Chapter 1 control

7/17/2019 Chapter 1 control

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Chapter:1

INTRODUCTION OF SIGNALS AND

SYSTEMS

Chapter 1

1UNIKL BMI- SIGNALS AND SYSTEMS - 2015


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1.0 Introduction of Signals and Systems

Defnition

i) A signal:

A variation o a !"antit# $e%&% pre''"re( te)perat"re( po'ition( *o+o"r( vo+ta&e( heart

rate( ne"ron irin&( et*%,

Conve#' inor)ation re&arin& the 'tate( the *hara*teri'ti*'( the *o)po'ition( thetra.e*tor#( the evo+"tion or the inten'ion o the 'o"r*e%

I' a )ean to *onve# inor)ation

ii) A system:

A )athe)ati*a+ )oe+ or a ph#'i*a+ pro*e'' that re+ate' the inp"t 'i&na+ to the o"tp"t

The o"tp"t )a# /e enhan*e( )anip"+ate( )oiie( an tran'or)e ver'ion o the

inp"t( or the inor)ation that i' etra*te ro) the inp"t%

2


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Mathematical model of signals and systems3

Model O Signals And Systems


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4

Application Areas Specific Uses of Signals and Systems

Telecommunications An'erin& )a*hine( )oe)( a )a*hine( *e++ phone'( 'peaer'

phone'%

Speech and Audio

Processing

3oi*e )ai+( 'peaer verii*ation( '#ntheti* 'pee*h( a"io

*o)pre''ion( 'pee*h re*o&nition( 'pee*h *oin&( et*%

Automotive En&ine *ontro+( anti+o* /reain& '#'te)'( a*tive '"'pen'ion( air/a&

*ontro+( '#'te) ia&no'i'( et*%

Medical 4eart irin&( hearin& ai'( re)ote )onitorin&( "+tra'o"n i)a&in&()a&neti* re'onan*e i)a&in& $MI,( et*%

Image Processing 6D ani)ation( i)a&e enhan*e)ent( i)a&e *o)pre''ion $78EG,(

vieo *o)pre''ion $M8EG,( hi&h einition T3( et*%

Control Systems 4ea po'itionin& in i'* rive'( +a'er *ontro+ ( ro/ot'( en&ine an

)otor *ontro+'( et*%

Military and Aerospace aar an 'onar( navi&ation '#'te)'( 'e*"re *o))"ni*ation'(

)i''i+e &"ian*e( /att+eie+ 'en'or'( et*%


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Continuos-Time (CT) signals Discrete-Time (DT) signals

Continuous-time signal isindicating the denition ofindependent variable x(t) , (t) iscontinuous value, such asamplitude, frequency, and phase

!he discrete-time signals aredened only at discrete time, andthe independent variable x[n], n]ta"es only a discrete set ofvalues # integer values

Continuos-Time (CT) signals Discrete-Time (DT) signals

Signals Classifiati!n

$


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%

Deterministic vs. Random

Periodic vs Aperiodic

here Ti' the perio in 'e*on


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1"# $asi Signals

1. Real-valued exponential signals:

The ti)e *on'tant i' the ti)e re!"ire or the

eponentia+ t0e*rea'e /# a a*tor o 19 e( hi*h i'

approi)ate+# 0%6:;


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3. Sinusoids:

Where :

- A is the amplitude (A, V, W)

- w0= 2pf0 is the angular frequency (rad/s),

- f0 is the carrier frequency (Hz),

- Tis the period (s) ithf0= 1/T,

-f is the phase (rad)!

"sing #uler$s equation:

We can deri%e:

&ampled sinusoid: n = ',,,!,*

Where Tsis the sampling time and fs= 1/Ts is the sampling frequency!

