The design of 2-D approximately linear phase filters using a direct approach

PROCESSING Signal Processing 57 (1997) 205-221

The design of 2-D approximately linear phase filters using a direct approach

Stuart Lawsona**, Martin Andersonb

a Department of Engineering, Universiy of Warwick, Coventv, CV4 7AL, UK

b Racal Instruments p/c, Slough, Berkshire, UK

Received 20 November 1995: revised 17 June 1996 and 14 October 1996

Abstract

A novel approach to the design of approximately linear phase recursive digital filters was recently described. The technique exploited a property of allpass transfer functions which allowed the creation of a set of linear simultaneous equations in the filter coefficients which could then be solved in the least-squares sense. This paper extends the design method to 2-D and multi-D filters. These filters find application in many areas including image processing, image coding, filter banks and seismic data processing. Several problems arise which are not present in the 1-D design and the paper looks at ways to overcome them. In addition, there is discussion on the stability issue. Several examples including circularly, elliptically and diamond symmetric lowpass filters and a fan filter are presented. Comparison is made between the proposed technique and those of Gu and Swamy (1994) and Toyoshima et al. (1990). 0 1997 Elsevier Science B.V.

Zusammenfassung

Kiirzlich wurde ein neuer Ansatz fiir den Entwurf von annlhernd linearphasigen rekursiven digitalen Filtern beschrieben. Diese Methode beniitzte eine Eigenschaft von Allpa&iibertragungsfunktionen, die die Formulierung eines linearen Gleichungssystems in den Filterkoeffizienten erlaubte; dieses Gleichungssystem konnte dann im Sinn eines minimalen quadratischen Fehlers gel&t werden. Der vorliegende Artikel verallgemeinert dieses Entwurfsverfahren auf zweidimensionale und mehrdimensionale Filter. Diese Filter werden in vielen Bereichen (z.B. Bildverarbeitung, Bildcodierung, Filterbgnke und seismische Signalverarbeitung) angewandt. Es existieren hier mehrere Probleme, die es beim eindimensionalen Entwurf nicht gibt; der Artikel untersucht MGglichkeiten, diese Probleme zu iiberwinden. Weiters werden Fragen der Stabilitlt diskutiert. Als Beispiele werden u.a. TiefpaD-Filter mit kreisfiirmiger, elliptischer und diamantenfijrmiger Symmetrie sowie ein Fgcherfilter gebracht. Die vorgeschlagene Methode wird mit den Methoden von Gu und Swamy (1994) sowie von Toyoshima et al. (1990) verglichen. @ 1997 Elsevier Science B.V.

RbumC

RCcemment, une nouvelle approche a ttt proposCe pour la conception de filtres numtriques rCcursifs & phase approximativement 1inCaire. La technique exploitait une propri&tC des fonctions de transfert passe-tout qui permettait la crtation d’un ensemble d’tquations 1inCaires simultan&es pour les coefficients du filtre, qui pouvait ensuite etre rksolu au sens des moindres carrts. Cet article &tend la mCthode de conception aux filtres 2-D et multi-dimensionnels. Ces filtres ont de nombreux champs d’application, comme le traitement d’images, le codage d’images, les banes de filtres et le traitement de donnttes sismiques. Plusieurs probltmes qui n’existent pas dans le cas 1-D se posent, et cet article cherche

*Corresponding author. Tel.: + 44 1203 523780: fax: + 44 1203 418922; e-mail: [email protected].

0165-1684/97/$17.00 Q 1997 Elsevier Science B.V. All rights reserved. PII SO165-1684(97)00004-2

206 S. Lawson, M. Anderson / Signal Processing 57 (1997) 205-221

des moyens de les resoudre. De plus, il faut discuter du probleme de la stabilite. Plusieurs exemples, incluant des filtres passe-bas symetriques circulairement, elliptiquement ou en diamant, ainsi qu’un filtre en &entail sont presentes. Une comparaison est faite entre la technique proposee et celles de Gu et Swamy (1994) d’un cot& et de Toyoshima et al. (1990) de l’autre. ~0 1997 Elsevier Science B.V.

Keywords: Recursive digital filters; Two-dimensional digital filters; Wave digital filters; Approximately linear phase filters; Fan filters; Image processing

1. Introduction

Due to the rapidly increasing interest in high-definition TV, multimedia, etc., the processing of still and motion, monochrome and colour pictures has become of great importance. 2-D linear phase FIR filters have been used particularly in scanning rate converters, PAL decoders and digital video codecs based on the discrete cosine transform (DCT), subband and pyramidal coding schemes [3]. PCAS filters offer strong competition to FIR filters because they require less computation per sample and thus will improve overall computational efficiency. In this way sophisticated signal processing can be performed on both audio and video signals such as subband coding for digital compression [27]. Although the PCAS structures are recursive, stability is assured given certain conditions and furthermore, unlike other recursive filters, approximately linear phase can be achieved easily [l, 21.

