# Fourier Analysis of Signals and PowerPoint Presentation, PPT - DocSlides 2018-12-18 2K 2 0 0

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Systems. Dr. Babul Islam. Dept. of Applied Physics and Electronic Engineering. University of Rajshahi. 1. Outline . Response of LTI system in time domain. Properties of LTI systems. Fourier analysis of signals. ID: 743270

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Slide1

Fourier Analysis of Signals and Systems

Dr. Babul IslamDept. of Applied Physics and Electronic EngineeringUniversity of Rajshahi

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Slide2

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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Slide3

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

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Linear System:

+

T

+

T

T

System,

T

is linear if and only if

i.e.,

T

satisfies the superposition principle.

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Time-Invariant System:

A system T is time invariant if and only if

T

implies that

T

Example: (a)

Since

, the system is time-invariant.

(b)

Since

, the system is time-variant.

5

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Any input signal x

(n) can be represented as follows:

Consider an LTI system

T

.

1

0

n

1

2

-1

-2

Graphical representation of unit impulse.

T

T

Now, the response of

T

to the unit impulse is

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.,

T

(LTI)

Applying the time-invariant property, we have

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Slide8

Properties of LTI systems

(Properties of convolution)Convolution is commutative

x[n]

 h[n] = h[n]  x[n]

Convolution is distributive

x[n]  (h

1

[n] + h

2

[n]) = x[n]  h

1

[n] + x[n]  h

2

[n]

8

Slide9

Convolution is Associative:

y[n] = h

1

[n] 

[

h

2[n]  x

[n] ] = [ h

1

[n] 

h

2

[n]

]

 x

[n]

h

2

x[n]

y[n]

h

1

h

2

x[n]

y[n]

h

1

=

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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,

This signal is completely characterized by three parameters:

A

= Amplitude of the sinusoid

= Angular frequency in radians/sec =

2

f

Fourier Series for Continuous-Time Periodic Signals:

A

A

cos

t

0

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Complex representation of sinusoidal signals:

Fourier series of any periodic signal is given by:

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

where

where

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Example:

0

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Power Density Spectrum of Continuous-Time Periodic Signal:

This is Parseval’s relation.

represents the power in the

n

-th harmonic component of the signal.

Power spectrum of a CT periodic signal.

If is real valued, then , i.e.,

Hence, the power spectrum is a symmetric function of frequency.

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Define as a periodic extension of

x

(

t

):

Fourier Transform for Continuous-Time

Aperiodic

Signal:

Assume

x

(

t

) has a finite duration.

Therefore, the Fourier series for :

where

Since for and outside this interval, then

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Now, defining the envelope of as

Therefore, can be expressed as

As

Therefore, we get

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Energy Density Spectrum of Continuous-Time Aperiodic Signal:

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.

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Fourier Series for Discrete-Time Periodic Signals:

Consider a discrete-time periodic signal with period

N

.

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

where

Since

Thus the spectrum of is also periodic with period

N

.

Consequently, any

N

consecutive samples of the signal or its spectrum provide a complete description of the signal in the time or frequency domains.

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Power Density Spectrum of Discrete-Time Periodic Signal:

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Fourier Transform for Discrete-Time Aperiodic Signals:

The Fourier transform of a discrete-time aperiodic signal is given by

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

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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:

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Energy Density Spectrum of Discrete-Time Aperiodic Signal:

represents the distribution of energy in the signal as a function of frequency, i.e., the energy density spectrum.

If is real, then

(even symmetry)

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

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Frequency Response of an LTI SystemFor continuous-time LTI system

For discrete-time LTI system

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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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