ECE 380 Introduction to Communication Systems
Homework Assignment 3 Due: No need to submit
Problem 1. Consider the periodic signal shown in Figure 1.
Figure 1: Signal for Problem 1.
a) Find the power spectral density of the signal.
b) Find the power and the autocorrelation function of the signal. Draw the autocorrelation function
(e.g., using Matlab) and discuss the result.
Problem 2. The unit impulse responses of two linear time-invariant systems are
h1(t) = 400πe−200πtu(t)
h2(t) = 400πe−200πt cos(20, 000πt)u(t).
a) Find the magnitude responses of these systems.
b) Determine the filter type and 3 dB cut-off frequency of the first system h1(t).
c) How about the second system h2(t)?
Problem 3. (Haykin & Moher Problem 2.31 with modifications) A system is a cascade of N identical
RC circuits, each has frequency response Hi(f) = 1/(1 + j2πfRC) as shown in Figure 2.
Figure 2: THe systematic diagram in Problem 3. (a) Determine the overall amplitude response of the system.
(b) Assume that τ0 = RC = T/(2π N). Show that as N approaches infinity, the amplitude response of the system approaches the Gaussian function e−f 2 T 2 /2 . Note: limx→∞ (1 + x1 )x = e.
Problem 4. a) The signal g(t) = e−|t| is input to a linear time-invariant system whose frequency re- sponse is: H(f) = 1 + (2πf)2. Calculate the energy of the output.
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ECE 380 Introduction to Communication Systems
b) How about the energy of the output when the signal is input to a linear time-invariant system whose frequency response is:
−j1+(2πf)2 0