ECE 132A: Introduction to Communication Systems
Homework #3
Winter 2025
Due: Tuesday, February 4, 2025 at 11:59 PM via Gradescope
1. (25 pts) Suppose the message signal
m(t) = 10 · cos(16πt)
is transmitted using FM modulation with κFM = 10, producing a transmit signal
(a) What is the two-sided bandwidth of m(t)?
(b) If xFM(t) is passed through an ideal band-pass filter centered at 2 kHz with a (total) passband bandwidth of 62 Hz, what is the power of the signal at the filter output? What percentage of the total signal power of xFM(t) made it through the filter?
(c) What should the passband bandwidth of the band-pass filter be to pass 90% of the total power of xFM(t)?
(d) Repeat (c) if κFM doubles.
2. (15 pts) A sinusoidal message m(t) with a frequency of fm = 15 kHz is transmitted using FM modulation with a so-called modulation index of βFM = Am · κFM/fm = 2.
(a) Find the approximate bandwidth of xFM(t) using Carson’s rule.
(b) What percentage of the total power of xFM(t) lies within the Carson’s rule bandwidth?
(c) Based on (b), would you say the Carson’s rule bandwidth is an adequate estimate of the signal bandwidth in this particular case?
3. (10 pts) Prove that the k-th order Bessel function of the first kind Jk(β) exhibits the property
J−k(β) = (−1)k
· Jk(β).
4. (20 pts) The FCC has just gifted us the band of spectrum from 200 MHz to 201 MHz to deploy a new FM radio broadcast system. Each channel is B Hz wide, and there is a guard band 20 kHz wide between neighboring channels. There are no other guard bands. All message signals have a one-sided bandwidth of at most 10 kHz and a magnitude of at most 2. The FM transmit signal is generated with a maximum frequency deviation of 50 kHz.
(a) Based on Carson’s rule, what should the bandwidth B of each channel be? Based on this, what is the maximum number of channels our band can support?
(b) Henceforth, consider a wideband FM bandwidth analysis, approximating the message signal as a sinusoid at 10 kHz. What is the smallest channel bandwidth B that contains at least 95% of the total modulated signal power?
(c) What is the largest frequency deviation such that 75% of the total signal power is contained in a channel whose passband bandwidth is B = 130 kHz? What percentage of the total signal power is contained in the adjacent guard bands?
5. (15 pts) Let x(t) be defined as follows, where fc = 10 kHz.
x(t) = 1[0,1)(t) · cos(2π(fc + 1000)t) + 1[1,2)(t) · cos(2π(fc − 1000)t) + 1[2,3)(t) · cos(2π(fc + 2000)t)
(a) Find and plot/sketch the magnitude spectrum of x(t), i.e., |x(f)|.
(b) Plot/sketch the instantaneous frequency of x(t) as a function of time t.
(c) What is the bandwidth of x(t)?
6. (35 pts) Let X be a discrete random variable taking +1 with probability 0.4 or −1 with probability 0.6. Let N ∼ N (0, 1) be an independent Gaussian random variable with mean zero and variance one. Let Y be the random variable defined as
Y =
√ρX + N,
where ρ > 0 is a constant. Let ˆX = sgn(Y ).
(a) Roughly sketch the PDF of N for ρ = 1.
(b) Roughly sketch the PDF of Y if X = +1 and that if X = −1, for ρ = 1.
(c) Repeat (a) and (b) for ρ = 10. What do you observe as ρ is increased?
(d) Let ρ = 1 for the remainder of this problem. What is the probability that Y = 1 or Y = −1?
(e) What is the probability that Y ≥ 2? Sketch the PDF of Y and indicate what this probability corresponds to on your PDF.
(f) What is the probability that |Y| ≤ 1? Sketch the PDF of Y and indicate what this probability corresponds to on your PDF.
(g) What is the probability that ˆX = −1? Sketch the PDF of Y and indicate what this probability corresponds to on your PDF.
(h) What is the probability that ˆXˆ = X if X = +1? Sketch the PDF of Y and indicate what this probability corresponds to on your PDF.
(i) What is the probability that ˆX = X if X = −1? Sketch the PDF of Y and indicate what this probability corresponds to on your PDF.
(j) What is the probability that ˆX = X?
(k) What is E[Y]? What is E [ˆX]?
(l) Suppose we redefine Y as
Y =
√ρX + N1 + N2,
where N1 ∼ N (0, 1) and N2 ∼ N (0, 2) are independent. What is the probability that ˆX = X?