Problem 1
Note: The solution to (f) contains error. The right solution is
Problem 2
Note:
Let Y (ejw)= dX(ejw)/dw. Then by the differentiation in frequency property, y[n] =
jnx[n].




By Parsevals theorem, 

(not
) or 6897.43.
Problem 3
Note: G
Homework # 2
Instructions:
You may solve the problems according to the following schedule:
1. After Lecture #3: Problems 1 4
2. After Lecture #4: Problems 5 8
3. After Lecture #5: Problems 9 12
1. Determine the DTFT of each of the following sequences:
(a
Name:
Section:
Laboratory Exercise 2
DISCRETETIME SYSTEMS: TIMEDOMAIN REPRESENTATION
2.1
SIMULATION OF DISCRETETIME SYSTEMS
Project 2.1
The Moving Average System
A copy of Program P2_1 is given below:
% Program P2_1
% Simulation of an Mpoint Moving Av
Name:
Section:
Laboratory Exercise 1
DISCRETETIME SIGNALS: TIMEDOMAIN REPRESENTATION
1.1
GENERATION OF SEQUENCES
Project 1.1
Unit sample and unit step sequences
A copy of Program P1_1 is given below.
% Program P1_1
% Generation of a Unit Sample Sequence
Problem 1
Correction: 3rd to last sample of (g) should be 28 (not 2.8).
Problem 2
Problem 3
Correction:
In (c), the average power is
1 K 2
1 K ( K 1)(2 K 1)
K ( K 1)
lim
n Klim 2K 1
6
6
K 2 K 1 n1
K
Px3 lim
In (e) the cos function in S should be change
Matlab Project 2
Instructions:
1. Download the lab manual titled Digital Signal Processing Laboratory Using Matlab from
http:/highered.mcgrawhill.com/sites/0072865466/student_view0/lab_manual.html
2. All Matlab commands used in the project are explained
Discrete Fourier Transform
The discrete Fourier transform (DFT) of a lengthN
sequence x[n], 0 n N1, is obtained by uniformly
sampling its DTFT X(ej) on the axis at k=2k /N
X [ k ] X (e
j
)
2 k/ N
N 1
x[n] e j 2 n k / N ,
0 k N 1
n0
Let WN=ej2 /N. Th
Matlab Project 1
Instructions:
1. Download the lab manual titled Digital Signal Processing Laboratory Using Matlab from
http:/highered.mcgrawhill.com/sites/0072865466/student_view0/lab_manual.html
2. All Matlab commands used in the project are explained
Digital Filter Structures
1
The convolution sum description of an LTI system can
be used, in principle, to implement the system
For an IIR finitedimensional system this approach is
impractical as the impulse response is of infinite length
However, a d
1
Simple FIR Digital Filters
Lowpass FIR Digital Filters
The simplest lowpass FIR digital filter is the 2point
movingaverage filter given by
z +1
H 0 ( z ) = 1 (1 + z 1 ) =
2
2z
The above transfer function has a zero at z = 1 and a
pole at z = 0
The
Analog Lowpass Filter Specifications
1
Typical magnitude response H a ( j) of an analog
lowpass filter may be given as indicated below
p : passband edge frequency
s : stopband edge frequency
p : passband peak ripple size
s : stopband peak ripple size
Homework # 1
Instructions:
You may solve the problems according to the following schedule:
1. After Lecture #1: Problems 1 4
2. After Lecture #2: Problems 5 8
1. Consider the following sequences
x[n] = 4 5 1 2 3 0 2 , 3 n 3
y[n] = 6 3 1 0 8 7 2 , 1 n 5
w
Spectral Transformations of IIR Digital Filters
1
Objective  Transform a given lowpass transfer function
GL (z ) to another digital transfer function GD (z ) that
could be a lowpass, highpass, bandpass or bandstop filter
Let z 1 denote the unit delay i
Ideal Filters
1
Filters are often classified according to their magnitude
response. Consider 4 types of ideal filters
Lowpass
Highpass
Bandpass
Bandstop
The frequencies c, c1, and c2 are cutoff frequencies
These ideal filters have a magnitude response
DiscreteTime Fourier Transform
1
Definition  The discretetime Fourier transform
(DTFT) X (e j ) of a sequence x[n] is given by
X (e
j
)=
x[n]e j n
n =
The DTFT is often called the Fourier spectrum
In general, X (e j ) is a complex function of the
1
Digital Processing of ContinuousTime Signals
Antialiasing
filter
S/H
A/D
Digital
processor
D/A
Reconstruction
filter
Digital processing of a continuoustime signal involves
the following basic steps:
Conversion of the continuoustime signal into a
di
zTransform
1
For sequence g[n], its ztransform G(z) is defined as
G( z )
g [n] z n
n
where z = Re(z) + jIm(z) is a complex variable
If we let z = rej, then the ztransform reduces to
G(r e
j
g [ n ] r n e j n
)
n
Hence, the ztransform is the DT
The Sampling Process
1
Often, a discretetime sequence x[n] is obtained by
uniformly sampling a continuoustime signal xa (t ) as
indicated below
x[n] = xa (t ) t =nT = xa (nT ),
n = K, 2, 1, 0,1, 2,K
Sampling frequency (in Hz):
FT = 1 / T
Sampling ang
Introduction
DSP is concerned with the
Representation
Transformation
manipulation
of digital signals and information they contain.
Advantages of DSP over analog processing:
Less sensitive to noise
Low manufacture cost
High accuracy (by increasing wor