SIGNALS & SYSTEMS

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SIGNALS & SYSTEMS

Contents of the Lecture


Signal & System?


Time
-
domain representation of LTI
system


Fourier transform and its application


Z transform and its application


Digital Filter & Its Application


Can you believe it?

Examples of System

1.
INTRODUCTION

What is a Signal?


(
DEF) Signal : A signal is formally
defined as a
function
of one or more
variables, which
conveys information

on the nature of physical phenomenon.

나는

무엇을

생각할까요
?

What is a System?


(
DEF) System : A system is formally
defined as an entity that manipulates
one or more signals to accomplish a
function, thereby yielding new signals.

system

output
signal

input
signal

Some Interesting Systems


Communication system


Control systems


Remote sensing system


Biomedical system(biomedical signal
processing)


Auditory system

Some Interesting Systems


Communication system

Some Interesting Systems


Control systems


Some Interesting Systems

Papero

Some Interesting Systems


Remote sensing system

Perspectival view of Mount Shasta (California), derived from a pair
of stereo radar images acquired from orbit with the shuttle Imaging
Radar (SIR
-
B). (Courtesy of Jet Propulsion Laboratory.)

Some Interesting Systems


Biomedical system(biomedical signal
processing)


Some Interesting Systems


Auditory system

Classification of Signals


Continuous and discrete
-
time signals


Continuous and discrete
-
valued signals


Even and odd signals


Periodic signals, non
-
periodic signals


Deterministic signals, random signals


Causal and anticausal signals


Right
-
handed and left
-
handed signals


Finite and infinite length

Continuous and discrete
-
time signals


Continuous signal


-

It is defined for all time t :
x
(
t
)


Discrete
-
time signal


-

It is defined only at discrete instants of
time :
x
[
n
]=
x
(
nT
)

Continuous and Discrete valued
singals


CV corresponds to a continuous y
-
axis


DV corresponds to a discrete y
-
axis

Digital signal

Even and odd signals


Even signals :
x
(
-
t
)=
x
(
t
)


Odd signals :
x
(
-
t
)=
-
x
(
t
)


Even and odd signal decomposition


x
e
(
t
)= 1/2∙(
x
(
t
)+
x
(
-
t
))

x
o
(
t
)= 1/2∙(
x
(
t
)
-
x
(
-
t
))




Periodic signals, non
-
periodic signals


Periodic signals


-

A function that satisfies the condition


x
(
t
)=
x
(
t
+
T
) for all
t


-

Fundamental frequency :
f
=1/
T


-

Angular frequency :

㴠2






Non
-
periodic signals

Deterministic signals,
random signals


Deterministic signals


-
There is no uncertainty with respect to its value
at any time. (ex) sin(3t)




Random signals


-

There is
uncertainty

before its actual occurrence
.

Causal and anticausal
Signals


Causal signals :
zero for all negative
time


Anticausal signals :
zero for all positive
time


Noncausal :
nozero values in both
positive and negative time

causal
signal

anticausal
signal

noncausal
signal

Right
-
handed and left
-
handed
Signals


Right
-
handed and left handed
-
signal :
zero between a given variable and
positive or negative infinity

Finite and infinite length



Finite
-
length signal : nonzero over a
finite interval
t
min
< t<
t
max


Infinite
-
length singal : nonzero over all
real numbers

Basic Operations on Signals


Operations performed on dependent
signals


Operations performed on the
independent signals


Operations performed on
dependent signals


Amplitude scaling


Addition


Multiplication


Differentiation


Integration

( ) ( )
y t cx t

1 2
( ) ( ) ( )
y t x t x t
 
1 2
( ) ( ) ( )
y t x t x t
 
( ) ( )
d
y t x t
dx

( ) ( )
t
y t x d
 



Operations performed on
the independent signals


Time scaling


a
>1 : compressed


0<
a
<1 : expanded

( ) ( )
y t x at

Operations performed on
the independent signals


Reflection

( ) ( )
y t x t
 
Operations performed on
the independent signals


Time shifting


-

Precedence Rule for time shifting & time
scaling




0
( ) ( )
y t x t t
 
( ) ( ) ( ( ))
b
y t x at b x a t
a
   
The incorrect way of applying the precedence rule. (a) Signal
x
(
t
).

(b) Time
-
scaled signal
v
(
t
) =
x
(2
t
). (c) Signal
y
(
t
) obtained by
shifting

v
(
t
) =
x
(2
t
) by 3 time units, which yields
y
(
t
) =
x
(2(
t

+ 3)).

The proper order in which the operations of time scaling and time
shifting (a) Rectangular pulse
x
(
t
) of amplitude 1.0 and duration 2.0,
symmetric about the origin. (b) Intermediate pulse
v
(
t
), representing
a time
-
shifted version of
x
(
t
). (c) Desired signal
y
(
t
), resulting from
the compression of
v
(
t
) by a factor of 2.

Elementary Signals


Exponential signals


Sinusoidal signals


Exponentially damped sinusoidal
signals

( )
at
x t Be

( ) cos( )
x t A t
 
 
( ) cos( )
at
x t Ae t
 
 
Elementary Signals


Step function

( ) ( )
x t u t

(a) Rectangular pulse
x
(
t
) of amplitude
A

and duration of 1 s,
symmetric about the origin. (b) Representation of
x
(
t
) as the
difference of two step functions of amplitude
A
, with one step
function shifted to the left by ½ and the other shifted to the right by
½; the two shifted signals are denoted by
x
1(
t
) and
x
2(
t
),
respectively. Note that
x
(
t
) =
x
1(
t
)


x
2(
t
).

