신호 및 시스템 Chapter 1. Introduction

신호및시스템 Chapter 1.

신호및시스템 [ Chapter 1. ]

Prof. Sung-Il Chien, School of EE, KNU 2

Introduction

1.1 What Is a Signal?

SignalsSpeech signalsImages or visual signalsInternetStock informationetc.

A signal is formally defined as a function of one or more variables, which

conveys information on the nature of a physical phenomenon.

variablesOne-dimensional – e.g. Speech

Multidimensional – e.g. Image



Introduction

1.2 What Is a System?

A system is formally defined as an entity that manipulates one or more

signals to accomplish a function, thereby yielding new signals.

SystemInput

signal

Output

signal



Introduction

Cf. modulation – the process of converting the message signal into a form that is

compatible with the transmission characteristics of the channel.

Analog communication

Digital communication

sampling, quantization, coding

Two modes of communications

Broadcasting

Point-to-point communication

pathfinder



Introduction

1.3 Overview of Specific Systems

Communication system

, distortion

noise introduced

interference

At the some time, high speed in communication system is required to cover

recent requirement of massive data.

Transmitter Channel Receiver

Messagesignal

Transmittedsignal

Receivedsignal

Estimateof message

signal

In channel



Introduction



Introduction

Control systems

FeedbackDigital control system

Controller Plant∑ ∑)(teReference

input )(tx

)(tv

Disturbance)(tv

Output)(ty

Sensor(s)Feedback signal)(tr



Introduction

Remote sensing

The process of acquiring information about an object of interest without

being in physical contact with it.

Usually, multiple sensors covering a large part of the electromagnetic

spectrum is required.

Radar sensors, Infrared sensors, X-ray

SAR satisfactory operation day and night and under all weather

conditions

still achieving high-resolution imaging capability instead requiring sophisticated signal-processing operation

e.g. Fourier transform (FT) (FFT)



Introduction

Biomedical signal processing

Biological signals – ECG (electrocardiogram)

Neurons EEG (electroencephalogram)

A record of fluctuations in the electrical activity of large

groups of neurons in the brain



Introduction

Auditory System

Analog Versus Digital Signal Processing



Introduction

1.4 Classification of Signals

* real-valued signal, complex-valued signal

Continuous-time and discrete-time signals

A discrete-time signal can be derived from a continuous-time signal by

sampling it at a uniform rate.

: sampling period

)(tx defined at discrete instants of time

K,2,1,0),(][ ±±== nnxnx

sampling quantization

Streams of bits

(to the world of computer)



Introduction

Even and odd signals

Even signalOdd signal

ttxtx allfor)()( =−ttxtx allfor)()( −=−

Ex. 1. 1 Is the signal and even or an odd function of time t ?

Solution: Replace t with –t yields

Hence, is an odd signal.

)(tx

( )

≤≤−

=otherwise,0

,sin)(

TtTtx T

tπ

( )

( ))(

otherwise,0

,sin

otherwise,0

,sin)(

txTtT

TtTtx

Tt

Tt

−=

≤≤−−

=

≤≤−−

=−

π

π

)(tx



Introduction

Arbitrary signal x(t)Even-odd decomposition

show that

)()()( txtxtx oe +=

)()(

)()(

txtx

txtx

oo

ee

−=−=−

[ ]

[ ])()(2

1)(

)()(2

1)(

txtxtx

txtxtx

o

e

−−=

−+=

tetx t cos)( 2−=Ex. 1.2

Solution:

tt

tetetx

tt

tetetx

tetetx

tto

tte

tt

cos)2sinh(

)coscos(21

)(

cos)2cosh(

)coscos(21

)(

)cos()cos()(

22

22

22

−=

−=

=

+=

=−=−

−

−



Introduction

A complex-valued signal is said to be conjugate symmetric if

Let , Then

)(tx

)()( * txtx =−

)()()( tjbtatx += )()()(* tjbtatx −= )()()( tjbtatx −+−=−)()( tata =− )()( tbtb −=−



Introduction

Periodic signals, nonperiodic signals

The smallest value of T that satisfied the above equation is called the

fundamental period of .

