University of California, San Diego
Spring 2016
ECE 45 Homework 1
Problem 1.1 Find the magnitude and phase of the following complex number:
(1 − j) (2e−jπ/3 ) (sin(1)) (j 2 )
(2 + 2j) (−j cos(1)) (ejπ )
Problem 1.2 Let X = 2 + j + 2e−j2π/3 − ejπ/2 .
(a) Find the real portion of X.
(b) Find the phase of the complex conjugate of X.
(c) Let Y = −2 + 2j. Plot X/Y on the complex plane (real and imaginary axes).
Problem 1.3 Represent the following sinusoidal functions as phasors.
(a) f1 (t) = 3 cos(4t) − 4 sin(4t)
(b) f2 (t) = 2(cos(ωt) + cos(ωt + π/4))
(c) f3 (t) = cos2 (t) − sin2 (t)
Problem 1.4 Find the voltage vr (t) in the circuit below, when
(a) vin (t) = 1/3
L
(b) vin (t) = sin(t)
(c) vin (t) = 1 + 2 sin(t − π).
where R = 1Ω, C = 1/2F , L = 2H.
vin(t)
+
−
C
R
+
vR(t)
−
Problem 1.5 Recall the Norton Equivalent of an RLC circuit is a current source in parallel with a
resistor and a capacitor or an inductor.
Find the value of C for which the Norton Equivalent is a current source in parallel with only a resistor
(i.e. the Thevenin Impedance is purely real). What is isc (t) and Rth is such a case?
R
+
v(t) +
−
C
L
i(t)
+
vo (t)
isc(t)
R th
−
vo (t)
−
where v(t) = cos(4t + π/3), i(t) = sin(4t + 5π/6), R = 1Ω, and L = 1/4H.
Problem 1.6 Suppose H(ω) is the transfer function of a linear system and H(ω) = 1 when |ω| < π
and H(ω) = 0 otherwise. If the input to the system is
∞
X
cos 3π
kt + πk
4
x(t) =
k
k=1
then what is the output y(t)?
Problem 1.7 Are the following steady-state input-output pairs consistent with the properties of LTI
systems?
(a) cos(2t) → H(ω) → 99 sin(2t − e)
(e) 4 → H(ω) → cos(3t)
(b) cos(4t) → H(ω) → 1 + 4 cos(4t)
(f) sin(πt) → H(ω) → cos(πt) + sin(πt)
(c) 4 → H(ω) → −8
(g) sin(πt) → H(ω) → sin2 (πt)
(d) 4 → H(ω) → 8j
(h) 0 → H(ω) → 5.
Problem 1.8
For each k = 1, 2, suppose yk (t) is the output when xk (t) is the input to an LTI system.

0≤t<1
 1
1 0≤t<2
−1 1 ≤ t < 2
y1 (t) =
y2 (t) =
0 else

0
else.
Determine a possible input to the LTI system (in terms of x1 (t) and x2 (t)) for the following outputs:
−1 2 ≤ t < 4
1 0≤t<1
1 t≥0
(a) z1 (t) =
(c) z3 (t) =
(e) z5 (t) =
0
else
0 else
0 else
1 1≤t<5
1 3≤t<4
(b) z2 (t) =
(d) z4 (t) =
(f) z6 (t) = 0.
0 else
0 else
Problem 1.9
An LTI system with input x(t) and output y(t) is given by the differential equation
3
d4 y(t)
d3 x(t)
d2 y(t)
−
2
+
y(t)
=
2x(t)
+
.
dt4
dt
dt
Find the output when x(t) = 1 + cos(t) + cos(2t).
Problem 1.10 Recall a low-pass filter only allows frequencies below some threshold, a high-pass
filter only allows frequencies above some threshold, a band-pass filter only allows frequencies in some
range, and a band-reject filter only allows frequencies outside of some range.
What type of filter are the following circuits? Justify your answer by finding the magnitude of the
transfer function of each circuit.
(a)
i in(t)
(b)
C
R
io(t)
i in(t)
io(t)
C
R
Problem 1.11 For each of the circuits in the previous problem, let RC = 100.
If the circuit is a low-pass filter, find the frequency ωc such that |H(ω)| < 0.01 for all ω > ωc .
If the circuit is a high-pass filter, find the frequency ωc such that |H(ω)| < 0.01 for all ω < ωc .
If RC = β for some β > 0, what is ωc in terms of β?