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.
input
signal
system
output
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 discretetime 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 yaxis
• 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
xe(t)= 1/2·(x(t)+x(-t)) xo(t)= 1/2·(x(t)-x(t))
Periodic signals, nonperiodic signals
• Periodic signals
- A function that satisfies the condition
x(t)=x(t+T) for all t
- Fundamental frequency : f=1/T
- Angular frequency : = 2/T
• 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 tmin< t< tmax
• 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 scalingy (t )  cx(t )
• Addition
y(t )  x1 (t )  x2 (t )
• Multiplication
y(t )  x1 (t )  x2 (t )
• Differentiation
• Integration
d
y (t ) 
x (t )
dx
t
y (t ) 
 x( )d

Operations performed on
the independent signals
• Time scaling y (t )  x(at )
a>1 : compressed
0<a<1 : expanded
Operations performed on
the independent signals
• Reflection y (t )  x(t )
Operations performed on
the independent signals
• Time shifting y(t )  x(t  t0 )
- Precedence Rule for time shifting & time
scaling
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(2t). (c) Signal y(t) obtained by
shifting
v(t) = x(2t) 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 x(t )  Beat
• Sinusoidal signals x(t )  A cos( t   )
• Exponentially damped sinusoidal
signals
x(t )  Aeat cos( 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 x1(t) and x2(t),
respectively. Note that x(t) = x1(t) – x2(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
input
signal
system
output
signal
Properties of Systems
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•
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.
t | x(t ) | M x    t | y(t ) | M y  
• Its Importance : the collapse of Tacoma
Narrows suspension bridge, pp.45
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 600ft 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]=rnx[n], where r>1
1
y[n]  x[n]  x[n  1]  x[n  2]
3
1
 (| x[n] |  | x[n  1] |  | x[ n  2] |)
3
1
 (M x  M x  M x )  M x
3
Memory
• Memory system : A system is said to
possess memory if its output signal
depends on past values of the input
signal
• Memoryless system
1 v(t )

i
(
t
)

• (example)
R
t
i (t )  1
L
 v( )d

 y[n]  x[n]  x[n  1]
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]+1/2x[n-1]
y[n]=x[n+1]+1/2x[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-1H=I
H-1
H
x(t)
y(t)
x(t)
Invertiblity(2)
• (Example)
-
t
y (t )  1
-
L

x( )d  x(t )  L

y(t )  x 2 (t )
d
y (t )
dt
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.
yi (t )  H {x(t  t0 )}
 H {S t 0{x(t )}}  HS t 0{x(t )}
y0 (t )  S t 0{ y(t )}
 S t 0{H {x(t )}}  S t 0 H {x(t )}
Time Invariance (2)
• Are following two systems equivalent?
St0
x(t-t0)
x(t)
H
x(t)
H
yi(t)
St0
y0(t)
Time Invariance (3)
• Examples
t
 y (t )  1
L
 x( )d

x(t )
 y (t ) 
R(t )
Linearity(1)
• Linear system : A system is said to be
linear if it satisfies the principle of
superposition.
N
x(t )   ai xi (t )
i 1
N
y (t )  H {x(t )}  H { ai xi (t )}
i 1
?
N
N
i 1
i 1
  ai H {xi (t )}   ai yi (t )
Linearity(2)
x1(t)
x2(t)
.
.
.
xN(t)
a1
H
a1
H
.
.
.
a2
.
.
.
H
aN
x1(t)
a2
.
.
.
aN
. 
.
.
H
x2(t)
y(t)
xN(t)
.
.
.

y(t)
Linearity(3)
• Examples
- y[n]  nx[n]
- y (t )  x(t ) x(t  1)
• Check superposition with simple two
inputs.
x(t )  a1 x1 (t )  a2 x2 (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, 3rd ed. Wiley and Sons, Inc,
2003.
• Kim Jin Young, “Handout”, IC & DSP
Research, EE Dept. Chonnam National
University, 2005.