Required Homework
1) Attach a cover sheet with your name, lab section, and homework assignment number.
2) Put your solution on US letter (8.5 x 11) plain white printer paper, engineering paper, or
ruled paper. Do not use A4 or legal size paper, torn sheet
SYSTEM DIAGRAMS OR REALIZATIONS
N
L
k =1
k =0
y[n] = ak y[n k ] + bk x[n k ]
For any causal, stable filter with real coefficients, all of the poles of the system transfer
function H ( z ) must lie inside the unit circle. The zeros may be located anywhere.
CH 8 Digital Signal Spectra
Spectrum of a signal is a detailed description of the frequency components the
signal contains.
The smooth transition in a signal comes from its low frequency elements.
Sharp edges and rapid changes come from its high frequency
Time Domain
Frequency Domain
where : discrete frequency : continuous frequency
Time domain
Fourier Transform pair (FT)
x ( t ) = X ( f ) e j 2 ft df
Fourier Series pair (FS)
x (t ) =
n =
1
Discrete time Fourier Transform
x n =
(DTFT)
2
[ ]
X ke
j 2 kt
T
F
Ch 11 DFT and FFT Processing
The DFT (Discrete Fourier Transform) identifies the frequency contents in a
signal from a set of the signals samples.
Most efficient implementation of DFT is FFT (Fast Fourier Transform)
The definition of DTFT was given
X () =
1
Circular convolution
The N -point circular convolution of two signals x1[n] and x2 [n] denoted by
x1[n] N x2 [n] is defined by the following:
N 1
(
)(
x1[n] N x2[n] = x1 n k mod N x2 k
k =0
)
(1)
N 1
(
)(
= x1 k x2 n k mod N
k =0
)
where ( x1 n k m
EE153
Midterm I
Last:
Name and ID: First:
Student ID #:
2009-10-01
* Make sure that your test has total 4 questions
Please write your initials on the right side of the each page number and if you use the
space on the back page then initial on the bottom r
EE153 [Fall 2009]
Last Name:
First Name:
St ID:
Please write your name on the back page too.
Quiz #1
2009 09 03
A function x ( t ) is given as
t 3
x ( t ) = 2
4
a) Plot x ( t )
b) Find the frequency response of x ( t ) and show your answer with sinc f
EE153
Midterm III
Last:
Name and ID: First:
Student ID #:
2010-05-06
* Make sure that your test has total 3 questions
Please write your initials on the right side of the each page number and if you use the
space on the back page then initial on the botto
EE153 [Sp 2010]
Last Name:
First Name:
St ID:
Please write your name on the back page too.
Quiz #1
2010 02 11
1. Fill the space with the proper equations
Time domain
Fourier Transform x ( t ) =
Fourier Series
x (t ) =
f =
1
Ts
Frequency domain
X ( f ) e j
EE153 [Fall 2009]
Last Name:
First Name:
St ID:
Please write your name on the back page too.
Quiz #2
2009 09 17
Two signals are given as
x1 ( t ) = cos ( 8 t )
x2 ( t ) = 2 cos ( 48 t )
y ( t ) = x1 ( t ) + x2 ( t )
The continuous function y ( t ) is samp
EE153 [Fall 2009]
Last Name:
First Name:
St ID:
Please write your name on the back page too.
Quiz #3
2009 10 20.
A system has following impulse response
h[ n] = u[ n] u[ n 5]
a) Plot the signal h[ n]
b) Is this filter realizable or non-realizable, why?
c)
EE153 [Fall 2009]
Last Name:
First Name:
St ID:
Please write your name on the back page too.
Quiz #4
2009 11 24.
Write Fourier transform pair (time and frequency domain) equations
For the signal, use x or X (example: x ( t ) or X ( f ) ) and make sure tha
EE153 [Sp 2010]
Last Name:
First Name:
St ID:
Please write your name on the back page too.
Quiz #2
2010 02 18
It is given that
x ( t ) = 2sin (100 t )
a) Write frequency response of x ( t ) , F cfw_ x ( t ) .[hint: F cfw_ A cos ( 2 f 0t ) =
A
( f f 0 ) +
EE153 [Sp 2010]
Last Name:
First Name:
St ID:
Please write your name on the back page too.
