COLLEGE OF ENGINEERING & TECHNOLOGY

Department:
Electronics and Communications Engineering
Course: Lecturer:
Sheet (1)
1- A message source generates one of four messages randomly every microsecond. The probabilities of these messages are 0.4, 0.3, 0.2 and 0.1. Each emitted message is independent of the other messages in the sequence. a)- What is the source entropy? b)- What is the rate of information generated by this source, if the source generates 10
4
character per second? 2- A standard television picture is composed of approximately 300,000 basic picture elements. Each of these elements can assume 10 distinguishable brightness levels with equal probability. Find the information content of TV picture frame. 3- A source emits seven messages with probabilities 1/2, 1/4, 1/8, 1/16, 1/32, 1/64 and 1/64 respectively. Find the entropy of the source and obtain the compact binary code. Determine the efficiency and redundancy. 4- A source emits seven messages with probabilities 1/3, 1/3, 1/9, 1/9, 1/27, 1/27 and 1/27 respectively. Find the entropy of the source and obtain the compact 3- ary code. Determine the efficiency and redundancy. 5- A source emits four messages with probabilities 0.5, 0.3, 0.1 and 0.1 respectively. Each emitted message is independent of the other messages in the sequence. Find a)- The entropy of the source. b)- The compact binary code. c)- Determine the efficiency and redundancy. d)- Repeat all previous for a compact ternary code.
6
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Consider Shannon’s third theorem, the Noisy Channel Coding Theorem for a continuous
communication channel having bandwidth W Hertz, perturbed by additive white Gaussian noise of power spectral density N0, and average transmitted power P. (a) Is there any limit to the capacity of such a channel if you increase its signal-to- noise ratio P/N0W without limit? If so, what is that limit? (b) Is there any limit to the capacity of such a channel if you can increase its spectral bandwidth W (in Hertz) without limit, while not changing N
0
or P? If so, what is that limit? (c) Consider a noisy analog communication channel of bandwidth W = 1 MHz, which is perturbed by additive white Gaussian noise whose total spectral power is N
0
W = 1. Continuous signals are transmitted across such a channel, with average transmitted power P = 1,000. Give a numerical estimate for the channel capacity, in bits per second, of this noisy channel. Then, for a channel having the same bandwidth W but whose signal-to-noise ratio P/N
0
W is four times better, repeat your numerical estimate of capacity in bits per second.

7- Several people at a party are trying to guess a 3-bit binary number. Alice is told that the number is odd; Bob is told that it is not a multiple of 3 (i.e., not 0, 3, or 6); Charlie is told that the number contains exactly two 1's; and Deb is given all three of these clues. How much information (in bits) did each player get about the number? 8- Calculate the bandwidth of black and white TV given that:

We have 10 probable levels of intensity.

We send 30 pictures per second.

SNR=30 dB.

We have 3*10
5
pixels per picture.
9
- A link is to be operated at a bandwidth efficiency of B=9, i.e. at a rate of 9 bps for each Hz of bandwidth. Obtain the minimum SNR required at the receiver to allow, in theory, error-free transmission with this bandwidth efficiency. Express your answer in dB. 10- Capacity of AWGN channel. The capacity in bits per second of an additive white Gaussian noise (AWGN) channel is where P is the received signal power, B is the signal bandwidth, and N
0
/2 is the noise power spectral density (PSD). The total noise power is N
0
B. Consider a wireless channel where received power falls off with distance d according to the formula Given d
0
=10m, transmitter power P
t
=1W, noise PSD N
0
=10
-9
W/Hz, and channel bandwidth B= 30 KHz, find the capacity of this channel for transmitter receiver distances of 100m and 1km. 11- Suppose that a low-pass communications system has a 1 MHz bandwidth. What bit rate is attainable using 8-level pulses? What is the Shannon capacity of this channel if the SNR is:

20 dB,

40 dB. 12- Suppose we wish to transmit at a rate of 64 kbps over a 3 kHz telephone channel. What is the minimum SNR required to accomplish this?