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    Design of 2D Finite-Impulse-Response (FIR) Filters

    2-D FIR Filter Design Problem

    Let

    and: Impulse response of ideal filter

    , 21

    ),( 21 nni

    consists of an infinite summation and cannot be realized in

    2211

    1 2

    ),(),( 2121njnj

    n n

    eenniI

    =

    ),( 21 I

    Approximate by a finite-degree polynomial function),( 21 I

    2211

    12 22njnj

    N N

    111 212

    computedbetotscoefficien

    2121 ,, Nn Nn= = 43421

    FIR Filter

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

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    Design of 2D Finite-Impulse-Response (FIR) Filters

    2-D FIR Filter design using windows

    Truncates infinite sum

    en ca o - proce ure

    Spatial or time-domain method

    Review of 1-D Procedure

    .

    2. Then

    .

    = dWIH )()(1)(

    3. The window w(n) should be chosen such that

    - it is N-point long

    -

    - w(n) should be conjugate symmetric if H() is to be linear-phase

    10);1()( * = NnnNwnw

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

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    Design of 2D Finite-Impulse-Response (FIR) Filters

    2-D FIR Filter design using windows

    The 2-D Procedure

    . e

    2. Then

    ,.,, 212121 nnwnnnn =

    =

    2122112121 ),(),(1

    ),( ddWIH

    3. The window w(n1,n2) should be chosen such that (Good

    Window criteria)

    - 1 2

    - its DTFT

    - w(n1,n2) should be conjugate symmetric if H(1,2) is to be

    -

    ),(),( 2121 W

    10

    10);1,1(),(

    22

    112211

    *

    21

    =

    Nn

    NnnNnNwnnw

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

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    Design of 2D Finite-Impulse-Response (FIR) Filters

    -

    Choosing The 2-D Window

    1-D windows often used for generating 2-D windows

    - Method 1:

    a 2D window formed as the outer roduct of Good 1-D windows

    )().(),( 221121 nwnwnnw =

    b) has a rectangular region of support

    c) Main lobe shape and side lobe heights can be calculated using

    1-D results =

    ),( 21 nnw

    - Method 2:

    ., 221121

    )(),(2

    2

    2

    121 nnwnnw +=

    a) w(n) is a Good 1-D continous window function

    b) w(n1,n2) is a sampled and approximately a circularly rotatedversion of the 1-D window.

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

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    Desi n of 2D Finite-Im ulse-Res onse FIR Filters

    2-D FIR Filter design using windows

    Example 1: 11x11 lowpass FIR filter design based on an outer product

    window and a 1-D Kaiser window

    Perspective Plot of Magnitude Response

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

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    Desi n of 2D Finite-Im ulse-Res onse FIR Filters 2-D FIR Filter design using windows

    Example 1: 11x11 lowpass FIR filter design based on an outer

    -

    Contour Plot of Magnitude Response

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

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    Desi n of 2D Finite-Im ulse-Res onse FIR Filters 2-D FIR Filter design using windows

    Example 2: 11x11 lowpass FIR filter design based on a rotated window

    -

    Perspective Plot of Magnitude Response

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

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    Desi n of 2D Finite-Im ulse-Res onse FIR Filters 2-D FIR Filter design using windows

    Example 2: 11x11 lowpass FIR filter design based on a rotated

    w n ow an a - a ser w n ow

    Contour Plot of Magnitude Response

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

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    Design of 2D Finite-Impulse-Response (FIR) Filters

    Optimal FIR Filter Design Error measure needed to assess how much the designed filter

    deviates from the desired ideal filter ),( 21 H),( 21 I

    ),(),(),( 212121 IHE =2211

    1 2

    ),(),( 2121njnj

    n n

    eennhI

    +=(n1,n2) in

    Choose the filter coefficients to minimize some function of this error L Norm:

    finite-extent ROS

    Lp Norm:

    2/1

    21

    2

    2122}),(

    4{ =

    ddEE

    L

    Norm:

    p

    p ddEE 21212 }),(4{ =

    ),(max EE =

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

    ),( 21 B

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    Design of 2D Finite-Impulse-Response (FIR) Filters

    Optimal Least-Squares Designs

    =

    22 1

    ddEE

    [ ] =1 2

    ;),(),( 22121n n

    nninnh

    4

    If R is the region of support of h(n1,n2):

    [ ] [ ] +=2

    21

    2

    2121

    2

    2 ),(),(),( n Rnn Rn nninninnhE

    To minimize E22, set

    = Rnnnni

    nnh),(,),(

    ),(2121

    21

    where w(n1,n2) is a rectangular window with support R.

    e se,),(),( 2121 nnnni =

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

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    Design of 2D Finite-Impulse-Response (FIR) Filters

    Optimal Design with Constraints

    ),(]exp[),(),( 21),(

    22112121

    21

    =Rnn

    InjnjnnhE

    We have been treating the filter coefficients as if they were independent.These may be constrained.

