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CTFT, DTFT and PropertiesMonday 12/04/2010

•Properties of CTFT•DTFS to DTFT transition•Discrete-time Fourier transforms (DTFT)•DTFT Properties•Relations among Fourier methods

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The “Uncertainty” PrincipleThe time and frequency scaling properties indicate that if a signal is expanded in one domain it is compressed in the other domain.This is called the “uncertainty principle” of Fourier analysis.

e−π t

2

2

F← → 2e−π 2 f( )2

e−πt2 F← → e−πf 2

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CTFT Properties

Transform of a Conjugate

x* t( ) F← → X * − f( )

x* t( ) F← → X * − jω( )

Multiplication-Convolution

Duality

x t( )∗y t( ) F← → X f( )Y f( )

x t( )∗y t( ) F← → X jω( )Y jω( )

x t( )y t( ) F← → X f( )∗ Y f( )

x t( )y t( ) F← →

12π

X jω( )∗Y jω( )

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Multiplication-convolution duality

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Transfer functionAn important consequence of multiplication-convolutionduality is the concept of the transfer function.

In the frequency domain, the cascade connection multipliesthe transfer functions instead of convolving the impulseresponses.

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CTFT Properties

Time Differentiation

ddt

x t( )( ) F← → j2πf X f( )

ddt

x t( )( ) F← → jω X jω( )

Modulation x t( )cos 2πf0t( ) F← →

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X f − f0( )+ X f + f0( )[ ]

x t( )cosω 0t( ) F← →

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X j ω − ω0( )( )+ X j ω +ω 0( )( )[ ]

Transforms ofPeriodic Signals

x t( )= X k[ ]e− j 2π kfF( )t

k =−∞

∞

∑ F← → X f( )= X k[ ]δ f − kf0( )k=−∞

∞

∑

x t( )= X k[ ]e− j kω F( )t

k =−∞

∞

∑ F← → X jω( )= 2π X k[ ]δ ω − kω0( )k=−∞

∞

∑

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CTFT Properties

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CTFT Properties

Parseval’s Theorem

x t( )2dt

−∞

∞

∫ = X f( )2df

−∞

∞

∫

x t( )2dt

−∞

∞

∫ =1

2πX jω( )2

df−∞

∞

∫

Integral Definitionof an Impulse

e− j2πxy

−∞

∞

∫ dy = δ x( )

Duality X t( ) F← → x − f( ) and X −t( ) F← → x f( )

X jt( ) F← → 2π x −ω( ) and X − jt( ) F← → 2π x ω( )

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CTFT Properties

Total-AreaIntegral

X 0( )= x t( )e− j2πftdt−∞

∞

∫

f →0

= x t( )dt−∞

∞

∫

x 0( )= X f( )e+ j 2πftdf−∞

∞

∫

t→0

= X f( )df−∞

∞

∫

X 0( )= x t( )e− jωtdt−∞

∞

∫

ω→0

= x t( )dt−∞

∞

∫

x 0( )=1

2πX jω( )e+ jωtdω

−∞

∞

∫

t→0

=1

2πX jω( )dω

−∞

∞

∫

Integration x λ( )dλ

−∞

t

∫F← →

X f( )j2πf

+12

X 0( )δ f( )

x λ( )dλ

−∞

t

∫F← →

X jω( )jω

+ π X 0( )δ ω( )

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CTFT Properties

x 0( )= X f( )df−∞

∞

∫

X 0( )= x t( )dt−∞

∞

∫

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CTFT Properties

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Extending the DTFS

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

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

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DT Pulse Train

This DT periodic rectangular-wave signal is analogous to theCT periodic rectangular-wave signal used to illustrate the transition from the CTFS to the CTFT.

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DTFS of DT Pulse Train

As the period of the rectangular wave increases, the period of the DTFS increases and the amplitude of the DTFS decreases.

