digital lattice filter

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AN EFFICIENT ALGORITHM FOR THE DESIGN OF LATTICE WAVE DIGITAL FILTERS

WITH SHORT COEFFICIENT WORDLENGTH

Juha Yli-Kaakinen and Tapio Saramki

Signal Processing LaboratoryTampere University of Technology

P. O. Box 553, FIN-33101 Tampere, Finland

ABSTRACT

This paper describes an efficient algorithm for the design of lat-tice wave digital filters with short coefficient wordlength. Theproposed algorithm guarantees that the optimum finite-wordlength solution can be found for both the fixed-point and themultiplierless coefficient representations. This is illustrated bymeans of several examples.

1. INTRODUCTION

The cost and the complexity of a digital filter depends heavily onthe necessary wordlengths of the coefficients. Therefore, thewordlength should be as short as possible but still sufficient to sat-isfy the filter specifications.

During the past three decades, a large number of algorithms fordesigning finite-wordlength recursive digital filters have been pro-posed [117]. In the past, the direct search by Hooke and Jeeves[18] was one of the most widely used discrete optimization algo-rithms [2, 3, 6, 10, 14, 15]. However, recently, the stochastic opti-mization algorithms such as genetic algorithms and simulatedannealing have received increasing attention [1217]. Althoughthese methods are characterized by a high computational and pro-

gramming cost, there is no guarantee that these methods will con-verge to the global optimum. This is because these methods arebased on probabilities [19].

The structures of the digital filter are very important for theefficiency of the filtering process. It is essential that filters can berealized using structures with a low coefficient sensitivity. Theimportance of the low-sensitivity structures is that if the effect ofthe deviation from the ideal coefficient value is small, then shortcoefficient wordlengths can be used without violating filter speci-fication, resulting in a faster, smaller, and less expensive hard-ware.

One of the best structures for implementing recursive digitalfilters are the lattice wave digital filters [2023] which are relatedto certain analog prototype networks. The lattice wave digital fil-ter consists of a parallel connections of two allpass filters. They

are characterized by many attractive properties, such as a reason-ably low coefficient sensitivity, a low roundoff noise level, and theabsence of parasitic oscillations.

The application of allpass sections produces an efficient real-ization in the terms of the number of multipliers for a given filterorder. It is well known that lattice wave digital filters can beimplemented with approximately half the multipliers required bythe canonic direct-form realizations. In addition, these allpass

subfilters can be realized by using first- and second-order sectionsas basic building blocks. The resulting filter structures are highlymodular which makes them suitable for signal processor andVLSI implementations [2426].

It is also possible to design lattice wave digital filters to have anapproximately linear phase in the passband [15, 2729]. Suchdesigns are especially suitable for applications where linear-phaseFIR filters would have an excessive signal delay.

In this paper, we propose an efficient algorithm for designing

lattice wave digital filters with short coefficient wordlength. Thealgorithm is based on the observation that by first finding the larg-est and smallest values for both the radius and the angle of all thecomplex-conjugate poles as well as the largest and smallest valuesfor the radius of a possible real pole in such a way that the givencriteria are still met, we are able to find a parameter space whichincludes the feasible space where the filter specifications are satis-fied. After finding this larger space, all what is needed is to checkwhether in this space there exist desired discrete values for thecoefficient representations.

The method is general but particularly efficient for filtersimplemented as a parallel connection of two allpass filters. This isbecause for these filters only the denominator coefficients of theallpass sections have to be quantized. Several examples are

included illustrating the efficiency of the proposed quantizationscheme.

2. LATTICE WAVE DIGITAL FILTERS

The transfer function for the overall filter as shown in Fig. 1 canbe expressed as

. (1)

Here, and are stable allpass filters of ordersMandN, respectively. In the case of lowpass filters, or

so that , the overall order of , is odd.

Fig. 1. Parallel connection of two allpass filters.

H z( )1

2--- A1 z( ) A2 z( )+[ ]=

A1 z( ) A2 z( )M N 1=

M N 1+= M N+ H z( )

1/ 2

X(z) Y(z )

A1(z )

A2(z )

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If and are implemented as a cascade of first- andsecond-order wave digital allpass structures and M and N areassumed to be odd and even, respectively, then andare expressible in terms of the adaptor coefficients as (see, e.g.,[22] or [24])

(2a)

and

, (2b)

where

. (2c)

If possesses a real pole at and m complex-con-jugate pole pairs at for and

possesses n complex-conjugate pole pairs atfor , then

(3a)

and

(3b)

In the highpass case, the corresponding transfer function isobtained by changing the sign of or . In the band-stop case,MandNare two times an odd integer and an even inte-ger, respectively, and or . Thecorresponding bandpass design can be generated by changing thesign of or . The main difference compared to low-pass and highpass cases is that a first-order section is absent.

