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Transcript of Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of...
Signals and SystemsProf. H. Sameti
Chapter 4: The Continuous Time Fourier Transform• Derivation of the CT Fourier Transform pair• Examples of Fourier Transforms Topic three• Fourier Transforms of Periodic Signals• Properties of the CT Fourier Transform• The Convolution Property of the CTFT • Frequency Response and LTI Systems Revisited • Multiplication Property and Parseval’s Relation• The DT Fourier Transform
Book Chapter4: Section1
2
Fourier’s Derivation of the CT Fourier Transform
x(t) - an aperiodic signalview it as the limit of a periodic signal as T → ∞
For a periodic signal, the harmonic components are spaced ω0 = 2π/T apart ...
As T → ∞, ω0 → 0, and harmonic components are spaced closer and closer in frequency
Computer Engineering Department, Signals and Systems
Fourier series Fourier integral
Book Chapter4: Section1
3
Motivating Example: Square wave
Computer Engineering Department, Signals and Systems
increaseskept fixed
Discrete frequency points become denser in ω as T increases
0
0 1
0
1
2sin( )
2sin( )
k
k
k
k Ta
k T
TTa
Book Chapter4: Section1
4
So, on with the derivation ...
Computer Engineering Department, Signals and Systems
For simplicity, assumex(t) has a finite duration.
( ),2 2( )
,2
T Tx t t
x tT
periodic t
As , ( ) ( ) for all T x t x t t
Book Chapter4: Section1
5
Derivation (continued)
Computer Engineering Department, Signals and Systems
0
0 0
0
0
2 2
2 2
0
2( ) ( )
1 1( ) ( )
( ) ( ) in this interval
1( ) (1)
If we define
( ) ( )
then Eq.(1)
( )
jk tk
k
T T
jk t jk tk
T T
jk t
j t
k
x t a eT
a x t e dt x t e dtT T
x t x t
x t e dtT
X j x t e dt
X jka
T
Book Chapter4: Section1
6
Derivation (continued)
Computer Engineering Department, Signals and Systems
0
0
0
0 0
0
Thus, for 2 2
1( ) ( ) ( )
1( )
2
As , , we get the CT Fourier Transform pair
1( ) ( ) Synthesis equation
2
( ) ( ) A
k
jk t
k
a
jk t
k
j t
j t
T Tt
x t x t X jk eT
X jk e
T d
x t X j e d
X j x t e dt
nalysis equation
Book Chapter4: Section1
7
For what kinds of signals can we do this?
(1) It works also even if x(t) is infinite duration, but satisfies:a) Finite energy
In this case, there is zero energy in the error
b) Dirichlet conditions
c) By allowing impulses in x(t)or in X(jω), we can represent even more Signals
E.g. It allows us to consider FT for periodic signals
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2( )x t dt
21( ) ( ) ( ) Then ( ) 0
2j te t x t X j e d e t dt
1 (i) ( ) ( ) at points of continuity
21
(ii) ( ) midpoint at discontinuity2
(iii) Gibb's phenomenon
j t
j t
X j e d x t
X j e d
Book Chapter4: Section1
8
Example #1
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0
0
0
( ) ( ) ( )
( ) ( ) 1
1( ) Synthesis equation for ( )
2( ) ( ) ( )
( ) ( )
j t
j t
j t
j t
a x t t
X j t e dt
t e d t
b x t t t
X j t t e dt
e
Book Chapter4: Section1
9
Example #2: Exponential function
Computer Engineering Department, Signals and Systems
( )0
( )
( ) ( ), 0
( ) ( )
1 1( )
0
a j t
at
j t at j t
e
a j t
x t e u t a
X j x t e dt e e dt
ea j a j
Even symmetry Odd symmetry
Book Chapter4: Section1
10
Example #3: A square pulse in the time-domain
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1
1
12sin( )
T j t
T
TX j e dt
Note the inverse relation between the two widths Uncertainty principle⇒
Book Chapter4: Section1
11
Useful facts about CTFT’s
Computer Engineering Department, Signals and Systems
1
(0) ( )
1(0) ( )
2
Example above: ( ) 2 (0)
1Ex. above: (0) ( )
21
(Area of the triangle)2
X x t dt
x X j d
x t dt T X
x X j d
Book Chapter4: Section1
12
Example #4:
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2
A Gaussian, important in probability, optics, etc.( ) atx t e2
2 2 2
22
2
[ ( ) ] ( )2 2
( )2 4
4
( )
[ ].
