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### Transcript of Fourier transform of continuous-time Fourier transform of continuous-time signals Spectral...

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Fourier transform of continuous-time signals

Spectral representation of non-periodic signals

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Fourier transform: aperiodic signals

• repetition of a finite-duration signal x(t)=> periodic signals.

• Periodic signal (T → ∞) => non-periodic signal x(t)

( ) ( ) ( ) ( ) ( ) ( )T k k

x t x t t x t t kT x t kT ∞ ∞

=−∞ =−∞ = ∗δ = ∗ δ − = −∑ ∑

( ) ( ) T

x t x t →∞ →

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Non-periodic signal & periodic signal, period T.

Non-periodic

Periodic

repetition T → ∞

( ) ( )1 11

0 otherwiseT , t T

x t p t ,

Δ ⎧ ≤ = = ⎨

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( ) ( ) T

x t x t →∞ →

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• analyze a non-periodic signal in the frequency domain: – using the frequency analysis of the

correspondent periodic signal

– and compute the limit for T →∞.

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• The periodic signal is non band-limited.

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The product T⋅ck & the envelope

X(ω) = envelope for T⋅ck

Relation between them?

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General case

• the Fourier coefficients of the periodic signal :

( ) 0 2

2

1 T jk t k

T c x t e dt

T − ω

− = ∫

( ) 0 2

2

1 T jk t k

T c x t e dt

T − ω

− = ∫

Equal on [-T/2, T/2]

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Fourier transform • outside [-T/2,T/2] the non-periodic signal =0

• With the function: ( ) 01 jk tkc x t e dtT

∞ − ω

−∞ = ∫

( ) ( ) j tX x t e dt ∞

− ω

−∞ ω = ∫

( )0 0 1 2 kc X k ,T T

π = ω ω =

(The envelope of T⋅ck )

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Square wave:different values of T

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Remarks

• The envelope is not affected by T. • Increase T ⇒ spectral components are “closer”. • T→∞ ⇒

– distance→0 – the discrete spectral representation becomes

continuous. – the periodic signal → non-periodic.

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Definition

• Fourier pair

( ) ( ) ( ) ( )1 2

j t j tx t X e d X x t e dt ∞ ∞

ω − ω

−∞ −∞ = ω ω ↔ ω =

π ∫ ∫

Fourier Transform (spectrum)

Inverse Fourier Transform

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Remarks

• periodic signals : spectral lines

• non-periodic signals spectra are continuous

( )0 0 1 2 kc X k ,T T

π = ω ω =

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CTFT for signals in the class L1

• for signals in L1, the Fourier transform is not necessarily from L1

• Reconstruction theorem!

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• the Fourier transform is convergent (the signal x(t)∈L1) but X(ω)∉L1.

• the reconstruction of the signal from its spectrum is not obvious.

( ) ω ωsin2ω τ=X( ) ( ) ( ) ( )ττ −−== tttptx σσ

( ) ∞

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1. Linearity

• If x(t) and y(t) ∈ L1 and have the Fourier transform X(ω) and Y(ω) then for any complex constants a and b the signal ax(t)+by(t) ∈ L1 and has the Fourier transform aX(ω)+bY(ω).

Homework: Prove it.

( ) ( ) ( ) ( )ωω bYaXtbytax +→+

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2. Time Shifting

• Time shifting -> modulation in frequency (multiplication with a complex exponential).

• Proof

( ) ( )00 j tx t t e Xω ω−− ↔

( ){ } ( ) ( ) ( ) ( ). F ωττ ωτω τ

ω Xedexdtettxttx tjtj tt

tj 00 0

00 1 −+−

∞−

−=∞

∞−

− =⋅=⋅−=− ∫∫

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Remarks

• Fourier transform: complex function.

• The Fourier transform H(ω) of the impulse response h(t) of a system: frequency response.

• frequency dependence of the magnitude of H(ω) = magnitude characteristic of the system |H(ω)|

• frequency dependence of the argument of H(ω) = phase characteristic of the system arg{H(ω)}

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3. Modulation

• Modulation in time -> shifting in frequency.

• Proof

( ) ( )0 0 .j te x t Xω ω ω→ −

( ){ } ( )

( ) ( ) ( )

( ) ( ).0

0

1

0

0

00

ωω

ωω

ω

ωω

ωωω

−→

−=⋅=

=⋅⋅=⋅

∞−

−−

− ∞

∞−

Xtxe

Xdtetx

dteetxetx

tj

tj

tjtjtjF

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Duality

• operation in time ⇒ another operation in frequency : – modulation ⇒ shifting (3rd property)

• 2nd operation in time ⇒ first operation in frequency. – time shifting ⇒ modulation (2nd property)

• This behavior is named duality.