Fundamental frequency

Angular frequency

As for periodic discrete-time signals

The smallest N fundamental period

Fundamental(angular) frequency

)0(allfor)()( >+= TtTtxtx

)(tx

)in(1

HzT

f =

)secondperradianin(2

T

πω =

nNnxnx integerallfor][][ +=

N

π2=Ω



Introduction

Deterministic signals, random signals

Energy signals, power signals

The instantaneous power dissipated in the resistor

in the signal analysis,

Now, the instantaneous power of the signal is expressed as

e.g. noise generated in the amplifier

)()(or)(

)( 22

tRitpR

tvtp ==

ohm1=R

)(or)()( 22 titvtp =

)()( 2 txtp =



Introduction

Total energy of

and its average power

Average power of a periodic signal

Similarly,

for a periodic signal

Usually

)(tx

∫ ∫−

∞

∞−∞→==

2/

2/

22 )()(limT

TTdttxdttxE

∫−∞→=

2

2

2 )(1

limT

TTdttx

TP

∫−=2/

2/

2 )(1 T

Tdttx

TP

∑

∑

−=∞→

∞

−∞=

=

=N

NnN

n

nxN

P

nxnxE

][2

1lim

])[ofenergytotal(][

2

2

∑−

=

=1

0

2 ][1 N

n

nxN

P

deterministic & nonperiodic signal energy signal

random and periodic power

)0( ∞<< E

)0( ∞<< P



Introduction

1.5 Basic Operations on Signals

Operations performed on dependent variables

Amplitude scaling

Addition

Multiplication

Differentiation

integration

][][or)()( ncxnytcxty ==

)()()( 21 txtxty +=

)()()( 21 txtxty =

)()( txdt

dty =

∫ ∞−=

tdxty ττ )()(



Introduction

Operations performed on the independent variableTime scaling: continuous-time signal

Time scaling: discrete-time signal

)()( atxty =

0][][ >= kknxny

expandedis)(10compressedis)(1

tyatya

<<>

2=k



Introduction

Reflection

Time shifting

)()( txty −=

Ex. 1.3

Cf. even signals an even signal is the same as its reflected version.)()( txtx =−

leftthetoshiftedis)(0rightthetoshiftedis)(0

0

0

txttxt

<>

)()( 0ttxty −=

Ex. 1.4



Introduction

Precedence rule for time shifting and time scaling

Correct way: is shifted to right b then is compressed as much as a

)()( batxty −=

)(tx )()( btxtv −=)(tv )()( atvty =

Ex. 1.5 )32()( += txty

)3()( += txtv )2()( tvty =The incorrect applying of the precedence leads to the result shown in below.

)(tx

( ))3(2)( += txty



Introduction

]32[][ += nxny

]3[][first += nxnv

Ex. 1.6

0]6[]3[1]4[]2[

1]2[]1[0]0[]0[

]2[][

=−=−−=−=−

=−=−==

=

vyvyvy

vy

nvny

Original signal at are lost2&2 −== nn



Introduction

1.6 Elementary Signals

Exponential signals

Discrete-time case

atBetx =)(

lexponentiadecaying0lexponentiagrowing0

<>

aa

nBrnx =][

galternatin0growing1decaying10

⇒<⇒>⇒<<

rr

r



Introduction

Sinusoidal signals

Discrete version

N is the number of samples contained in a cycle

ωπφω 2

)cos()( =+= TtAtx

( )

)()cos()2cos()cos(

)(cos)(

txtAtATtA

TtATtx

=+=++=++=

++=+

φωφπωφωω

φω

periodic

)cos(][ φ+Ω= nAnx

( ))cos(

)(cos][φφ

+Ω+Ω=++Ω=+

NnANnANnx

)cleradians/cy(2

radians)(2

N

m

mN

ππ

=Ω

=Ω



Introduction

Ex. 1.7

]5cos[3][

]5sin[][

2

1

nnx

nnx

ππ

=

=

(a) Find common fundamental period.

Solution:Angular freq. of x1[n] and x2[n] is Ω = 5π radians/cycle

525

2

2

m

m

mN

=

=

Ω=

ππ

π

For x1[n] and x2[n] to be periodic, N must be an integer

N = 2, 4, 6, …



Introduction

(b) Express the composite sinusoidal signal ][][][ 21 nxnxny +=

in the form of y[n] = Acos(Ωn + φ) and evaluate the amplitude A and phase φ.

)sin()sin()cos()cos()cos( φφφ nAnAnA Ω−Ω=+ΩSolution:

Letting Ω = 5πthe right-hand side of identity is of the same form as x1[n] + x2[n]

Ex. 1.7

]5cos[3][

]5sin[][

2

1

nnx

nnx

ππ

=

=

1)sin( −=φA 3)cos( =φA

3

1

][ofamplitude

][ofamplitude

)cos(

)sin()tan(

2

1 −===nx

nx

φφφ radians3/πφ −=

1)3/sin()sin( −=−= πφ AA 2=A

−=∴

35cos2][

ππnny



Introduction

Use the following identity:

)cos(sincos 022 φθθθ −+=+ BABA

A

B10 tan−=φ

1,3 == BA

33

1tan2 0

22 πφ ===+∴ BA



Introduction

The sum of two periodic signals is also periodic.

The product of two periodic signals is also periodic.

N1: a period of x1(n) and N2: a period of x2(n)

The fundamental period is

)()()( 21 nxnxnx +=

),gcd( 21

21

NN

NNN =

)()()( 21 nxnxnx =

The period is also),gcd( 21

21

NN

NNN =

However, the fundamental period may be smaller by the following properties.

)sin()sin(cossin

)cos()cos(coscos

)cos()cos(sinsin

21

21

21

BABABA

BABABA

BABABA

++−=

++−=

+−−=



Introduction

Relation between sinusoidal and complex exponential signals

)identitysEuler'(sincos θθθ je j +=

tjtjj

tj

BeeAeAetAtx

ωωφ

φωφωReRe

Re)cos()( )(

===+= +

φjAeB =

)sin()cos()( φωφωφωω +++== + tjAtAAeBe tjtj

Similarly Re)cos( njBenA Ω=+Ω φIm)sin( njBenA Ω=+Ω φ

K,2,1,0&4

for ==ΩΩ ne nj π



Introduction

Exponentially damped sinusoidal signals

Step function

0)sin()( >+= − αφωα tAetx t

)()( tutx =

<≥

=00

01][

n

nnu



Introduction

Impulse function

≠==

0001

)(nn

nδ

0for0)( ≠= ttδ ∫∞

∞−=1)( dttδ

Dirac delta functions

One way to visualize is to view it as the limiting form of a rectangular pulse of unit area.)(tδ

)(lim)(0

tgt TT→

=δ sifting property : )()()( 00 txtttx =−∫∞

∞−δ

0)(1

)( >= ata

at δδ (Proof is included in the text.)

)62.1()()( tudt

dt =δ

Conversely, )63.1()()( ∫ ∞−= tdtu ττδ



Introduction

Ex. 1.9The capacitor C is assumed to be initially uncharged.

At t = 0, the switch connecting the dc voltage

source V0 to the RC circuit is closed.

Find the voltage v(t) across the capacitor for t ≥ 0.

Solution:The switching operation is represented by a step function V0u(t)

as shown in the equivalence circuit.

The capacitor cannot charge suddenly, so, with it being initially uncharged, we have

(equivalence circuit)

.0)0( =v

For t = ∞, the capacitor becomes fully charged; hence,

.)( 0Vv =∞Recognizing that the voltage across the capacitor increases exponentially with

a time constant RC, we may thus express v(t) as

).()1()( )/(0 tueVtv RCt−−=



Introduction

Ex. 1.10

The capacitor is initially uncharged

and the switch connecting it to the dc voltage

source V0 is suddenly closed at time t = 0.

Determine the current i(t) that flows through the capacitor for t ≥ 0.

