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A DI RECT APPROACH T O TH E DESIGN OF MULTIDIMENSIONAL PCAS DIGITAL FIL TERSSATISFYING GAI N AND GROUP DELAY SPECIFICATI ONS SIMULTANEOUSLY

M artin Anderson and Stuart L awson'

Digital fil ters with linear or almost li near phase are required in many applications includi ng video, sonarand image processing. A direct design approach is reported here which can be applied to 1-D and M -Dfi lter design so long as the filter structure isaparallel combination of allpass subfi lters(PCA S). PCA Sfilters are used because of their low complexity and roundoff noise as well as their abili ty to realise non-mini mum phase transfer functions. The fil ter design wil l be shown to reduce to the solution of two sets oflinear simultaneous equations. Examples are given in both the I -D and 2-D casetoil lustrate the method.Phase Function of M-D A llpass Subfi lter To simplify notation, the bold type is used to represent vectorsand matrices. The M-D 2 transform is defined using the vector z = L I , 2, .., 2,t{T. T he M-D frequencyis defined using the vector u =[ w ~ , w z ,. , w M ] ~ .

The 2 transfer function of an M -D allpass digi tal fi lter of order N=(.Vl, N2, .,NM ] s given by

whereN is the set of all M x 1 integer vectors up to and including NNoting that N ( r )=D ( z - ' ) z - ~ , he phase is given by

Using eqn.(2) it can be shown[l] that

where [o] s the D x 1null vector and ~ ( [ o ] )1 .Eqn.(3) is linear in the filter coefficients a ( n ) and so can form the basis of a set of simultaneous linearequations to solve for ~ ( n ) .he set is generated by choosing anumber of frequency p0int.s in the bandsof interest. The number of frequency poi nts wil l, in general, be greater than the number of coeff ici ents.The resulti ng overdetermined equation set can be solved in the least squares sense.The equations to be solved can be wri tten in the following matrix form,

A a = b (4 )where a=[a(n)lT,

A= [sin{nu- (4~p Nu) }] ,b=[sin{f(4ap+N W) }]~. for n E N- o]E ' RALP .'Department of EngiueeringJ J nivemity of WarwicL,Coventry,CV47AL

2 1993 The Institution of Electrical EngineersPrinted and published by th e IEE. Savoy Place. London WCZR OBL. UK.

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T he weighted least squaressolution to eqn.(4) isa=[ A T W A ] - A T W b (5)

where W is a positive definite weighting matrix. For filter design, this matrix will, in most cases, bediagonal. We have used a simplified case in which W is set to be the unit matrix. This solution isunique if A ~W As non singular.PC A S Transfer Function L et the transfer function of AP1 be H1 and that of A P2 be H2 then, becauseof their allpass nature, it is well known that

where M(w) ,4 (w)and ~ ( w ) TO^ 702, ..,TOM] are the magnitude, phase and group delay of the overalltransfer function H =${H I +H z}, and & , T , = [711, T I P , ..,TIM] and 4 2 , T, =[m, 22. ..,T ~ M ] re thephase and group delay(s) of A PI and A P2 respectively.In the passband, M (w )=1 and so,using eqn.(6), 414 (w)=-kw , where k=[kl, k2, ..,kM]. Hence, in the passband, both 41 and 4 2 should approximate toIn the stopband, M (w )=0 and so , using eqn.(6),41=42+x . There is no A L P conditionDesign A lgori thm Using the results of the previous section, the AL P design algorithm for an M-Dlowpass filter of order N can now be formulated.Firstly the order of the allpass subfil ters must be determined. I t is known that in the I -D case, theirorders must differ by one . Good multidimensional results have also beeii obtained by ensuring theirorders also differ by one in each dimension, so the calculation is straightforward. In addition, the orderof AP1 must be less than that of AP2.

42 . For A LP it is also required that- kw.

The design procedure concentrates on AP I first.In the passband, the phase $1 must approximate -kw , so that eqn.(3) is used over a set of points inthe passband region,R, . The point w =[o] s not used as it yields a zero row in the matrix A .The over-determined set of linear equations is then solved for the subfilter coefficients using eqn.(5).The next stage is to design A P2.In this case, both the passband region R, and the stopband region R, have to be considered.In the passband region R,, 42must also approximate -kw .In the stopband region a, , t must approximate $1- .Eqn.(3) is used to generate a set of simultaneous linear equations over X p and 72.. for A P2.A weighted least squares solution for the equations isobtained using eqn.(5) , giving the fi lter coefficientsfor A P2.General Properties