In this contribution a technique that has been extensively applied to 1-D filter design is extended to the 2-D and multi-D cases. This technique ex- ploits a property of allpass functions which allows a set of linear equations to be set up in the filter coefficients. These equations can be solved in the least-squares sense. The filter structure used is a parallel combination of allpass subjilters (PCAS). Because of the nature of the overall transfer function it is easy to simultaneously meet magnitude and phase constraints. This represents both a more efficient and more elegant technique than cascading a minimum-phase filter with a delay equaliser. Non-recursive filters can be designed to have a linear phase or constant delay characteristic. Recursive filters, on the other hand, can in practical cases have only approximately linear phase (ALP) characteristics. However by increasing the filter order, the ALP performance can move closer to that of the

non-recursive filter class. In practice, as with the magnitude response, specifications give a tolerance for the delay so it is not necessary to have perfect linear phase everywhere in the passband.

Allpass filters have been in use for many years. In both the continuous and discrete time domains they have been invaluable as phase and delay equalisers [4,8,13,23,28-J. An extensive theory for digital allpass networks has been developed in recent years showing links with classical analogue network theory with ideas such as passivity, losslessness and low sensitivity [20,21]. There are also strong links with wave digital filter theory [8].

The first use of PCAS filters came in 1974 when the structure for WDF lattice networks was introduced [7]. In 1986, two papers introduced the PCAS structure in which one of its branches was a pure delay [12,22]. A more general approach to allpass filters was described in [20] which aimed to survey and unify all the various treatments so far. A design method using optimisation for general PCAS networks was described in 1992 [ 151. In 1992, the direct design approach to PCAS filter design was introduced. This avoided the need for time consum- ing optimisation, instead using a property of allpass functions discovered by Gregorian and Temes many years before [S]. As a result, a set of over-determined linear simultaneous equations in the allpass coefficients was derived, the solution of which could be found using the method of least squares [13, 141. This early work used the simplest PCAS structure; a more general structure was used in [16].

In 1990, Japanese researchers extended the PCAS concept to two dimensions [25]. Again they used the simplest structure, i.e. with a pure delay in one of the branches, and determined the coefficients using non-linear optimisation. They appreciated the problems of the 2-D allpass phase function in that certain spatial points were fixed at zero or

S. Lawson, M. Anderson / Signal Processing 57 (1997) 205-221 201

unity and, in the circularly symmetric case, a diagonal passband was created. They developed a technique to overcome this by cascading the designed filter with another whose purpose was to cancel the unwanted passband. Their technique was restricted to lowpass and bandstop filter specifications. A later paper aimed to overcome the limitations of the first technique by allowing complex coefficients was published in 1991 [26]. In a recent paper by Gu and Swamy, a new, more complex, PCAS structure was introduced which allowed a greater range of responses by adopting a multi-rate strategy [9]. Design is accomplished in 1-D with a transforma- tion to 2-D. We will use some of their design specifications for our examples.

In our contribution, which is a more general discussion than contained in [l], we extend the direct design approach from 1-D to 2-D design using the technique of Toyoshima and co-workers to overcome the fixed points and lines in the spatial frequency response. Our proposed algorithm is easy to use, fast and the filter responses are as good and, in some cases, superior to those obtained elsewhere. The PCAS structure we use allows a large range of responses to be realised but its low complexity allows economical hardware realisation.

2. The allpass transfer function

The following discussion will be carried out on the basis of 2-D functions but much of the argument can be extended to higher dimensions. The transfer function of an N1 x N2 order allpass filter with first quadrant support is given by

N(z) -gl=, IfLo a,,Z;N~+mZ;N2+n HAP (z) = m = cpio -y;zo amnZ;mZ;” ’

(1) where aoo = 1 and z = {zi, z2}.

The transfer function of Eq. (1) is allpass, the numerator and denominator functions are related by the following expression:

N(z) = D(Z)z; Nl z; N2) (2)

where i = _iz; ‘, zii). A s a consequence, an expression for the phase of an allpass filter can easily be derived. The phase 4,&O) is given by the following expression:

4AP(@) = 2arctan i

CEI=, C,“:. amn sin(m0i + n&)

cc:, x;Lo urn* cos@iei + n&) 1

- NIB, - NdA, (3)

where 0 = (6’ 0 1, 2). Before discussion of filter structure, it is instructive to consider the behaviour of 2-D allpass phase functions. The phase for different values of Ni and Nz can be evaluated at some special frequency points in a straightforward way. The values are independent of coefficient values and so can be thought of as fixed points. They will have important repercussions when design is considered in the next section. In fact, looking at Eq. (3), when 6, = 0,~ or - 71, j = 1, 2, it can be seen easily that 4Ap(@) = - N1ol - N2132.

Another important property is found by evaluat- ing the phase function of Eq. (3) along the line H1 + o2 = 0. It is found that

4AP(@) = zarctan

i c*,n hnn Csin(m - 01) c m.n %n (cos(m - 4 02 > I

- WI - N2) 81. (4)

For transfer functions with circular symmetry,

amn = and Ni = Na, so Eq. (4) reduces to 4Ap(@y’ - (N, - NJ 8i = 0. This phenomenon will also have implications in filter design to be discussed in a later section.

