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 fs = (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 fu, then x(t) can be recovered from its ideally sampled version ms(t) by band pass filtering if fs = 2fu/k, where k is the largest integer not exceeding fu/B. All higher sampling rates are not necessarily usable unless they exceed 2fu. 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 ms(t) when fs = 25,45,50 kHz. Indicate if and how the analog signal can be recovered from its samples.
Q3. A binary channel with bit rate Rb = 36 kbps is available for PCM voice transmission. Find appropriate values for the sampling rate fs, 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(2πfct), where we assume that m(t) ≥ 0 for all t. Assuming that the highest frequency content of m(t) is much less than fc, 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.