Fourier Analysis(Ta3520)

Fourier – p. 1/16

Fourier AnalysisContinuous Fourier transformDiscrete Fourier Transform and Sampling TheoremLinear Time-Invariant (LTI) systems andConvolutionConvolution TheoremFiltersCorrelationDeconvolution

Fourier – p. 2/16

Continuous Fourier transformDefinition:

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 cosinesFourier – p. 3/16

Decomposition into cosines

0

50

100

150

200

250

0 10 20 30

frequency

time

Fourier – p. 4/16

Decomposition into sines and cosines

Fourier – p. 5/16

Discretisation of signalDiscretisation 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 spectraAcontinuous

A D

0

01∆t

A D

01∆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 coefficientn = 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 = 1Nyquist criterion: fN = 1

2∆t Fourier – p. 9/16

Linear Time-Invariant (LTI) systemsand convolution

Output of linear time-invariant system = convolution ofinput 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 30 400

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 30 400

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 30 400

1.0

frequency

| A(f)

|

OUTPUT OF FILTER

Fourier – p. 12/16

Filters: original Wassenaar data

0

0.5

1.0

1.5

time

(s)

100 200 300offset (m)

Fourier – p. 13/16

Filters: Wassenaar data0

0.5

1.0

1.5

time

(s)

100 300distance (m)

0

50

100

150

200

250

300

350

400

450

500

frequ

ency

(Hz)

100 300distance (m)

0

50

100

150

200

250

300

350

400

450

500

frequ

ency

(Hz)

100 300distance (m)

0

0.5

1.0

1.5

time

(s)

100 300distance (m)

Fourier – p. 14/16

Filters: filtered Wassenaar data

0

0.5

1.0

1.5

time

(s)

100 200 300offset (m)

Fourier – p. 15/16

CorrelationAuto-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