TSDT14 Signal Theory€¦ · Classification of Systems 2016-10-06 7 TSDT14 Signal Theory - Lecture 11 LSI: Bothlinearand space-invariant. Similarilyfor space-discretesystems Linearity:

TSDT14 Signal TheoryLecture 11

Multi-Dimensional Processes – Primarily 2-D

Mikael Olofsson

Department of EE (ISY)

Div. of Communication Systems

Example: 2-D Filtering

2016-10-06 2

TSDT14 Signal Theory - Lecture 11

Source: http://www.grand-illusions.com/opticalillusions/angry_and_calm/

Mr.

Angry

Mrs.

Calm

Example: 2-D Filtering

2016-10-06 3


Source: http://www.grand-illusions.com/opticalillusions/angry_and_calm/

Mr.

Angry

Mrs.

Calm? ?

Multi-Dimensional Signals & Systems

2016-10-06 4


Two-Dimensional Signals

2016-10-06 5


),( 21 aax

A function of two variables

(space-discrete)

(space-continuous)

],[ 21 nnx

Separable signals:

][][],[ 221121 nxnxnnx ⋅=

)()(),( 221121 axaxaax ⋅=

−1.5

−1

−0.5

0

0.5

1

1.5

−1.5

−1

−0.5

0

0.5

1

1.5

−1

−0.5

0

0.5

1

2a1a

1a−1.5 −1 −0.5 0 0.5 1 1.5

−1.5

−1

−0.5

0

0.5

1

1.5

2a

Two-Dimensional Stochastic Processes

2016-10-06 6


{ }),(E),( 2121 aaXaamX =Mean:

Auto-correlation: ( ) { }),(),(E,;, 221121221121 babaXaaXbabaaarX ++=++

),( 21 aa

Wide-sense stationarity:

Both ),( 21 aamX( )221121 ,;, babaaarX ++and independent of .

Simplified notation: Xm ( )21,bbrXand

Classification of Systems

2016-10-06 7


LSI: Both linear and space-invariant.

Similarily for space-discrete systems

( )211 , aaxLinearity: Input gives output ( )211 ,aay

( ) ( )212211 ,, aaxbaaxa ⋅+⋅

( ) ( )212211 ,, aaybaaya ⋅+⋅

( )212 ,aaxInput gives output ( )212 ,aay

Then input

gives output

.

.

.

Space-invariance: ( )21,aaxInput gives output ( )21,aay

( )2211 , babax −−Then input

gives output ( )2211 , babay −−

.

.

Two-Dimensional Convolution

2016-10-06 8


222222111111 )()()()( dbbahbxdbbahbx −−= ∫∫∞

∞−

∞

∞−

))(())(( 222111 ahxahx ∗⋅∗=

For LSI systems, the output is given by a two-dimensional convolution ofthe input and the impulse response of the filter.

Separable signals: ( )( ) ( ) ( ) ( ) ( ) 21222111221121, dbdbbahbahbxbxaahx −−= ∫ ∫∞

∞−

∞

∞−

*

Definition: ( )( ) ( ) ( ) 2122112121 ,,, dbdbbabahbbxaahx −−= ∫ ∫∞

∞−

∞

∞−

*

Space-discrete: ( )[ ] [ ] [ ]22112121 ,,,1 2

knknhkkxnnhxk k

−−⋅= ∑∑*

Non-Separable Convolution 1(2)

2016-10-06 9


( ) ( ) <+

==elsewhere,0

1,1,,

2

2

2

1

2121

aaaahaax

−2

−1

0

1

2

−2

−1

0

1

2

0

0.2

0.4

0.6

0.8

1

Non-separable signals−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Non-Separable Convolution 2(2)

2016-10-06 10


−2

−1

0

1

2

−2

−1

0

1

2

0

0.5

1

1.5

2

2.5

3

3.5

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

<+=−−

=elsewhere,0

4,)2/(1)2/arccos(2 2

2

2

1

22aaaaaa

( ) ( )( )2121 ,, aahxaay = *

Separable Convolution 1(3)

2016-10-06 11


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2

−1

0

1

2

−2

−1

0

1

2

0

0.2

0.4

0.6

0.8

1

Separable signals

( ) ( ) == 2121 ,, aahaax

<<

=elsewhere,0

21and2/1,1 21 /aa

( ) ( )21 azaz ⋅=

( ) ( ) <

==elsewhere,0

2/1,1rect

aaazwith


2016-10-06 12


{ } { }21 1,0max1,0max aa −⋅−=

( ) ( )21 triangletriangle aa ⋅=

−2

−1

0

1

2

−2

−1

0

1

2

0

0.2

0.4

0.6

0.8

1

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

( ) ( )( )2121 ,, aahxaay = *


2016-10-06 13


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Scaledgraphic

−2

−1

0

1

2

−2

−1

0

1

2

0

0.2

0.4

0.6

0.8

1

{ } { }21 1,0max1,0max aa −⋅−=

( ) ( )( )2121 ,, aahxaay = *

( ) ( )21 triangletriangle aa ⋅=

2-D Space-Continuous Fourier Transform

2016-10-06 14


( ) ( ){ } 21

)(2

2121212211),(,, dadaeaaxaaxffX

afafj +−

∞

∞−

∞

∞−

∫ ∫==π

FDefinition:

