Signals and Systemsce.sharif.edu/courses/98-99/1/ce242-2/resources/root... · 2020. 9. 7. · Signals and Systems Chapter 4: The Continuous Time Fourier Transform • Derivation of

Signals and SystemsChapter 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

Fourier’s Derivation of the CT Fourier Transform

x(t) - an aperiodic signal

view 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

Book Chapter4: Section1

Computer Engineering Department, Signals and Systems 2

Fourier series Fourier integral

Motivating Example: Square wave



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

So, on with the derivation ...



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

Derivation (continued)



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 t

k

k

T T

jk t jk t

k

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

Derivation (continued)



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

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



2( )x t dt

21( ) ( ) ( ) Then ( ) 0

2

j te t x t X j e d e t dt

1 (i) ( ) ( ) at points of continuity

2

1 (ii) ( ) midpoint at discontinuity

2

(iii) Gibb's phenomenon

j t

j t

X j e d x t

X j e d

Example #1



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

Example #2: Exponential function



( )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

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



1

1

12sin( )

Tj t

T

TX j e dt

Note the inverse relation between the two widths ⇒ Uncertainty principle

Useful facts about CTFT’s



1

(0) ( )

1(0) ( )

2

Example above: ( ) 2 (0)

1Ex. above: (0) ( )

2

1(Area of the triangle)

2

X x t dt

x X j d

x t dt T X

x X j d

Example #4:



2

A Gaussian, important in probability, optics, etc.( ) atx t e

2

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 makeboth ∆t and ∆ω arbitrarily small.

CT Fourier Transforms of Periodic Signals



0

0

0

0

0

0 0

periodic in with freq

Suppose

( ) ( )

1 1( ) ( )

2 2

That 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

0

0( ) ( ) 2 ( )jk t

k k

k k

t a e X j a k

Example #5:



0 0

0

0 0

1 1( ) cos

2 2

( ) ( ) ( )

j t j tx t t e e

X j

“Line Spectrum”

Example #6:



Sampling function( ) ( )n

x t t nT

0

0

2

2

2

1 1( ) ( )

2 2( ) ( )

k

Tjk t

kT

n

a 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!

Properties of the CT Fourier Transform



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

0

0

( ) ( )

Linear change in phase

( ( )) ( )

j t

j t

e X j X j

FT

e X j X j t

Properties (continued)



*

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

The Properties Keep on Coming ...



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

−𝑋∗(𝑗𝜔)

The CT Fourier Transform Pair



)𝑥(𝑡) ↔ 𝑋(𝑗𝜔

𝑋(𝑗𝜔) = න

−∞

∞

𝑥(𝑡)𝑒−𝑗𝜔𝑡 𝑑𝑡

𝑥(𝑡) =1

2𝜋න

−∞

∞

𝑋(𝑗𝜔)𝑒𝑗𝜔𝑡𝑑𝜔

─ FT(Analysis Equation)

─ Inverse FT(Synthesis Equation)

Last lecture: some propertiesToday: further exploration

)𝑦(𝑡) = ℎ(𝑡) ∗ 𝑥(𝑡) ↔ 𝑌(𝑗𝜔) = 𝐻(𝑗𝜔)𝑋(𝑗𝜔)𝑤ℎ𝑒𝑟𝑒 ℎ(𝑡) ↔ 𝐻(𝑗𝜔

𝑥(𝑡) =

−∞

∞1

2𝜋𝑋(𝑗𝜔)𝑑𝜔 𝑒𝑗𝜔𝑡

Coefficient a

𝑎𝑒𝑗𝜔𝑡 → → 𝐻(𝑗𝜔)𝑎𝑒𝑗𝜔𝑡

𝑦(𝑡) =

−∞

∞

𝐻(𝑗𝜔)1

2𝜋𝑋(𝑗𝜔)𝑑𝜔 𝑒𝑗𝜔𝑡

=1

2𝜋න−∞

∞

𝐻(𝑗𝜔)𝑋(𝑗𝜔)𝑒𝑗𝜔𝑡𝑑𝜔

Y(jω)


Computer Engineering Department, Signal and Systems 20

Convolution Property

A consequence of the eigenfunction property :

h(t)

H(jω).a

Synthesis equation for y(t)

The Frequency Response Revisited



h(t) )𝑥(𝑡) → → 𝑦(𝑡) = ℎ(𝑡) ∗ 𝑥(𝑡

impulse response

)𝑌(𝑗𝜔) = 𝐻(𝑗𝜔)𝑋(𝑗𝜔

frequency response

The frequency response of a CT LTI system is simply the Fourier transform of its impulse response

Example #1: ൯𝑥 𝑡 = 𝑒𝑗𝜔0𝑡 → → 𝑦(𝑡H(jω)

൯𝑒𝑗𝜔0𝑡 ↔ 2𝜋𝛿(𝜔 − 𝜔0Recall

)𝑌(𝑗𝜔) = 𝐻(𝑗𝜔)𝑋(𝑗𝜔) = 𝐻(𝑗𝜔)2𝜋𝛿(𝜔 − 𝜔0) = 2𝜋𝐻(𝑗𝜔0)𝛿(𝜔 − 𝜔0

⇓𝑦(𝑡) = 𝐻(𝑗𝜔0)𝑒

𝑗𝜔0𝑡

inverse FT

Example #2 A differentiator

𝑑

𝑑𝑡sin𝜔0𝑡 = 𝜔0cos𝜔0𝑡 = 𝜔0sin(𝜔0𝑡 +

𝜋

2)

𝑑

𝑑𝑡cos𝜔0𝑡 = − 𝜔0sin𝜔0𝑡 = 𝜔0cos(𝜔0𝑡 +

𝜋

2)



𝑦(𝑡) =)𝑑𝑥(𝑡

𝑑𝑡- an LTI system

Differentiation property: )𝑌(𝑗𝜔) = 𝑗𝜔𝑋(𝑗𝜔

⇓

𝐻(𝑗𝜔) = 𝑗𝜔

1) Amplifies high frequencies (enhances sharp edges)

Larger at high ωo phase shift2) +π/2 phase shift ( j = ejπ/2)

Example #3: Impulse Response of an Ideal Lowpass Filter



ℎ(𝑡) =1

2𝜋න−𝜔𝑐

𝜔𝑐

𝑒𝑗𝜔𝑡𝑑𝜔

=sin𝜔𝑐𝑡

𝜋𝑡

=𝜔𝑐𝜋sin𝑐

𝜔𝑐𝑡

𝜋

Define: sinc(θ) =sin𝜋𝜃

𝜋𝜃

Questions:1) Is this a causal system? No.

2) What is h(0)?

ℎ(0) =1

2𝜋න−∞

∞

𝐻(𝑗𝜔)𝑑𝜔 =2𝜔𝑐2𝜋

=𝜔𝑐𝜋

3) What is the steady-state value of the step response, i.e. s(∞)?

𝑠(𝑡) = න−∞

𝑡

ℎ(𝑡)𝑑𝑡

𝑠(∞) = න−∞

∞

ℎ(𝑡)𝑑𝑡 = 𝐻(𝑗0) = 1

Example #4: Cascading filtering operations



)𝑒. 𝑔. 𝐻1(𝑗𝜔) = 𝐻2(𝑗𝜔

𝐻(𝑗𝜔) = 𝐻12(𝑗𝜔) ℎ𝑎𝑠 𝑎

𝑠ℎ𝑎𝑟𝑝𝑒𝑟 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦

⇒



Example #5:

⇓

⇓

sin4𝜋𝑡

𝜋𝑡∗sin8𝜋𝑡

𝜋𝑡= ?

h(t)x(t)

)𝑌(𝑗𝜔) = 𝑋(𝑗𝜔)⇒ 𝑦(𝑡) = 𝑥(𝑡

Example #6: 𝑒−𝑎𝑡2∗ 𝑒−𝑏𝑡

2= ?

𝜋

𝑎 + 𝑏. 𝑒

−𝑎𝑏𝑎+𝑏 𝑡2

𝜋

𝑎𝑒−

𝜔2

4𝑎 ×𝜋

𝑏𝑒−

−𝜔2

4𝑏 =𝜋

𝑎𝑏𝑒−𝜔2

41𝑎+1𝑏

Gaussian × Gaussian = Gaussian ⇒ Gaussian ∗ Gaussian = Gaussian

Example #2 from last lecture



𝑥(𝑡) = 𝑒−𝑎𝑡𝑢(𝑡) , 𝑎 > 0

𝑋(𝑗𝜔) =

−∞

∞

𝑥(𝑡)𝑒−𝑗𝜔𝑡𝑑𝑡 = න0

∞

𝑒−𝑎𝑡𝑒−j𝜔𝑡𝑑𝑡

𝑒 )−(𝑎+𝑗𝜔

= −1

𝑎 + 𝑗𝜔𝑒− 𝑎+𝑗𝜔 𝑡]

∞0=

1

𝑎 + 𝑗𝜔



Example #7:

⇓

⇓)𝑦(𝑡) = [𝑒−𝑡 − 𝑒−2𝑡]𝑢(𝑡

)ℎ(𝑡) = 𝑒−𝑡𝑢(𝑡) , 𝑥(𝑡) = 𝑒−2𝑡𝑢(𝑡)𝑦(𝑡) = ℎ(𝑡) ∗ 𝑥(𝑡

⇓

𝑌(𝑗𝜔) = 𝐻(𝑗𝜔)𝑋(𝑗𝜔) =1

1 + 𝑗𝜔.

