2‐D discrete signals2 D discrete signals

2‐D Signal2 D Signal

2‐D unit sample sequence2 D unit sample sequence

Separable signalsSeparable signals

2‐D line impulse2 D line impulse

2‐D unit step2 D unit step

Exponential sequenceExponential sequence

Finite extent sequenceFinite extent sequence

Rectangular periodic sequencesRectangular periodic sequences

• ConstraintsConstraints

A 2‐D periodic sequenceN1=3, N2=3

Vertical PeriodPeriod

Horizontal Period

N1,N2=?N1,N2 ?

General periodicity constraintGeneral periodicity constraint

• Given the signalGiven the signal• Constraints

• Where• The ordered pairs (N11 N21) and (N12 N22) areThe ordered pairs (N11,N21) and (N12,N22) are vectors N1 and N2

– Represent displacement– Represent displacement

Periodicity in 2D signalsPeriodicity in 2D signals

Periodicity vectorsN1 = (7,2)’1 ( , )N2 = (‐2,4)’

Multi‐dimensional systemsMulti dimensional systems

SystemSystem

• The operator T[.] can be a rule or set of rules p [ ]for mapping input to an output

Fundamental operations on l d l lmultidimensional signals

• AdditionAddition• Multiplication by a constant

• Shifting

Fundamental operations on ltidi i l i lmultidimensional signals

• Multiplication with another sequence

• Also known as spatially varying gain

2D convolution2D convolution

• Short hand notation

• Vector notationVector notation

2D LTI system2D LTI system

• Linear?Linear?• Shift invariant?

Flipping of a 2‐D sequenceFlipping of a 2 D sequence

Convolution exampleConvolution example

• Unit step sequence is convolved with finiteUnit step sequence is convolved with finite extent sequence

Convolution exampleConvolution example

Convolution exampleConvolution example

Convolution exampleConvolution example

Convolution exampleConvolution example

Convolution exampleConvolution example

Frequency response of 2D LTI systemFrequency response of 2D LTI system

• Let input beLet input be– Are the horizontal and vertical frequencies

• Output using convolution is1 2,

• Output using convolution is

• Frequency response is

Frequency response‐ exampleFrequency response example

• GivenGiven 

Frequency response‐ example

Frequency response‐ exampleFrequency response example