Assignment #1
Due Date: 21 – July - 2016
1. Consider a discrete memoryless source with alphabet {s0, s1, s2} and statistics {0.7,
0.15, 0.15} for its output.
a. Apply the Huffman algorithm to this source and compute the average code-word
length of the code.
b. Let the source be extended by encoding two sources alphabet at a time. Apply the
Huffman algorithm to the resulting extended source and compute the average
code-word length of the new code.
c. Compare the efficiency of the code in part (b) with the efficiency of the original
source.
2. Consider the following binary sequence
11101001100010110100…
Use the Lempel-Ziv algorithm to encode this sequence.
3. The probability of any letter occurring in English text is estimated as the frequency of
occurrence of that letter in a collection of sample text. A table of the frequency of
occurrence of the letters of a memoryless model of English language based on
inspection of many samples of the English text is shown in the Table below. (Hint:
Use Matlab to solve this question)
a. Computer the entropy of this memoryless source.
b. Compute the efficiency if ASCII code is used to encode the letters.
c. What do you observe from the above results?
Source symbol ai Probability pi Source symbol ai Probability pi
Space 0.186 N 0.057
A 0.064 O 0.063
B 0.013 P 0.015
C 0.022 Q 0.001
D 0.032 R 0.048
E 0.103 S 0.051
F 0.021 T 0.080
G 0.015 U 0.023
H 0.047 V 0.008
I 0.058 W 0.017
J 0.001 X 0.001
K 0.005 Y 0.016
L 0.032 Z 0.001
M 0.020
4. Consider the four signal waveforms as shown in Figure below.
a. Determine a set of basis functions for these waveforms.
b. Use the basis functions to represent the four waveforms by vectors.
c. Determine the minimum distance between all the vector pairs.
5. Consider the sequence 01001011100111110100 to be transmitted from a transmitter
a. Assuming a 4-ary pulse-amplitude modulation system, draw the waveform
generated by it when transmitting. State all your assumptions clearly and use
Matlab the plots.
b. Repeat (a) for the case that 4-Phase Shift Keying is used.
6. Consider the octal signal point constellation shown in Figure below.
a. The nearest neighbor signal points in the 8-QAM signal constellation are separated
by a distance of A. Determine the radii a and b of the inner and outer circles.
b. The adjacent signal points in the 8-PSK are separated by a distance of A. Determine
the radius r of the circle.
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c. Determine the average transmitter powers for the two signal constellations and
compare the two powers. What is the relative power advantage of one
constellation over the other? (Assume that all signal points are equally probable.)
d. Is it possible to assign three data bits to each point of the signal constellation such
that nearest neighbor (adjacent) points differ in only one-bit position?
e. Determine the symbol rate if the desired bit rate is 90 Mbps.
7. Consider a 4-PSK constellation with dmin = √2. What is the additional energy required
to send one extra bit (8-PSK) while keeping the same minimum distance (and thus
with the same bit error probability)?
8. A spacecraft located 100,000 km from the earth is sending data at a rate of R bps. The
frequency band is centered at 2 GHz, and the transmitted power is 10 W. The
spacecraft has an antenna with a gain of 10 dB while the earth station uses a parabolic
antenna that is 50 m in diameter and has an illumination efficiency of 50%. The noise
temperature of the receiver front end is 300 K.
a. Determine the received power level at the receiver front end?
b. If the desired Eb/N0 = 10 dB, determine the maximum bit rate that the spacecraft
can transmit.
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9. Consider the waveform shown in Figure below. Using Matlab and any sinusoidal
function as a modulating signal
a. Plot the ASK modulated waveform
b. Plot the PSK modulated waveform
c. Plot the FSK modulated waveform of the original signal
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