If circular symmetry is imposed on the filter coefficients and N = N1 = Nz at the start then Eq. (3) becomes, after some manipulation,

$Ap(@) = %arctan

i

E=02=0,mZn 2a,, sin[J(m + n)(6, + 0J]cos [f(m - n)(0, - &)I + CEzoa,, sin[m(e, + 6,)]

CL0 CL 0,mZn2a,,cosCf(m + n)(e, + UlcosC4C~ -464 - &)I +C~=oammcosCWA + WI 1

- N(B1 + 0,). (5)


As before, along the diagonal 8i + o2 = 0, it is easy to see that Eq. (5) yields zero phase. In contrast, we can see that the phase is not zero in general along the diagonal 0, - Q2 = 0.

In the 1-D case, an allpass transfer function can be factorised into first and second-order allpass factors. However, in the 2-D case this is not generally possible. The method to be described finds the overall allpass transfer functions of the upper and lower branch subfilters.

3. The peas filter structure

The PCAS structure which is to be considered, is shown symbolically in Fig. 1. Considerable research on the 1-D PCAS structure has been carried out over a period of some 20 years since the wave digital lattice$filter was introduced. However, it is possible to consider the PCAS structure without recourse to wave digital jilter (WDF) theory.

If the transfer functions of the upper and lower branch subfilters are Gi (z) and G2(z) then, from Fig. 1,

Hi (4 = ~iiGi(z) + Y~zGz(z)>

Hz (4 = ~~iGr(z) + ~22G2Cz) >

(6)

or in matrix form H = TG. The 2 x 2 complex matrix r is normally defined so that it satisfies the power complementary condition ) HI (@)I2 + lH2(0)12 = 1. It may also satisfy the allpass complementary condition 1 H 1 (0) + H2(0)( = 1 [20,21]. 0 is the two-dimensional frequency vector c 1 ‘0 , Cl,}, Bi = wiTi. For the remainder

Fig. 1. Block diagram of PCAS filter structure.

of this contribution, the matrix r will take the following value:

r=ll l [ 1 21 -1

With this value for F, both the power and allpass complementary conditions will be satisfied.

The magnitude and phase of the overall transfer functions H,(z), Hz(z) are given by the following:

Ml(@) = lcos{(4,(@) - 42W)P)l>

M2(@) = Isin{bhW - 42(@W)l 3

(8)

&I(@) = (4,(@) + 42(@W,

42(O) = (dJl(@) + 42w + 74/2,

where h W, +2 (0) are the phase functions of the upper and lower branches, respectively. In the 1-D case, lowpass and bandstop filters can be realised using H 1 (z), whereas H2 (z) will realise highpass and bandpass designs. The form of the expressions in Eq. (8) allows the meeting of magnitude and phase constraints simultaneously. For example, in the passband of a filter, M,(O)r 1. This condition implies that $J 1 (0) z I$~ (0). Using the phase relation of Eq. (8) and applying the linear phase constraint 4,(O) = -K@, whereK= (k,,k,),itcan beseen that approximately constant magnitude and linear phase can be achieved in the passband if the phase of the upper and lower branch subfilters both approximate -KOT. Because the subfilters are allpass, the design process is essentially phase approximation.

The results of this and the previous section can be combined to give a picture of the magnitude characteristics possible with PCAS filters. Fig. 2 summarises the various fixed points and lines for M1 in the 2-D case. For Ml, the positions of the passbands and stopbands are reversed. It might appear that this figure rules out the PCAS structure for realising useful filters, however it will be seen that, by cascading with another PCAS filter, the fixed lines can be removed. Fixed points can be removed with notch filters. In this discussion we have assumed that the orders of the upper and lower branch subfilters differ by one. This is a property required in the design of lowpass and highpass

. passband

1

Fig. 2. Fixed points and line for M,

filters. In other cases, such as in the design of fan filters, other possibilities for fixed points and lines are manifest.

4. Design algorithm

4. I. The allpass linear equation

4.1. I. First quadrant support The phase function of Eq. (3) can be further

developed by applying the tangent function to both sides and rationalising the expression on the right- hand side. With some algebraic manipulation, it is possible to obtain the following equation in which the phase 4AP is now implicit:

: ? amn m=O n=O-(0.0)

x sin (m0, + n0 2 - i (4.M + NlOl + N202))

= sin {f (4AP + NlOi + N2B2)} . (9)

Eq. (9) is linear in the unknown coefficient matrix A assuming that 4AP is specified. In general there are (N, + 1)(N2 + 1) - 1 coefficients to be determined (aoo = 1) and so as many equations would be required to find the values of these coefficients. If

the filter is to have circular symmetry, then the number of independent coefficients reduces to jN(N + 3) if

N1=N2=N.


To form a set of linear simultaneous equations, a grid of spatial frequencies is generated with points in both passband and stopband. In practice there will be more equations than unknowns and so the system will be solved in the least-square sense. The grid shape will depend on the type of 2-D filter to be designed. For general use, a 180” sector is sufficient (Fig. 3). This sector is at an angle of 45” to the 8i axis so as to include the line 19~ + I!?~ = 0 which has important properties.

