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:
18/21
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
20/21
Atousa Hajshirmohammadi, SFU
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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Atousa Hajshirmohammadi, SFU
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