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Fourier Fourier Representation Representation of Signals and of Signals and LTI Systems.LTI Systems.

CHAPTER CHAPTER 33

EKT 232 EKT 232

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Signals are represented as superposition's of complex sinusoids which leads to a useful expression for the system output and provide a characterization of signals and systems.

Example in music, the orchestra is a superposition of sounds generated by different equipment having different frequency range such as string, base, violin and ect. The same example applied the choir team.

Study of signals and systems using sinusoidal Study of signals and systems using sinusoidal representation is termed as representation is termed as Fourier AnalysisFourier Analysis introduced by Joseph Fourier (1768-1830).introduced by Joseph Fourier (1768-1830).

There are four distinct Fourier representations, each applicable to different class of signals.

3.1 Introduction.3.1 Introduction.

3

Fourier Series

Discrete Time Fourier series (DTFS)

0

0

12 / 2 /1

x X X xF

F F

F

n Nj kn N j kn N

Fk N n nF

n k e k n eN

where is the representation time and the notation means

a summation over any range of consecutive ’s exactly in length.F

F k N

F

N

k N

Fourier SeriesNotice that in

2 /x X F

F

j kn NF

k N

n k e

the summation is over exactly one period, a finite summation. This is because of the periodicity of the complex sinusoid,

2 / Fj kn Ne

in harmonic number . That is, if is increased by any integer

multiple of the complex sinusoid does not change. F

k k

N

2 /2 / , ( an integer)F FF j k mN n Nj kn Ne e m

This occurs because discrete time n is always an integer.

5

Fourier Series

0 0

0 0

2 / 2 /

0

1x X X xj kn N j kn N

k N n N

n k e k n eN

F S

0

In the very common case in which the representation time is

taken as the fundamental period the DTFS isN

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CT Fourier Series Definition

0 0

The Fourier series representation x t of a signal x( )

over a time isF

F

t

t t t T

2x X Fj kf tF

k

t k e

where X[k] is the , is the

and 1/ . The harmonic function

can be found from the signal asF F

k

f Tharmonic function harmonic

number

0

0

21X x

F

F

t Tj kf t

F t

k t e dtT

The signal and its harmonic function form a

indicated by the notation x X .t k

Fourier series

pair F S

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CTFS Properties

Linearity

x y X Yt t k k F S

0

0

Let a signal x( ) have a fundamental period and let a

signal y( ) have a fundamental period . Let the CTFS

harmonic functions, each using a common period as the

representation time, be X[ ] a

x

y

F

t T

t T

T

k nd Y[ ]. Then the following

properties apply.

k

Dr. Abid Yahya

04/20/23 8

CTFS Properties

Time Shifting 0 02

0x Xj kf tt t e k F S

0 00x Xjk tt t e k F S

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CTFS Properties

Frequency Shifting (Harmonic Number

Shifting)

0 020x Xj k f te t k k F S

0 00x Xjk te t k k F S

A shift in frequency (harmonic number) corresponds to multiplication of the time function by a complex exponential.

Time Reversal x Xt k F S

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CTFS Properties

Change of Representation Time

0With , x XF xT T t k F S

X / , / an integerX

0 , otherwisem

k m k mk

(m is any positive integer)

0With , x XF x mT mT t k F S

Dr. Abid Yahya

04/20/23 11

CTFS PropertiesChange of Representation Time

04/20/23 12

CTFS Properties

Time Differentiation

0

0

x 2 X

x X

dt j kf k

dtd

t jk kdt

F S

F S

F S

F S

04/20/23 . J. Roberts - All Rights Reserved

13

Time Integration

Case 1. X 0 0

0

Xx

2

t kd

j kf

F S

0

Xx

t kd

j k

F S

Case 2. X 0 0

xt

d is not periodic

CTFS PropertiesCase 1 Case 2

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CTFS PropertiesMultiplication-Convolution Duality

x y X Yt t k k F S

(The harmonic functions, X[ ] and Y[ ], must be based

on the same representation period .)F

k k

T

0x y X Yt t T k kF S#

0

x y x yT

t t t d #

x t y t xap t y t where xap t is any single period of x t

The symbol indicates .

Periodic convolution is defined mathematically by

periodic convolution#

04/20/23 15

Fourier Series(DTFS)

0

0

12 / 2 /1

x X X xF

F F

F

n Nj kn N j kn N

Fk N n nF

n k e k n eN

where is the representation time and the notation means

a summation over any range of consecutive ’s exactly in length.F

F k N

F

N

k N

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Notice that in

2 /x X F

F

j kn NF

k N

n k e

the summation is over exactly one period, a finite summation. This is because of the periodicity of the complex sinusoid,

2 / Fj kn Ne

in harmonic number . That is, if is increased by any integer

multiple of the complex sinusoid does not change. F

k k

N

2 /2 / , ( an integer)F FF j k mN n Nj kn Ne e m

This occurs because discrete time n is always an integer.

