Ch00_Introduction Signals Systems

8/2/2019 Ch00_Introduction Signals Systems

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Signals and Systems

Introduction


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Textbook

Simon Haykin and Barry Van Veenand Systems, Second Edition, Joh

Sons, 2003.

40%,60%,


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Introductions to Signals

Signal:

a function of one or more independentvariables; typically contains information about

the behavior or nature of some physicalphenomena.

e.g. Voice (audio), TV (audio + video), light, voltage,

current, stock price, etc.


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Introductions to System

System:

responds to a particular signal input byproducing other signal.

e.g. Biological sensory system, electronic circuits,automobile, etc.

SystemInput signal Output signal


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Objectives

System characterization

how it responds to input signal

(e.g. human auditory system)

System design

to process signal in a particular way

(e.g. signal restoration, signal identification,image processing)


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Examples of Signals

Audio (intensity vs. time)


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Examples of Signals

TV signal (voltage vs. time)modulated picture signal + audio signal; carrier signal;

system involves :antenna, tuner, CRT

Horizontal

Sync Pulse

Active Video

Signal

Color Burst


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Examples of Signals (cont.)

Biomedical signal (voltage vs. time)

e.g. Electrocardiogram

Traffic flow (quantity vs. time) volume, composition, pattern, mobility;

system involved: traffic lights, roads, junctions

Network Throughput

# of packets/sec., pattern


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Stock price (index vs. time; $ vs. time)


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Picture (intensity vs. space)

on film -- continuous space, continuous tone

on computer -- discrete, quantized


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Continuous vs. Discrete Signal

x(t)

t

x[n]

n

Discrete-time Signal(independent variable: n)

Continuous-time Signal(independent variable: t)

x[n] is also referred as a sequence. Any particularone in x[n] is called a sample.

(a).

(b).


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Transformation of

independent variable

(1). Reflection

x(t) x(-t)

x[n] x[-n] x(t)

t

x[n]

n

x[-n]

n

t

x(-t)


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Transformation of

independent variable (cont.)

(2). Scaling

x(t) x(ct)

x(t)

t

x(t/2)

t

x(2t)

t


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Transformation of


(3). Time-shift

x[n] x[n-n0]

x[n]

n

x[n-2]

n


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Transformation of


x(t)

1 2 t

x(t+1)

-1 1 2 t

-2/3 2/3

x(1.5t+1)

1 2 t

To perform transformation x(t)

x(a t+b)

You have to do time-shiftingthenscaling.


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Even and Odd functions

A signal is called an even signal (function) if

A signal is called an odd signal (function) if

x(t) = - x(-t)x[n] = - x[-n]

x(t) = x(-t)x[n] = x[-n]


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Even and Odd functions (cont.)

Any signal can be broken into sum of one evenand one odd signal.

x(t)= Ev{x(t)} + Od {x(t)}

Q: How to find the even (odd) part of a signal.

A: Ev{x(t)} = (1/2) [x(t) + x(-t)]

Od {x(t)} = (1/2) [x(t) - x(-t)]


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Even and Odd functions (cont.)

Ex:

x[n]

n-3 -2 -1 0 1 2 3

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x[n]

n-3 -2 -1 0 1 2 3

1

Ev{x[n]} = (1/2) [x[n] + x[-n]] Od {x[n]} = (1/2) [x[n] - x[-n]]

x[n]

n0 1 2 3

1/2

-1/2


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Periodicand AperiodicSignal

IF x(t) is periodic with period T, then

x(t) = x(t+T)

IF x[n] is periodic with period N, then

x[n] = x[n+mN] ; m an integerFundamental period (T0 or N0) :

the smallest positive value of (T or N) for which

the above equation holds.Aperiodicis also called Non-periodic


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Unit Step and Unit Impulse

function

Unit Step function:

u(t) = 0, t0

Unit Impulse function:

d(t) = d u(t) / dt

also

* A system can be characterized by its unit step response

or unit impulse response.

