Engineer Maths - Digital Communication

Page 2 of 4

Question 1

(a) A communication channel with a bandwidth of 5 MHz has a channel capacity of bits/s at a SNR of 15 dB. Analyze the above channel and determine the new

SNR (in dB) required to support the same channel capacity when the bandwidth

is doubled.

(10 marks)

(b) A digital source generates equi-probable symbols each with probability 0.01. Examine the above digital source and determine . Compute the entropy of this

digital source.

(10 marks)

(c) Given the signal ( ) = 20000 Sa(20000 ). Sketch the magnitude spectrum and find the minimum time interval required between samples of ( ) in order to avoid aliasing.

(5 marks)

Question 2

(a) The power spectral density (PSD) of a zero-mean ergodic process ( ) is given by ( ) = 10 Λ 1000 + Π 4000 .

(i) Analyze ( ) and formulate an expression for the corresponding autocorrelation function ( ).

(5 marks)

(ii) Compute the variance of ( ). (5 marks)

(b) A signal ( ) = Λ is corrupted with an additive noise whose power spectral density is given by ( ) = Sa( ), where is a real-valued scalar. This corrupted signal is fed into a causal LTI filter with frequency response ( ). Solve for ( ) such that the SNR at the output of the filter is maximized. Assume the filter gain to be = 1. Let ℎ( ) denote the impulse response of the above

Page 3 of 4

filter. Illustrate ℎ( ) with a sketch and give its mathematical expression. Note: Appropriately choose the delay .

(10 marks)

(c) An LTI system ( ) consists of two LTI systems with transfer functions ( ) = 3 + and ( ) = 2 − 6 connected in cascade. The power spectral density (PSD) of the input signal to the LTI system ( ) is 10. Compute the output signal’s PSD.

(5 marks)

Question 3

(a) A communication system employs a 32-level line code to transmit data over a channel with absolute bandwidth of 20 MHz. The overall transmission system

(i.e., transmitter, channel and receiver) has a raised cosine filter characteristic

with a rolloff factor of = 0.5. Compute the following parameters:

(i) Maximum baud rate supported by the system. (5 marks)

(ii) Maximum bit rate supported by the system. (5 marks)

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

(b) A wireless transmitter is tasked to transmit the message signal ( ) given by ( ) = 1.2 cos(2000 ).

The carrier frequency is 10 Hz and its positive amplitude is +2 V. Formulate the mathematical expression for the following modulated signals carrying ( ):

(i) FM signal with frequency deviation constant of 10 rad/V-s. (5 marks)

(ii) AM signal. (3 marks)

(iii) PM signal with phase deviation constant of 15 rad/V. (2 marks)

Page 4 of 4

Question 4

In a binary communication system, the receiver statistic consists of a unipolar signal corrupted by an additive noise . The unipolar signal is generated such that the binary ‘0’ is represented by = 0 volts and the binary ‘1’ is represented by = + volts, where > 0. The probabilities of transmitting ‘0’ and ‘1’ are and , respectively. The noise follows a Gaussian probability density function (PDF) ( ) with mean >0, > and variance as follows ( ) = 1√2 ( ) /( ).

(a) Describe the conditional PDFs for the receiver statistic , i.e., ( | ) and ( | ), using suitable mathematical expressions. Sketch the conditional PDFs ( | ) and ( | ). Clearly label the sketch with appropriate values.

(7 marks)

(b) Use probability theory to show that the optimum detection threshold is given by = 2 + − ln .

(8 marks)

(c) Rate the performance of the above system by formulating an expression for the bit error rate using the optimum detection threshold obtained in Question 4(b).

(5 marks)

(d) Given that = = 0.5, = 5, = 0.5 and = 1. Solve for and . (5 marks)