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 fundam
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.
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
Solution to Assignment 3
EE 216.3
20152016 Term II
Instructor: N. Chowdhury
Note: Red numbers indicate marks allotted to the questions
Marker: Farah Deeba
1.
a) Given that, x=8.5, the Lagrange polyno
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
Solution to Assignment 7
EE 216.3
20152016 Term II
Instructor: N. Chowdhury
Marker: Farah Deeba
1.
[2+1.5+1.5]
(a):
Number of class intervals = 1+3.3log(60) = 6.87
Class width = (5719)/6.87 = 5.53 r
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
Solution to Assignment 1
EE 216.3
20152016 Term II
Instructor: N. Chowdhury
Note: Red numbers indicate marks allotted to the questions
T/A: Farah Deeba
1.
(7)
The hand calculation result is the same
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
Solution to Assignment 10
EE 216.3
20152016 Term II
Instructor: N. Chowdhury
Marker: Farah Deeba
Note: 1. Red numbers indicate marks allotted to the questions
2. You lose marks if you didnt show all
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
Solution to Assignment 9
EE 216.3
20152016 Term II
Instructor: N. Chowdhury
Marker: Farah Deeba
Note: 1. Blue numbers indicate marks allotted to the questions
2. You lose marks if you didnt show all
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
Solution to Assignment 2
EE 216
2015  2016 Term II
Instructor: N. Chowdhury
T.A.: Farah Deeba
Note: Blue numbers indicate marks allotted to the questions
1.
(10)
a) GaussJordan Elimination:
(3)
8.47
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
Solution to Assignment 5
EE 216.3
201520016 Term II
Instructor: N. Chowdhury
Marker: Farah Deeba
1.
a). Eulers method
f ( x, y ) y x 2 1
y n 1 y n hf ( x n , y n )
[2]
x0 1.0,
y 0 0.6
x1 x0 h 1.0 0.1
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
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
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
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
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
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 )
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
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 s
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
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 <
Mathematical Methods for Electrical and Computer Engineering
MATH 267

Fall 2011
Solution to Assignment 4
EE 216.3
20152016 Term II
Instructor: N. Chowdhury
Note: Blue numbers indicate marks allotted to the questions
Marker: Farah Deeba
1.
[5]
Applying the Forward Difference Form