Question 1

(a) A digital source generates 32 symbols. One symbol has a probability of 0.5 and another symbol has a probability of 0.1. The remaining 30 symbols each have a probability of p. Compute the value for and then the source entropy. Discuss the steps involved in the above computati on.

(b) A digital source generates 1024 equi-probable symbols. Use information theory to determine the minimum number of bits required to represent each symbol.

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

(d) It is required to support a data rate of 512 Mbps over a link with 128 MHz bandwidth. Estimate the minimum signal-to-noise ratio (SNR) in dB scale required to support the above data rate.

Question 2

(a) A digital communication system using multilevel signaling transmits one of the 32 possible levels in 0.4 ms. For the above system, calculate the number of bits corresponding to each level, the baud rate and the bit rate.

(b) Random processes and are zero-mean, independent and uncorrelated. The power spectral density (PSD) of x(t) and y(t) are P_x(f) = 2π(f/B) and P_y(f) = 3π(f/B). respectively. Random process z(t) is obtained as z(t) = x(t) + y(t). Therefore, the autocorrelation of z(t) is given by R_z(τ) = R_x(τ) + R_y(τ), where R_x(τ) and R_y(τ) are the autocorrelation functions of random processes x(t) and y(t), respectively.

(i) The PSD of random process z(t) is denoted by P_z(f). Illustrate P_z(f) with a sketch and a mathematical expression. Clearly label the amplitude and frequency values in the sketch.

(ii) Formulate an expression for the autocorrelation function R_z(τ) using the mathematical expression obtained in Question 2(b)(i).

Question 3

(a) Construct the Polar RZ signalling waveform for the binary data sequence 0 1 1 0 1. Assume the bit duration to be 2 ms.

(b) An analog signal is converted into PCM signal that is a binary polar NRZ line code. The PCM quantizer has 64 steps and the overall equivalent system transfer function is of the raised cosine-rolloff type with The PCM bit rate is 12 kbps.

(i) Estimate the maximum bandwidth allowable for the analog signal.

(ii) Test whether a channel that is absolutely bandlimitted to 9 kHz can be used to transmit the above PCM signal without introducing intersymbol interference (ISI).

(c) Signal x(t)  is given by x(t) = s(t)  + n(t), where s(t)  is the desired signal and n(t) is the additive white Gaussian noise whose power spectral density (PSD) is P_n(f) = N₀/2. The signal is illustrated in Figure Q3(c). Assuming that K = N₀/2, construct a matched filter h(t)  to filter the signal x(t) such that the signal-to-noise ratio at the filter output is maximized. Sketch the impulse response of the matched filter. The sampling time should be such that the matched filter is causal.

Figure Q3(c)

Question 4

In a binary communication system, the receiver statistic r0 corresponding to a polar transmitted signal is corrupted by an additive noise (n₀) which has triangular distribution f(n₀) as given below. Figure Q4 graphically illustrates the triangular distribution function f(n₀).

Figure Q4

The mean µ and variance σ₀² for n₀ are given as follows:

µ = (a + b + c) / 3
σ₀² = (a² + b² + c² -ab - ac - bc) / 18.

The transmitted polar signals have values s₁ = +A and s₂ = -A, where A  is a positive real number.

(a) Given that a = -5, b = 5, c = 0 and A = 2.

(i) Sketch the noise probability density function (PDF) f(n₀) and the conditional PDFs for the receiver statistic r₀ , i.e., f(r₀/s₁) and f(r₀/s₂). Clearly label the sketch with appropriate values.

(ii) Estimate the optimum detection threshold V_T for the above system, if the polar signals are equally likely.

(iii) Rate the system performance by computing the bit error rate (BER) corresponding to the optimum detection threshold V_T obtained in Question 4(a)(ii).

(b) Given that a = -5, b = 5, c = 0 and A =5. Sketch the conditional PDFs f(r₀/s₁) and f(r₀/s₂). Use the conditional PDFs and propose a strategy to detect the transmitted polar signals from the receiver statistic r₀. Will there be any detection errors?