1

DESIGN OF LINEAR-PHASE LATTICEWAVE DIGITAL FILTERS

H�kan Johansson and Lars WanhammarDepartment of Electrical Engineering, Link�ping University

S-581 83 Link�ping, SwedenE-mail: hakanj@isy.liu.se, larsw@isy.liu.se

ABSTRACT

Lattice wave digital filters can be realized to have approximately linearphase in the passband by letting one of their allpass branches be a puredelay. In this paper, an algorithm for designing these filters is described.Several design examples using this algorithm are also given.

1. INTRODUCTION

A major advantage of wave digital filters (WDFs) over most other recursivefilters is that they can maintain stability under finite-arithmeticconditions. A particularly favourable type of wave digital filter is the latticewave digital filter. Using lattice WDFs, highly modular and parallel filteralgorithms can be obtained. This is advantageous from an implementationpoint of view.

To each wave digital filter there is a corresponding filter in a referencedomain. The design of wave digital filters can therefore be carried out inthe analog domain using classical filter approximations, whereupon atransformation from the analog domain into the digital domain, applyingcertain transformation rules, can be performed. For lattice wave digitalfilters, it is possible to use explicit formulas to directly compute the adaptorcoefficients, as given in [Gazs-85]. However, for filters satisfying bothmagnitude and phase requirements, there exist no closed form solutions.For these cases numerical optimization techniques must be adopted [Kuno-88, Abo-Z-95, Leeb-91].

In this paper we describe an algorithm for design of linear-phase latticeWDFs. The algorithm was introduced by Renfors and Saram�ki [Renf-86]for design of filters composed of two allpass filters in parallel, of which thelattice WDF is a special case. Approximately linear phase is obtained by

2

letting one of the allpass branches be a pure delay. We show how theadaptor coefficients can be computed from the result of the algorithm, forsome different WDF realizations of the allpass branches. Several designexamples using the described algorithm are given.

2 LATTICE WAVE DIGITAL FILTERS

A lattice wave digital filter is a two-branch structure where each branchrealizes an allpass filter [Fett-74]. These allpass filters can be realized inseveral ways [Fett-86]. One approach that yields parallel and modularfilter algorithms is to use cascaded first- and second-order sections, asshown in Fig. 2.1. The first- and second-order sections are here realizedusing symmetric two-port adaptors. These can easily be replaced by two-port series- or parallel adaptors using certain equivalence transformations[Fett-86, Gazs-85]. The second-order sections can also be realized usingthree-port series- or parallel adaptors [Ande-95]. Another approach is torealize the allpass filters using Richards' structures [Fett-86].

a3

a4

T

T

x(n)

a7

a8

T

T

a0

T

T

a1 a5 a9

a6

T T

a2

T T T

a10

y(n)1/2

Figure 2.1. An 11th-order lattice wave digital filter.

3

The transfer function of a lattice WDF can be written as

H (z) =

12

H0 (z) + H1(z)( ) (2.1)

where H0(z) and H1(z) are allpass filters The overall frequency responsecan therefore be written as

H (e jwT ) =

12

e jF0 (wT) + e jF1(wT)( ) (2.2)

where F0(wT) and F1(wT) are the phase responses of H0(z) and H 1(z),respectively. The magnitude of the overall filter is thus limited by

H (ejwT ) £ 1

The transfer function of a lattice WDF and its complementary transferfunction are power complementary, i.e.,

H (ejwT )

2+ Hc (e jwT )

2= 1 (2.3)

where

H z H z H zc( ) ( ) ( )= -( )1

2 0 1 (2.4)

This means that, if H(z), for example, is a lowpass filter, then a highpassfilter is obtained by simply changing the sign of the allpass filter H1(z).

An attenuation zero corresponds to an angle w0T at which themagnitude function reaches its maximum value. For lattice WDFs thisoccur when

H e j T( )w0 1=

A transmission zero corresponds to an angle w1T at which themagnitude function is zero, i.e. when

H e j T( )w1 0=

At an attenuation zero, the phase responses of the branches must havethe same value, i.e.

4

F0 (w0T ) = F1(w0T ) (2.5)

Hence, in the passband of the filter the phase responses must beapproximately equal. At a transmission zero the difference in phasebetween the two branches must be

F0 (w1T ) - F1(w1T ) = ±p (2.6)

Thus, the difference in phase between the two branches must approximate±p in the stopband of the filter. To make sure that only one passband andone stopband occur, the orders of H0(z) and H1(z) must differ by one.

