Fourier Analysis - TU Delft OCW Fourier Analysis Continuous Fourier transform Discrete Fourier...

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Transcript of Fourier Analysis - TU Delft OCW Fourier Analysis Continuous Fourier transform Discrete Fourier...

  • Fourier Analysis (Ta3520)

    Fourier – p. 1/16

  • Fourier Analysis Continuous Fourier transform Discrete Fourier Transform and Sampling Theorem Linear Time-Invariant (LTI) systems and Convolution Convolution Theorem Filters Correlation Deconvolution

    Fourier – p. 2/16

  • Continuous Fourier transform Definition:

    A(f) =

    ∫ +∞

    −∞

    a(t) exp(−2πift) dt

    Inverse transform:

    a(t) =

    ∫ +∞

    −∞

    A(f) exp(2πift) df

    f = frequency (Hz); t = time (s)

    Decomposition of signal into sines and cosines Fourier – p. 3/16

  • Decomposition into cosines

    0

    50

    100

    150

    200

    250

    0 10 20 30

    frequency

    tim e

    Fourier – p. 4/16

  • Decomposition into sines and cosines

    Fourier – p. 5/16

  • Discretisation of signal Discretisation in time makes spectrum periodic:

    ADiscrete(f) = +∞ ∑

    m=−∞

    AContinuous(f + m

    ∆t )

    At least two samples per frequency/wavelength.

    Otherwise: aliasing.

    Fourier – p. 6/16

  • Aliasing in time domain

    0

    0

    0

    (a)

    (b)

    (c)

    Fourier – p. 7/16

  • Aliasing = overlapping spectra Acontinuous

    A D

    0

    0 1 ∆t

    A D

    0 1 ∆t

    1 ∆t

    1 ∆t1 1

    2 2

    (a)

    (b)

    (c)

    f

    f

    f

    Fourier – p. 8/16

  • Discrete Fourier transform

    An = ∆t N−1 ∑

    k=0

    ak exp(−2πink/N) for n = 0, 1, ..., N − 1

    k = index for time-domain coefficient n = index for Fourier-domain coefficient

    ak = ∆f N−1 ∑

    n=0

    An exp(2πink/N) for k = 0, 1, ..., N − 1

    Basic relation:N∆t∆f = 1 Nyquist criterion: fN = 12∆t Fourier – p. 9/16

  • Linear Time-Invariant (LTI) systems and convolution

    Output of linear time-invariant system = convolution of input with impulse response of system:

    h(t)

    h(t)

    h(t)

    h(t)

    δ(t)

    δ(t-τ)

    s(τ)δ(t-τ)

    s(τ)δ(t-τ) dτ

    h(t)

    h(t-τ)

    s(τ)h(t-τ)

    s(τ)h(t-τ) dτ

    time-invariance :

    linearity :

    linearity :

    = s(t)

    Fourier – p. 10/16

  • Convolution theorem

    Ft

    ( ∫ +∞

    −∞

    h(t′)g(t − t′)dt′ )

    = Ft (h(t) ∗ g(t))

    = H(f)G(f)

    Fourier – p. 11/16

  • Filters

    0 10 20 30 40 0

    0.5

    1.0

    frequency

    | A (f)

    |

    0 0.5 1.0 1.5 -2

    -1

    0

    1

    2

    time

    a( t)

    FT

    0 10 20 30 40 0

    0.5

    1.0

    | A (f)

    |

    frequency

    0 0.5 1.0 1.5 -2

    -1

    0

    1

    2

    time

    a( t)

    -1FT

    INPUT TO FILTER

    0 10 20 30 40 0

    1.0

    frequency

    | A (f)

    |

    OUTPUT OF FILTER

    Fourier – p. 12/16

  • Filters: original Wassenaar data

    0

    0.5

    1.0

    1.5

    tim e

    (s )

    100 200 300 offset (m)

    Fourier – p. 13/16

  • Filters: Wassenaar data 0

    0.5

    1.0

    1.5

    tim e

    (s )

    100 300 distance (m)

    0

    50

    100

    150

    200

    250

    300

    350

    400

    450

    500

    fre qu

    en cy

    (H z)

    100 300 distance (m)

    0

    50

    100

    150

    200

    250

    300

    350

    400

    450

    500

    fre qu

    en cy

    (H z)

    100 300 distance (m)

    0

    0.5

    1.0

    1.5

    tim e

    (s )

    100 300 distance (m)

    Fourier – p. 14/16

  • Filters: filtered Wassenaar data

    0

    0.5

    1.0

    1.5

    tim e

    (s )

    100 200 300 offset (m)

    Fourier – p. 15/16

  • Correlation Auto-correlation:

    Ft

    ( ∫ +∞

    −∞

    a(τ)a∗(τ − t)dτ

    )

    = A(f)A∗(f) = |A(f)|2

    Cross-correlation:

    Ft

    ( ∫ +∞

    −∞

    a(τ)b∗(τ − t)dτ

    )

    = A(f)B∗(f)

    Fourier – p. 16/16

    Fourier Analysis Continuous Fourier transform Decomposition into cosines Decomposition into sines and cosines Discretisation of signal Aliasing in time domain Aliasing = overlapping spectra Discrete Fourier transform Linear Time-Invariant (LTI) systems and convolution Convolution theorem Filters Filters: original Wassenaar data Filters: Wassenaar data Filters: filtered Wassenaar data Correlation