An equation has been established which, in prin- ciple, can generate a linear system set Cx = d to be solved for x in the least-squares sense. Thus, when C is not square, the weighted least-squares solution is as follows:

x = (CT WC)-’ CTWd, (10)

where Wis a positive definite weighting matrix. For filter design this matrix will be diagonal and good results have been obtained with W = I. However, improved stopband performance can be obtained by weighting passband and stopband differently.

Fig. 3. Lowpass filter design grid (elliptical symmetry).


4.2. Upper branch subjilter design We have found that a 10 to 1 weighting ratio in favour of the stopband has been beneficial. If the ratio is increased further then, in some cases, insta- bility arises. Other weighting strategies are possible. For example, the number of passband and stopband regions can be increased, each with its own weight [25]. In the examples of Section 5, we will use various weighting strategies. The solution vector in Eq. (10) is unique if CT WC is non-singular. Eq. (10) has been used successfully for the design of 1-D filters [13,14,16] and we show in this paper that good designs are possible in two dimensions as well. We conjecture that the method can be extended to three and higher dimensions.

Assuming a lowpass filter is to be designed with passband edges of %,, and ep2 along the axes %r and g2 respectively (Fig. 3) then the first part of the design procedure is as follows and concerns only the upper branch subfilter. For a suitable grid of points within the passband, set up a linear equation set using Eq. (9) with the phase 4AP set to - ki%i - k2g2, i.e. linear. The equation set can

then be solved for the unknown coefficients {a,,). If circular symmetry is required then we further im- pose that N1 = Nz and reduce the number of equations to be solved. Since the design grid is the passband only, no differential weighting is required.

4.1.2. Other 2-D allpass jilter supports

Eq. (9) refers to 2-D allpass filters with general first quadrant support. This can be also rewritten for 2-D allpass filters with octagonal symmetry or with non-symmetric half plane support (NSHP).

For the case of NSHP allpass filters, Eq. (3) is replaced by

4.3. Lower branch subjilter design

Having determined the phase function b,(O) by the method of least squares, it can now be applied to the formation of the function 4*(O). However it

HAP = 2arctan CEl=,C~&a,,sinW1 +n%,)+C~I=,C~~,,a-,,sin( -meI +nW cN1=

m 0 cNz

n Damn cosW4 + n%,) + CzI= 1 CfZ 1 a-mn cos( - m%, + n%J

_Nlel _ N g 2 29

and Eq. (9) is replaced by

+ Nl%l + N202))

N, Nz

+ C 1 a_,,sin{ -m%, +n%2-+(4,4p m=l n=l

+ Nl%l + N2%2))

= sin(+ ($AP + Ni%i + N2%*)). (12)

Eq. (12) will generate a set of linear simultaneous equations in {a,,} if %i, g2 are varied. For the case of 2-D allpass filters with octagonal symmetry, we use the form of Eq. (3) in which amn = a,,,,, and N1 = N2 = N, i.e. Eq. (5).

(11)

is now necessary to include information about the stopband so the grid will cover a larger area than for the upper branch subfilter (Fig. 3). Within the passband the condition is the same, i.e. to approximate linear phase. In the stopband, the phase of & must be rc out of phase with &. Since 4i is known, this can easily be imposed. The equation set to be solved is the sum of the passband and stopband sets. Here, a differential weighting is used.

4.4. Unwanted passbands

In earlier sections, it was observed that the phase along the line %r + g2 = 0 was identically zero for any allpass subfilter so by virtue of Eq. (8), the magnitude of any lowpass PCAS filter would be unity and thus a passband. This diagonal passband can be removed by cascading the filter with another


whose stopband is along the same diagonal. If the transfer function of the first filter is H(zl,z,) then that of the second filter is H(z; ‘, z2) [25]. Thus it is not necessary to design this second filter and, although it doubles the computation, the overall cascade still represents a competitive solution to the 2-D IIR design problem. It must be noted that the second filter has a diagonal passband too, along 0, - O2 = 0, but this is removed by the first filter.

As can be observed from Fig. 2, in the lowpass case, passbands exist in the vicinity of the spatial frequency points { +_ K, +_ rc 3. These can be easily removed with 1-D notch filters, the design of which is discussed in a later section.

4.5. Stability

It has been shown that for the 1-D direct design method, stability is assured in most cases if the grid points are equally spaced [4]. Unfortunately, an extension of this property to 2-D and higher dimensions has been found not to be possible and so additional measures must be taken. The situation is made more complex because for circularly symmetric designs, as was seen in Section 2, the phase is fixed at certain points and along the line o1 + I32 = 0. Thus any grid design must take ac- count of this by including point and line passbands into the passband grid. If this is done then stability, whilst not being certain, will be more likely. Fig. 4 shows the whole of the grid area including the unwanted diagonal passband for a typical specification. In practice a finite area around the diagonal passband is chosen such that 10, + t12( 6 v where q > 0. Various values for v have been tried and good results have been achieved with the value 0.27~. In the stopband, the grid must exclude the diagonal passband region because, of course, within it, the phase is close to zero. In practice the condition J0i + 19~1 3 y is applied where rl = 0.37~. A further stopband region is defined around the line f3r - t!& = 0 because it will be necessary to weight it more strongly than the rest of the stopband so that on cascading with the 90” rotation H(z; ‘, z,), stopband attenuation is maximised [25]. Although no proof exists it is believed that stable designs will always be found if the design grids are chosen

Regions

1 (Passband)

7. passband)

3 (sfopband)

- 4(S-d)

Fig. 4. Modified lowpass filter grid showing diagonal passband.

appropriately as discussed above and as shown in Fig. 4.

