Digital Signal Processing I
Midterm Exam (A) Spring 2015
Instructions:
1) The exam is open book. Textbook, notes and any reference materials are allowed.
A hand
calculator is also allowed. No other electronic devices are permitted.
2) Work out your soluti
Digital Signal Processing I
Midterm Exam (B) Spring 2015
Instructions:
1) The exam is open book. Textbook, notes and any reference materials are allowed.
A hand
calculator is also allowed. No other electronic devices are permitted.
2) Work out your soluti
EL6113
Hwk #2 Selected Solutions
3.1
a)
z
z 1/ 2
; ROC
z >1 / 2
b)
z
z 1/ 2
; ROC
z <1 / 2
n
n
1
1
1
c) Note if U (n) = x(n 1) where x(n) = U (n 1) . Therefore, using the
2
2
2
n
1
1
z
1
time shifting theorem we get U (n) z 1 X ( z ) = z 1
; ROC
2
2
70 Discrete-Time Signals and Systems Chap. 2
That is for a zero-mean white-noise input the cross~correlation between input and
output of a linear system is proportional to the impulse resPonse of the system. Similarly,
the power spectrum of a white-noise
1
Exercises in Digital Signal Processing
Ivan W. Selesnick
The Discrete Fourier Transform
1.1 Compute the DFT of the 2-point signal by hand (without a calculator or
computer).
September 1, 2013
x = [20,
Contents
1 The Discrete Fourier Transform
5]
1.2 Com
Introduction to Matlab
Ela Pekalska, Marjolein van der Glas
Pattern Recognition Group, Faculty of Applied Sciences Delft University of Technology January 2002
Send comments to ela@ph.tn.tudelft.nl
Contents
Introduction Preliminaries . . . . . . . . . . .
EL6113
Solution #11
1. By using the bilinear transformation we wish to design an IIR digital low pass filter
with the following specifications:
1 H ( e j ) 1 ,
H ( e j ) ,
0.3
0.5
For this problem let = 0.01 .
Using a butterworth analog prototype, desig
Problems
345
5.14. Determine the group delay for 0 <
W
<
If
for each of the following sequences:
(a)
x1[n]
ponse.
I
n-l
9 - n.,
0,
=
(b)
x2[n]
1 S n S 5,
5 < n S 9,
otherwise.
1)11l-11
= (-
2
(l)lnl .
+ -
2
5.15. Consider the class of discrete-time filter
Chapter 4
238
Sampling of Continuous-Time Signal
4.3. The continuous-time signal
= cos C4000nt)
to obtain the discrete-time signal
xcCt)
is sampled with a sampling period
T
:3 .
x[n] = cos (nn)
(a) Determine a choice for T consistent with this information
EL6113
Hwk #7 Solution
1. A set of samples, f (nT ) , is given below. All samples that are not shown are zero.
f (nT )
1
2
T
T
t
2
Find the unique fuction, f (t ) , whose bandwidth satisfies / T that passes
through all of these samples. Your answer may co
Homework #10
Note: The group delay of a filter is the slope of the phase at 0 . This term is used on
problem 5.19.
Scanned Textbook Problems:
5.15, 5.16, 5.19, 5.20, 5.76
7.5, 7.6, 7.16
Additional Problems:
1. For each of the low pass filter specification
Homework #8b
Scanned Homework Problems:
2.85, 4.8, 4.29.
Additional Problems:
1. Run the matlab program square_pulse.m (posted in this website). This program
computes 128 samples of the DTFT of the sequence shown below and then compares
the result to the
EL6113
Homework #8
Scanned Homework Problems:
2.44, 2.56, 2.79.