+n practice, fs> 2 f0( yquist heorem)!(

)sin()( 0 += tAtx

)sin()cos( ajae ja =

)(

')sin( 000

tjtjee

jt

= )(

')cos( 000

tjtjeet +=

),sin()( 0 += snTfAny


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4. Step unction !"eaveside #nit unction)

Signals Analogue signal Digital signal

)tep function

Signals Analogue signal Digital signal

)tep function

*


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5. Rectangular Pulse

Signals Analogue signals Digital signals

+ectangularpulse

Signals Analogue signals Digital signals

+ectangular

pulse

1


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11

6. Unit Impulse function ( Dirc!delt impulse"

.roperties of irac-delta function:

i) #%en function : ,

ii) &ampling .roperty: =


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12

#. Si$n function


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13

1.4 $nergy-type vs Po%er-type

Signal energy

nergy signal .

/eterministic aperiodic signals are energy-type signals

Po!er signal

0eriodic and random signals are poer-type signals


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14

&.' $ven and (dd

i x(- t) = x(t)or a++ ( then the 'i&na+ i' even%

i x(-t) = -x(t)or a++ t( then the 'i&na+ i' o%


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1$

1"% Transf!r&ati!ns !f the In'epen'ent (aria)le

central concept in signal and system analysis is that of thetransformation of a signal

i) Time-shit

time-shift in continuous-time signal, hich to signals that are

identical in shape, but that are displaced or shifted relative to each

other here is represents a delayed if is positive, or advanced if

is negative


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1%

ii" Time #eversal%

Startin& ro) the 'i&na+ in *ontin"o"' ti)e( ti)e rever'a+ reer' to the

operation that &ive' "' the 'i&na+ that i' a re+e*tion o a/o"t


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1'

ii" Time Scaling%

Startin& ro) the 'i&na+ in *ontin"o"' ti)e( ti)e '*a+in& reer' to the

operation that &ive' "' the 'i&na+ that i'=

linearly stressed if ,

linearly compressed if and

reversed in time if


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1(

Examle


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1*

C!ntin*!*s+Ti&e S,ste&s

system is a mathematical model or a physical process that relates the input signal to

the output signal!

continuous-time system is a system in hich continuous-time input signals are

applied and result in continuous-time output signals! discrete-time system is a

system in hich discrete-time signals are applied and result in discrete-time output

signals!

he input %oltage produces an output %oltage!

1

E(t)V0(t)

input

output

&ysteminput outputx(t) y(t)


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2

Inter!nneti!n !f s,ste&s

'! &eries or cascade interconnection! he output of &ystem ' is the output to &ystem !

! .arallel interconnection

he same input signal is applied to &ystem ' and &ystem

)ystem 1 &ystem utput5ntput

&ystem '

&ystem

utput5ntput 2


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21

! 3om4ination of 4oth cascade and parallel interconnection!

5! 6eed4ac7 interconnection! he output of &ystem is fed 4ac7 and added to the e8ternal

input to produce the actual input to &ystem '

&ystem

9utput+ntput2

&ystem ' &ystem

&ystem 5

&ystem '

&ystem

9utput+ntput

2


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22

-r!perties !f S,ste&

- o sho that the system has one of the properties, it is generally necessary to sho

that the property holds for all possi4le input signals!

1" S,ste&s .ith an' .ith!*t &e&!r,

Me&!r,less s,ste&s!

he output at instant tfor continuous-time systems depends only on the

%alue of

input at the same instant t!

6or e8ample :

S,ste&s .ith &e&!r,!he output at instant tfor continuous-time systems depends not only on the

%alue

of input at the same instant t, 4ut also on past or future %alues of the input!

6or e8ample: !


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23

23

#8ample:

'! y(t)=3x(t) 4x2(t-1)

Sine t!e "#tp#t $epen$s "n #%%ent x(t) an$ past &a'#e "f inp#ts (x(t-1)) syste

is n"t e"%y'ess "% syste wit! e"%y

2 y(t) = 4x(t-1) * +x (t-2)

Sine t!e "#tp#t $epen$s "n past &a'#es "f inp#ts, syste is n"t e"%y'ess0


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#" Ca*sal S,ste&

system is causal if (and only if) the current output is only a function of present and

past inputs!

the current output does not depend on future inputs(or outputs)!

