Lahore University of Management Sciences SBA School of Science and Engineering, Department of Electrical Engineering EE574 - Discrete-Time Designs for Wireless Communications Spring 2015 Homework 1 - Review of Discrete-Time Signals and Systems
Date Assigned: February 9th , 2015
Date Due: February 16th , 2015 (In class)
Note: In order to get the maximum benefit, try to solve the problems without any consultations. The thought process behind searching for the correct approach all on your own will be invaluable. Feel free to talk to me after you have some insights. 1. (Oppenheim and Schafer, 4.7.) Consider a signal sc (t) transmitted over a communication channel. Under a simplified model of a multipath channel, the output xc (t) is related to the input by xc (t) = sc (t) + αsc (t − τ ). The received signal is sampled with a sampling period T to obtain the sequence x[n] = xc (nT ). Assume that sc (t) is band-limited such that Sc (jΩ) = 0 for |Ω| ≥ Tπ . (a) Determine the Fourier transform of xc (t) and the Fourier transform of x[n] in terms of Sc (jΩ). (b) Suppose we wish to simulate the multipath system in a computer, which can only be done with discrete-time signal processing. In particular, given the samples s[n] = sc (nT ) of the channel input, our objective is to design a discrete-time LTI system with frequency response H(ejw ) such that with s[n] as the system input, the system output r[n] should be equivalent to the samples of the channel output, i.e., r[n] = xc (nT ). Determine H(ejw ) that satisfies this condition. (c) Determine the impulse response h[n] of the system for (i) τ = T , (ii) τ = T /2. 2. (Oppenheim and Schafer, 4.22.) A complex-valued continuous-time signal xc (t) is pass-band, i.e., Xc (jΩ) = 0 for Ω > ΩH and Ω < OmegaL . (a) What is the lowest sampling frequency that can be used without incurring any aliasing distortion, i.e., so that xc (t) can be recovered from x[n]? (b) Draw the block diagram of a system that can be used to recover xc (t) from xc [n] if the sampling rate is greater than or equal to the rate determined in the previous part. Assume that (complex) ideal filters are available. 3. (Discrete-time AM Demodulator) Consider a band-limited audio signal s(t) that has the highest frequency of 4 kHz. The audio signal is modulated to a carrier of frequency 100 kHz and transmitted. You are required to build a demodulator in order to listen to the transmitted audio signal. However, for doing so, you are only required to use an ideal ADC and a DAC (ideal in the sense of having infinite resolution) that are parametrized by their sampling frequencies fADC and fDAC , respectively. Ignoring the effect of any channel distortion, is it possible to reconstruct a copy of s(t) using just the ADC and DAC block? If your answer is yes, explain with proper justifications, what minimum values of fADC and fDAC will guarantee recovery of s(t)? If your answer is no, explain your reasoning.
4. (Discrete-time Differentiator) Consider a band-limited signal xc (t) with Xc (jΩ) = 0 for |Ω| > ΩN . Let x[n] be the samples of xc (t) sampled at above the Nyquist rate. Suppose we wish to emulate continuous-time differentiation in the discrete-time. In particular, we wish to design a discrete-time LTI system with impulse response h[n] such d that y[n] = x[n] ∗ h[n] corresponds to samples of yc (t) = dt xc (t), i.e., y[n] = yc (nT ). Determine h[n] that satisfies this property. 5. (Discrete-time FM Demodulator) Problem 3.40 of textbook.
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