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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:

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