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ECE113, Fall 2022

Midterm

Digital Signal Processing

1. Problem 1 (20 points)

A discrete-time system is represented by a difference equation:

n y[n] = 6(x[n − 3] + x[2n + 2] + x[ 3 ])

(a) The input of the system is generated by sampling an analog signal x(t) = 2 cos(t) using two different sampling periods T₁ = 1/6 seconds and T₂ = π/6 seconds. If we want the discrete-time signal x[n] to be periodic, which of the two sampling periods would you use? Why?

For the chosen sampling period what would be the fundamental period of x[n]?

(b) If x[n] is periodic, would the output of the system also be periodic? If yes, what would be the fundamental period? If no, show your reasoning.

2. Problem 2 (20 points)

Now we have the following sequences:

2π π

x₁[n] = cos( 3 n + 6 )

π x₂[n] = 2 sin( 4 n)

x₃[n] = x₁[n] + x₂[n]

(a) What is the period for each signal?

(b) What is the energy for x₃[n]?

Hint: sin² x = ¹ (1 − cos 2x), cos² x = ¹ (1 + cos 2x), cos x sin y = ¹ (sin(x + y) − sin(x − y))

3. Problem 3 (20 points)

Assume x[n] = 0 for n < 0:

(a) What is the even part x_e[n] in terms of x[n]?

(b) Show that x[n] can be expressed in terms of unit step function u[n] and its even part

x_e[n] alone.

4. Problem 4 (20 points)

A system is described by the block diagram shown in the figure below with x[n] denoting the input sequence, y[n] denoting the output sequence, w[n] = e^jω0n, and v[n] = e⁻^jω0n. h[n] is known to be a linear, time-invariant, stable, and causal system.