In the sequel, we concentrate mainly on designing finite-

wordlength lowpass filters. The design of a finite-wordlengthbandpass filter will be considered in connection with examples toemphasize the applicability of the proposed quantization schemeto designing also filters of other kinds.

3. COEFFICIENT REPRESENTATION FORMS

UNDER CONSIDERATION

This contribution considers the coefficient quantization in thefixed-point arithmetic in two cases. In the first case, the coeffi-cient values are expressed as the following fixed-point binarynumbers:

,

where for is either 0 or 1. The goal is to findall the filter coefficient values in a such a way that the filter meetsthe given specifications and all the coefficient values are express-ible in the above form with the minimized value ofR. Filters ofthis kind are useful for signal processor implementations.

The second case considers VLSI implementations where ageneral multiplier element is very costly. To get around this prob-lem, it is attractive to carry out the multiplication of a data sampleby a filter coefficient value by using a sequence of shifts and adds.In this case, it is desired to express the coefficient values in theform

,

where each of the s is either 1 or 1 and the s are integersin the increasing order. In this case, the goal is to find all the coef-ficient values in such a way that, first,R is made as small as possi-ble and, secondly, is made as small as possible.

4. STATEMENT OF THE PROBLEM

Before stating the optimization problem, we denote the transferfunction of the filter by , where is the adjustableparameter vector

. (4)

The amplitude specifications for the filter1 are stated as follows:

(5a)

. (5b)

Alternatively, these criteria can be expressed as

(6a)

where

, (6b)

with

(6c)

The optimization problem under consideration is:

Optimization problem: Find the adjustable parameter vectoras given by Eq. (4) in such a way that

1. meets the criteria given by Eq. (5) or Eq. (6)

2. The coefficients values for areoptimized according to one of the two above-mentioned cri-teria.

5. FILTER OPTIMIZATION

The solution to the stated optimization problem can be found intwo steps. In the first step, a non-linear optimization algorithm isused for determining a parameter space of the infinite-precisioncoefficients including the feasible space where the filter meets the

1These specifications are typical of most recursive filters built using all-pass subfilters as building blocks. In these cases, the filter structure con-strains the maximum of the amplitude response to be unity [21].

A1 z( ) A2 z( )

A1 z( ) A2 z( )

A1 z( )0 z 1+

1 0z 1----------------------

2l 1 2l 2 l 1 1( )z 1 z 2+ +1 + 2 l 2 l 1 1( )z 1 2 l 1 z 2-----------------------------------------------------------------------------

l 1=

m

=

A2 z( )2 l 1 2l 2l 1 1( )z 1 z 2+ +

1 + 2 l 2 l 1 1( )z 1 2l 1 z 2-----------------------------------------------------------------------------

l m 1+=

m n+

=

m M 1( ) 2 , n N 2= =

A1 z( ) z r0=z rl jl( )exp= l 1 2 m, , ,=

A2 z( ) z =rl jl( )exp l m 1+ m 2+ m n+, , ,=

0 r0=

2 l 1 rl2= 2l,

2rl l( )cos1 rl

2+------------------------- for l 1 2 m n.+, , ,= =

A1 z( ) A2 z( )

M N 2= M N 2+=

A1 z( ) A2 z( )

a0 ar2 rr 1=

R

+

ar r 0 1 R, , ,=

ar2Pr

r 1=

R

ar Pr

Pr

H z,( )

r0 rm n+ 1 m n+, , , , ,[ ]=

1 p H ej,( ) 1 for 0 p,[ ]

H ej,( ) s for s ,[ ]

E ,( ) 1 for 0 p,[ ] s ,[ ]

E ,( ) W ( ) H ej,( ) D ( )[ ]=

D ( )1 0 p,[ ],

0 s ,[ ],,

W ( )1 p 0 p,[ ],

1 s s ,[ ].,

==

H z,( )l l 0 1 M N 1+, , ,=

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given criteria. The second step involves then finding the filterparameters in this space such that the resulting filter meets thegiven criteria with the simplest coefficient representation forms.

