Signals and Linear Systems

7/31/2019 Signals and Linear Systems

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9/3/2009 Dr. Ir. Risanuri Hidayat, M.Sc. 1

Signals and LinearSignals and LinearSystemsSystems

Dr.Dr. RisanuriRisanuri HidayatHidayat

9/3/2009 2Dr. Ir. Risanuri Hidayat, M.Sc.

Preview Preview review the linear systems analysis used in thereview the linear systems analysis used in theanalysis of communication systems.analysis of communication systems.

linear timelinear time--invariant (LTI) systemsinvariant (LTI) systemsFrequency domainFrequency domainFourier series and transforms;Fourier series and transforms;power and energy concepts,power and energy concepts,the sampling theorem,the sampling theorem,lowpasslowpass representation of representation of bandpassbandpass signals.signals.


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System

The input-output relation of a linear time-invariantsystem is integral defined by

h(t) denotes the impulse response of the system, x(t)is the input signal, and y (t) is the output signal


System

t f j Aet x 02)( =

)()(

)(

)()(

0

00

0

2

22

)(2

t xd he

ed he A

d h Aet y

f j

t f j f j

t f j

=

=

=

+

+

+


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Fourier Series

Any periodic signallx(t) with period T0can be expressed as

+

=

=n

T nt jne X t x 0

/ 2)(

the X n are called the Fourierseries coefficients of thesignal x(t) and are given by

+

=0

0 / 2

0

)(1

T T nt j

n dt et x

T

X


Fourier SeriesHere is an arbitrary constant chosen in such a

way that the computation of the integral issimplified.

The frequency f 0 =1/T 0 is called thefundamental frequency of the periodic signal,and the frequency f n = nf 0 is called the n-th

harmonic.In most cases either = 0 or =T 0/2 is a goodchoice.


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Deret FourierDeret Fourier

Jika Jika x(t)x(t) real, makareal, maka x x * * (t) =(t) =x(t)x(t)

GantiGanti n n dengandengan n n ,,didapatkandidapatkan a a - - n n =a =a * * n n atauataua a n n =a =a * * - - n n

+

=

+

=

=

==

n

t jnn

n

t jnn

e X t x

e X t xt x

0

0

*

**

)(

)()(

00

1

*

0

10

/ 2

)(

)(

00

00

T

e X e X at x

e X e X at x

n

t jn

n

t jn

n

n

t jnn

t jnn

=

++=

++=

+

=

+

=



{ }+

=

+=1

00Re2)(

n

t nk ne X at x

Penjumlahan konjugatePenjumlahan konjugatekompleks menghasilkankompleks menghasilkan

Jika Jika X X n n = A= A n n e e j j n n

Jika Jika X X n n = B= B n n + j C + j C n n

( )+

=

++=1

00 2)(n

nn t nCos Aat x

[ ]+

=

+=1

000 )()(2)(n

nn t nSinC t nCos Bat x


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+

=

+

=

+

=

=

=

=

n

T t k n j

n

T t jk

T

n

t jk t jnn

T t jk

n

t jk t jnn

t jk

dt e X dt et x

dt ee X dt et x

ee X et x

0

0

0

0

0

00

0

0

000

0

)(

0

00

).(

.).(

.).(



=

=0

0

0

0)(

,0

,T t k n jk n

k nT dt e

+

=

=

=

=

0

0

0

0

0

0

0

0

00

0

0

0

)(

0

).(1

.).(

).(

T t jk

k

k

T t jk

n

T t k n j

n

T t jk

dt et xT

X

sehingga

T X dt et x

dt e X dt et x


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KoefisienKoefisien a a kk disebut koefisien deret Fourier ataudisebut koefisien deret Fourier ataukoefisien spektralkoefisien spektral

Komponen dc = X Komponen dc = X 00 ::

+

=

=

=

0

0

0

00

).(1

.)(

T t jn

n

n

t jnn

dt et xT

X

e X t x

=0

)(10

0

T

dt t xT

X



Assuming A=1,T=4, and t 0= l ,1. Determine the Fourier series coefficients of x(t) in exponential and

trigonometric form.2. Plot the discrete spectrum of x(t).


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[ ]

x x xc

nc

een j

dt e X

n jn j

nt jn

)sin()(sin

2sin

2

1

2

1

4

1

4 / 24 / 2

1

1

4 / 2

=

=

=

=

+




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The discrete spectrum of the signal-10 -5 0 5 10

0

0.1

0.2

0.3

0.4

0.5



n=n= --10:10;10:10;y=0.5*sinc(pi*n/2);y=0.5*sinc(pi*n/2);stem(n,y); grid on;stem(n,y); grid on;

-10 -5 0 5 10-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-10 -5 0 5 100

0.1

0.2

0.3

0.4

0.5


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[The Magnitude and the Phase Spectra] Determine and plot themagnitude and the phase spectra of a periodic signal with aperiod equal to 12 (in the interval [-6, 6]) that is given by



The magnitude and phase spectra


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Periodic Signals and LTI Systems

When a periodic signal x (t) is passed through a lineartime-invariant (LTI) system, the output signal y(t) isalso periodic,usually with the same period as the input signal,therefore it has a Fourier series expansion.




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x(t) h(t)



The discrete spectrum of the signal x(t)


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The magnitude of H (n/2)



The discrete spectrum of the output


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Fourier Transforms

The inverse Fourier transform of X(f) is x(t)

If x (t) is a real signal, then X (f) satisfies theHermitian symmetry,


Fourier transform propertiesLinearity

Duality

Time shift:

Scaling


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Fourier transform propertiesModulation

Differentiation

Convolution


Fourier transform table


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Fourier transform table


Fourier transform examplePlot the magnitude and the phase spectra of signalsx1(t) and x2(t)


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Fourier transform example

|X1(f)| = |X2(f)|


Frequency-Domain Analysis of LTISystem

The output of an LTI system with impulse response h(t) whenthe input signal is x(t) is given by the convolution integral


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Power and Energy

A signal with finite energy is called an energy-typesignal, and a signal with positive andfinitepowerisapower-typesignal.x(t)= lI(t)is an example of an energy-type signal,

whereas x(t) = cos(t) is an example of a power-typesignal.

All periodic signals are power-type signals.


Energy and Power SignalsEnergy and Power SignalsConsiderConsider v(t v(t ) to be the voltage across a resistor ) to be the voltage across a resistorR producing a currentR producing a current i(ti(t ). The instantaneous ). The instantaneouspower p( t) per ohm is defined aspower p( t) per ohm is defined as

Total energy E and average power P on a per Total energy E and average power P on a per --ohm basis areohm basis are


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Energy and Power SignalsEnergy and Power Signals

For an arbitrary continuousFor an arbitrary continuous --time signaltime signal x(tx(t ), the ), thenormalized energy content E of normalized energy content E of x(tx(t ) is defined as ) is defined as

The normalized average power P of The normalized average power P of x(tx(t ) is ) isdefined asdefined as


Energy and Power SignalsEnergy and Power Signalsx(tx(t ) (or ) (or x[nx[n ]) is said to be an energy signal (or ]) is said to be an energy signal (orsequence) if and only if 0 < E


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Power and Energy The energy spectral density of an energy-type signalgives the distribution of energy at various frequenciesof the signal


Power and Energy

Time-average autocorrelation function

the autocorrelation function of x (t), defined as


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Power and Energy

The total power is the integral of the power-spectral density given by

The power-spectral density is in general given by

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Signals and Linear Systems