Elementary Signals


Impulse function

( ) ( )
x t t


(a) Evolution of a rectangular pulse of unit area into an impulse of unit
strength (i.e., unit impulse). (b) Graphical symbol for unit impulse.

(c) Representation of an impulse of strength
a

that results from allowing
the duration Δ of a rectangular pulse of area
a

to approach zero.

Elementary Signals


Ramp function

( ) ( )
x t r t

Systems Viewed as
Interconnection of
Operations

system

output
signal

input
signal

Properties of Systems


Stability


Memory


Invertibility


Time Invariance


Linearity

Stability(1)


BIBO stable : A system is said to be
bounded
-
input bounded
-
output stable

iff every bounded input results in a
bounded output.



Its Importance : the collapse of Tacoma
Narrows suspension bridge,
pp
.45

| ( ) | | ( ) |
x y
t x t M t y t M
    
Dramatic photographs
showing the collapse of
the Tacoma Narrows
suspension bridge on
November 7, 1940. (a)
Photograph showing the
twisting motion of the
bridge’s center span just
before failure.

(b) A few minutes after
the first piece of concrete
fell, this second
photograph shows a 600
-
ft section of the bridge
breaking out of the
suspension span and
turning upside down as it
crashed in Puget Sound,
Washington. Note the car
in the top right
-
hand
corner of the photograph.


Stability(2)


Example
pp
.46


-

y
[
n
]=1/3(
x
[
n
]+
x
[
n
-
1]+
x
[
n
-
2])


-

y
[
n
]=
r
n
x
[
n
], where
r
>1

1
[ ] [ ] [ 1] [ 2]
3
1
(| [ ] | | [ 1] | | [ 2] |)
3
1
( )
3
x x x x
y n x n x n x n
x n x n x n
M M M M
    
    
   
Memory


Memory system : A system is said to
possess
memory

if its output signal
depends on past values of the input
signal


Memoryless system


(example)



1
( ) ( )
1
( ) ( )
[ ] [ ] [ 1]
t
i t v t
R
i t v d
L
y n x n x n
 

 
 
   

Memory or memoryless?

Causality


Causal system : A system is said to be
causal

if the present value of the output
signal depends only on the present
and/or past values of the input signal.


Non
-
causal system


(example)


y
[
n
]=
x
[
n
]+1/2
x
[
n
-
1]


y
[
n
]=
x
[
n
+1]+1/2
x
[
n
-
1]



Invertiblity(1)


Invertible system : A system is said to
be
invertible

if the input of the system
can be recovered from the system
output.


H:
x

y
, H
-
1
:
y

x


H
-
1
{
y
(
t
)}=
H
-
1
{H{
x
(
t
)}},
H
-
1
H=I

H

H
-
1

x
(
t
)

x
(
t
)

y
(
t
)

Invertiblity(2)


(
Example)


-




-

1
( ) ( ) ( ) ( )
t
d
y t x d x t L y t
L
dt
 

  

2
( ) ( )
y t x t

Time Invariance (1)


Time invariant system : A system is
said to be
time invariant

if a time delay
or time advance of the input signal
leads to a identical time shift in the
output signal.

0
0 0
( ) { ( )}
{ { ( )}} { ( )}
i
t t
y t H x t t
H S x t HS x t
 
 
0
0
0 0
( ) { ( )}
{ { ( )}} { ( )}
t
t t
y t S y t
S H x t S H x t

 
Time Invariance (2)

S
t
0

H

x
(
t
)

y
i
(
t
)

x
(
t
-
t
0
)

H

S
t
0

x
(
t
)

y
0
(
t
)


Are following two systems equivalent?

Time Invariance (3)


Examples

1
( ) ( )
( )
( )
( )
t
y t x d
L
x t
y t
R t
 

 
 

Linearity(1)


Linear system : A system is said to be
linear

if it satisfies the
principle of
superposition
.

1
1
?
1 1
( ) ( )
( ) { ( )} { ( )}
{ ( )} ( )
N
i i
i
N
i i
i
N N
i i i i
i i
x t a x t
y t H x t H a x t
a H x t a y t


 

 
 


 
Linearity(2)

a
1

a
2

a
N

.

.

.



.

.

.

H

x
1
(t)

x
2
(t)

x
N
(t)

.

.

.

y(
t)

H

H

H

.

.

.

a
1

a
2

a
N



.

.

.

.

.

.

x
1
(t)

x
2
(t)

x
N
(t)

y(
t)

Linearity(3)


Examples


-


-



Check superposition with simple two
inputs.


[ ] [ ]
y n nx n

( ) ( ) ( 1)
y t x t x t
 
1 1 2 2
( ) ( ) ( )
x t a x t a x t
 
Theme Examples

Example of multiple propagation paths in a wireless
communication environment.

Tapped
-
delay
-
line model of a linear communication
channel, assumed to be time
-
invariant

Stock Price : filtering


(a) Fluctuations in the closing stock price of Intel over a three
-
year
period.

(b) Output of a four
-
point moving
-
average system.

References


S. Haykin and B. Van Veen,
Signals and
Systems
, 3
rd

ed. Wiley and Sons, Inc,
2003.


Kim Jin Young, “Handout”, IC & DSP
Research, EE Dept. Chonnam National
University, 2005.