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Signals and Systems Prof. 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

Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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Page 1: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Page 2: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Page 3: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Page 4: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Page 5: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Page 6: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Page 7: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Computer Engineering Department, Signals and Systems

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

Page 8: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

Book Chapter4: Section1

8

Example #1

Computer Engineering Department, Signals and Systems

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

Page 9: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Page 10: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

Book Chapter4: Section1

10

Example #3: A square pulse in the time-domain

Computer Engineering Department, Signals and Systems

1

1

12sin( )

T j t

T

TX j e dt

Note the inverse relation between the two widths Uncertainty principle⇒

Page 11: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Page 12: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

Book Chapter4: Section1

12

Example #4:

Computer Engineering Department, Signals and Systems

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.

Page 13: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

Book Chapter4: Section1

13

CT Fourier Transforms of Periodic Signals

Computer Engineering Department, Signals and Systems

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

Page 14: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

Book Chapter4: Section1

14

Example #5:

Computer Engineering Department, Signals and Systems

0 00

0 0

1 1( ) cos

2 2

( ) ( ) ( )

j t j tx t t e e

X j

“Line Spectrum”

Page 15: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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!

Page 16: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Page 17: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

Book Chapter4: Section1

17

Properties (continued)

Computer Engineering Department, Signals and Systems

*

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

Page 18: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Page 19: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Page 20: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

Book Chapter4: Section2

20

Coefficient a

Y(jω)

Computer Engineering Department, Signal and Systems

Convolution Property

A consequence of the eigenfunction property :

h(t)

H(jω).a

Synthesis equation for y(t)

Page 21: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Page 22: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

Book Chapter4: Section2

22

Example #2 A differentiator

Computer Engineering Department, Signal and Systems

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)

Page 23: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Page 24: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

Book Chapter4: Section2

24

Example #4: Cascading filtering operations

Computer Engineering Department, Signal and Systems

¿

𝐻 ( 𝑗 𝜔)=𝐻12( 𝑗𝜔 ) h𝑎𝑠 𝑎

𝑠h𝑎𝑟𝑝𝑒𝑟 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦

Page 25: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Page 26: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

Book Chapter4: Section2

26

Example #2 from last lecture

Computer Engineering Department, Signal and Systems

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

Page 27: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Page 28: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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)

Page 29: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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

Page 30: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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) ...

Page 31: Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms

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