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Normalized DTFS of DT Pulse Train

By multiplying the DTFS by its period and plotting versus instead of k, the amplitude of the DTFS stays the same as the period increases and the period of the normalized DTFS stays at one.

kF0

22X0.5

1/22

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DTFS-to-DTFT TransitionThe normalized DTFS approaches this limit as the DTperiod approaches infinity.

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Definition of the DTFT

x n[ ]= X F( )e j 2πFndF

1∫F← → X F( )= x n[ ]e− j 2πFn

n=−∞

∞

∑

F Form

x n[ ]=

12π

X jΩ( )e jΩndΩ2π∫

F← → X jΩ( )= x n[ ]e− jΩn

n=−∞

∞

∑

Ω Form

ForwardInverse

ForwardInverse

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Linearity

α x n[ ]+ β y n[ ] F← → α X F( )+ β Y F( )

α x n[ ]+ β y n[ ] F← → α X jΩ( )+ β Y jΩ( )

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Time Shifting

x n − n0[ ] F← → e− jΩn0 X jΩ( )

x n − n0[ ] F← → e− j 2πFn0 X F( )

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Frequency Shifting

ej2πF0n x n[ ] F← → X F − F0( )

ejΩ0n x n[ ] F← → X j Ω − Ω0( )( )

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Time Reversal

x −n[ ] F← → X −F( )

x −n[ ] F← → X − jΩ( )

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Differencing

x n[ ]− x n −1[ ] F← → 1− e− j 2πF( )X F( )

x n[ ]− x n −1[ ] F← → 1− e− jΩ( )X jΩ( )

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Accumulation

x m[ ]m=−∞

n

∑ F← → X F( )

1− e− j2πF +12

X 0( )comb F( )

x m[ ]

m=−∞

n

∑ F← → X jΩ( )

1− e− jΩ +12

X 0( )combΩ2π

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Multiplication-Convolution Duality

As in other transforms, convolution in the time domain is equivalent to multiplication in the frequency domain

x n[ ]∗y n[ ] F← → X F( )Y F( ) x n[ ]∗y n[ ] F← → X jΩ( )Y jΩ( )

x n[ ]y n[ ] F← →

12π

X jΩ( ) Y jΩ( ) x n[ ]y n[ ] F← → X F( ) Y F( )

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DTFT Properties

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DTFT Properties

AccumulationDefinition of a Comb Function

e j 2πFn

n=−∞

∞

∑ = comb F( )

The signal energy is proportional to the integral of the squared magnitude of the DTFT of the signal over one period.

Parseval’sTheorem x n[ ]2

n=−∞

∞

∑ =1

2πX jΩ( )2

dΩ2π∫

x n[ ]2

n=−∞

∞

∑ = X F( )2dF

1∫

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The Four Fourier Methods

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Relations Among Fourier MethodsDiscrete Frequency Continuous Frequency

CT

DT

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CTFT - CTFS Relationship

X f( )= X k[ ]δ f − kf0( )k=−∞

∞

∑

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CTFT - CTFS Relationship

X p k[ ]= fp X kfp( )

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CTFT - DTFT Relationship

Let xδ t( )= x t( ) 1Ts

combt

Ts

= x nTs( )δ t − nTs( )n=−∞

∞

∑

and let x n[ ]= x nTs( )

X DTFT F( )= Xδ fs F( ) Xδ f( )= XDTFT

f

fs

X DTFT F( )= fs XCTFT fs F − k( )( )k =−∞

∞

∑

There is an “information equivalence” between and . They are both completely described bythe same set of numbers.

xδ t( )x n[ ]

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CTFT - DTFT Relationship

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DTFS - DTFT Relationship

X F( )= X k[ ]δ F − kF0( )k=−∞

∞

∑

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DTFS - DTFT Relationship

X p k[ ]=1

Np

X kFp( )

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Duties for next lecture 24/04/2010

• No lecture on Wednesday• No class next week• Study exercises with Fourier transforms

from the book chapter 5.9 (pg 353 - 370)