2.1. LINEAR-PHASE LATTICE WDFS

It is possible to obtain a lattice WDF (and more generally, a filter composedof two allpass filters in parallel) with approximately linear phase by lettingone of the branches consist of pure delays [Kuno-88, Renf-86]. A linear-phase lattice WDF is shown in Fig. 2.2. The transfer function of a linear-phase lattice WDF is

H (z) =

12

H0 (z) + z-M( ) (2.7)

The transfer function H0(z) corresponds to a general allpass function andcan consequently be written as

H z

b z

b z

i i

i

i i

i

N

NN00

0

( ) = =

-

=

å

å(2.8)

For lowpass and highpass filters N and M must be selected such that N= M± 1. The selection N = M + 1 gives the best result in most cases. This will beillustrated in section 4.

The overall frequency response can be expressed as

H e e ej T j jT M T( ) ( )w w w= +( )-1

20F (2.9)

In the passband, the phase response of branch zero, F0(wT), mustapproximate the phase response of the other branch, which in this case islinear. This forces the overall phase response to be approximately linear in

5

the passband. There exist no closed form solutions for design of linear-phase lattice WDFs. Therefore, numerical optimization algorithms have tobe used.

x(n)

a3

a2a1

y(n)1/2

T

T

T

5T

Figure 2.2. An 11th-order linear-phase lattice WDF

3. ALGORITHM

In this section we describe in detail an algorithm for design of linear-phaselattice WDFs. It was introduced by Renfors and Saram�ki [Renf-86], fordesign of approximately linear-phase filters composed of two allpass filtersin parallel, of which the linear-phase lattice WDF is a special case. Here,we consider design of lowpass filters. Let wcT and wsT denote the passbandand stopband edges respectively, and let the magnitude specification be

1- dc (wT ) £ H (e jwT ) £ 1 , wT Î 0, wcT[ ] (3.1a)

H e T T Tj s sT( ) ,w d w w w p£ Î[ ]( ) , (3.1b)

The overall frequency response can be written as

H e e e

e e e

eT M T

j T j j

j j j

j

T M T

T M T T M T T M T

T M T

( )

cos( )

( )

( ( ) / ( ( ) / ( ( ) /

( ( ) /

) ) )

)

w w w

w w w w w w

w w w w

= +( ) =

= +( )

=+æ

èöø

-

- + - +

-

1212

2

0

0 0 0

0 0

2 2 2

2

F

F F F

F F

(3.2)

The magnitude function is consequently given by

6

H (e jwT ) = cos

F0 (wT ) + MwT2

æè

öø

(3.3)

The design of the overall filter can therefore be performed with the aid ofthe phase response F0(wT). To obtain a lowpass filter the differencebetween F0(wT) and ÐMwT must approximate zero in the passband and Ðp(N = M + 1) in the stopband. Using Eq.(3.3) we see that the specification ofEq.(3.1) is met if

L(wT ) £ F0 (wT ) £U (wT ) , wT Î 0, wcT[ ]È wsT , p[ ] (3.4a)

where

L(wT ) =-MwT - 2cos-1(1- dc (wT )) , wT Î 0, wcT[ ]

-MwT - p - 2sin-1(ds (wT )) , wT Î wsT , p[ ]

ìíï

îï(3.4b)

U (wT ) =-MwT + 2cos-1(1- dc (wT )) , wT Î 0, wcT[ ]

-MwT - p + 2sin-1(ds (wT )) , wT Î wsT , p[ ]

ìíï

îï(3.4c)

Next, an error function that is to be minimized is defined as

E(wT ) = W (wT ) F0 (wT ) - D(wT )[ ] (3.5a)

where

W (wT ) =

12cos-1(1- dc (wT ))

, wT Î 0, wcT[ ]

12sin-1(ds (wT ))

, wT Î wsT , p[ ]

ì

í

ïï

î

ïï

(3.5b)

and

D(wT ) =-MwT , wT Î 0, wcT[ ]

-MwT - p , wT Î wsT , p[ ]

ìíï

îï(3.5c)

The algorithm works in a way that is similar to Remez multipleexchange algorithm for polynomial approximation problems. Thisalgorithm is often used in, e.g., design of linear-phase FIR filters [Oppe-89].

In the first step of the algorithm, N + 1 extremal points are selected onthe union of the passband and stopband regions. Next, the coefficients bi of

7

H0(z), as given by Eq.(2.8), and d are computed such that the error functionE(wT) alternatingly equals ±d at these extremal points. Then, N + 1 newextremal points are determined by finding the N + 1 angles at which E(wT)has its local extrema. The process is then repeated until the new extremalpoints are the real extremal points of E(wT). Finally the adaptorcoefficients are computed. The algorithm is thus performed in the followingfive steps:

1) Select N + 1 initial extremal points

W = [ ] Î[ ]È[ ]+w w w w w w p1 2 1 0T T T T T TN c s, , , , , (3.6)

2) Solve the system of N + 1 equations

E T W T T D T i Ni i i i i( ) ( ) ( ) ( ) ( ) , , ,w w w w d= -[ ] = - = +F0 1 1 1 L

(3.7)for d and the filter coefficients bi of H0(z).