Additional Problems:
1. Prove equation (8.2) of lecture 8. To do this,
a) Show that
1
2
e
j ( k n)
1 ,
d = ( k n ) =
0 ,
if k = n
if k n
b) Multipy (1.1) on both sides by e j k and integrate
2004 Polytechnic University
DISCRETE TIME SIGNALS
A discrete time signal, x(n) , is a sequence of numbers, real or complex. In general the signal
can be bi-infinite with the time index, n , running over all integers from to , as
. , x(2), x(1), x(0), x(1)
FIR Linear Phase Filters
Consider the ideal low pass filter frequency response
H (e j )
1
c
c
The impulse response of this filter is
1
h( n) =
2
H (e
j
)e
j n
1
d =
2
c
(1) e j n d
c
Which works out to be
h( n) =
sin c n c sin c n
=
.
n
c n
(5.1)
h(n)
EL6113
1.
Final Exam
Spring 2014
h(n)
x(n)
8
1
01 2 3 4 5 6 7
1
01 2 3 4 5 6 7
n
n
a) Consider the two signals above to be extended periodically with period 8. Let z (n) be the
periodic convolution of h(n) and x(n) . Find and sketch z (n) .
b) Let X (k )
2004 Polytechnic University
MINIMUM PHASE SYSTEMS
A causal, rational system is stable if all its poles are inside the unit circle. Such a system
is called called minimum phase if, in addition, all its zeros are inside the unit circle.
Equivalently it has
2004 Polytechnic University
MAGNITUDE CHARACTERISTICS FOR REAL RATIONAL TRANSFER
FUNCTIONS
In this section we will restrict ourselves to causal, stable systems H (z ) . Recall that such
systems always possess a frequency response, i.e. the unit circle is
DIGITAL PROCESSING OF ANALOG SIGNALS
The C/D converter (Continuous to Discrete)
x(n) = xc (nT )
xc (t )
C/D
T
Let xc (t ) X c ( ) , so the input analog spectrum is X c ( ) . The
output DTFT is found by (with replaced by T )
X (e
j T
)=
x ( n) e
j n T
n
EL6113
Midterm Exam
Spring 2014
1) A linear (but not necessarily time invariant) system has the following input-output
pair (i.e. the given input results in the corresponding output).
y1 (n)
x1 (n)
Linear
system
1
0 1
1
n
1
0 0
1 2 3
0
Linear
system
1
10 Discrete-Time Signals and Systems Chap. 2
That is‘ for a zero-mean white-noise input. the cross—correlation between input and
output of a linear system is proportional to the impulse resPonse of the system. Similarly,
the power spectrum of a white-nois
EL6113
Hwk #1 Selected Solutions
2.1 a)
y (n) T cfw_x(n) g (n) x (n)
with g (n) given.
To check for stability, let the input be bounded with
y (n) T cfw_x(n) g (n) x(n) g (n) x (n) M g (n) .
x( n) M .
Then
We can see that if g (n) is
bounded, say by g (n)
EL6113
Homework #11
Spring 2014
1. By using the bilinear transformation we wish to design an IIR digital filter low pass
filter with the following specifications:
1 H ( e j ) 1 ,
H ( e j ) ,
0.3
0.5
For this problem let = 0.01 .
Using a butterworth anal
EL6113
Hwk #6
1. Show that the set of functions
jn
1
n (t ) =
e
T
2
t
T
;
for 0 t T
is orthonormal.
2. Suppose a function, f (t ) , can be represented in terms of a given orthonormal basis
set cfw_ n (t ) as
f (t ) =
f n n (t ) .
n =1
Now suppose we wan
Homework #9a
Scanned Homework Problems:
4.31, 4.32
Homework #9b
Note 1: In these problems, wherever the authors use the term Discrete Fourier Series
(DFS), you may replace it with Discrete Fourier Transform (DFT). They use the term
DFS when they are viewi
EL6113 - Instructions
1) THE EXAM IS MULTIPLE CHOICE. YOU MUST CIRCLE THE LETTER OF THE CORRECT
ANSWER TO EACH QUESTION ON THE QUESTION SHEET, EXCEPT FOR QUESTION
NUMBER 8, IN WHICH YOU ARE TO WRITE THE CORRECT ANWSERS IN THE TABLE.
Y O U MUST HAND IN THE