)
s 2 (t )  3 cos(2  1010 t)
0  t  10 6
10
(b) s1 (t )  3 cos( 2  10 t)
(10%)
s 2 (t )  0
0  t  10 6
6. (20%) For each of the following cases, find the required null-to-null bandwidth to
achieve a bit rate of 1 Mbps.
(a) 16-PSK.
(5%)
1
.
2Ts
2
(c) Noncoherent 8-FSK, with tone spacing of
.
Ts
(b) Coherent 16-FSK, with tone spacing of
(d) 16-QAM.
(5%)
(5%)
(5%)