Fourier Analysis of Signals and

Systems

Dr. Babul Islam

Dept. of Applied Physics and Electronic

Engineering

University of Rajshahi1

Outline

• Response of LTI system in time domain

• Properties of LTI systems

• Fourier analysis of signals

• Frequency response of LTI system

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• A system satisfying both the linearity and the time-invariance properties.

• LTI systems are mathematically easy to analyze and characterize, and consequently, easy to design.

• Highly useful signal processing algorithms have been developed utilizing this class of systems over the last several decades.

• They possess superposition theorem.

Linear Time-Invariant (LTI) Systems

3

• Linear System:

+ T

)(1 nx

)(2 nx

1a

2a

][][)( 2211 nxanxany T

][][)( 2211 nxanxany TT +

)(1 nx

)(2 nx

1a

2aT

T

System, T is linear if and only if

i.e., T satisfies the superposition principle.

)()( nyny 4

• Time-Invariant System:A system T is time invariant if and only if

)(nx T )(ny

implies that)( knx T )(),( knykny

Example: (a)

)1()()(

)1()(),(

)1()()(

knxknxkny

knxknxkny

nxnxny

Since )(),( knykny , the system is time-invariant.

(b)

][)()(

][),(

][)(

knxknkny

knnxkny

nnxny

Since )(),( knykny , the system is time-variant. 5

• Any input signal x(n) can be represented as follows:

k

knkxnx )()()(

• Consider an LTI system T. 1

0for ,0

0for ,1][

n

nn

0 n1 2-1-2 ……

Graphical representation of unit impulse.

)( kn T ),( knh

)(n T )(nh

• Now, the response of T to the unit impulse is

)(nx T ),()(][)( knhkxnxnyk

T

• Applying linearity properties, we have

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• LTI system can be completely characterized by it’s impulse response.

• Knowing the impulse response one can compute the output of the system for any arbitrary input.

• Output of an LTI system in time domain is convolution of impulse response and input signal, i.e.,

)()()()()( khkxknhkxnyk

)(nx T(LTI)

)()(),()()( knhkxknhkxnykk

• Applying the time-invariant property, we have

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Properties of LTI systems (Properties of convolution)

• Convolution is commutative

x[n] h[n] = h[n] x[n]

• Convolution is distributive

x[n] (h1[n] + h2[n]) = x[n] h1[n] + x[n] h2[n]

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• Convolution is Associative:

y[n] = h1[n] [ h2[n] x[n] ] = [ h1[n] h2[n] ] x[n]

h2x[n] y[n]

h1h2x[n] y[n]

h1

=

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Frequency Analysis of Signals

• Fourier Series

• Fourier Transform

• Decomposition of signals in terms of sinusoidal or complex exponential components.

• With such a decomposition a signal is said to be represented in the frequency domain.

• For the class of periodic signals, such a decomposition is called a Fourier series.

• For the class of finite energy signals (aperiodic), the decomposition is called the Fourier transform.

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Consider a continuous-time sinusoidal signal,

)cos()( tAty

This signal is completely characterized by three parameters:

A = Amplitude of the sinusoid

= Angular frequency in radians/sec = 2f

= Phase in radians

• Fourier Series for Continuous-Time Periodic Signals:

AAcos

t

)cos()( tAty

0

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Complex representation of sinusoidal signals:

,2

)cos()( )()( tjtj eeA

tAty sincos je j

Fourier series of any periodic signal is given by:

1 1

000 cossin)(n n

nn tnbtnaatx

Fourier series of any periodic signal can also be expressed as:

n

tjnnectx 0)(

where

Tn

Tn

T

tdtntxT

b

tdtntxT

a

dttxT

a

0

0

0

cos)(2

sin)(2

)(1

where T

tjnn dtetxT

c 0)(1

12

Example:

T

n

T

tdtntxT

a

dttxT

a

0

00

0sin)(2

0)(1

11, 7, ,3for ,4

9, 5, ,1for ,4

2sin

4cos)(

20

nn

nnn

ntdtntx

Tb

T

n

02

T

2

T TT t

)(tx1

1

ttttx

5cos

5

13cos

3

1cos

4)(

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• Power Density Spectrum of Continuous-Time Periodic Signal:

n

nTcdttx

TP

22)(

1

• This is Parseval’s relation.

