ECE132A: Introduction to Communication Systems
Midterm Exam
Wednesday, February 7, 2024
1. True/false questions. Write T or F on the line preceding each question. One point for the correct answer and one for the correct explanation.
(a) (2 points) If z is the product of two zero-mean independent random variables x and y, then Var(z) = Var(x)Var(y).
(b) (2 points) If x(t) is a stationary Gaussian random process with power spectral density Sx (f) = rect(f), then x(1) and x(2) are dependent random variables.
(c) (2 points) If x(t) is a stationary Gaussian random process with autocorrelation function Rx(τ) = (πτ)/sin(πτ), then y(t) = a + x(t), with a a deterministic constant, is a WSS process.
(d) (2 points) Let x(t) be a wide-sense stationary random process with autocorrelation Rx(τ). The autocorrelation of its Hilbert transform is different from the autocorrelation of x(t).
(e) (2 points) Sx (f) = tri(f) - 2/1 is a valid power spectral density of a wide-sense stationary process.
2. Stationarity. Let x(t) = acos(2πf0t + θ), where a is a random variable taking values 0 and 1 with equal probability, θ is a uniform random variable taking values in [0, 2π); a and θ are statistically independent.
(a) (6 points) Find the mean function, m(t), and the autocorrelation function, Rx (t + τ, t), of x(t).
(b) (2 points) Is x(t) a wide-sense stationary process, a cyclostationary process, or neither? Explain.
3. Constellation. Define the following unit-energy pulse:
Consider the following signal set:
where 0 < t < 4.
(a) (1 point) Plot the four waveforms as functions of time, t.
(b) (6 points) Use the Gram-Schmidt orthogonalization procedure to find a basis for this signal set.
(c) (2 points) Compute the coordinates of the four signals using the basis you found.
(d) (1 point) What is the dimensionality of this constellation?
(e) (2 points) Find the energy per signal and average energy of the constellation, assuming the signals in the set are equiprobable.
(f) (2 points) Compute the distances and angles between each pair of signals in the set. (Recall that the inner product between vectors a and b is related to the angle between them as follows: a · b = ⅡaⅡⅡbⅡcos(θ).)
4. Hilbert transform. Consider a signal x(t) with Fourier transform X(f). The Hilbert transform.
of x(t), and its Fourier transform, can be expressed as follows:
in the time and frequency domains, respectively, with
(a) (4 points) Show that the Hilbert transform of ˆ(x)(t) is equal to —x(t).
(b) (4 points) Using the previous part, show that