Engineer Maths - Digital Communication

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Question 1

(a) A digital source generates 16 symbols. One-fourth of the symbols have a probability of for each symbol, where is a positive real number such that 0 ≤ ≤ 1. Another one-fourth of the symbols have a probability of 2 for each symbol and the remaining symbols each have a probability of 3 . Examine the above digital source and calculate the value of and the digital source’s entropy.

(10 marks)

(b) Let signal ) be a sinusoid defined as ) = cos ω + ϕ), where A is the amplitude, is the angular frequency and ϕ is the phase shift. Analyze the signal ) by formulating an expression for its autocorrelation and power spectral density (PSD).

(10 marks)

(c) The power transfer function ) of a system is given by ) = )) , where ) and ) denote the power spectral density of the input and output signals of the system, respectively. If the system transfer function is

given by , then solve for ). (5 marks)

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Question 2

(a) The magnitude spectrum | )| of a baseband signal ) is shown in Figure Q2(a), where = 5 MHz. The bandpass signal ) is obtained by translating the baseband signal ) to a carrier frequency = 2.5 GHz. Sketch the magnitude spectrum and power spectral density of the bandpass signal ). Clearly label the frequency and magnitude values in your sketch.

Figure Q2(a)

(4 marks)

(b) A zero-mean white noise with power spectral density (PSD) ) = 2⁄ is given as input to a linear time-invariant filter whose transfer function ) is given by

) = 1 − | |), | | ≤ 0, otherwise where 0. Calculate the variance of the filter’s output signal in terms of .

(9 marks)

(c) Multilevel data with an equivalent bit rate of 36 Mbits/s is sent over a channel using a eight-level line code that has a rectangular pulse shape at the output of

the transmitter. The overall transmission system (i.e., the transmitter, channel

and receiver) has a raised cosine filter characteristic with a rolloff factor of = 0.5. Compute the following parameters:

(i) Baud rate of the received signal. (4 marks)

(ii) System’s 6-dB bandwidth. (4 marks)

(iii) System’s absolute bandwidth. (4 marks)

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Question 3

(a) The PM signal ) generated by a transmitter is given by ) = 200 cos 2 + 10sin 10 ) + 5),

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

(2 marks)

(b) A QPSK signal is used to transmit data at a rate of 30 Mbits/s over a satellite transponder. The satellite link has a bandwidth (transmission bandwidth) of

27 MHz.

(i) The satellite signal is equalized to have an equivalent raised cosine filter characteristic. Determine the rolloff factor for the above scenario.

(9 marks)

(ii) There is an urgent need to support a data rate of 55 Mbits/s over the above satellite link. Test whether the 27 MHz satellite link can support

this data rate or not by finding a suitable value for the rolloff factor r.

(9 marks)

(c) Sketch the Manchester RZ signalling for the binary data sequence 1 1 0 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 consists of a polar signal plus noise . The polar signal has values = + and = − . The noise has zero- mean and an RMS value equal to .

(a) Assume the polar signals to be equally likely and the noise ) to have Laplacian distribution given by ) = √ √ | | . Use probability and statistical models to determine the optimum value for the

detection threshold . Construct an expression for the probability of error as a function of ⁄ .

(10 marks)

(b) If = 10 and the noise variance is 0.25, then rate the probability of error using the expression obtained in Question 4(a).

(2 marks)

(c) Assume that the polar signals are not equally likely. The probability of transmitting the polar signals and are and , respectively. The noise has the same Laplacian distribution given in Question 4(a). The mean and

variance of the noise are 0 and 0.25, respectively. If = 1, = 3 4 and = 1 4 , then determine the optimum value for the detection threshold and find the corresponding probability of error .

(Hint: If | + | − | − | = where | | ≤ 2 , then = /2) . (13 marks)