1

t

1

t'dt)(t'u(t) t

d


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Discrete Time Unit Step and

Unit Impulse Sequence

Unit Step function:

u[n] = 0, n


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Discrete Time Unit Step and

Unit Impulse Sequence (cont.)

d[n] = u[n]-u[n-1] : first difference of unit step

u[n]= : running sum of unit sample

or equivalently u[n]=

n

m

m][d

0

][k

knd

x[n]

n-3 -2 -1 0 1 2 3

1

......x[n]

n-3 -2 -1 0 1 2 3

x[n-1]

n-3 -2 -1 0 1 2 3

= +

n

x[n-2]

-3 -2 -1 0 1 2 3

+ + ...


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Overview of System

System: any process that results in thetransformation of signal.

x(t) y(t)

Continuous-time

x[n] y[n]

Discrete-time

A system is continuous-time if both input and outputare continuous-time.A system is discrete-time if both input and outputare discrete-time.


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Interaction of Multiple Systems

(a) Cascade (Series)

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

System #1 System #2

#1

#2

(b) Parallel

I t ti f M lti l S t


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Interaction of Multiple Systems

(cont.)

(c) Series/Parallel

#2

#3

#1

x2

( )2( )2

+

-

Ex. y[n]=(2x[n]-x[n]2)2


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Properties of System

Memory/Memoryless

Invertibility and Inverse System

CausalityStability

Time-invariance

Linearity


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(1). Memory and Memoryless

A system is memoryless if the output onlydepends on input at the same time

Ex. (i) Resister is a memoryless component.

y(t)=R x(t) where y(t) : voltage

x(t) : current

R: resistance

k

kxny ][][(ii) With memory : e.g.

Capacitor is also with memory.

(2)


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(2). Invertibility and Inverse

System

A system is invertible if distinct inputs lead todistinct outputs.

x(t) y(t) z(t) = x(t)

System #1 System #2

If z(t)=x(t), then system #2 is the inverse system

of system #1.E.g. y(t)=2 x(t) then z(t) =0.5 y(t)

]1[][][][][

n-ynynzthenkxnyn

k


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(3). Causality

A system is causal if the output at anytime only dependson the input at the present time and before.

E.g. y[n]=x[n]-x[n-1] : causal

y(t)=x(t+1) : non-causal

Causal property is more important for real-timeprocessing.

But for some applications, such as image-processing, no

need to process the data causally.E.g.

M

Mk

knxM

ny ][12

1][

Q:

Memoryless Causal ?


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(4). Stability

A system is stable if bounded input givesbounded output.

--- BIBO stable

Eg. x(t) : the horizontal force; y(t) : vertical displacement

x(t)

y(t)

Stable System

x(t)y(t)

Unstable System


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E.g. ][)1(][][ nunkunyn

k

][)1( nun

Input: u[n], boundedOutput: (n+1)u[n], non-bounded as

][lim nyn

(4). Stability (cont.)


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(5). Time-invariance

A system is time-invariant if a time shift in theinput only causes a time shift in the output.

i.e. If x[n] y[n], then x[n-n0] y[n-n0]

Ex.1 y(t)=sin(x(t))

Let y1(t)= sin(x1(t)), x2(t)=x1(t-t0)

Then y2(t)=sin(x2(t))=sin(x1(t-t0))

= y1(t- t0)

time-invariant (T.I.)


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(5). Time-invariance (cont.)

Ex.2 y[n]=n x[n]

Let y1[n]= n x1[n] & x2[n]=x1[n-n0]

Then y2[n]=n x2[n]=n x1[n-n0]

However y1[n-n0]=(n-n0) x1[n-n0]

y2[n]

not time-invariant (T.I.)


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(6). Linearity

A system is linear if

1. the response to x1(t)+x2(t) is y1(t) +y2(t)

---additivity

2. the response to ax1

(t) is ay1

(t), where a isany complex constant.

---scaling

Combine the above two properties, we can conclude

ax1(t)+ bx2(t) ay1(t) + by2(t)---superposition property

For discrete-time :

ax1[n]+ bx2[n] ay1[n] + by2[n]

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