Introduction to Fourier Transform - Linear Systems

Image Processing - Lesson 4

• Linear Systems

• Definitions & Properties

• Shift Invariant Linear Systems

• Linear Systems and Convolutions

• Linear Systems and sinusoids

• Complex Numbers and Complex Exponentials

• Linear Systems - Frequency Response

Introduction to Fourier Transform - Linear Systems

Linear Systems

• A linear system T gets an input f(t)and produces an output g(t):

• In the discrete caes:– input : f[n] , n = 0,1,2,…– output: g[n] , n = 0,1,2,…

Tf(t) g(t)

( ){ }tfTtg =)(

[ ] )](f[g nTn =

T

Linear System Properties

• A linear system must satisfy two conditions:

– Homogeneity:

– Additivity:

[ ]{ } [ ]{ }nfTanfaT =

[ ] [ ]{ } [ ]{ } [ ]{ }nfTnfTnfnfT 2121 +=+

T

T

T

Homogeneity

Additivity

T

Linear System - Example

• Contrast change by grayscale stretching around 0:

– Homogeneity:

– Additivity:

T{bf(x)} = abf(x) = baf(x) = bT{f(x)}

T{f(x)} = af(x)

T{f1(x)+f2(x)} = a(f1(x)+f2(x)) = af1(x)+af2(x)= T{f1(x)}+ T{f2(x)}

Linear System - Example

• Convolution:

– Homogeneity:

– Additivity:

T{bf(x)} = (bf)*a = b(f*a) = bT{f(x)}

T{f(x)} = f*a

T{f1(x)+f2(x)} = (f1+f2)*a = f1*a+f2*a= T{f1(x)}+ T{f2(x)}

Shift-Invariant Linear System

• Assume T is a linear system satisfying

• T is a shift-invariant linear system iff:

T

Shift Invariant

( ){ }tfTtg =)(

( ){ }00 )( ttfTttg −=−

T

Shift-Invariant Linear System - Example

• Contrast change by grayscale stretching around 0:

– Shift Invariant:

• Convolution:

– Shift Invariant:

T{f(x-x0)} = af(x-x0) =g(x-x0)

T{f(x)} = af(x) = g(x)

T{f(x-x0)} = f(x-x0)*a

T{f(x)} = f(x)*a =g(x)

∑ −−∑ =−−=j

0i

0 )xjx(a)j(f)ix(a)xi(f

= g(x-x0)

Matrix Multiplication as a Linear System

• Assume f is an input vector and T is a matrix multiplying f:

• g is an output vector. • Claim: A matrix multiplication is a

linear system:

– Homogeneity– Additivity

• Note that a matrix multiplication is not necessarily shift-invariant.

Tfg =

( ) 2121 TfTfffT +=+( ) aTfafT =

Impulse Sequence

• An impulse signal is defined as follows:

• Any signal can be represented as a linear sum of scales and shifted impulses:

[ ]

=≠

=−knwhereknwhere

kn1

0d

[ ] [ ] [ ]jnjfnfj

−= ∑∞

−∞=

δ

= + +

Shift-Invariant Linear System is a Convolution

Proof: – f[n] input sequence– g[n] output sequence– h[n] the system impulse response:

h[n]=T{δ[n]}

[ ] [ ]{ } [ ] [ ]

[ ] [ ]{ }

[ ] [ ]

hf

nariancceshiftfromjnhjf

linearityfromjnTjf

jnjfTnfTng

j

j

j

∗=

−−=

−=

−==

∑

∑

∑

∞

−∞=

∞

−∞=

∞

−∞=

)i(

)(δ

δ

The output is a sum of scaled and shifted copies of impulse responses.

Convolution as a Matrix Multiplication

• The convolution (wrap around):

can be represented as a matrix multiplication:

– The matrix rows are flipped and shifted copies of the impulse response.

– The matrix columns are shifted copies of the impulse response.

[ ] [ ] [ ]283256123210021 −−−=∗−−

−−−

=

−−

283256

210021

210003321000032100003210000321100032

CirculantMatrix

Convolution Properties

• Commutative:

– Only shift-invariant systems are commutative.– Only circulant matrices are commutative.

• Associative:

– Any linear system is associative.

• Distributive:

– Any linear system is distributive.

fTTfTT ∗∗=∗∗ 1221

( ) ( )fTTfTT ∗∗=∗∗ 2121

( )( ) 2121

2121

fTfTffTandfTfTfTT∗+∗=+∗

∗+∗=∗+

Complex Numbers

The Complex Plane

Real

Imaginary

(a,b)

a

b

• Two kind of representations for a point (a,b) in the complex plane

– The Cartesian representation:

12 −=+= iwherebiaZ– The Polar representation:

θiZ Re=

R

θ

• Conversions:

– Polar to Cartesian:

– Cartesian to Polar

( ) ( )θθθ sincosRe iRRi +=( )abiebabia /tan22 1−

+=+

(Complex exponential)

• Conjugate of Z is Z*:

– Cartesian rep.

– Polar rep.

( ) ibaiba −=+ ∗

( ) θθ ii −∗= ReRe

Reala

b

θ

-θ

-b

θibia Re=+

θibia −=− Re

R

R

Imaginary

Algebraic operations:

• addition/subtraction:

• multiplication:

• Norm:

)()()()( dbicaidciba +++=+++

( )( ) ( ) ( )adbcibdacidciba ++−=++

( )ßaßa += iii ABeBeAe

( ) ( ) 222 baibaibaiba +=++=+ ∗

( ) 22ReReReReRe Riiiii === −∗ θθθθθ

• The (Co-)Sinusoid as complex exponential:

Or

( )2eexcos

ixix −+=

( )i2eexsin

ixix −−=

The (Co-) Sinusoid

( ) ( )ixex Realcos =

( ) ( )ixex Imagsin =

x1

1

cos(x)

sin(x)eix

x

( )xπω2sin

ω

– The wavelength of is .

– The frequency is .

1

x1

1

cos(x)

sin(x)eix

The (Co-) Sinusoid- function

( )xπω2sinω1

x

( )xA πω2sin

A

– Changing Amplitude:

– Changing Phase:

x

( )ϕπω +xA 2sin

( ) )(Imag2sin 2 xii eAexA πωϕϕπω =+

Scaling and shifting can be represented as a multiplication with ϕiAe

Frequency Analysis• If a function f(x) can be expressed as a

linear sum of scaled and shifted sinusoids:

it is possible to predict the system response to f(x):

• The Fourier Transform:It is possible to express any signal as a sum of shifted and scaled sinusoids at different frequencies.

( ) xieFxf πω

ω

ω 2)( ∑=

( ){ } ( ) ( ) xieFHxfTxg πω

ω

ωω 2)( ∑==

( )∑= xieFxf πωω 2)(

( ) ωωω

πω deFxf xi∫= 2)(Or

3 sin(x)

+ 1 sin(3x)

+ 0.8 sin(5x)

+ 0.4 sin(7x)

=

Every function equals a sum of scaled and shifted Sines

+

+

+

Linear System Logic

Input Signal

Express as sum of scaled

and shifted impulses

Express as sum of scaled

and shifted sinusoids

Calculate the response to

each impulse

Calculate the response to

each sinusoid

Sum the impulse

responses to determine the

output

Sum the sinusoidal

responses to determine the

output

Frequency Method

space/time Method

)()()( xhxfxg ∗=)()()( ωωω HFG =