Fourier Series(DTFS)

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0 0

0 0

2 / 2 /

0

1x X X xj kn N j kn N

k N n N

n k e k n eN

F S

0

In the very common case in which the representation time is

taken as the fundamental period the DTFS isN

Fourier Series(DTFS)

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DTFS Properties0

0

Let a signal x[ ] have a fundamental period and let a

signal y[ ] have a fundamental period . Let the DTFS

harmonic functions, each using a common period as

the representation time, be X[ ] a

x

y

F

n N

n N

N

k nd Y[ ]. Then the following

properties apply.

k

Linearity

x y X Yt t k k F S

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DTFS Properties

Time Shifting 02 /0x XFj kn Nn n e k F S

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DTFS Properties

x Xn k Time Reversal F S

Frequency Shifting(Harmonic Number

Shifting) 02 /

0x XFj k n Ne n k k F S

* * x Xn k Conjugation F S

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DTFS PropertiesTime Scaling

Let z x , 0n an a If a is not an integer, some values of z[n] are undefinedand no DTFS can be found. If a is an integer (other than1) then z[n] is a decimated version of x[n] with some values missing and there cannot be a unique relationshipbetween their harmonic functions. However, if

x / , / an integerz

0 , otherwise

n m n mn

then

0Z 1/ X , Fk m k N mN

04/20/23 22

DTFS Properties

Change of Representation Time

0

0

With , x X

With , x X

F x

F x q

N N n k

N qN n k

F S

F S

(q is any positive integer)

X / , / an integerX

0 , otherwiseq

k q k qk

04/20/23 23

DTFS Properties

First Backward Difference

2 /x x 1 1 XFj k Nn n e k F S

Multiplication-Convolution

Duality

0

0

x y Y X Y X

x y Y X

q N

n n k k q k q

n n N k k

F S

F S

#

#

Dr. Abid Yahya

The Fourier Transform

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Extending the CTFS

• The CTFS is a good analysis tool for systems with periodic excitation but the CTFS cannot represent an aperiodic signal for all time

• The continuous-time Fourier transform (CTFT) can represent an aperiodic signal for all time

Dr. Abid Yahya

04/20/23 26

Forward Inverse

2X x x j ftf t t e dt

F -1 2x X X j ftt f f e df

F

f form

X x j tj t x t e dt

F -1 1x X X

2j tt j j e d

F

formForward Inverse

Definition of the CTFT

x Xt fF x Xt jFor

Commonly-used notation:

04/20/23 27

Some Remarkable Implications of the Fourier Transform

The CTFT expresses a finite-amplitude, real-valued, aperiodic signal which can also, in general, be time-limited, as a summation (an integral) of an infinite continuum of weighted, infinitesimal-amplitude, complex sinusoids, each of which is unlimited in time. (Time limited means “having non-zero values only for a finite time.”)

The Discrete-Time Fourier Transform

04/20/23 29

Extending the DTFS

• Analogous to the CTFS, the DTFS is a good analysis tool for systems with periodic excitation but cannot represent an aperiodic signal for all time

• The discrete-time Fourier transform (DTFT) can represent an aperiodic signal for all time

Dr. Abid Yahya

04/20/23 30

Definition of the DTFT

2 2

1x X X xj Fn j Fn

n

n F e dF F n e

F

F Form

2

1x X X x

2j j n j j n

n

n e e d e n e

F

Form

ForwardInverse

ForwardInverse

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The Four Fourier Methods

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Relations Among Fourier Methods

Discrete Frequency Continuous Frequency

Continuous Time x y X Y x y X Y

Discrete Time x y Y X x y X Y

t t k k t t f f

n n k k n n F F

F S F

F S F# #

0

0

Discrete Frequency Continuous Frequency

Continuous Time x y X Y x y X Y

Discrete Time x y Y X x y X Y

t t T k k t t f f

n n N k k n n F F

F S F

F S F

#

#

Multiplication-Convolution Duality

04/20/23 33

Relations Among Fourier Methods

0 0 0

0 0 0

0 0

0 0

Discrete Frequency Continuous Frequency

Continuous Time x X x X

Discrete Time x X x X

j k t j t

j k n j nj

t t k e t t j e

n n k e n n e e

F S F

F S F

Time and Frequency Shifting

0 0 0

0 0 00

0 0

0

Discrete Frequency Continuous Frequency

Continuous Time x X x X

Discrete Time x X x X

j k t j t

j k n jj n

t e k k t e j

n e k k n e e

F S F

F S F

Dr. Abid Yahya

04/20/23 34

Tutorials1. Compute the CTFS:

( ) 4cos(500 )x t t 1/ 50FT

,

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2.2. Find the Find the frequency-domainfrequency-domain representation of the representation of the signal in Figure 3.1 below.signal in Figure 3.1 below.

Figure 3.1: Time Domain Signal.Figure 3.1: Time Domain Signal.

Solution:Solution:Step 1Step 1: Determine N and : Determine N and ..

The signal has period N=5, so =2/5.

Also the signal has odd symmetry, so we sum over n = -2 to n = 2 from equation

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Step 2Step 2: Solve for the frequency-domain, : Solve for the frequency-domain, XX[[kk].].From step 1, we found the fundamental frequency, N

=5, and we sum over n = -2 to n = 2 .

5/45/205/25/4

5/22

2

1

0

210125

1

5

1

1

jkjkjjkjk

njk

n

njkN

n

exexexexex

enxkX

enxN

kX o

Cont’d…Cont’d…

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From the value of x{n} we get,

5/2sin15

1

2

1

2

11

5

1 5/25/2

kj

eekX jkjk

Cont’d…Cont’d…