Anti+a*sal s,ste&

&ystem is anti-causal if is not causal!

3urrent output is depends on the future input!

;ost system are causal!

3ausality is important hen dealing ith online system 4ecause the system does

not 7no the future %alue! hus, it is impossi4le to compute the un7non %alue!

Hoe%er non-causal system is not a pro4lem to an offline system here the input

signal has 4een stored earlier!


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2$

#8ample:

'!y(t) = x(t*1)T!is syste is n"na#sa' sine t!e &a'#e y(t) "f t!e "#tp#t at tie t $epen$s "n t!e

&a'#e x(t*1) "f t!e inp#t at tie (t*1 )

y()=x(.), y $epen$s "n f#t#%e n"na#sa'

2 y(t) = x2

(t-1)

T!is syste is a#sa' sine t!e &a'#e "f t!e "#tp#t at tie t $epen$s "n'y "n t!e

&a'#e "f t!e inp#t at tie (t-1)


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2%

/" Ti&e+in0ariant an' ti&e+0ar,ing s,ste&s

Ti&e+in0ariant TI2 s,ste&s!

+ system is one in hich if y(t)is the output hen the input x(t)is applied, then

y(t t0) is the output hen x(t-t0) is applied!

system is TIif the 4eha%iour and the characteristics of the system are fi8ed o%er

time! 6or e8ample the system y(t) = ax(t)!

Ti&e+0ar,ing s,ste&s!

time-%arying system is one in hich if y(t)is the output hen the input x(t) is

applied, then y(t t0) is not necessarily the output hen x(t-t0)is applied!

6or e8ample consider the system y(t) = ax(t)+n that case if y1(t) is the output to

the input x1(t), then the output to the input

!

hus, the system is not time-in%ariant!


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2'

!he are to 6 transformation7 involved.

i8 )hifting a signalii8 0utting the signal through the system

system91:t8 y1:t8

shiftingy1:t-t8

92:t8

)hifting of 91:t8

system y2:t8;y2:t-t891:t-t8

8(t) y(t)

8(t-t0

) y(t-t0

)

system

3 Linear an' n!n+linear s,ste&s


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2(

3" Linear an' n!n+linear s,ste&s

Linear s,ste&s"

linear system is one that possesses the property of superposition, i!e!, if the input

consists of the sum of se%eral signals, then the output is the sum of the responses ofthe system to each of those signals!

linear system possesses the folloing to properties!

i) A$$iti&ity p%"pe%ty+f the response to x1(t) is y1(t)and the response to

x2(t)is y2(t)then the response to the signal x1(t) *x2(t) is y'(t) 2 y(t)

ii) ""eneity p%"pe%ty+f the response to x1(t)isy1(t), then the response

to the signal x1(t) is y1(t) here is any comple8 constant!

N!n+linear s,ste&s!

t least one of the a4o%e properties does not hold!

linear system is a system that posses the superposition property!


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2*

linear system is a system that posses the superposition property!

:


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3

'! &7etch the folloing functions!

a) x(t) = ) f(t) = t

c) y =-2t $) f(t) = 2-t

e)

=

0

)(tx

"t!e%wise

tt

0,0


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! &7etch the transformation of the signal as depicted 6igure '>

(a) (4)

6igure '

a) )( tx

4) )( +tx

c) )( +tx

d) )( + tx

e) 8(t/ -')

0 ' -'-

'

-'

x(t)

t0 ' -'-

'

-'

! he trapezoidal pulse 8(t) shon in 6igure 5 is defined 4y


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32

+

=

,0

?

,'

,?

)(t

t

tx

"t!e%wise

t

t

t

5?

55

?5

etermine the total energy of 8(t)!

6igure 5

5! etermine if the folloing systems are memoryless, causal, time-in%ariant or linear!

a) y(t) = x(4t - 3)

4) y(t) = sin (8(t))

c) )sin()'()( ++= ttxty hen 0

d) ))()(cos()( t#txty = !

e) )()()( = txtxty

g) )@(cos

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Chapter 1 control