ELEC 3100 Assignment
Homework 3
Deadline: May 10, 6pm
Full mark: 100 points
1
• This homework is due on May 10, 6pm.
• Please submit your solution as ONE pdf file (A4, Portrait, 1-in-1) to Canvas.
• NO other format will be considered.
• NO late submission will be allowed.
1
Channel Coding
Question 1 (20 points)
The generation matrix for a linear binary code is


1 0 0 1 1 1 0
G =  0 1 0 0 1 1 1 .
0 0 1 1 1 0 1
a) Determine the parity check matrix H for the code. (5 points)
b) Let u = [1 1 1] be the data bits. Determine the corresponding codeword c. (5
points)
c) Assuming the codeword in b) is transmitted and there is only one-bit error, construct
the table of syndromes for the code. (10 points)
2
2
Baseband Communication and Noise
Question 2 (30 points)
Consider a binary communication over an AWGN channel with a noise power spectral density
N0 /2. Let
A, 0 ≤ t ≤ T
−A, 0 ≤ t ≤ T
s1 (t) =
, and s2 (t) =
0, otherwise
0,
otherwise
represent the transmitted waveforms for bit ”0” and bit ”1”, respectively. The integrateand-dump structure shown below is utilized as the receiver. Bit ”0” and bit ”1” are sent
with probability p and 1 − p, respectively.
For the following questions, please show your mathematical derivation.
a) If p = 0.5 and the decision threshold is V = 0, please determine bit error rate for this
communication system. (15 points)
b) If p ̸= 0.5, how to determine the decision threshold to minimize the average bit error
rate? (Hint: Write down the expression of bit error rate with respect to p, and try to
find the optimal condition and the optimizer.) (15 points)
3
3
Optimal Receiver
Question 3 (30 points)
Consider a binary communication over an AWGN channel with a noise power spectral density
N0 /2. Let
A
A
t,
0
≤
t
≤
T
− T t, 0 ≤ t ≤ T
T
s1 (t) =
, and s2 (t) =
0, otherwise
0,
otherwise
represent the transmitted waveforms for bit ”0” and bit ”1”, respectively.
a) Calculate the energy for the two signals. (5 points)
b) Design a matched filter for the two signals, and give a block diagram for the matched
filter. (10 points)
c) Sketch the waveform at the output of the matched filter. (10 points)
d) Compute the probability of error for this signaling scheme. (5 points)
4
4
Digital Modulation
Quesion 4 (20 points)
A binary FSK system operates with carrier frequencies of 40 kHz and 80 kHz. The bit rate
is 20000 bits per second and the transmitted signals are equally probable. The received
noise is AWGN with PSD of N0 /2 = 1µW/Hz and the matched filter is used in the receiver.
Now the receiver receives a modulated signal as shown below. Please answer the following
questions.
a) What is the bit sequence after demodulation? State your assumption. (10 points)
b) Write down the expression of s0 (t) and s1 (t) that represent transmission of bit ”0”
and an bit ”1”, respectively. (5 points)
c) Write down the mathematical expression of the impulse response of the matched filter.
(5 points)