    Exam le: Desi n of a zero- hase filter with real coefficients

    (i.e., H(1, 2) Symmetric) ),(),( 2121 nnhnnh =

    ,,,, 21),(

    22112121

    21

    =Rnn

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

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    Design of 2D Finite-Impulse-Response (FIR) Filters

    Example: Design of a zero-phase filter with real coefficients (continued)

    => H(1, 2) & h(n1,n2) even symmetric

    ),(]cos[),(2)0,0(),( 2122112121 ++= InnnnhhE

    where (assuming filter is (2N+1)x(2N+1))

    )(),(),()( 211

    21 = = IiaF

    i

    i

    ),( 21 Rnn

    2N+1

    F= # of degrees of freedom = # samples in region above1)12( 22 ++N

    a(i) = ith free parameter

    = th

    2==

    +++=

    ==

    1)12(,),(2

    1,)0,0(

    1221 nnNinnh

    ih

    Note: For any linear constraints, the error can be written in the form (*); but F,

    a i will differ for different choices of constraints.

    ,21i

    1)12(),cos( 122211 +++=+= nnNinn

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

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    Design of 2D Finite-Impulse-Response (FIR) Filters

    p ma es gn w ons ra n s

    where

    ),(),()(),( 211

    2121 IiaEF

    i

    i=

    =

    F= # of degrees of freedom = # of free parameters

    a(i) = ith free parameter

    =ith basis function),( 21 i

    Example: Suppose we want to design a 5x5 circularly symmetric

    lowpass FIR filter. The ideal frequency response is given by:

    + R,1 2222

    1

    otherwise,0, 21

    2

    2

    2

    1

    2

    2

    2

    121 offunctionJINC"")(),( nnnnfnni +=+=

    So, we would like to have:

    ,,,22

    nnhnnhnnnnh ==+=

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

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    Design of 2D Finite-Impulse-Response (FIR) Filters

    Optimal Design with Constraints

    Example: Design of a 5x5 circularly symmetric lowpass filter (continued)

    ),(),()(),( 12212

    2

    2

    121 nnhnnhnnfnnh ==+=

    )1,2()1,1()0,1()1,1()1,2(

    )2,2()1,2()0,2()1,2()2,2(

    hhhhh

    hhhhhonly 6 out of the 25parameters are free.

    )2,2()1,2()0,2()1,2()2,2(

    )1,2()1,1()0,1()1,1()1,2(

    hhhhh

    hhhhh

    = ),( 21 H

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

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    Design of 2D Finite-Impulse-Response (FIR) Filters

    Optimal Design with Constraints

    Example: Design of a 5x5 circularly symmetric lowpass filter (continued)

    , 21

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

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    Design of 2D Finite-Impulse-Response (FIR) Filters

    Optimal Design with Constraints

    Example: Design of a 5x5 circularly symmetric lowpass filter (continued)

    F

    ==

    =i

    iiaH

    1

    2121 ),)),

    where F=

    =

    =

    )2(

    )1(

    a

    a

    =

    =

    =

    ),(

    ),(

    212

    211

    =)(Fa

    M

    =),(

    ,

    21

    213

    F

    M

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

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    Design of 2D Finite-Impulse-Response (FIR) Filters

    -

    Takin the derivative of E 2 with res ect to the each of the a i and settin the

    =

    =

    21

    2

    21

    1

    212

    2

    2 ),(),()(4

    1ddIiaE

    F

    i

    i

    result to zero gives the following set of Flinear equations:

    =

    ==F

    i

    Kik FKIia1

    ,...,2,1;)(

    =

    2121212 ),(),(4

    ddKiiK

    =

    2121212 ),(),(1 ddII KK Note: If orthogonal basis , then

    KiiK = ;0ii

    iIia

    = )(

    )},({ 21 i

    Issues:

    Integral may be difficult to solve

    No possibility of frequency weighting.

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

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    Design of 2D Finite-Impulse-Response (FIR) Filters

    Optimal Designs with Constraints

    Discrete Solutions

    Previous difficulties can be partially alleviated by replacing E2by

    )],(),([ 21212 mmm mmm IHWE

    )],(),()([ 2121 mmmmi

    F

    m IiaW =

    - Weights

    - Error can be controlled with the values of the weights and thedensity/locations of the samples.

    0MW

    1m i=

    - The coefficients that minimize E2are given by

    =

    ==F

    i

    Kik FKIia1

    ,...,2,1;)(

    w ere),(),( 21

    *

    21 mmKmmi

    m

    miK W =

    ,,*

    IWI =

    Copyright by Prof. Lina Karam

    EEE 507 - Lecture 12

    m