ECE 132A: Introduction to Communication Systems
Homework #2
Winter 2025
Due: Monday, January 27, 2025 at 11:59 PM via Gradescope
1. (10 pts) Let z(t) be a complex-valued baseband signal with two-sided baseband bandwidth B, and let fc ≫ B. Show that the energy of is equal to that of z(t).
2. (10 pts) Let x(t) be any real-valued passband signal centered at fc, occupying a passband bandwidth B ≪ fc. Rigorously show that if then x(t) can be written as
3. (5 pts) Prove that the bandwidth of a signal output by an LTI system cannot be greater than that of its input signal.
4. (20 pts) Consider conventional AM transmission of the following message signal at a carrier frequency of 1 MHz.
m(t) = cos(4000πt) − 3 · sin(6000πt)
(a) Find and sketch/plot the spectrum of the transmit signal xAM(t), with appropriate labels on the horizontal and vertical axes. What is the passband bandwidth of xAM(t)?
(b) What is the best-case transmit power efficiency, under the assumption an ideal envelope detector can be used to perfectly recover the message?
(c) Suppose the received conventional AM signal (a time-varying voltage) is a perfect copy of the transmitted signal xAM(t) and is passed through an ideal full-wave rectifier followed by a simple RC circuit in an attempt to recover the message m(t). Using MATLAB or Python, plot the recovered message ˆm(t), i.e., the output voltage of the RC circuit, as a function of time t, comparing it against the original message m(t). What values of R and C yield satisfactory demodulation? Hint: For plotting, use a numerical approximation of the differential equation describing the output voltage at time t.
5. (20 pts) Suppose we construct the transmit signal
where m1(t) and m2(t) are independent, real-valued baseband signals with zero mean, each of which occupies a two-sided baseband bandwidth much less than fc. This signal has a construction different from what we have talked about in class but resembles some sort of combination of two DSB signals.
(a) What is x(f) in terms of m1(f) and m2(f)?
(b) Is x(f) conjugate symmetric about f = 0? Is it locally conjugate symmetric about fc?
(c) Mathematically show what happens if we apply a DSB demodulator to x(t).
(d) Can we recover both m1(t) and m2(t) from x(t)? If so, how? Draw a block diagram of your technique and mathematically show that it works.
(e) What takeaways might you draw from (d)?
6. (10 pts) Prove that the definition of the Hilbert transform. of a signal m(t) presented in class
is equivalent to the convolution
7. (10 pts) Below, I have given you part of the spectra for a real-valued signal m(t). Fill in the rest, including all relevant labels on the horizontal/vertical axes, with ˘m(t) the Hilbert transform. of m(t).
8. (10 pts) Find the Hilbert transform. of m(t) = 20 · sinc(10t).
9. (25 pts) Good news! The US Federal Communications Commission (FCC) has just gifted me exclusive rights to the electromagnetic spectrum from 30 kHz to 80 kHz. After auctioning off licenses to this precious spectrum, I have granted four parties rights to their own exclusive channel within this 50 kHz band. Each of these four parties will operate their own radio station to broadcast music, news, talk shows, etc., but each station must abide by a few rules.
• Each station is assigned to broadcast within a channel that is 8 kHz wide.
• The carrier frequencies of neighboring channels will differ by 12 kHz.
• Each station must broadcast conventional AM transmissions.
All four of these stations are now on the air broadcasting, and I have directly sampled the signal received by an antenna tuned to this 50 kHz band. Your task is to engineer a software-based receiver in MATLAB/Python/etc. Create a script. to process the file radio.mat posted under Week 3 of the Modules tab on Bruin Learn. This file contains two fields:
• y: samples of the received signal
• fs: the rate at which the received signal was sampled (in units of samples per second).
(a) Using the fast Fourier transform. (FFT), plot the magnitude of the (discrete) Fourier transform.|y(f)|2 of the received signal with the vertical axis in linear units and the horizontal axis in units of kHz.
(b) Are you familiar with decibels (dB)? It is a common way engineers compress very large and very small numbers into a more manageable range, and it also makes calculating products and ratios much easier by transforming multiplication into addition via the logarithm. Feel free to read more about dB on your own; we may cover it more later in this course. Repeat (a) with the vertical axis in dB. This can be accomplished by simply plotting f vs. where
[X]dB = 10 log10(X).
(c) What carrier frequencies have each of the four channels been assigned?
(d) Sketch a simple “band plan” of this spectrum, with frequency on the horizontal axis and labels showing where each of the four assigned channels, their bandwidth, and their carrier frequencies.
(e) Draw a receiver block diagram (similar to those drawn in class) with an antenna, followed by the necessary components/processes to recover any one station’s broadcast message.
(f) Process and demodulate the received signal in order to “tune in” to a particular channel. Following successful demodulation, listen to the message signal (e.g., using the soundsc(m,fs) function in MATLAB). What is being played on each of the four channels?
(g) What might the reason be for a gap of 12 kHz between channels, considering the channels are only 8 kHz wide?