3) Find the N + 1 local extrema of E(wT) on 0, wcT[ ]È wsT , p[ ] with thecondition that the maxima and minima alternate. Let the angles atwhich these extrema occur be ¢ = ¢ ¢ ¢[ ]+W w w w1 2 1T T TN, ,

4) If w w ei iT T i N- ¢ £ = +, for 1 1, ,L then go to step 5. Otherwise, set W= W ' and go to step 2.

5) Compute the adaptor coefficients.

The specification of Eq.(3.1) is satisfied if d £ 1. The final problem isconsequently to find the minimum N such that d £ 1.

In the first step of the algorithm N + 1 initial extremal points must beselected. The locations of these extremal points are not as crucial for theconvergence of the algorithm as is the distribution of these to the passbandand stopband. A good starting point is to select the number of points in thepassband as wcT(N + 1)/(wcT + p Ð wsT) rounded to the nearest integer[Renf-86]. The extremal points in the passband and stopband can then bedistributed equidistantly in each respective band.

In step two, Eq.(3.7) is to be solved. For each value of d the coefficients bi

can be computed as functions of d. This can be done conveniently in the Y-domain using simple recurrence formulas [Henk-81, Renf-87, Joha-96a].

We first rewrite Eq.(3.7) as

8

F0

11 1( )

( )( )

( ) , , ,wd

wwi

i

iiT

W TD T i N=

-= ++ L (3.8)

The problem now is to determine the coefficients of the allpass filter H0(z)of order N such that Eq.(3.8) is satisfied, i.e. design H0(z) such that

arg ( ( ) , , ,H e T i Nj iiT0 0 1w w( ) = =F L (3.9)

The corresponding allpass transfer function in the Y-domain can beexpressed as

H

PP

N

N0( )

( )( )

YYY

=-

(3.10)

The numerator and denominator polynomials of this function contributewith the same amount to the phase response. It is therefore sufficient tofind a numerator polynomial PN(Y) having half the desired phase responsevalues at the extremal points. Thus, using Richards' variable as given by

Y =

z - 1z + 1

(3.11)

the equations that must be satisfied are

arg ( )

( ), , ,P j

Ti NN i

iWF( ) = =0

21

w L (3.12a)

where

Wi

iT i N= æè

öø

=tan , , ,w2

1 L (3.12b)

The polynomial PN(Y) is generated using recurrence formulas. First, theparameters b b b1 2, , , L N are computed using the following recursivecontinued fraction formula:

9

b

w

bb

w

bb

b

bb

1

0 1

1

22

12 1 2

0 2

21

2 12 2

2 2

22

12

1

2

1

2

1

1

11

1

2

2 3

=

æè

öø

-( ) = -æè

öø

-( ) = --( )

--( )

--( )

-

-- -

- -

tan( )

tan( )

F

W

W WW

F

W WW W

W W

W WW

T

T

i i ii i i

i i i

i

M

M

ii

iT

i N

tan( )

,

F02

2

wæè

öø

< £

(3.13)

The polynomial PN(Y) can then be derived using the recurrence formula

PP

P P P i Ni i i i i

0

1 1

1 21

2

11

22

( )( )

( ) ( ) ( ) ,

Y

Y Y

Y Y Y W Y

=

= +

= + +( ) £ £- - -

b

b

(3.14)

The coefficients bi in Eq.(2.8) can be computed from PN(Y) via thesubstitution of Eq.(3.11). The remaining equation of Eq.(3.8), afterrewriting it, can now be expressed as

f b b b fN( , ( ), ( ), , ( )) ( )d d d d d1 2 0L = = (3.15)

Equation (3.15) can be solved using standard methods for solving f(x) = 0.