5.1. Optimization of Infinite-Precision Filters

It has turned out that a very straightforward quantization schemefor the filter coefficients is obtained by first finding the largest andsmallest values for both the radius and the angle of all the com-

plex-conjugate pole pairs as well as the largest and smallest val-ues for the radius of the real pole in such a way that the givencriteria are still met. Therefore, there are problemsof the following form: Find the adjustable parameter vector tominimize subject to the conditions of Eq. (6a). For these prob-lems, is and as well as , , , and for

.

To solve these problems, we discretize the passband and stop-band regions into the frequency points

and , . Theresulting discrete minimization problem is to find to minimize

subject to

. (7)

The above problems can be solved conveniently by using the sec-ond algorithm of Dutta and Vidyasagar [30].

The use of the above algorithm gives for the real pole thelower and upper values, denoted by and , as well asthe lower and upper values for both the radius and angle of the l thcomplex-conjugate poles, denoted by , , , and

for . Based in these limits, the upper andlower limits for the adaptor coefficients can be determined as fol-lows:

, (8a)

, (8b)

, (8c)

, (8d)

, (8e)

and

. (8f)

Figure 2 shows the upper and lower limits for both the radius andthe angle of the innermost complex-conjugate pole of a 7th-orderlowpass filter to be considered later on in Example 1 as well as thecorresponding upper and lower limits for the adaptor coefficients.The dots indicate the allowable locations for the poles when 7fractional bits are used for the adaptor coefficients.

5.2. Optimization of Finite-Precision Filters

It has been proved experimentally that the parameter spacedefined above forms a space including the feasible space where

the filter specifications are satisfied. After finding this largerspace, all what is needed is to check whether in this space thereexist the desired discrete values for the coefficient representa-tions.

For the fixed-point coefficients, this search can be done in astraightforward manner by first determining the upper and lowerlimits of the adaptor coefficient forto the fixed number of bits as

, (9)

where stands for the integer part ofx and for the small-est integer larger than or equal tox. Here, withR beingthe desired wordlength or the number of fractional bits.

For each , the value of the coefficient is then increased fromto using a quantization step of size equal to q. The

magnitude response is evaluated for each combination of discreteadaptor coefficient values to check whether the filter meets themagnitude specifications. However, as can be seen from the Fig.2, there are, particularly for the innermost complex-conjugatepole, regions where the angle of the pole corresponding to quan-tized values of and is smaller than or larger than

. For this reason, it is advisable to check if the angle of the

discrete pole is in prespecified region in order to avoid the vainevaluation of the corresponding magnitude response.

For the multiplierless coefficient representation, the search canbe conveniently accomplished by first preparing a look-up tableincluding all possible values for a given wordlength R and thegiven maximum number of powers of two. In this case, the searchis performed only for those combinations of adaptor coefficientsthat belong simultaneously to both the discrete fixed-point spaceand the look-up table.

It should be pointed out that for a certain wordlength, there aretypically several solutions which will meet the magnitude specifi-cations. Therefore, it is advisable to find first all the solutions andthen to choose, for example, the one with the best attenuationcharacteristics or the minimum number of adders required toimplement the multiplier coefficients for the given wordlength.

Fig. 2. Search space of the innermost complex-conjugate pole of a 7th-or-

der lowpass filter to be considered in Example 1. The dots correspond to

those locations obtained by using 7 fractional bits for the adaptor coeffi-

cients.

Figure 3 shows the search space for a 7th-order lowpass filterwith 8-bit coefficients ( ). The smaller dots represent thepole locations in the search space, while the larger dots represent

2 4 m n+( )+

r0 1 r0 1 rl rl l l

l 1 2 m n+, , ,=

i 0 p,[ ], i 1 2, ,= Np, i s ,[ ] i Np 1 Np 2, Np Ns+,+,+=

E i,( ) 1 0 for i 1 2 Np Ns+, , ,=

r0r0

low( ) r0up( )

rllow( ) rl

up( ) llow( )

lup( ) l 1 2 m n+, , ,=

0low( ) r0

up( )=

0up( ) r0

low( )=

2 l 1low( )

rlup( )

( )2 for

l 1 2 m n+, , ,= =

2 l 1up( ) rl

low( )( )2 for l 1 2 m n+, , ,= =

2 llow( ) 2rl

low( ) lup( )( )cos

1 rllow( )( )2+

---------------------------------------- for l 1 2 m n+, , ,= =

2 lup( ) 2rl

up( ) llow( )( )cos

1 rlup( )( )2+

---------------------------------------- for l 1 2 m n+, , ,= =

l l 0 1 ,, ,= M N 1+

q llow( ) q l low( ) q , qlup( ) q l up( ) q==

x xq 2 R=

lq l

low( ) q lup( )

