Engineer Maths - Digital Communication

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Question 1

(a)

A digital source with entropy of 20 bits has 𝑛

alphabets

with equal probability

of occurrence. Examine the concept of information theory and obtain the value

for 𝑝−𝑛𝑝, where 𝑝

is the probability of occurrence of any

alphabet.

(10 marks) (b)

A communication link supports a maximum data rate of 20 Mbps at a

signal-to-

noise ratio (SNR) of 10 dB. Analyze the communication link and estimate its

channel capacity when the SNR is doubled in the linear scale. (10 marks)

(c)

Signal 𝑥(𝑡)

is given as input to a system whose transfer function is denoted

by 𝐻1(𝑓) = 1

1+3𝑗 . The output from this system is given as input to another

system whose transfer function and output signal are denoted by 𝐻2(𝑓) = −𝑗 3+𝑗

and 𝑦(𝑡), respectively. Solve for power transfer function 𝐺ℎ(𝑓) of the system

which is given by 𝐺ℎ(𝑓) = 𝑃𝑦(𝑓)

𝑃𝑥(𝑓) , where 𝑃𝑥(𝑓) and 𝑃𝑦(𝑓) denote the power

spectral density of 𝑥(𝑡) and 𝑦(𝑡), respectively. (5 marks)

Question 2

(a) An ergodic process 𝑥(𝑡) is given by 𝑥(𝑡) = 𝑎 + 𝑦(𝑡) + 𝑧(𝑡), where 𝑎 is a real- valued scalar and 𝑦(𝑡) and 𝑧(𝑡) are zero-mean and uncorrelated ergodic processes. Determine the autocorrelation function of 𝑥(𝑡).

(5 marks)

(b) The autocorrelation function 𝑅𝑥(𝜏) of a wide-sense stationary random process

𝑥(𝑡) is given by 𝑅𝑥(𝜏) = 𝛼 2𝐴 Sa(2𝜋𝐴𝜏) + 𝛽 𝐴 2

Sa2 �𝜋𝐴𝜏 2 �, where 𝐴, 𝛼 and 𝛽

are positive real-valued scalars and 𝛽 > 𝛼. Analyze the autocorrelation function 𝑅𝑥(𝜏) and obtain the power spectral density function 𝑃𝑥(𝑓) of the random process 𝑥(𝑡). Sketch the power spectral density function 𝑃𝑥(𝑓).

(10 marks)

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(c) A signal (𝑓) = 0.5 𝑇 Sa2(𝜋𝑇𝑓) , given in the frequency domain, is corrupted by white-noise whose power spectral density is given by 𝑃𝑛(𝑓) = 𝑁0 2⁄ . Design a causal matched filter to maximize the signal-to-noise ratio (SNR). Assume the constant 𝐾 to be 1. Sketch the impulse response of the matched filter.

(10 marks)

Question 3 (a) An eight-level line code that has a rectangular pulse shape at the output of the

transmitter is transmitted through a channel at the rate of 24 M symbols/s. The overall transmission system (i.e., the transmitter, channel and receiver) has a raised cosine filter characteristic with a rolloff factor of 𝑟 = 0.5. Compute the following parameters: (i) System’s supported bit rate.

(5 marks) (ii) System’s 6-dB bandwidth.

(5 marks) (iii) System’s absolute bandwidth.

(5 marks)

(b) The PM signal 𝑠(𝑡) generated by a transmitter is given by

𝑠(𝑡) = 200 sin(2𝜋𝑓𝑐𝑡 + 10 sin(10𝜋𝑡)),

where 𝑓𝑐 = 2.5 GHz. If the phase deviation constant is 50 rad/V, then deduce the modulating signal and compute its peak value.

(5 marks)

(c) Sketch the Bipolar NRZ signalling for the binary data sequence 1 0 1 1 0. Assume the bit duration to be 4 ms. Clearly indicate the time durations in the sketch.

(5 marks)

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Question 4 In a binary communication system, the receiver statistic 𝑟0 consists of a polar signal plus noise 𝑛0. The polar signal has values 𝑠1 = +𝐵 and 𝑠2 = −𝐵, where 𝐵 is a positive real-valued scalar. The noise (𝑛0) follows sinusoidal distribution given by 𝑓(𝑛0) = 0 for |𝑛0| ≥ 𝜎0 and

𝑓(𝑛0) = 1

𝜋�𝜎0 2−𝑛0

2 for |𝑛0| ≤ 𝜎0.

Figure Q4 illustrates the sinusoidal distribution.

Figure Q4 Sinusoidal distribution

The mean and variance of 𝑛0 are zero and 𝜎02/2 , respectively, where 𝜎0 is a real-valued scalar. Use probability and statistical models to answer the following questions. (a) Assuming that 𝐵 > 𝜎0 , sketch the conditional probability density functions

𝑓(𝑟0|𝑠1) and 𝑓(𝑟0|𝑠2). (5 marks)

(b) If 𝐵 > 𝜎0 and the detection threshold is 𝑉𝑇 = 0, then obtain the probability of

error 𝑃𝑒. (5 marks)

(c) In Question 4(b), is the detection threshold 𝑉𝑇 = 0 an optimum value? Is there

any other detection threshold which gives the same probability of error (𝑃𝑒) obtained for Question 4(b)?

(5 marks) (d) Let 𝐵 = 1, 𝜎0 = 1.5, detection threshold 𝑉𝑇 = 0 and the probability of polar

signals 𝑠1 and 𝑠2 be 0.25 and 0.75, respectively. Rate the corresponding probability of error. (Hint: Use the cumulative distribution function.)

(10 marks)