at j t
j ja t j t a
a a a
ja t
a a
a
a
X j e e dt
e dt
e dt e
ea
Also a Gaussian!
(Pulse width in t)•(Pulse width in ω) ∆⇒ t•∆ω ~ (1/a1/2)•(a1/2) = 1
Uncertainty Principle! Cannot make both ∆t and ∆ω arbitrarily small.
Book Chapter4: Section1
13
CT Fourier Transforms of Periodic Signals
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0
0
0
0
0
0 0
periodic in with freq
Suppose
( ) ( )
1 1( ) ( )
2 2That i
All the energy is concentrated in one fr
ue
s
2 ( )
More generall
equency
ncy
y
j tj t
j t
X j
x t t e d e
e
x
00( ) ( ) 2 ( )jk t
k kk k
t a e X j a k
Book Chapter4: Section1
14
Example #5:
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0 00
0 0
1 1( ) cos
2 2
( ) ( ) ( )
j t j tx t t e e
X j
“Line Spectrum”
Book Chapter4: Section1
15
Example #6:
Computer Engineering Department, Signals and Systems
Sampling function( ) ( )n
x t t nT
0
0
2
2
2
1 1( ) ( )
2 2( ) ( )
k
T jk tk T
na k
x t a x t e dtT T
kX j
T T
x(t)
Same function in the frequency-domain!
Note: (period in t) T ⇔ (period in ω) 2π/T Inverse relationship again!
Book Chapter4: Section1
16
Properties of the CT Fourier Transform
Computer Engineering Department, Signals and Systems
0
0
0
0
( )
1) Linearity ( ) ( ) ( ) ( )
2) Time Shifting ( ) ( )
Proof: ( ) ( )
magnitude unchanged
j t
j tj t j t
tX j
ax t by t aX j bY j
x t t e X j
x t t e dt e x t e dt
FT
0
00
( ) ( )
Linear change in phase
( ( )) ( )
j t
j t
e X j X j
FT
e X j X j t
Book Chapter4: Section1
17
Properties (continued)
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*
Conjugate Symmetry
( ) real ( ) ( )
( ) ( )
( ) ( )
{ ( )} { ( )}
{ ( )} { ( )}
Even
x t X j X j
X j X j
X j X j
Re X j Re X j
Im X j Im X j
Odd
Od
n
d
Eve
Book Chapter4: Section1
18
The Properties Keep on Coming ...