Inverse: ( ) ( ){ } 21

)(2

2121212211),(,, dfdfeffXffXaax

afafj +

∞

∞−

∞

∞−

∫ ∫==π1-

F

Properties: ( ){ } ( ){ }{ } ( ){ }{ }2112212121 ,,, aaxaaxaax FFFFF ==

( ) ( ){ } ( ){ } ( ){ }2221112211 axaxaxax FFF ⋅=⋅

*( ) ( ){ } ( )212121 ,)(,, ffHXaahaax =⋅F

( ){ } ( ) ( )212121 ,,,)( ffHffXaahx ⋅=F *

2-D Space-Discrete Fourier Transform

2016-10-06 15


[ ] [ ]{ } [ ] )(2

2121212211

1 2

,,,nnj

n n

ennxnnxXθθπ

θθ+−∑∑==FDefinition:

Properties:

Inverse: [ ] [ ]{ } [ ] 21

)(2

21

1

0

1

0

21212211,,, θθθθθθ

θθπddeXXaax

nnj +

∫ ∫==1-

F

[ ]{ } [ ]{ }{ } [ ]{ }{ }2112212121 ,,, nnxnnxnnx FFFFF ==

[ ] [ ]{ } [ ]{ } [ ]{ }2221112211 nxnxnxnx FFF ⋅=⋅

*[ ] [ ]{ } [ ]212121 ,)(,, θθHXnnhnnx =⋅F

[ ]{ } [ ] [ ]212121 ,,,)( θθθθ HXnnhx ⋅=F *

[ ] [ ]221121 ,, mmXX ++= θθθθ for integers and .1m 2m

(periodic conv)

Power and Power Spectral Density

2016-10-06 16


( ) ( ){ } ( ) 21

)(2

2121212211,,, dbdbebbrbbrffR

bfbfj

XXX

+−

∞

∞−

∞

∞−

∫ ∫==π

FPSD:

Mean: )0,0(),( 2121 Hmdadaaahmm XXY ⋅== ∫ ∫∞

∞−

∞

∞−

Output: ),)((),( 2121 aahXaaY = *

ACF: ),)(~

(),( 2121 bbrhhbbr XY = * * with ),(),(~

2121 aahaah −−=

Output of LSI-system if input is WSS:

PSD: ( ) ( ) ( )21

2

2121 ,,, ffRffHffR XY =

Power: ( ){ } ( ) ( ) 212121

2 ,0,0, dfdfffRraaXEP XXX ∫ ∫∞

∞−

∞

∞−

===

Two-Dimensional Sampling

2016-10-06 17


[ ] ( )221121 ,, AnAnxnny =Sampling:

Probabilistic (for rectangular grid):

Deterministic (for rectangular grid):

Spectrum: [ ] ∑∑

−−=

1 2 2

22

1

11

21

21 ,1

,k k A

k

A

kX

AAY

θθθθ

1A 2ASampling periods: and

Sampling: [ ] ( )221121 ,, AnAnXnnY =

PSD: [ ] ∑∑

−−=

1 2 2

22

1

11

21

21 ,1

,k k

XYA

k

A

kR

AAR

θθθθ

Two-Dimensional PAM

2016-10-06 18


Probabilistic (for rectangular grid):

Deterministic (for rectangular grid):

Spectrum: ( ) ( ) [ ]22112121 ,,, AfAfYffPffZ ⋅=

PSD: ( ) ( ) [ ]2211

2

21

21

21 ,,1

, AfAfRffPAA

ffR YZ =

PAM: ( ) [ ] ( )∑∑ −−=

1 2

2221112121 ,,,n n

AnaAnapnnyaaz

PAM: ( ) [ ] ( )∑∑ Ψ−−Ψ−−=

1 2

222211112121 ,,,n n

AnaAnapnnYaaZ

1Ψ 2Ψuniform on [ )1,0 A [ )2,0 Auniform onand

both independent of and of each other.[ ]21,nnY

www.liu.se

Mikael Olofsson

ISY/CommSys

Rounding Up the Course

2016-10-06 20


Stochastic processes: Stationarity, ergodicity, mean, ACF, PSD…

LTI filtering: Mean, ACF, PSD.

Cross-correlation and cross-spectrum. Joint stationarity.

Poisson processes.

Prediction.

Non-linearities: Squaring and such, saturation, quantization.

Modulation: AM, FM, PM, noise.

Estimation (only on laborations).

Linear mappings: Sampling, PAM, reconstruction.

Two-dimensional: Signals, systems,…

Written Examination

2016-10-06 21


When: Friday 2015-10-30, 14.00-18.00. Sign up!

Allowed aids:

Olofsson: Tables and Formulas for Signal Theory

Henriksson/Lindman: Formelsamling i Signalteori

Pocket calculator with empty memory

A German 10 mark note of the fourth series (1991-2001)

What:

A three-part introductory task (simple, 2/3 must be OK).

Five problems – 5 points each, pass is 10 points.

Written Examination – cont’d

2016-10-06 22


A German 10 mark note of the fourth series (1991-2001)

Good Practices at Exams

2016-10-06 23


Rules according to the exam cover:

• Only one task on the same piece ofpaper.

• Use only one side of the paper.

• Number the pages. (see common sense → )

• Do not use a red pen(cil). (that’s my color)

Common sense:

1. Solve the exam problems.

2. Sort the papers according totask numbering.

3. Number the pages last!

4. Now hand in your exam.

Do not do it in any other order!

Let me add:

• Hand in readable solutions.

• Do not hand in scriblings!

Finally:

• Always provide solid arguments for steps taken in your solutions.

TSDT14 Signal Theory€¦ · Classification of Systems 2016-10-06 7 TSDT14 Signal Theory - Lecture 11 LSI: Bothlinearand space-invariant. Similarilyfor space-discretesystems Linearity:

Documents