ASSIGNMENT - INTRODUCTION TO COMMUNICATIONS SYSTEMS

Q1. Consider a signal x(t) = cos(ω₀t), where ω₀ = 2πf₀. Sketch the effect of under sampling x(t) in both the frequency and time domains for a sampling rate f_s = (3/2)f₀.

Q2. The band pass sampling theorem states that if a band pass signal has a spectrum with bandwidth B (in Hz) and an upper frequency limit f_u, then x(t) can be recovered from its ideally sampled version m_s(t) by band pass filtering if f_s = 2f_u/k, where k is the largest integer not exceeding f_u/B. All higher sampling rates are not necessarily usable unless they exceed 2f_u. Consider the band pass signal m(t) with the spectrum shown below. Check the validity of the band pass sampling theorem by sketching the spectrum of the ideally sampled signal m_s(t) when f_s = 25,45,50 kHz. Indicate if and how the analog signal can be recovered from its samples.

Q3. A binary channel with bit rate R_b = 36 kbps is available for PCM voice transmission. Find appropriate values for the sampling rate f_s, the quantizing level L, and the binary digits n, assuming that the bandwidth of the voice signal is 3.2 kHz.

Q4. In a binary PCM system, the output signal-to-quantizing noise ratio is to be held to a minimum of 40 dB. Determine the number of required levels and find the actual corresponding output signal-to-quantizing noise ratio.

Q5. Consider a full wave rectified AM signal s(t) = m(t)cos(2πf_ct), where we assume that m(t) ≥ 0 for all t. Assuming that the highest frequency content of m(t) is much less than f_c, can s(t) be considered the approximate result of one of the pulse modulation methods that we have studied applied to m(t)? If so, which one? Explain in detail.

Q6. This is a question in Probability & Random Variables. Let Y = aX + b, where X and Y are random variables and a and b are constants. Assume that the random variable X is Gaussian with a mean value μ and a variance σ². How is the random variable Y distributed and what is its mean and variance? Write its probability density function.