2l 1 2 l low( )

up( )

r low( )

2 l 1 2 l 1up( ) , 2l 2l

low( )= =

low( )

2l 1 2l 1low( ) , 2 l 2 l

low( )= =

up( )

rup( )

2 l 1 2l 1up( )

, 2l 2 lup( )

= =

R 8=

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the pole locations where the filter specifications are met. In addi-tion, the pole positions after the infinite-precision optimizationare shown in Fig. 3. The crosses numbered from 1 to 4 correspondto the smallest and largest values for both the radius and the angleof the outermost pole. The locations of the inner poles are indi-cated in a similar manner.

The proposed quantization scheme provides significant advan-tages over those based on the use of simulated annealing orgenetic algorithms. First of all, it is always guaranteed that the

optimum solution can be found. Secondly, the computationalworkload to arrive at the optimum discrete-valued solution is inmost cases significantly smaller.

Fig. 3. Search space with 8-bit fixed-point adaptor coefficients (R=8).

6. NUMERICAL EXAMPLES

Several example illustrating the efficiency and flexibility of theproposed quantization scheme have been given in [31]. Here weconsider four examples.

6.1. Example 1

It is desired to design a lowpass filter with the passband edge atand stopband edge at . The maximum

allowable passband ripple and the required stopband attenuationare 0.2 dB ( ) and 60 dB ( ), respectively.The minimum order of a lattice wave digital filter to meet thegiven amplitude criteria is seven. The optimized discrete-valuedadaptor coefficients of of order forare , , and , respectively. The adaptor

coefficients of of order for are, , , and , respectively. In this

case, only seven fractional bits are needed for the adaptor coeffi-cients.

The magnitude response as well as the passband details of thequantized filter are shown in Fig. 4. A plot of pole locations in the

z-plane is depicted in Fig. 6(a). Among the 28 solutions satisfyingthe overall specifications the one with the best magnitude charac-teristics has been selected. The coefficient quantization took

57.57 CPU minutes on an AlphaServer 4100.

Fig. 4. Magnitude response and zero-pole plot for the quantized lattice

wave digital filter with 7-bit adaptor coefficients.

6.2. Example 2

The criteria are the same as in Example 1 except that the multipli-erless representation is used for the adaptor coefficients. Like inExample 1, 7 fractional bits are required to meet the magnitudespecifications. The optimized discrete-valued adaptor coefficients

of of order for are ,, and , respectively. The adaptor

coefficients of of order for are, , , and ,

respectively. In this case, all the coefficients can represented astwo or three powers of two. A total of only eleven adders arerequired to implement all the multipliers.

Fig. 5. Magnitude response for the quantized lattice wave digital filter with

7-bit multiplierless adaptor coefficients.

0.1 0.2 0.3 0.4 0.5 0.6

0

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r(up)

r(low)

(up)

(low)

1

2

3

4

5

6

7

8

9

10

11

12

13 14

p 0.4= s 0.5=

p 0.0228= s 10 3=

l A1 z( ) M 3= l 0 1 2, ,=56 2 7 84 2 7 48 2 7

l A2 z( ) N 4= l 3 4 5 6, , ,=45 2 7 75 2 7 115 2 7 35 2 7

100

90

80

70

60

50

40

30

20

10

0

10

Angular Frequency

AmplitudeindB

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.08

0.06

0.04

0.02

0

AmplitudeindB

0 0.1 0.2 0.3 0.4

l A1 z( ) M 3= l 0 1 2, ,= 2 1 2 52 1 2 3 2 6 2 1 2 3 2 5

l

A2

z( ) N 4= l 3 4 5 6, , ,=2 1 2 3+ 2 1 2 5 2 7+ + 1 2 3 2 6+ 2 2 2 6+

100

90

80

70

60

50

40

30

20

10

0

10

Angular Frequency

AmplitudeindB

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.2

0.15

0.1

0.05

0

AmplitudeindB

0 0.1 0.2 0.3 0.4

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Fig. 6. Search spaces for 7th-order lowpass filters with 7-bit adaptor coef-

ficients. (a) Fixed-point coefficients. (b) Multiplierless coefficients.

The magnitude response of the multiplierless implementationis shown in Fig. 5. As can be seen from the Fig. 6(b), there is onlyone solution which meets the magnitude specifications for a givencoefficient representation. The coefficient quantization took 15.62CPU minutes.