Quiz #4
2010 05 04
Time domain sequence x [ n ] is given below.
x [ n ] = 3 [ n ] + 2 [ n 1] + 1 [ n 2]
Find DFT of x [ n ] , X [ k ]
X [ k ] =Wx
1
= 1
1
clc; clear
CH9 Finite impulse response filters (Nonrecursive filters)
Rely only on past input information
Never on past output information
A nonrecursive filter, the difference equation takes the form
y n = b0 x n + b1x n 1 + b2 x n 2 + . + bM x n M
M
y n = b
By using transfer function in z domain, the same result will be achieved
more easily.
1
Ex 4.7
2
3
Finding inverse z transform
1. Simple inverse z transforms by using Table
2. Inverse z transforms by long division
3. Inverse z transforms by partial fracti
Recursive difference equations
N
M
k =1
k =0
y [ n ] = ak y [ n k ] + b [ k ] x [ n k ]
= a1 y [ n 1] a2 y [ n 2] . aN y [ n N ]
+ b0 x [ n ] + b1 x [ n -1] + b2 x [ n - 2] + . + bM x [ n - M ]
1
The following steps develop a direct form II realization.
N
Superposition
x1 [ n ]
x2 [ n ]
h [ n]
y [ n]
xN [ n ]
y[n] = h[n] x[n]
x[n] = x1[n] + x2 [n] + .xN [n]
(1)
y[n] = h[n] ( x1[n] + x2 [n] + .xN [n])
y[n] Y ( f )
h[n] H ( f )
(2)
Y( f )= H( f )X ( f )
= H ( f ) ( X 1 ( f ) + X 2 ( f ) + . + X N ( f ) )
= H
HW06: 5.20
6.11 6.14
6.1(b,c) 6.2(a,b,c),6.4(a,b,c) 6.5(a)
6.15(b,c,e,h,k) 6.17 6.23(a,c) 6.27
6.7(a,b,c) 6.10
6.28(b,d)
5.20 (a)
The easiest way to deduce the impulse response is to identify impulse
response samples one at a time from the table below.
x[
1
x (t )
2
i.
4
6
Find two functions that produce x ( t ) after convolution.
x ( t ) = z1 ( t ) z2 ( t )
ii.
Find the frequency response of x ( t )
iii.
Find the frequency response of ( x ( t ) x ( t ) ) and plot magnitude and
phase
2
i.
t
t
t
t
Fi
1. Let x ( t ) be a square wave, as shown in the Figure below
Find Fourier coefficient xk and plot it where [ k = 10 to 10]
2. Consider a triangular wave, as shown in the figure below
Find the Fourier coefficients and plot, xk , where [ k = 5 to 5]
Text b
2.15 When the signal is sampled at 600 Hz, images of the sine waves frequency on
either side of every multiple of the sampling frequency. Since the aliased frequency is
150 Hz, copies appear at 0 150, 600 150, 1200 150, etc. Only 150, 450 and 750 Hz
lie b
1. The triangle function x ( t ) can be defined as
t 4
t
x ( t ) = 2
= 2 ( t 4 )
2
2
Let
t
s ( t ) = 2 .
2
Since the width of the x ( t ) is 4, the width of z ( t ) has to be half of x ( t ) .
s (t )
2
2
2
t
t
It is easy to see that the convolution of
1. Let x ( t ) be a square wave, as shown in the Figure below
Find Fourier coefficient ak and plot it where [ k = 10 to 10]
Lets assume that the Fourier series coefficient is
2. Consider a triangular wave, as shown in the figure below
Find the Fourier coe
1
Hearing test
CD sampling rate: 44100 samples/second
DAT sampling rate: 48000 samples/second
clc;
clear all;
f = 200;% frequency that I want to hear.
fs = 44100; % sampling frequency
time_dur = 3; % totoal time that you want to hear.
n = 0:fs*time_dur; %
1
Examples of sampling process
2
Fourier series representation of an impulse train is denoted by
1
0.8
0.6
0.4
0.2
0
-0.2
-3
-2
-1
0
1
2
3
Figure 1: Pulse Train with 2 samples/second
1
The Fig. 1 shows the pulse train with Ts = 0.5 or f s = = 2
Ts
Mathem