Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
Math 267 Homework Assignment 1 SOLUTIONS
1. Solve the initial value problems and sketch the solutions:
(i)
y 6y + 9y = 0,
y (0) = 0,
y (0) = 2.
(ii)
y + 4y = 3 sin 2t,
y (0) = 2,
y (0) = 1.
SOLUTION.
These problems should be familiar from your previous co
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
MATH 267 Homework Assignment 6 Solutions
1. (Frequency Shifting) Show that if g (t) = ei0 t f (t) then g ( ) = f ( 0 ).
Solution. By denition
g (t)eit dt =
g ( ) =
ei0 t f (t)eit dt =
f (t)ei(0 )t dt = f ( 0 )
2. Find the Fourier transform of the followin
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
MATH 267 Homework Assignment 5 Solutions
1. Consider the RL circuit
x(t)
y (t)
1H
1
A current source produces an input current x(t) and the system output is considered to be the current,
y (t), owing through the inductor.
(a) Find the dierential equation
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
MATH 267 Homework Assignment 4 Solutions
1. (a) Let f (t) = t for 2 t 2 and extend f to be periodic with period 4. Write down the Fourier
series for f in complex exponential form.
(b) Let g (t) = t for 2 t 2 and extend g to be periodic with period 4. Wr
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
MATH 267 Homework Assignment 2 Solutions
1. Find the Fourier series and sketch three periods of the graph of the following function, which is assumed
to have period 2 :
x + if x 0
f (x) =
if 0 < x <
Solution 1 (using the real form). The period 2 = 2 , so
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
MATH 267 Homework Assignment 3 Solutions
1. (a) Sketch the graph of two periods of the odd periodic function F o (x) dened on < x < , with
period 2, which extends the function f (x) that is dened on 0 x 1,
f (x) =
if 0 x < 0.5
x
0.5 if 0.5 x 1
(b) Find th
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
MATH 267 Homework Assignment 7 Solutions
1. Consider the circuit:
+
R
+
x(t)
L y (t)
C
The input is a timevarying voltage x(t) and the output is the voltage y (t) measured across L (or C ).
(a) The signals described above are related by a constantcoecien
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
MATH 267 Homework Assignment 8 Solutions
1. Suppose that the input and the output of a system are related by the dierential equation
y y 2y = x(t).
(a) Find the impulse response function H (t) of the system.
(b) Write the response y (t) of this system for
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
Math 267 Midterm 4 Solutions
1. All parts have the same impulse response function.
t
1
From the formula sheet, the Fourier transform of t sin 2 is rect( ).
1
By scaling, the Fourier transform of (1 ) sin t is rect , for any > 0. Call this ().
t
2
1
1
Appl
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
Math 267 Midterm Solutions
1. The given function has period 2 so that
g (t) =
= 1 and
1
cn eint
with
cn =
1
2
n=
For n = 0, since t is an odd function, c0 =
cn =
2. This time
=
1
2
1
1
t dt = 0. For n = 0, eint = (1)n for both t = 1 so that
int
t
in e
1
2
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
MATH 267 Homework Assignment 11 Solutions
1. The output y [n] generated by a system for the input x[n] is
y [n] =
g [n 3m] x[m]
m=
where
g [n] = u[n] u[n 6] =
(a)
(b)
(c)
(d)
0 if n < 0
1 if 0 n < 6
0 if 6 n
Determine y [n] when x[n] = [n].
Determine y [n
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
MATH 267 Homework Assignment 10 Solutions
1. Compute, using the dening sum x( ) = x[n]ein directly, the discretetime Fourier transform
n=
of each of the following functions. Also determine the fundamental period (the smallest positive period)
of the Four