Because of the allpass nature of the subfilters, it is assumed that there are no essential singularities of the 2nd kind [17, IS].

4.6. The phase slopes

Another important aspect of the design process is concerned with the values of the phase slopes k, and k2. These are initially both set equal to the lowest of the subfilter orders. Then a trial and error approach is used to obtain the optimal values, increasing and/or decreasing the values of kI and k2 by small amounts until the errors are minimised. Alterna- tively a standard numerical optimisation technique such as steepest descent or simulated annealing could be employed to carry out the procedure. In the case of fan filter designs, it has been found that the value of kI must remain equal to the order of the lower subfilter for stability. This will be illustrated in the example of Section 5.5.

4.7. 1-D PCAS notch jilter

The removal of the single-point passbands in the spatial frequency response can be achieved using 1-D filters [25]. We propose to use a PCAS notch filter consisting of a 1st order allpass section in parallel with a direct connection branch [22]. The


magnitude response of first order notch filter 21 I b I , ,c I I I I 1

0 I I I , 1 I I I

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1 freq(hz)

phase response of first order notch filter 2 I l I I I 1 I I I

-2. I I I I I , , I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9

freq(hz)

Fig. 5. Magnitude and phase response of 1st order 1-D PCAS notch filter.

overall transfer function is

H(z) = 1 + p+z-’

1 + flz-’ = (1 + p) ll$i . (13)

By choosing /3 = 0.95, a very sharp notch at o = rt can be achieved (Fig. 5). The filter also has a very good phase response (Fig. 5). This solution is a considerable improvement over that used in [25].

5. Design examples

5. I. Circularly symmetric lowpass jilter

The first example is the design of a lowpass filter with circular symmetry such that in passband,

dm < QP, lH(z,,z,)l = 1 and in stopband,

dm 3 8,, lH(z,,z,)l = 0, where 8, = 0.5rt and 6, = 0.7~. In addition, the phase must be approximately linear. The filter order was chosen

to be 5 x 5, the design grid was 20 x 20 and the analysis grid was 32 x 32. The ideal phase slopes (k, and k2) were both 2.4. The method described in an earlier section can be used and the magnitude and phase responses are shown in Figs. 6 and 7. Stability analysis of the design was carried out and the root locus was entirely within the unit circle.

5.2. Elliptically symmetric lowpass filter

The design algorithm can be easily extended to cover filters with elliptical symmetry in which case we have the following passband and stopband conditions:

(14)


Fig. 6. Example 1. Magnitude versus spatial frequency response plot.

-0.5

Fig. 7. Example 1. Phase versus spatial frequency response plot.

The specification is taken from the paper by Gu and Swamy [9] as follows: f,r = 0.3, fPz = 0.2, fsI = 0.4275,fs2 = 0.285. The order is 7 x 7 and the delay slopes are both 3.5. Grid and analysis densities

are the same as in the previous example. Figs. 8 and 9 show the contour plot and root map, respectively. The responses can be refined by altering the delay slopes.

214 S. Lawson, M. Anderson 1 Signal Processing 57 (1997) 205-221

:._

_0.4_ . . ; .:. .i . . .._

-0.5 ; L I I I I I I -0.6 -0.4 -0.2 0 0.2 0.4 0.6

Fig. 8. Example 2. Contour plot of magnitude response showing elliptical symmetry.

-0.5 0 0.5 1

Fig. 9. Example 2. Stability plot of overall PCAS transfer function.

5.3. Pure delay in one branch

If one of the branches of the PCAS structure is a pure delay then the design procedure simplifies

significantly because there is only one subfilter to design. Early work for 1-D filter design used this configuration [12, 221 as did Toy- oshima and co-workers in the 2-D case [25, 261. The 2-D transfer function can be written as follows:

H(zi,zz) = $ (G(zi,zJ + z;~,z;~~), (15)

where G(zi,zJ is the transfer function of the allpass subfilter of order N1 x Nz. We will assume that M, = M, = M and N1 = N2 = N. In addition M=N-1.

The phase of the allpass function G(zl,zz) has already been given in Eq. (3). The overall phase of the filter defined by Eq. (15) is as follows:

&, = +(& - Mioi - Mz’&). (16)

As an example we consider the specification of 5.1 with N = 5, M = 4 and kl = kz = 4. The contour response is shown in Fig. 10 from which it is clearly evident that circular symmetry is highly accurate. The stability of the overall filter is determined by the denominator of G.


Fig. 10. Example 3. Contour plot of magnitude response of PCAS filter with delay-only branch, specification as Example 1.

