ECE 2050 Practice Midterm 1
1.) Find numerical values for each of the following:
For parts (a) and (b) convert each complex number to polar form (express angle in degrees) :
a) (3 pts): —50 + j200
b) (3 pts): —4 — j3
For parts (c–e) convert each complex number to cartesian form.
c) (3pts): 2.5 exp(—jπ/9) (note angle is in radians)
d) (3 pts): (100 / 180o ) + (200 / — 145o)
e) (3 pts): (2 + j14/7) · 200 exp (j35o)
f) (5 pts): Use phasors to find A and θA for
Acos(120πt + θA ) = 50 cos(120πt + 140o) — 250 sin(120πt — 55o)
2.) (20 pts): A continuous time signal, f(t) = 10 cos(240πt + 145o ) is sampled at Ts = △t = 0.022 sec so that the sampled signal is f[n] = 10 cos(ˆ(ω)on + θo ) where ˆ(ω)o is the normalized radial frequency for the principal zone (principal alias) description of the sampled signal. Findˆ(ω)o and θo for the sampled signal. Identify whether the signal is oversampled or undersampled and justify why.
3.) (30 points): In each discrete time system below, the input signal is x[n] and the output signal is y[n]. For each system determine if the system is linear or nonlinear and if the system is time-invariant or not time-invariant. Briefly justify your answers:
a.) y[n] = 2ejπn/2 · x[n]
b.) y[n] = 100x[n] — 40x[n — 10]
c.) y[n] = 100 cos(2πx[n]/100)
4.) (30 points): A LTI system starts at rest (no stored values) and has an impulse response
h[n] = 1.25δ[n] — 0.5δ[n — 1] + δ[n — 3]
and an input signal
x[n] = 10 cos(0.25πn + 15o )(u[n — 1] — u[n — 3))
Find a closed form (analytic expression) for the output of the system, y[n].