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4. Time Scaling

• If x(t) ∈ L1 ⇒ its scaled version x(t/a) ∈ L1 and the spectrum of x(t/a) is a frequency scaled version of the spectrum of x(t).

• the scaling is an auto-dual operation.

( ) 1 .x at X a a

ω⎛ ⎞→ ⎜ ⎟ ⎝ ⎠

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• Proof

( ){ } ( ) ( )

( ) .1

;111

⎟ ⎠ ⎞

⎜ ⎝ ⎛→

⎟ ⎠ ⎞

⎜ ⎝ ⎛=⋅=⋅=

−∞

∞−

∞−

= − ∫∫

a X

a atx

a X

a dex

a dteatxatx a

jattj

ω

ωττ τωτωF

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Example: the square wave

• spectrum

• its time-scaled variant , a=2:

• a=1/2 ( )2

1 2 2 2212 2 2

sin sinp t p tτ τ ωτ ωτ⎛ ⎞ = ↔ =⎜ ⎟ ω ω⎝ ⎠

( ) 2 sinp tτ ωτ

↔ ω

( ) ( ) 2

22 2 22 = 2

2

sin sin p t p tτ τ

ω ωτ τ = ↔

ω ω

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time compression → frequency dilation

time dilation → frequency compression

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CTFT of the constant distribution

( ) ( ) F

1 2t πδ ω↔

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Proof

• the constant distribution can be approximated:

• We know that

( ) ( )ttp 1lim = ∞→

τ τ

( ) ω ωτω

τ sin2=⋅∫

∞−

− dtetp tj

( ) ω ωτ

τ ω

τ τ

sin2limlim ∞→

− ∞

∞−∞→ =⋅∫ dtetp tj

( ) ⎩ ⎨ ⎧

≠ =∞

==⋅ ∞→

− ∞

∞− ∫ 0,0

0,sin2lim1 ω ω

ωτ ωττ

τ ω dtet tj

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• The area under the graphical representation of the spectrum:

• So:

• and:

( ) πω ωτ

ωττ 24sinsin2sin2 0

0 =∞=⎟

⎟ ⎠

⎞ ⎜ ⎜ ⎝

⎛ +== ∫∫∫

∞−

∞− Sidu

u udu

u udA

( ) ⎩ ⎨ ⎧

≠ =∞

=⋅ − ∞

∞− ∫ 0,0

0, 1

ω ωω dtet tj

π2=A

( ) ( ) ( ) ( )ωπδωπδω 2121 F ↔⇔=⋅ −

∞− ∫ tdtet tj

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• An immediate consequence: a new representative string for the Dirac distribution:

( )sinlim τ→∞

ωτ = δ ω

πω

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5. Complex Conjugation

• complex conjugation in time -> reversal and complex conjugation in frequency.

• Proof

( ) ( )* *x t X ω↔ − 1F

( ){ } ( ) ( ) ( ) ( )

( ) ( )ω

ωωω

−↔

−= ⎥ ⎥ ⎦

⎢ ⎢ ⎣

⎡ ⋅=⋅= ∫∫

∞−

−−− ∞

∞−

**

* *

*1

1

*

Xtx

Xdtetxdtetxtx tjtj

F

F

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6. Time Reversal

• Time reversal -> reversal in frequency.

• Homework. Prove it.

( ) ( )ω−↔− Xtx 1F

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7. Signal’s Derivation

• Time differentiation -> multiplication with jω in frequency.

( ) ( )ωω Xjtx ⋅↔ 1

' F

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Proof:

Integrating by parts:

the signal is in L1 :

So:

( ){ } ( ) dtetxtx tjω− ∞

∞− ⋅= ∫ ''1F

( ){ } ( ) ( ) dtetxjetxtx tjtj ωω ω − ∞

∞−

∞− − ⋅+⋅= ∫'1F

( ) ( ) 0limlim ==⋅ ±∞→

±∞→ txetx

t tj

t ω

( ) ( )ωω Xjtx ⋅↔ 1

' F

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8. Signal’s Integration

• For x(t) ∈ L1 with X(0)=0 (no DC component), its integral ∈ L1

• Time integration -> multiplication with

1/ jω in frequency

( ) ( ) ( ) 1F

for 0 0 t X

x d X j ω

τ τ ω−∞

↔ =∫

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Proof

• We have: • Apply for y(t) the differentiation property:

• Y defined in 0 :

• So:

( ) ( ) ττ dxty t ∫ ∞−

=

( ) ( ) ( ) (