5.4. Diamond symmetry 5.5. Fan jiltering

Gu and Swamy have discussed diamond symmetry arguing of their importance in geophysical, video and picture processing [9]. They achieved diamond symmetry in their designs by a trans- formation technique. The approach taken here is to construct suitable passband and stopband regions in the manner of previous examples. The passband and stopband grids are bounded by various straight lines as illustrated in Fig. 11, creating three regions. The line @I + e2 = 0 must be used again to ensure stability. A summary of the regions for passband and stopband is given in Eq. (17). Boundary contour for

931: 0, < 0, O2 > 0 is F - $ < 1, P2 PI

%2: O1 > 0, e2 2 0 is F + P2

F < 1, PI

!Jb: O1 < 0, e2 > 0 is $ - G 1. Pl

3 P2

(17)

As an example we use again a specification from [9] in which fpI = 0.3,fp2 = 0.2&r = 0.45 andfsz = 0.3.

Fig. 11. Example 4. Lowpass filter design grid (diamond symmetry).

The allpass subfilter was of order 7 x 7 and grid densities were the same as in previous examples. The magnitude response is shown in Fig. 12.

Fan filters are used in seismic signal processing to remove unwanted components in the seismic image. Fan-type responses can be approximated with just one 2-D PCAS filter with non-symmetric half plane (NSHP) support [l].

An ideal fan bandpass magnitude specification is given as

wh, b) = 1 for el, t12+5910,

0 otherwise, (18)

where the sector angle !I& is defined as

or

for fan responses whose axis lies along the axes e1 or 612, respectively.

It is necessary to determine what PCAS subfilter orders give the correct fixed magnitude restrictions. The correct fixed values for a fan bandpass filter along the tI1 axis are achieved when N1 - M, = 0

or 2 and N2 - M2 = 1. For a fan bandpass filter along the e2 axis, the correct fixed values are


achieved when N, - Ml = 1 and Nz - M2 = 0 or 2. The bandstop versions are obtained using the complementary output of the above.

The correct restrictions can also be achieved by using subfilters with equal orders along both dimensions, that is N1 = N2 and Ml = M2 and plac- ingaz;’ or z; ’ delay in one of the branches. This is the approach taken here.

Because only one 2-D PCAS filter is required, the approximation domain is quite simple, only consisting of a passband region ‘!I$, and a stopband region 912, as shown in Fig. 13. The approximation domain takes up half of the frequency plane. However, because the allpass filters have NSHP support, the phase is not as restricted as with first quadrant support filters. In fact, it is required that the phase be non-zero along the diagonal e1 + O2 = 0. Negative phase can be obtained in the region where 0r + Q2 < 0 and so the frequency plane is split down the t12 axis (when the lower branch contains a z; r delay) or down the 13~ axis (when the lower branch contains a z; ’ delay).

0.5

Fig. 12. Example 4. Magnitude response of PCAS filter with diamond symmetry.

Fig. 13. Example 5. Approximation domain for fan filter.

The direct design method for NSHP PCAS filters was used to design a 7 x 7 order filter having a transition width of 0.2n: with kl fixed at 3.0 and k2 ranging from 3.0 to 3.90. The approximation


Table 1

Relationship between filter performance and kz for fan filter design

3.00 0.0072 0.0360 0.8070 0.6276 23.74 18.17

3.10 0.0069 0.0334 0.7854 0.6167 23.10 17.76

3.20 0.0066 0.0307 0.1677 0.6121 22.57 17.53

3.30 0.0066 0.0280 0.7535 0.6323 22.14 17.92

3.31 0.0066 0.0277 0.1523 0.6369 22.10 18.02

3.32 0.0067 0.0275 0.7512 0.6417 22.07 18.12

3.33 0.0067 0.0272 0.7501 0.6466 22.03 18.23

3.34 0.0067 0.0273 0.7490 0.6517 22.00 18.34

3.35 0.0067 0.0275 0.7480 0.6571 21.97 18.45

3.40 0.0068 0.0290 0.7436 0.6869 21.82 19.12

3.50 0.0070 0.0332 0.7366 0.7603 21.59 20.71

3.60 0.0073 0.0396 0.7275 0.8538 21.25 22.65

3.70 0.0075 0.0518 0.7089 0.9633 20.57 24.76

3.80 0.0075 0.0564 0.6703 1.0737 19.24 26.63

3.90 0.0076 0.0786 0.6557 1.1823 18.72 28.35

0.8

Fig. 14. Example 5. Magnitude response of fan filter.

domain consisted of 560 points. It was found that k, must be equal to the lower subfilter order for the filter to remain stable. Table 1 shows that 6,i de- creases as k2 increases, whilst 6,, increases with k2. Both 6, and 6, decrease as k2 rises to a value of 3.33, and then increase as k2 rises above 3.33. The magnitude response of the filter designed with kl = kz = 3.33, is shown in Fig. 14.