The phase of a multidimensional allpass subfi lter is fixed at zero when D(z)=D( z - ) z -~.hephase is also fixed at C=lN,w, when w, E { - 7 r , o , m } , {i=l ,M }. This results in either fixed pass-bands or stopbands at these frequencies. When D(z)=D ( z - ) z - ~ccurs depends on the region of

M

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-31.032.033.034.035.036.037.038.039.040.041.0

- .0004 66.880.0004 68.530.0003 70.030.0003 71.670.0002 73.640.0001 75.980.0001 78.610.0001 81.420.0001 84.320.0002 87.27T.0002 90.21 4.394.293.973.533.102.802.662.682.843.093.42-able 1: Example 1: variation of various parameters with k

support of the polynomial D(z) and the polynomial coefficients. For responses which are symmetric inall dimensions with n; >0, the al lpass subfi lters have zero phase when W I +w2+..+3~=0. T hedesired filter response is obtained by using the technique described in [l]. In some cases the desiredfixed response at certain frequencies can be obtained by inserting a delay z -ND n the upper subfilterbranch ,which then modifies the branch phase toT he value of k is crucial in ensuring asatisfactory solution. In extreme cases, the design could be unsta-ble. Upper and lower bounds for the 1-D case have been reported [5]. We have found for each design, arange for k that yields acceptable results. T he number of frequency points on R, andR, s avariableparameter too. However, wi th the various filters designed, the effect of increasing the number of pointsappears not to be significant. T he minimum number is equal to the number of fil ter coefficients in A P2,but in general is chosen to be much greater because of the large frequency space w .Example I:]-D lowpass An A L P lowpass fil ter was designed using the following narrowband specifi -cation: f, =0.05, a =0.1,ap=0.1 dB and a, =70 dB, where a, a, are the passband and stopbandtolerances, respectively. T he fi lter order used was 19.In order to measure the success of the design method, the percentage delay error, d e . is used and isdefined as follows:

-(Ni +Np)w,.

1 - Xd e =200 (-)+ X (7)where X =T , , , ~ ~ / T , , , ~ ~ .his error was chosen because it does not depend on the actual values of maxi mumand mini mum delay, only on their ratio. Thi s error is calculated over the passband of the designed f il ter.In addition, the passband ripple and minimum stopband attenuation, both in dB's, are recorded.Various values of k were tried and the results are summarised in Table 1. T he grid consisted of only 30equi-spaced frequency points. This contrasts with 50 used in [2]. From the table, i t is clear that thecase when k =33 yields a design that meets the specification. T he resulting delay error is 3.97%. It ispossible to perform asingle variable optimisation on k to find the minimum delay error subject to theconstraints on apand as . I t turns out that the optimum value of k , in this case is 37.39 at which thedelay error is 2.65%.Example II :2-D lowpass fan filter A n A L P 45" lowpass fan filter was designed wi th the following speci-fication

Rp E I 4 2b2l72, E lull +0.2n5 2 1 4 (8)

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Figure 1: gth x 9h order 2-D fan filter responseT he orders of the upper and lower branches were set to N= [4,4] and N= [5,5]respectively. giving anoverall filter order of N= [9.9]. T he difficult specifi cation required the subfil ter polynomials to havenon-symmetric half plane support[4]. T hat is, the allpasssubfil ters had support in the f irst and secondquadrants but were still recursively computable.T he correct f ixed PC A S response i t s described in section 4. was obtained by inserting an extra t; delayin the upper branch. I i ith k= [4.4] and R, plus R, consisting of approximately 400 points, the filterachieved the following specificationap =0.0382 dB, a, =40.97 dB ,del =13.56% and de2 =15.96% with average passband group delays70, =4.508 and 7002=4.0207. T he magnitude response i s shown in f igure 3 .Acknowledgements T he authors would like to thank both SE RC(U K ) and the Isle of Man Governmentfor supporting this workReferences[ I ] M.S.Anderson and S.S.Lawson, Direct design of ALP 2D I IR digital filters, Electronics L etters, Vo1.29,[2] M.Lang and T.I .Laakso, Design of Allpass Filters for Phase Approximation and Equalization Using LSEE[3] S.S.Lawson, A New Direct Design Techniquefor ALP Recursive Digital Fi lters. Proc. I SCAS-93,Chicago,[4] M.Ekstrom, R.Twogood and J .Woods, Two -dimensional recursive fi lter design- a spectral factorisation[5] Z.J ing, A new method for digital dpass filter design, I EEE Trans. acoustic. , Speech 4 S:g.Proc., vo1.35,

804-5, 1993Error Criterion, Proc. ISCAS-92, San Diego, IEEE, 2417- 2420. 1992.IEEE, 499- 502, 1993.approach, IEEE Trans. ASSP, Vo1.28, 16-26, 1980.1557- 1564. 1987.

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