Engineer Maths - Digital Communication

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Question 1

(a)

A computer terminal has 128 characters on its keyboard and each character is transmitted using binary words. How fast can the characters be sent (characters/s) over a telephone line having a bandwidth of 3.2 kHz and an SNR of 20 dB?

(10 marks)

(b)

A source transmits binary words of length 10 bits. One fourth of the possible

words have a probability of being transmitted that is �1 2 � 12

for each word. The

remaining three fourths have probabilities equal to 5 �1 2 � 12

. Find the entropy of

the source. (10 marks)

(c) A waveform, 0.23 + 1.69 sin(300𝑡 + 30𝑜), is to be sampled and reproduced

from these values. Determine the maximum allowable interval between the sample values.

(5 marks) Question 2 (a) The power spectral density (PSD) of a zero-mean ergodic process 𝑦(𝑡) is given

by

𝑃𝑦(𝑓) = �

1 𝐵

(𝐵 − |𝑓|), |𝑓| ≤ 𝐵

0, 𝑓 otherwise

�

Find the variance of 𝑦(𝑡) in terms of 𝐵, where 𝐵 > 0. (5 marks)

(b) An analog signal is to be converted into a PCM signal that is a binary polar

NRZ line code. The signal is transmitted over a channel that is absolutely bandlimited to 8 kHz. The PCM quantizer has 32 steps. The overall equivalent transfer function is of the raised cosine-rolloff type with 𝑟 = 0.25 .

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(i) Find the maximum bit rate that can be supported by the above PCM

system without introducing intersymbol interference (ISI). (5 marks)

(ii) Find the maximum bandwidth that can be permitted for the analog signal.

(5 marks) (c) The input signal 𝑟(𝑡) = 𝑠(𝑡) + 𝑛(𝑡) is given as input to a matched filter with

impulse response ℎ(𝑡) . The input noise 𝑛(𝑡) is white noise whose power spectral density (PSD) is given by 𝑃𝑛(𝑓) = 1. The Fourier transform of signal 𝑠(𝑡) is given by

𝑆(𝑓) = 𝑇 � sin(𝜋𝑓𝑇)

(𝜋𝑓𝑇) � 2

𝑒−𝑗2𝜋𝑓𝑇.

The matched filter output signal attains the peak value at 𝑡0 = 2𝑇. Determine the impulse response ℎ(𝑡) of the matched filter and sketch it out for 𝑇 = 1 ms and assuming the matched filter’s real-valued constant 𝐾 is equal to 1.

(10 marks) Question 3 (a) Represent the data sequence 0 1 1 0 1 0 1 using Manchester NRZ signaling.

Assume the duration of each bit to be 4 ms. (7 marks)

(b) A transmitter generates signal 𝑠(𝑡) given by

𝑠(𝑡) = 10 cos[2𝜋𝑓𝑐𝑡 + 6 sin(2000𝜋𝑡)] where 𝑓𝑐 = 100 MHz. (i) If the phase deviation constant is 50 rad/V, find the mathematical

expression for the corresponding phase modulation voltage 𝑚(𝑡) . Specify the peak value and frequency of this phase modulation voltage.

(9 marks)

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(ii) If the frequency deviation constant is 105 rad/V-s, find the mathematical

expression for the corresponding FM voltage 𝑚(𝑡). Specify the peak value and frequency of this FM voltage.

(9 marks) Question 4 The following baseband signaling waveforms are used for the transmission of binary data generated by a digital source

𝑠1(𝑡) = 𝑎, 0 < 𝑡 ≤ 𝑇 (binary 1) 𝑠2(𝑡) = 𝑏, 0 < 𝑡 ≤ 𝑇 (binary 0)

where 𝑎, 𝑏 are real-valued constants and 𝑎 > 𝑏. The signal at the output of the receiver filter is a sum of baseband signal and white Gaussian noise whose mean and variance are 0 and 𝜎2, respectively. (a) Show that the optimum detection threshold level (𝑉𝑇) for detecting the

transmitted data is given by

𝑉𝑇 = 2𝜎2 ln�𝑝2 𝑝1� � + 𝑎

2 − 𝑏2

2(𝑎 − 𝑏) .

(10 marks) (b) Find the BER for the above baseband signaling system when 𝑎 = 10, 𝑏 = 5,

𝑝1 = 3 4� , 𝑝2 = 1

4� and 𝜎 2 = 0.5. Refer to Appendix A for the Q function

table. (9 marks)

(c) Find the entropy of the digital source, if the optimum detection threshold level

(𝑉𝑇) given in part(a) is

𝑉𝑇 = 𝑎 + 𝑏

2 .

(6 marks)

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APPENDIX A