# SIGNALS & SYSTEMS

AI and Robotics

Nov 24, 2013 (4 years and 5 months ago)

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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
-
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

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.