6. Discussion

6.1. Comparison with other design approaches

The previous sections have shown that the 2-D PCAS filter is capable of realising very good approximations to circularly and elliptically symmetric lowpass responses and various fan-type responses,

218 S. Lawson, M. Anderson / Signal Processing 57 (I 997) 205-221

for ALP specifications. There is considerable evid- ence to suggest that the best 1-D filter structure is the PCAS filter, however a comparison between 2-D PCAS filters and other 2-D filters meeting the same specification is needed to decide whether or not the 2-D PCAS filter is also the ‘best’ 2-D filter structure. This subsection presents comparisons between 2-D PCAS filters, FIR filters and other IIR filters designed to meet the same or similar specifications. Design methods for 2-D filters approximating fan type responses have been presented by several authors. This type of response has been used as a benchmark test for 2-D digital filter design techniques [S].

Toyoshima and Takahashi [26] proposed a design method for complex 2-D PCAS filters approximating a fan-type response with the use of analytic signal processing. The overall structure requires two analytic signal transformers and two complex 2-D allpass filters. They considered the design of a 43.6’ fan filter, with a transition bandwidth of 0.2rc, passband tolerance 6, = 0.006, stopband tolerance 6, = 0.078 and linear phase tolerance 6, = 0.107. The specification was met with a filter of order 5 x 5, having a 3 x 3 order complex allpass filter and a 2 x 2 delay-only branch. The resulting structure had 32 complex multipliers which can be equated to 128 real multipliers. The delay elements must store complex data and must have double the storage of real registers of the same wordlength.

Lu and Antoniou [18] proposed a design technique for 2-D FIR filters approximating fan-type responses using singular-value decomposition. They considered the design of a 62” fan with a transition bandwidth of 0.1429~ Their SVD-LUD realisation had a passband tolerance of 6, = 0.0411, a stopband tolerance of 6, = 0.0281 and used 198 multipliers.

The design example in Section 5.5 for a 45’ fan using a 2-D PCAS filter also with a transition bandwidth of 0.2x can be fairly compared with the above designs. This 7 x 7 order 2-D PCAS filter had a smaller stopband tolerance of 0.0429 and a passband tolerance of 0.0077 and a phase tolerance of 0.0693. Only one PCAS filter is required and this has 40 coefficients. The complementary output is also available.

Harris [lo] developed a recursive lattice structure that was well suited to the realisation of fan filters.

This lattice structure was designed to realise a 90” fan response with a transition bandwidth of n/8. The resulting filter has 55 independent parameters and a passband ripple of 0.01 and a stopband attenuation of 34 dB (0.02). The phase response was not considered.

Ekstrom et al. [S] also developed design algorithms for the design of fan filters using 2-D IIR filters. They achieved some excellent designs using spectral factorisation. One notable example was a 90’ fan filter realised with a 3 x 3 order first quadrant numerator and a 3 x 3 NSHP denominator with a magnitude-squared passband tolerance of 6, = 0.046464 and a stopband tolerance of 6, = 0.063 given a transition bandwidth of 0.2rr. The squared magnitude was considered because a zero-phase implementation was assumed. This consists of two filters, each having 41 coefficients.

An NSHP 2-D PCAS filter designed in [1] satisfies both specifications. This filter has only 40 coefficients and has the extra advantages of produ- cing a complementary output for free and low coefficient sensitivity. Furthermore, because it has ALP, a zero-phase implementation is not required.

The circularly symmetric lowpass response is probably the most widely published 2-D filter response and a fair comparison between 2-D filters must include this type of response.

Harris and Mersereau [ 111 made a comparison of algorithms used for the design of optimal 2-D linear phase FIR filters. Four design examples for a circularly symmetric response with passband edge 0, = 0.47r and 8, = 0.6n and 6, = 6, are listed in Table 2.

These filters have eight-fold symmetry which reduces the number of independent coefficients dramatically, especially for high orders. It is difficult

Table 2 Optimal FIR filters

Impulse size

5x5

1x7

9x9

II x 11

No. of No. of indep.

coeff. coeff. 6,, 6,

25 6 0.2670

49 10 0.1269

81 13 0.1141

121 21 0.0569

Table 3

S. Lawson, 111. Anderson / Signal Processing 57 (1997) 205-221 219

Comparison between Toyoshima’s technique and direct design

Toyoshima’s technique Direct design technique

Simplified PCAS General PCAS Simplified PCAS Simplified PCAS

Filter order Passband error Stopband error Weightings No. of coeffs.

5X5 5x5 9x9 5x5 0.02 0.030 0.002 0.007 0.03 0.025 0.017 0.034 5,1,5,50 1,4,1,40 1,1,1,30 5,5,1,100 19 29 41 19

to make a completely fair comparison between a zero phase FIR filter and an ALP PCAS filter because in some applications ALP is adequate whilst in others zero phase is required. Also in some applications the phase response may not be of any importance, so that a magnitude-only design can be used.

In the paper by Toyoshima et al. [25], there is discussion of ALP design but only one example was presented. The PCAS structure with a pure delay in one of the branches was used and the circular symmetry obtained is fairly accurate. Examples 1 and 3 have used the same magnitude specification as given in [25]. The results are collected in Table 3 for comparison purposes.