6.3. Example 3

We consider next the design of a tenth-order bandpass filter withtransition bands from 0.2 to 0.3 and from 0.5 to 0.6. Therequired passband ripple is 0.2 dB ( ), while thedesired stopband attenuation is at least 60 dB ( ) on [0,0.2] and at least 50 dB ( ) on [0.6, ].

In this case, 6 fractional bits are required for the amplituderesponse to stay within the given specifications. The optimizeddiscrete-valued adaptor coefficients of of orderfor are , , , ,

, and , respectively. The adaptor coefficientsof of order for are ,

, , and , respectively.

The magnitude response and the passband details for the quan-tized bandpass filter are depicted in Fig. 7. A plot of pole loca-tions in thez-plane is illustrated in Fig. 9(a). In this case, there are6 solutions which satisfy the overall specifications, from whichthe one with the best attenuation characteristics has been chosen.

6.4. Example 4

The criteria are the same as in Example 3 except that the multipli-erless representation is used for the adaptor coefficients. Like inExample 3, 6 fractional bits are required for the adaptor coeffi-cients. The optimized discrete-valued adaptor coefficients of

of order for are 21 23 , 22

+26 , 1+23 25 , 21 +23 26 , 1+23 25 , and 24 ,respectively. The adaptor coefficients of of order

for are 1+22 +25 , 25 , 1+22

+26 , and 21 +26 , respectively. In this case 14 additions areneeded to implement all the adaptor coefficients.

From Fig. 9(b), it is seen that there are two solutions satisfyingthe overall specifications. The magnitude response as well as thepassband details of the solution requiring the smallest number ofadders are shown in Fig. 8.

Fig. 7. Magnitude response for the quantized 10th-order lattice wave digi-

tal filter with 6-bit adaptor coefficients.

7. CONCLUSION

A straightforward two-step scheme has been developed fordesigning lattice wave digital filters with short coefficientwordlength. The first step determines a parameter space of theinfinite-precision coefficients including the feasible space wherethe filter meets the given criteria. The second step involves thenfinding the coefficients in this space such that the given criteriaare met by the simplest representation forms. The efficiency of theproposed procedure has been demonstrated by several examples.

Future work is devoted to making the search of the solutionssatisfying the given amplitude criteria faster. Instead of goingthrough all the combinations, there exist several alternatives ofcombining infinite-precision optimization and the actual searchprocedure, resulting in a significantly faster overall quantizationscheme.

Fig. 8. Magnitude response for the quantized 10th-order lattice wave digi-

tal filter with 6-bit multiplierless adaptor coefficients.

0.2 0.3 0.4 0.5 0.6

0

0.2

0.4

0.6

0.8

1

0.2 0.3 0.4 0.5 0.6

0

0.2

0.4

0.6

0.8

1

(a) (b)

p 0.0228=s 10 3=

s 0.0032=

l A1 z( ) M 6=l 1 2 6, , ,= 39 2 6 20 2 6 58 2 6 40 2 6

58 2 6 2 2 6 lA2 z( ) N 4= l 7 8 9 10, , ,= 45 2 6

5 2 6 47 2 6 35 2 6

lA1 z( ) M 6= l 1 2 6, , ,=

l A2 z( )N 4= l 7 8 9 10, , ,=

100

90

80

70

60

50

40

30

20

10

0

10

Angular Frequency

AmplitudeindB

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.2

0.15

0.1

0.05

0

AmplitudeindB

0.3 0.4 0.5

100

90

80

70

60

50

40

30

20

10

0

10

Angular Frequency

AmplitudeindB

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.2

0.15

0.1

0.05

0

AmplitudeindB

0.3 0.4 0.5

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Fig. 9. Search spaces for 10th-order bandpass filters with 6-bit adaptor co-

efficients. (a) Fixed-point coefficients. (b) Multiplierless coefficients.

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sive Filters, Diploma Engineer Thesis, Department of ElectricalEngineering, Tampere University of Technology, Finland, 1997.[30] S. R. K. Dutta and M. Vidyasagar, New algorithm for con-

strained minimax optimization, Mathematical Programming,vol. 13, pp. 140155, 1977.

[31] J. Yli-Kaakinen, Optimization of Recursive Digital Filter forPractical Implementation, Diploma Engineer Thesis, Depart-ment of Electrical Engineering, Tampere University of Technol-ogy, Finland, 1998.

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digital lattice filter

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