Computer Engineering Department, Signals and Systems
1Time-Scaling ( ) ( )
1 E.g. 1
( ) ( ) compressed in time stretched in frequency
a) ( ) real and even
x at X ja a
a a at t
x t X j
x t
*
*
( ) ( )
( ) ( ) ( ) Real & even
b) ( ) real and odd ( ) ( )
( ) ( ) ( ) Purely imaginary &:
c) ( ) { ( )}+ { ( )}
x t x t
X j X j X j
x t x t x t
X j X j X j
X j Re X j jIm X j
( ) { ( )} { ( )}x t Ev x t Od x t
For real
Book Chapter4: Section2
19
The CT Fourier Transform Pair
Computer Engineering Department, Signals and Systems
─ FT(Analysis Equation)
─ Inverse FT(Synthesis Equation)
Last lecture: some propertiesToday: further exploration
Book Chapter4: Section2
20
Coefficient a
Y(jω)
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Convolution Property
A consequence of the eigenfunction property :
h(t)
H(jω).a
Synthesis equation for y(t)
Book Chapter4: Section2
21
The Frequency Response Revisited
Computer Engineering Department, Signal and Systems
h(t)
¿impulse response
¿frequency response
The frequency response of a CT LTI system is simply the Fourier transform of its impulse response
Example #1: ¿H(jω)
¿Recall
¿⇓𝑦 (𝑡)=𝐻 ( 𝑗 𝜔0)𝑒
𝑗 𝜔0𝑡inverse FT
Book Chapter4: Section2
22
Example #2 A differentiator
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𝑦 (𝑡)=¿¿ - an LTI system
Differentiation property:¿⇓
𝐻 ( 𝑗 𝜔)= 𝑗 𝜔1) Amplifies high frequencies (enhances sharp edges)
Larger at high ωo phase shift 2) +π/2 phase shift ( j = ejπ/2)
Book Chapter4: Section2
23
Example #3: Impulse Response of an Ideal Lowpass Filter
Computer Engineering Department, Signal and Systems
h (𝑡)=1
2𝜋 −𝜔𝑐
𝜔𝑐
𝑒 𝑗 𝜔𝑡 𝑑𝜔
¿sin𝜔𝑐𝑡𝜋 𝑡
¿𝜔𝑐
𝜋sin𝑐 (𝜔𝑐𝑡
𝜋 ) Define: sinc(θ) ¿
sin 𝜋𝜃𝜋𝜃
Questions:1) Is this a causal system? No.
2) What is h(0)?
h (0)= 12𝜋 −∞
∞
𝐻 ( 𝑗 𝜔)𝑑𝜔=2𝜔𝑐
2𝜋=𝜔𝑐
𝜋3) What is the steady-state value of the step response, i.e. s(∞)?
𝑠(𝑡)=− ∞
𝑡
h (𝑡)𝑑𝑡
𝑠(∞)=− ∞
∞
h (𝑡)𝑑𝑡=𝐻 ( 𝑗 0)=1
Book Chapter4: Section2
24
Example #4: Cascading filtering operations
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¿
𝐻 ( 𝑗 𝜔)=𝐻12( 𝑗𝜔 ) h𝑎𝑠 𝑎
𝑠h𝑎𝑟𝑝𝑒𝑟 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦
⇒
Book Chapter4: Section2
25Computer Engineering Department, Signal and Systems