• represents the power in the n-th harmonic component of the signal.2

nc

2

nc

2 323 0

Power spectrum of a CT periodic signal.

• If is real valued, then , i.e., )(tx *nn cc

22

nn cc

• Hence, the power spectrum is a symmetric function

of frequency.

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2

22)(

)(~T

tperiodic

Tt

Ttx

tx

• Define as a periodic extension of x(t):)(~ tx

n

tjnnectx 0)(~

2/

2/

0)(~1 T

T

tjnn dtetxT

c

dtetxT

dtetxT

c tjnT

T

tjnn

00 )(1

)(1 2/

2/

• Fourier Transform for Continuous-Time Aperiodic Signal:

• Assume x(t) has a finite duration.

• Therefore, the Fourier series for :)(~ tx

where

• Since for and outside this interval, then

)()(~ txtx 22 TtT 0)( tx

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.)( toapproaches )(~ and variable)s(continuou ,0, 00 txtxnT

dtetxT

X tj )(1

)(

• Now, defining the envelope of as)(X nTc

)(1

0nXT

cn

n

tjn

n

tjn enXenXT

tx 00000 )(

2

1)(

1)(~

• Therefore, can be expressed as)(~ tx

• As

• Therefore, we get

deXtx tj)(

2

1)(

dtetxT

X tj )(1

)(

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• Energy Density Spectrum of Continuous-Time Aperiodic Signal:

dXdttxE

22)()(

dXXdX

dtetxdX

deXdttx

dttxtxE

tj

tj

2*

*

*

*

)()()(

)(2

1)(

)(2

1)(

)()(

• This is Parseval’s relation which agrees

the principle of conservation of energy in

time and frequency domains.

• represents the distribution of

energy in the signal as a function of

frequency, i.e., the energy density

spectrum.

2)(X

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• Fourier Series for Discrete-Time Periodic Signals:• Consider a discrete-time periodic signal with period N. )(nx

nnxNnx allfor )()(

• Now, the Fourier series representation for this signal is given by

1

0

/2)(N

k

Nknjkecnx

where

1

0

/2)(1 N

n

Nknjk enxN

c

• Since k

N

n

NknjN

n

NnNkjNk cenx

Nenx

Nc

1

0

/21

0

/)(2 )(1

)(1

• Thus the spectrum of is also periodic with period N. )(nx

• Consequently, any N consecutive samples of the signal or its spectrum provide a complete description of the signal in the time or frequency domains. 18

• Power Density Spectrum of Discrete-Time Periodic Signal:

k

kn

cnxN

P2

0

2)(

1

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• Fourier Transform for Discrete-Time Aperiodic Signals:• The Fourier transform of a discrete-time aperiodic signal is given by

n

njenxX )()(

• Two basic differences between the Fourier transforms of a DT and

CT aperiodic signals.

• First, for a CT signal, the spectrum has a frequency range of

In contrast, the frequency range for a DT signal is unique over the

range since

.,

,2,0 i.e., ,,

)()()(

)()()2(

2

)2()2(

Xenxeenx

enxenxkX

n

nj

n

knjnj

n

nkj

n

nkj

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• Second, since the signal is discrete in time, the Fourier transform

involves a summation of terms instead of an integral as in the case

of CT signals.

• Now can be expressed in terms of as follows:)(nx )(X

nm

nmmxdenx

deenxdeX

nmj

n

mj

n

njmj

,0

),(2)(

)()(

)(

deXnx nj)(2

1)(

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• Energy Density Spectrum of Discrete-Time Aperiodic Signal:

dXnxE

n

22)(

2

1)(

• represents the distribution of energy in the signal as a function of

frequency, i.e., the energy density spectrum.

2)(X

• If is real, then)(nx .)()(* XX

)()( XX (even symmetry)

• Therefore, the frequency range of a real DT signal can be limited further to

the range .0

22

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Frequency Response of an LTI System

• For continuous-time LTI system

• For discrete-time LTI system

][nhnje njeH

n cos HnH cos

)(th

tje tjeH

HtH cos t cos

Conclusion

• The response of LTI systems in time domain has been examined.

• The properties of convolution has been studied.

• The response of LTI systems in frequency domain has been analyzed.

• Frequency analysis of signals has been introduced.

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