Table 3 shows that the simplified PCAS structure, i.e with one branch a pure delay, gives the best result using either Toyoshima’s or the direct design methods. However, the direct design method does not require non-linear optimisation and further gives improved performance. The weightings for Toyoshima’s method relate to four separate regions used in the approximation. We have used the same regions but the weightings will be different because they are related specifically to the design algorithm used. The last row of the table shows the total number of independent multi- piers required in a realisation of a complete PCAS filter, i.e. H(z,, z2) cascaded with H(z;’ ,z2) and a 1-D notch filter. Comparing with the results of Table 2 for the same transition bandwidth, it can be seen that to achieve a better passband and stopband error would require a much larger order of FIR filter. The hardware implications of this are significant as considerably more delays

and adders would be required even though the number of independent multipliers may not be significantly greater.

In the paper by Gu and Swamy [9], a design technique was presented which could be applied to a wide range of 2-D recursive filters. The PCAS structure used consisted of cascades of 1 x 1 allpass sections so that designs would be expressed in these low order building blocks. In order to achieve a wide range of specifications it was necessary to incorporate a multirate sampling scheme. In addition, ALP behaviour was not possible. It is therefore difficult to compare the direct design method with that of Gu and Swamy. The diamond symmetry approximation of Example 4 is a trade-off between magnitude and phase behaviour since we desire to keep the filter order low. In [9], very good magnitude approximations are achieved because the phase is not constrained to be approximately linear.

With the exception of the fan filter, all the recursive filters mentioned so far are non-causal. In this paper and [25], a causal filter is designed and then cascaded with a non-causal filter to eliminate the unwanted passbands. In [9], the structure used from the outset is non-causal. A causal PCAS structure is possible but so far the responses achieved have not been as good as those obtained using methods discussed here [24].

4.2. Future work

The present technique can design only lowpass and bandstop filters. The line e1 + 19~ = 0 on which

270 S. Lawson. M. Anderson / Signal Processing 57 (1997) 205-221

the phase is zero prevents the realisation of highpass and bandpass designs. Toyoshima et al. have discussed the use of 2-D PCAS filters with complex coefficients as a means to overcome the problem. It is hoped to consider using this approach in conjunction with the direct design method.

The link between the 2-D PCAS structure and 2-D wave digitalJilters (WDFs) is an important one to make because of the well documented theory, particularly in regard to stability [6]. This is the subject of ongoing research.

Much of what has been said about 2-D design can be applied to multi-dimensional design. The direct design technique is easily extendable although, of course, the matrix equation to solve will be considerably larger than in the 2-D case and the stability problem will be more difficult to resolve [ 11. However, establishing a link with multi-dimensional WDFs would be an essential step.

The direct design technique can be used for multi-dimensional filters with arbitrary phase since, in Eq. (9), the allpass subfilter phase function $*r is essentially a free parameter. However it must be set appropriately so that the delay is positive, i.e. - d4,,/d% > 0.

7. Conclusions

The direct design technique has been extended to the two-dimensional case and a general approach has been described, enabling a range of filters to be designed. The method is fast because it is essentially the solution of a set of linear simultaneous equations, once the approximation domain has been established. The paper has discussed the zero phase along the line 8, + %2 = 0 and how this problem may be overcome. In addition, we have described the use of a simple 1-D notch filter to remove the unwanted responses at ( 5 n, + n). The technique proposed offers a simple approach to include approximately linear phase which is often required in 2-D applications. However, because of the nature of the formulation, the technique can be applied to specifications with arbitrary phase. Further work will concentrate on establishing a strong link with WDF theory.

Acknowledgements

The authors would like to acknowledge the support of the Engineering and Physical Sciences Re- search Council (UK) under grant GR/J/16534.

Notation

NI, Nz 2-D filter orders N used if filter orders are the same

HAP(Z) transfer function of allpass subfilter

N(z) numerator function

D(z) denominator function

amn filter coefficient Zl> a2 z-transform variables

I z = pi, z2 ) vector of z variables -1 z= (Zi , z; 11 vector of inverse z variables

01, %2 normalized frequency variables

4317 ($2 passband edge normalised frequencies

&l, f&2 stopband edge normalised frequencies

_&l&,2 passband edge frequencies

.Ll,.LZ stopband edge normalised frequencies

b,, 6, passband and stopband tolerances

&1, L delay tolerances

%r, error, delay errors as percentages

%r?. errOr 0 = (%, , %,) vector of normalised frequency

variables

4AP(@) phase of allpass subfilter G,(z), G2(z) 2-D transfer functions of upper and

lower branch subfilters H,(z), Hz(z) overall 2-D transfer functions H, G vectors of transfer functions

‘ii j coefficients of matrix r r’ 2 x 2 matrix

wt> 02 angular frequency variables T,> 7-z sampling periods

Ml, Mz magnitude of overall transfer functions

:::; phase of overall transfer functions phase slopes

K = {k,, k2: vector of phase slopes x pi B coefficient of first-order notch filter

S. Lawson, M. Anderson J Signal Processing 57 (1997) 205-221 221

References

derivative of phase wtih respect to normalised frequency

summation variables matrix and vector variables used in

the solution of the least-squares equation

transpose of matrix C used in design technique transfer function of 1-D first-order

notch filter region defined by

Cl1

PI

c31

c41

c51

C61

c71

PI

c91

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The design of 2-D approximately linear phase filters using a direct approach