Example #5:
sin 4 𝜋𝑡𝜋 𝑡
∗sin 8𝜋𝑡𝜋 𝑡
=?
h(t)x(t)
¿
Example #6: 𝑒−𝑎𝑡 2
∗𝑒−𝑏𝑡 2
= ? √ 𝜋𝑎+𝑏
.𝑒−( 𝑎𝑏𝑎+𝑏 )𝑡2
√ 𝜋𝑎 𝑒− 𝜔2
4𝑎×√ 𝜋𝑏 𝑒− −𝜔2
4𝑏 = 𝜋√𝑎𝑏
𝑒− 𝜔2
4 ( 1𝑎+ 1𝑏)
Gaussian × Gaussian = Gaussian ⇒ Gaussian ∗ Gaussian = Gaussian
Book Chapter4: Section2
26
Example #2 from last lecture
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𝑥 (𝑡)=𝑒−𝑎𝑡𝑢 (𝑡) , 𝑎>0
𝑋 ( 𝑗 𝜔)= −∞
∞
𝑥 (𝑡)𝑒− 𝑗 𝜔𝑡 𝑑𝑡 =0
∞
𝑒−𝑎𝑡𝑒− j 𝜔𝑡𝑑𝑡𝑒¿ ¿
¿ −( 1𝑎+ 𝑗 𝜔 )𝑒− (𝑎+ 𝑗 𝜔 ) 𝑡 ¿∞
0= 1𝑎+ 𝑗𝜔
Book Chapter4: Section2
27Computer Engineering Department, Signal and Systems
Example #7:
¿- a rational function of jω, ratio of polynomials of jω
Partial fraction expansion
𝑌 ( 𝑗 𝜔)=1
1+ 𝑗 𝜔−
12+ 𝑗 𝜔
Inverse FT
Book Chapter4: Section2
28
Example #8: LTI Systems Described by LCCDE’s (Linear-constant-coefficient differential equations)
Computer Engineering Department, Signal and Systems
∑𝑘=0
𝑁
𝑎𝑘¿¿¿
Using the Differentiation Property¿¿
⇓Transform both sides of the equation
∑𝑘=0
𝑁
𝑎𝑘 . ( 𝑗 𝜔 )𝑘𝑌 ( 𝑗 𝜔 )=∑𝑘=0
𝑀
¿¿1) Rational, can use
PFE to get h(t)
2) If X(jω) is rationale.g. x(t)=Σcie-at u(t)then Y(jω) is also rational
𝑌 ( 𝑗 𝜔)=[∑𝑘=0
𝑀
𝑏𝑘 ( 𝑗𝜔 )𝑘
∑𝑘=0
𝑁
𝑎𝑘 ( 𝑗 𝜔 )𝑘 ]𝑋 ( 𝑗𝜔 )
H(jω)
⇓
Book Chapter4: Section2
29
Parseval’s Relation
Computer Engineering Department, Signal and Systems
− ∞
∞
¿ 𝑥 (𝑡)¿2𝑑𝑡= 12𝜋 −∞
∞
¿ 𝑋 ( 𝑗𝜔)¿2𝑑𝜔
Total energyin the time-domain
Total energyin the time-domain
12𝜋∨𝑋 ( 𝑗 𝜔)¿2
- Spectral density
Multiplication Property FT is highly symmetric,
𝑥 (𝑡)´́𝐹 −1 1
2𝜋 −∞
∞
𝑋 ( 𝑗𝜔)𝑒 𝑗 𝜔𝑡 𝑑𝜔 , 𝑋 ( 𝑗 𝜔) ´́𝐹−∞
∞
𝑥(𝑡)𝑒− 𝑗 𝜔𝑡 𝑑𝑡
We already know that:
Then it isn’t asurprise that:
¿𝑥 (𝑡). 𝑦 (𝑡)↔
12𝜋
𝑋 ( 𝑗 𝜔)∗𝑌 ( 𝑗 𝜔 )Convolution in ω
¿ 12𝜋 − ∞
∞
𝑋 ( 𝑗 𝜃)𝑌 ( 𝑗 (𝜔−𝜃 ) )𝑑𝜃
— A consequence of Duality
Book Chapter4: Section2
30
Examples of the Multiplication Property
Computer Engineering Department, Signal and Systems
𝑟 (𝑡)=𝑠(𝑡 ).𝑝 (𝑡)↔𝑅 ( 𝑗 𝜔 )= 12𝜋
[𝑆 ( 𝑗𝜔 )∗𝑃 ( 𝑗 𝜔 ) ]
𝐹𝑜𝑟 𝑝 (𝑡)=cos𝜔0 𝑡↔𝑃 ( 𝑗𝜔 )=𝜋𝛿 (𝜔−𝜔0 )+𝜋𝛿 (𝜔+𝜔0 )
𝑅 ( 𝑗 𝜔 )=12𝑆 ( 𝑗 (𝜔−𝜔0 ))+
12𝑆 ( 𝑗 (𝜔+𝜔0 ))
For any s(t) ...
⇓
Book Chapter4: Section2
31
Example (continued)
Computer Engineering Department, Signal and Systems
𝑟 (𝑡)=𝑠(𝑡 ).cos (𝜔0𝑡 )Amplitude modulation(AM)
𝑅 ( 𝑗 𝜔)=12 [𝑆 ( 𝑗 (𝜔−𝜔0 ) )+𝑆 ( 𝑗 (𝜔+𝜔0 )) ]
Drawn assuming:𝜔0 −𝜔1>0𝑖 .𝑒 . 𝜔0>𝜔1