Signals Through Linear Shift-Invariant Systems

Definitions

A system (operator) H is called linear if for every two signals and constants a,b:

A system (operator) H is called shift-invariant (or time-invariant) if for each signal :

Operators can be both linear and SI, or neither. It is also possible for an operator to be SI but not linear, or linear but not SI.

1 2( ), ( )t t

1 2 1 2( ) ( ) ( ) ( )H a t b t a H t b H t

( )t

0 0( ) ( )t tH T t T H t

The Impulse Response of an LSI System

The impulse response of an LSI system H, is the result of H operating on a delta function . It is often marked ( )LSIH t ( )LSIh t

( )LSIh t( )t H

Proporties of LSI Systems

Every LSI operation can be expressed as a convolution between the input signal and the system's impulse response . Mathematically: for each LSI system H and for each input signal :

Proof: Let H be an LSI system and let be its input signal.

( )t ( )LSIh t( )t

( ) ( ) ( )LSIH t t h t ( )t

(1) (2) (3)

(4) (5)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )LSI LSI

H t H t d H t d H t t d

h t d t h t

( ) ( )LSIh t t( )t H

Proporties of LSI Systems

Let H1,H2 be two LSI systems, then:

Proof:

1 2 2 1( ) ( )H H t H H t

1 2 1 2 1 2 1 2(1) (2) (3)

2 1 2 1 2 1 2 1(4) (5) (6) (7)

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

H H t H H t H h t h h t

h h t H h t H H t H H t

LSI Operators On Orthonormal Families

Let be an orthonormal family.

Let . We want to check if are orthonormal.

If by some miracle this family satisfies such that

then the inner product would be:

we say that the functions are eigenfuctions of H and are their eigenvalues.

0( ), ( ) ( )

N

k l i it t t

( ) , ( )k lH t H t

1 i n ( )H i

( ) ( ) ( )i H iH t i t

(1)

(2)

( ) , ( ) ( ) ( ) ( ) ( ) ( ) ( )

1( ) ( ) ( ) ( ) ( ) ( )

0

k l k l H k H l

H H k l H H

H t H t H t H t dt k t l t dt

k lk l t t dt k l

k l

0( )

N

i it

( )H i

0( )

N

i it

The Majestic Family

Let H be an LSI system, then for each

, where

Proof:

( ) ( ) ( )k H kH t k t ( ) ( ) ( )H k F h t k

2 2 ( )

(1) (2)

2 2

(3)

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

i k i k tk k

i k t i kk

H t h t d e h t d e h d

e e h d t F h t

The Majestic Family

As a result, when the signal (the representation of the signal in the Fourier basis) goes through an LSI system, the result is also a linear combination of with new coefficients :

Another important conclusion is the following theorem:

Proof:

0

ˆ( ) ( )N

i ii

t t

( )t

0( )

N

i it

( )i i H i

(1) (2)

0 0 0

( ) ( ) ( ) ( )N N N

i i i i i H ii i i

H t H t i t

( ) ( ) ( ) ( )k kF t h t F t F h t

(1) (2) (3)( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )k H k H k kF t h t F k t k F t F t F h t

Example of Medical Usage of LSI Ops.

Laplacian filter of bone scan (a)

Sharpened version of bone scan achieved by subtracting (a) and (b) Sobel filter of bone

scan (a)

(a)

(b)

(c)

(d)

Meet Bob!

Example of Medical Usage

Sharpened image which is sum of (a) and (f)

Result of applying a power-law trans. to (g)

(e)

(f)

(g)

(h)

The product of (c) and (e) which will be used as a mask

Image (d) smoothed with a 5*5 averaging filter

BEFORE: AFTER: