Review of Fourier Analysis
Nicholas Kyriakopoulos
Department of Electrical and Computer Engineering
Fourier Analysis - NK
1
x(t )
x(t )
x(t )
0
T
t)
0
k
x(t )e
jk
bk sin(k
)
T0 /2
T0 /2
0
0t
dt
t ) where
Fourier Series Representation
ak cos(k
k
1
T0
k 1
J
Signals and Their Representation
Nicholas Kyriakopoulos
Department of Electrical and Computer Engineering
x(t)
x*(t)
y*(t)
Digital
processing
Analog processing
Analog
processing
D/A
conversion
y(t)
What Is Digital Signal Processing?
x(t)
Ideal
sampling
y(
Sampling and Reconstruction of Signals
Nicholas Kyriakopoulos
Department of Electrical and Computer Engineering
Sampling/Reconstruction - NK
1
x(0)
k
k
x(t ) (t
0
kTs
x(t ) (t kTs )
(t kTs )
)dt
Ideal Sampling
t
The sampling property of impulses
x(t ) (t
Discrete Systems: Difference Equations
Finite Impulse Response Filters (FIR)
Nicholas Kyriakopoulos
Department of Electrical and Computer Engineering
Discrete Variables and Operators
y (kTs )
y (kTs )
y (kTs )
x(k n)Ts )
x(k 1)Ts )
(nTs )-delay operator
Frequency Response of Discrete-time
Systems
Nicholas Kyriakopoulos
Department of Electrical and Computer
Engineering
Frequency Response-NK
1
Filtering of Discrete-time Signals - 1
c
k
0
0.1T0
) / 2] sinc[(
,
c
2
2
k
) ] sinc[(
T0 2
Tc
k
0
) / 2]
2
k
) ]
T
Review of Fourier Analysis
Nicholas Kyriakopoulos
Department of Electrical and Computer Engineering
Fourier Analysis - NK
1
x(t )
x(t )
x(t )
0
T
t)
0
k
x(t )e
jk
bk sin(k
)
T0 /2
T0 /2
0
0t
dt
t ) where
Fourier Series Representation
ak cos(k
k
1
T0
k 1
J
The George Washington University
Department of Electrical and Computer Engineering
ECE 3220-10 Introduction to Digital Signal Processing
Instructor: Dr. Nicholas Kyriakopoulos; 202 994 6797; kyriak@gwu.edu
Teaching Assistants: Yiwen Zhou, 202 907 1087; zh