3.1. COMPUTATION OF ADAPTOR COEFFICIENTS

The allpass function H0(z) can be realized in several ways. For theapproach of cascaded first- and second-order sections using symmetric two-port adaptors, according to Fig. 3.1, the transfer functions are, for the first-order section

-a0z + 1z - a0

(3.16)

and for the second-order section

10

-a1z2 - a2 (1- a1)z + 1z2 - a2 (1- a1)z - a1

(3.17)

The adaptor coefficients can be derived either from the transfer functionH0(z) as given by Eq.(2.8), or from the numerator polynomial PN(Y). Here,we use the latter approach. The numerator polynomial is then firstfactored into first- and second-order factors of the form

-Y + a0i

and

Y2 - a1 jY + a0 j

where the indices indicate the first-order factor i and second-order factor j,respectively. The coefficients for the first- and second-order sections cannow be computed as

a0i =

1- a0i1+ a0i

(3.18)

and

a1

1 0

1 0

11j

j j

j j

a aa a

=- -

+ +(3.19a)

a2

11

0

0j

aa

j

j=

-

+(3.19b)

respectively.

y(n)x(n)

a1

T

a) b)

x(n) y(n)

a1

a2

T

T

Figure 3.1. Realizations of first- and second-order allpass sections using symmetric two-port adaptors. a) First-order section. b) Second-order section.

11

The second-order sections can also be realized with three-port series (orparallel) adaptors as shown in Fig. 3.2. In this case the transfer function is

( ( )( )

)g g g gg g g g

1 2 2 2 12 2 1 1 2

1 11

+ - + - ++ - + + -

z zz z

(3.20)

The coefficients of second-order section j can be computed as

g 1

1 0

21j

j ja a=

+ +(3.21a)

g 2

0

1 0

21j

j

j j

aa a

=+ +

(3.21b)

The coefficients can also be computed using the following relations:

g a a1 1 2

121 1j j j= - +( )( ) (3.22a)

g a a2 1 2

121 1j j j= - -( )( ) (3.22b)

An alternative to using cascaded first- and second-order sections is theRichards' structure shown in Fig. 3.3. The adaptor coefficients of thisstructure can be computed from the port resistances of the two-portadaptors, which can be derived iteratively from PN(Y) by using Richardstheorem [Erik-78], or by using Schur parametrization [Neir-79]. Here, weuse the algorithm in Box 1 which is based on the Schur parametrization[Joha-96b]. The algorithm derives the coefficients iteratively given thecoefficients of the polynomial PN(Y), where

P aN i

i

Ni( )Y Y= +

=å11

(3.23)

T T

x(n) y(n)

-1

-g2-g1

-1

Figure 3.2. A second-order allpass section using a three-port series adaptor.

12

aN

T TT

a2 a1

x(n)

y(n)

Figure 3.3. An Nth-order Richards' allpass section.

c a

c a

for i to Nc a c

c a c

for j i downto

ci

j i ja c c

i i i i i

i i i i

i j i j i i j i j

1 0 1

1 1 1

1 1

0 1 0

1 1 1

1

1

2

1

1 1

1

,

,

, ,

, ,

, , ,

( )

!!( )!

( )

= -

= +

=

= +

= - +

= -

=-

- + +

- -

-

- - - -

endend

bc

aj N

for i N downtoif i

for j i downto i

bb b b

b

b

NN

Njj

m mm

i ji j i i i i j

i i

i i j

,,

,, , ,

,

,

( ), , , ,

/

=

+ -

=

=

>

= - ë û +

=-

-

=

=

--

- -

å1 11 2

22

1 2 1

1

1

1

1 2

L

bb b bb

endendif i even

bb

b

endend

b j N

i i j i i i j

i i

i ii i

i i

j j j

, , ,

,

, /, /

,

, , , , ,

-

-

-

-

=+

= - =

1

1

1 2

2

1 22

a L

Box 1. Algorithm for adaptor coefficients of Richards' structures.

13

4. EXAMPLES

In this section we design some linear-phase lattice WDFs using thealgorithm described in section 3. The allpass filter H0(z) is realized usingcascaded first- and second-order sections according to Fig. 3.1.

Example 1Consider the following specification: wcT = 18°, wsT = 36°, and Amin = 40dB. For the ripple in the passband we use three different specifications,Amax = 0.1 dB, 0.001 dB, and 0.00001 dB. The orders of the filters meetingthese requirements become 15, 25, and 33. As a comparison, a linear-phase FIR filter, using equiripple approximation, requires a filter order of44 for Amax = 0.1 dB.

The adaptor coefficients for the three different filters are compiled inTable 1. The magnitude and group delay for the filters are shown in Figs.4.1- 4.3. As can be seen, the passband ripple must be very small in order toobtain a small group delay variation. The required coefficient word lengthafter rounding is for all three filters 9. This is under the assumption thatthe stopband attenuation is still larger than 35 dB. The passband ripplesfor the three different filters, using quantized coefficients, then become0.058 dB, 0.0019 dB, and 0.00056 dB, respectively.

