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

xe(t)= 1/2·(x(t)+x(-t)) xo(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 : = 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 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 tdx

( ) ( )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 ta

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• Sinusoidal signals• Exponentially damped sinusoidal

signals

( ) atx t Be( ) cos( )x t A t

( ) cos( )atx 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 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

Operationssystem output

signalinput 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 yt 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]=rnx[n], where r>1

1[ ] [ ] [ 1] [ 2]

31

(| [ ] | | [ 1] | | [ 2] |)31

( )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 tR

i t v dL

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]+1/2x[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:xy, H-1:yx

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

H H-1

x(t) x(t)y(t)

Invertiblity(2)

• (Example)

-

-

1( ) ( ) ( ) ( )t d

y t x d x t L y tL 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

00

0 0

( ) { ( )}

{ { ( )}} { ( )}

t

t t

y t S y t

S H x t S H x t

Time Invariance (2)

St0 H

x(t) yi(t)x(t-t0)

H St0

x(t) y0(t)

• Are following two systems equivalent?

Time Invariance (3)

• Examples

1( ) ( )

( )( )

( )

t

y t x dL

x ty 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 ii

N

i ii

N N

i i i ii i

x t a x t

y t H x t H a x t

a H x t a y t

Linearity(2)

a1

a2

aN

.

.

.

.

H

x1(t)

x2(t)

xN(t)

.

.

y(t)

H

H

H

.

.

a1

a2

aN

.

...

x1(t)

x2(t)

xN(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 ExamplesExample 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.