1

2 + 𝑗𝜔

- a rational function of jω, ratio of polynomials of jω

Partial fraction expansion

𝑌(𝑗𝜔) =1

1 + 𝑗𝜔−

1

2 + 𝑗𝜔

Inverse FT

Example #8: LTI Systems Described by LCCDE’s(Linear-constant-coefficient differential equations)



𝑘=0

𝑁

𝑎𝑘൯𝑑𝑘𝑦(𝑡

𝑑𝑡𝑘=

𝑘=0

𝑀

𝑏𝑘൯𝑑𝑘𝑥(𝑡

𝑑𝑡𝑘

Using the Differentiation Property

൯𝑑𝑘𝑥(𝑡

𝑑𝑡𝑘↔ 𝑗𝜔 𝑘𝑋 𝑗𝜔

⇓ Transform both sides of the equation

𝑘=0

𝑁

𝑎𝑘 . 𝑗𝜔𝑘𝑌 𝑗𝜔 =

𝑘=0

𝑀

൯𝑏𝑘 . 𝑗𝜔𝑘𝑋(𝑗𝜔

1) Rational, can usePFE to get h(t)

2) If X(jω) is rationale.g. x(t)=Σcie-at u(t)

then Y(jω) is also rational

𝑌(𝑗𝜔) =

𝑘=0

𝑀𝑏𝑘 𝑗𝜔 𝑘

σ𝑘=0𝑁 𝑎𝑘 𝑗𝜔 𝑘

𝑋 𝑗𝜔

H(jω)

⇓

Parseval’s Relation



න−∞

∞

|𝑥(𝑡)|2𝑑𝑡 =1

2𝜋න−∞

∞

|𝑋(𝑗𝜔)|2𝑑𝜔

Total energyin the time-domain

Total energyin the time-domain

1

2𝜋|𝑋(𝑗𝜔)|2

- Spectral density

Multiplication PropertyFT is highly symmetric,

𝑥(𝑡)𝐹−11

2𝜋න−∞

∞

𝑋(𝑗𝜔)𝑒𝑗𝜔𝑡𝑑𝜔, 𝑋(𝑗𝜔)ഫഫ𝐹න−∞

∞

𝑥(𝑡)𝑒−𝑗𝜔𝑡𝑑𝑡

We already know that:

Then it isn’t asurprise that:

)𝑥(𝑡) ∗ 𝑦(𝑡) ↔ 𝑋(𝑗𝜔). 𝑌(𝑗𝜔

𝑥(𝑡). 𝑦(𝑡) ↔1

2𝜋𝑋(𝑗𝜔) ∗ 𝑌 𝑗𝜔

Convolution in ω=

1

2𝜋න−∞

∞

𝑋(𝑗𝜃)𝑌 𝑗 𝜔 − 𝜃 𝑑𝜃

— A consequence of Duality

Examples of the Multiplication Property



𝑟(𝑡) = 𝑠(𝑡). 𝑝(𝑡) ↔ 𝑅 𝑗𝜔 =1

2𝜋𝑆(𝑗𝜔) ∗ 𝑃 𝑗𝜔

𝐹𝑜𝑟 𝑝(𝑡) = cos𝜔0𝑡 ↔ 𝑃 𝑗𝜔 = 𝜋𝛿 𝜔 − 𝜔0 + 𝜋𝛿 𝜔 + 𝜔0

𝑅 𝑗𝜔 =1

2𝑆 𝑗 𝜔 − 𝜔0 +

1

2𝑆 𝑗 𝜔 + 𝜔0

For any s(t) ...

⇓

Example (continued)



𝑟(𝑡) = 𝑠(𝑡). cos 𝜔0𝑡

Amplitude modulation(AM)

𝑅(𝑗𝜔)

=1

2ൣ𝑆 𝑗 𝜔 − 𝜔0

Drawn assuming:

𝜔0 − 𝜔1 > 0𝑖. 𝑒. 𝜔0 > 𝜔1

Documents

Signals and Systemsce.sharif.edu/courses/98-99/1/ce242-2/resources/root... · 2020. 9. 7. · Signals and Systems Chapter 4: The Continuous Time Fourier Transform • Derivation of