Carleton University
Department of Systems and Computer Engineering
Principles of Digital Communications
Assignment #3
Due on Wednesday, November 26, 2014
SYSC 5504
ELG 6154
Fall 2014/15
1. Consider the multipath communication model shown in Fig. (a). Depending on the message, m, the transmitted
signal is defined by
s(t) = (1 − 2m)p(t) =
p(t) if m = 0
,
−p(t) if m = 1
where
p(t) =
5 cos 2π103 t for 0 ≤ t ≤ 3 × 10−3
.
0
elsewhere.
Assume that Pr{m = 0} = 21 .
The three paths
Attenuation:
Delay:
Noise PSD:
of the channel are characterized by the following
α1 = 0.2
α2 = 0.4
τ1 = 1 msec
τ2 = 1.5 msec
Φn1 (f ) = 0.002 W/Hz Φn2 (f ) = 0.006 W/Hz
parameters:
α3 = 0.6
τ3 = 2 msec
Φn3 (f ) = 0.004 W/Hz
(a) Show that the optimum receiver can be realized in the form illustrated in Fig. (b). Determine h1 (t), T1 , and
the specification of the decision device.
(b) Calculate the probability of a bit error.
α1
n1 (t)
Delay
τ1
α2
m
n2 (t)
s(t)
r(t)
Delay
τ2
Transmitter
α3
n3 (t)
Delay
τ3
(a)
r(t)
h1 (t)
r
t = T1
Decision
Device
m
b
(b)
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2. Determine the capacity of the channel shown below, where
δ = Pr {Y = 2|X = 0} = Pr {Y = 2|X = 1} .
1−δ
0
0
δ
2
X
Y
δ
1
1−δ
1
3. Suppose the linear (n, k) block code defined by the generator matrix

1
0

G=
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1
1
0
1
1
1
1
0

0
1

1
1
is used for error detection. A binary symmetric channel with a cross-over probability of ρ is used to transmit the
code bits.
(a) What is n, k, and the rate of this code?
(b) In a table, show which code words are transmitted for each of the possible messages.
(c) Find the minimum distance of the code.
(d) What is the random-error detection capability of this code?
(e) Give the parity check matrix for the code.
(f) What is the syndrome if r = 1010111 is received.
(g) Find PC , the probability that a code word is received correctly.
(h) Find Pε , the probability that a code word is received incorrectly, and no errors are detected.
(i) Find Pd , the probability that an error is detected.
4. Suppose the linear block code with generator matrix
G=
10101
01110
is used for error correction.
(a) In a table, show which code words are transmitted for each of the possible message words.
(b) Construct the standard array.
(c) For each of the 32 possible error patterns, determine the number of message bits that would be in error if
standard array decoding is used. Show your results in the standard array beside each error pattern. Assume
that the all-zero codeword was transmitted.
(d) Determine the probability of a message bit error after error correction has been attempted. Express your
answer in terms of the channel crossover probability, p.
Hint: Because the code is linear, the probability of error does not depend on the transmitted message word.
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Fall 2014/15
5. For the continuous ARQ protocol,
(a) Show that the average number of transmitted blocks per correctly received block is
N=
1 + (N − 1)Pd
,
1 − Pd
where N − 1 is the number of blocks transmitted during a round-trip delay, and Pd is the probability that a
block needs to be retransmitted. Use the identity
∞
X
n=0
αn =
1
1−α
∀ |α| < 1 .
(b) What is the effective transmission rate if the nominal rate is Rm = 1000 bits/sec, the round-trip delay is
Trt = 4 msec, and the code given in Question 3 is used for error detection. Assume the code bit transmission
rate is Rc = Rm /R and the energy per code bit is Ecb = REmb , where R is the rate of the code. BPSK
transmission over an AWGN channel is used to transmit the code bits, with a SNR of Emb /N0 kdB = 7 dB.
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Fall 2014/15