)()( 0 tuVtv =

Solution:

The voltage across the capacitor is expressed as

By definition, the current flowing through the capacitor is

.)(

)(dt

tdvCti =

Hence, for the problem at hand, we have

)(

)()(

0

0

tCVdt

tduCVti

δ=

=



Introduction

Derivatives of Impulse

doublet:

)()( )1( ttdt

d δδ =

( ) )70.1()2/()2/(lim)( 1

0

)1( ∆−−∆+= ∆→∆ttt δδδ

∫∞

∞−= )71.1(;0)()1( dttδ

∫∞

∞− ==− )72.1(.|)()()(00

)1(tttf

dt

ddttttf δ

Generally, ∫∞

∞− ==− .|)()()(00

)(ttn

nn tf

dt

ddttttf δ



Introduction

Ramp function

<≥

=00

0)(

t

tttr

)()( ttutr =

<≥

=00

0][

n

nnnr

][][ nnunr =



Introduction

1.7 Systems Viewed as Interconnections of Operations

)()( txHty = ][][ nxHny =

e.g. discrete-time shift operator kS

Ex. 1.8 Moving average system

)1(3

1

]2[]1[][3

1][

2SSH

nxnxnxny

++=

−+−+=

Cascade form Parallel form

kS][nx ][ knx −



Introduction

1.8 Properties of Systems

Stability

A system is said to be bounded input-bounded output (BIBO) stable if and

only if every bounded inputs results in a bounded output.

where

tMty y allfor|)(| ∞<≤

∞<≤ xMtx |)(|

Ex. 1.14 1ifunstableis][][ >= rnxrny n

Solution: nMnx x allfor][ ∞<≤

xnnn Mrnxrnxrny ≤== ][][][Then

unstable1if →∞→> nrr



Introduction

Memory

Causality

A system is said to be causal if the present value of the output signal depends

only on the present and/or past value of the input signal.

memory)(1

)(

memoryless)(1

)(

→=

→=

∫ ∞−

tdv

Lti

tvR

ti

ττ

noncausal]1[][]1[3

1][

causal]2[]1[][3

1][

→−+++=

→−+−+=

nxnxnxny

nxnxnxny



Introduction

Invertibility

not invertible

Time invariance

Two operators commute with each other !

H)(tx )(ty invH)(tx

[ ] )()()( txHHtxHHtyH invinvinv ==operatoridentity →= IHH inv

)(2 txy =

)()( txHty = )()( 00 txSttx t=−

0tS)(tx )( 0ttx −

H)(tyi H

)(tx0tS

)(tyo

)()()( 00 txHSttxHty t

i =−= )()()( 00 txHStySty tto ==

invarianttime00 −→= HSHS tt



Introduction

Assume a system is time invariant

and

Question? Find for input

H)(tx )(ty

)(ty )( 0ttx −

H)( 0ttx − )(ty

)()(

)(

)(

)()(

0

0

0

0

0

ttytyS

txHS

txHS

ttxHty

t

t

t

−==

=

=

−=



Introduction

Linearity

A system is said to be linear if it satisfies the principles of superpositionnonlinear

==

=

∑

∑

=

=N

iii

N

iii

txaHtHxty

txatx

1

1

)()()(

)()(

if the system is linear )()(where)()(1

txHtytyaty ii

N

iii ==∑

=

∑=

=N

iii txHaty

1

)()(

Ex. 1.19

Sol.

linear?][][ nnxny =

∑=

=N

iii nxanx

1

][][if

linear][][][][y111

→=== ∑∑∑===

n

iii

n

iii

N

iii nyannxanxann



Introduction

1.9 NoiseExternal sources of noiseInternal sources of noise



Introduction

Thermal NoiseTime-average-squared value (v(t): thermal noise voltage)

k: Boltzman’s constant

Tabs: absolute temperature

R: resistance

∆f: width of the frequency band in Hertz over which v(t) is measured

∫−∞→=

T

TTdttv

Tv )95.1(.)(

2

1lim 22

22 vlots4 fRkTv abs ∆=

where,



Introduction

1.10 Theme ExampleMoving-Average Systems

∑−

=−=

1

0

][1

][N

k

knxN

ny



Introduction



Introduction



Introduction

신호 및 시스템 Chapter 1. Introduction

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