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

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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 ∫ ∞−

    =

    ( ) ( ) ( ) (