Amax ai a1j, a2j

0.1 Ð Ð0.285796, Ð0.758443

Ð0.296030, Ð0.256958

Ð0.339827, 0.415722

Ð0.815232, 0.915477

0.001 0.743100 Ð0.314339, Ð0.846186

Ð0.411574, Ð0.712056

Ð0.438589, Ð0.323692

Ð0.457513, 0.148652

Ð0.498529, 0.577198

Ð0.850167, 0.892194

0.00001 Ð0.279675 Ð0.522581, Ð0.926784

Ð0.530127, Ð0.756600

Ð0.538759, Ð0.456192

Ð0.546794, Ð0.0842540

Ð0.557383, 0.295048

Ð0.586155, 0.623725

Ð0.868893, 0.877157

Ð0.557033, 0.945335

Table 1. Adaptor coefficients for the filters in example 1.

14

Figure 4.1. Magnitude and group delay for a 15th-order linear-phase lattice WDF.

Figure 4.2. Magnitude and group delay for a 25th-order linear-phase lattice WDF.

Figure 4.3. Magnitude and group delay for a 33rd-order linear-phase lattice WDF.

Example 2The condition for approximately linear-phase IIR filters is similar to that ofFIR filters. That is, for the cases in which the transition bands are verynarrow, the filter orders will be very high, even if they generally will be

15

lower than the orders of the corresponding FIR filters. For example,reducing the stopband edge from 36° to 27° in the example above (Amax =0.1 dB) increases the filter order from 15 to 31. The magnitude and groupdelay of this filter are shown in Fig 4.4. For a linear-phase FIR filter, usingequiripple approximation, the required order is 86.

Figure 4.4. Magnitude and group delay for a 31st-order linear-phase lattice WDF.

Example 3We have so far in this paper only considered the case where the order ofthe filter H0(z) larger than the order of the delay branch (N = M + 1). Thereason for this is that the overall filter order in this case is much less thanfor the case where the order of H0(z) is lower than the delay branch (N = MÐ 1). To illustrate this, we have designed these two different types of filtersfor the following specification: wcT = 36°, wsT = 90°, Amax = 0.001 dB, andAmin = 40 dB. The orders of the filters become 9 for the case where N = M +1, and 21 for the case where N = M Ð 1. The required order for the latterfilter type is thus more than two times larger than for the former filtertype. The magnitude and group delay for the two filters are shown in Figs.4.5 and 4.6, respectively. From the figures we can see that the group delayvariation is better for the former filter type.

Figure 4.5. Magnitude and group delay for a 9th-order linear-phase lattice WDF, N = M +1.

16

Figure 4.6. Magnitude and group delay for a 21st-order linear-phase lattice WDF, N = M Ð1.

Example 4From Figs. 4.1-4.6, we see that the group delay has a small variation forlow frequencies and deteriorates at the passband edge. The variation islargely dependent on the passband ripple. One way to increase thestopband attenuation for a given filter order, without increasing the groupdelay variation, is therefore to allow a larger ripple for low frequencies. Asimple way to do this is to let dc(wT) decrease linearly in the passband fromKdc to dc, i.e.,

d w d d w wc c c cT K K T T( ) ( ) /= + -1 (3.24)

for some constant K.Consider the following specification: w cT = 36°, w sT = 72°, Amax =

0.0000086 dB, and Amin = 40 dB. The specification is met by a 17th-orderfilter. We use K = 1 and K = 15. The magnitude and group delay for the twocases are shown in Figs. 4.7 and 4.8. The group delay variations are in bothcases 0.126 in the passband. The stopband attenuations are 40 dB for K =1, and almost 44 dB for K = 15.

Figure 4.7. Magnitude and group delay for the 17th-order linear-phase lattice WDF whenK = 1.

17

Figure 4.8. Magnitude and group delay for the 17th-order linear-phase lattice WDF, whenK = 15.

5. CONCLUSIONS

In this paper an algorithm for designing linear-phase lattice WDFs hasbeen described in detail. The filters consist of a parallel connection of twoallpass filters of which one corresponds to a pure delay. Several designexamples using the algorithm were given. It was observed that in order toobtain a filter with a small group delay variation in the passband, thepassband ripple must be very small. Further, for a narrow transition bandspecification, the filter order becomes high. It was also demonstrated thatthe overall filter order is minimized if the order of the filter thatcorresponds to a pure delay is less than the order of the other allpass filter.We also suggested a simple way to increase the stopband attenuation for agiven filter order, without increasing the group delay variation in thepassband.

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18

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[Oppe-89] Oppenheim A.V. and Schafer R.W.: Discrete-Time Signal Processing,Prentice Hall, 1989.