ENSC380Lecture 16

Objectives:

• Learn the properties of CTFT

• Find CTFT of different signals using these properties

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Atousa Hajshirmohammadi, SFU

CTFT Properties

• Here we list the properties of CTFT.

• Using these properties and the FT pairs listed in Appendix E, we should beable to find the CTFT of many signals.

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Atousa Hajshirmohammadi, SFU

CTFT PropertiesIf F x t( )( )= X f( ) or X jω( ) and F y t( )( )= Y f( ) or Y jω( )then the following properties can be proven.

Linearity α x t( )+ β y t( ) F← → ⎯ α X f( )+ β Y f( )

α x t( )+ β y t( ) F← → ⎯ α X jω( )+ β Y jω( )

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CTFT Properties

Time Shifting

x t − t0( ) F← → ⎯ X f( )e− j 2πft0

x t − t0( ) F← → ⎯ X jω( )e− jωt0

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CTFT Properties

x t( )e+ j2πf0t F← → ⎯ X f − f0( )

Frequency Shifting

x t( )e+ jω0t F← → ⎯ X ω − ω0( )

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CTFT Properties

Time Scaling x at( ) F← → ⎯

1a

Xfa

⎛ ⎝

⎞ ⎠

x at( ) F← → ⎯

1a

X jωa

⎛ ⎝

⎞ ⎠

Frequency Scaling

1a

xta

⎛ ⎝

⎞ ⎠

F← → ⎯ X af( )

1a

xta

⎛ ⎝

⎞ ⎠

F← → ⎯ X jaω( )

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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−πt2 F← → ⎯ e−πf 2

e−π t

2⎛ ⎝

⎞ ⎠

2

F← → ⎯ 2e−π 2 f( )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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CTFT Properties 9/21

CTFT PropertiesAn 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← → ⎯

12

X f − f0( )+ X f + f0( )[ ]

x t( )cos ω 0t( ) F← → ⎯

12

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 kωF( )t

k =−∞

∞

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

∞

∑

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CTFT Properties 12/21

CTFT Properties

Parseval’s Theorem

x t( )2 dt−∞

∞

∫ = X f( )2 df−∞

∞

∫

x t( )2 dt−∞

∞

∫ =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 PropertiesX 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−∞

∞

∫Total-Area

Integral

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 16/21

Example 1

If x(t)F←→ X(f), what are the FTs of x(2(t− 1)) and x(2t− 1)?

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Atousa Hajshirmohammadi, SFU

Example 2

Find the FT of 10 sin(t) ∗ 2δ(t + 4), using two different approaches:

• Approach 1: Find the convolution first and then the FT:

• Approach 2: Use the convolution property of FT:

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Atousa Hajshirmohammadi, SFU

Example 3

• Find the FT of δ(t) directly (do not use the table).

• Now find the FT of

x(t) =∞∑

k=−∞

δ(t− kT )

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Atousa Hajshirmohammadi, SFU

Example 4

What is the Inverse FT (IFT) of e−4|f |?

– p. 6/7

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Example5

We have previously seen that the impulse response of an RC lowpass filter ish(t) = e−t/τ u(t). If the input voltage to this filter is vi(t) = e−btu(t) (b > 0), find theoutput voltage vo(t). Use the convolution property of FT.

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