ARTICLE IN PRESS

Contents lists available at ScienceDirect

Signal Processing

Signal Processing 90 (2010) 679–683

0165-16

doi:10.1

$ Thi� Cor

E-m

ecrwan

journal homepage: www.elsevier.com/locate/sigpro

Fast communication

An improved orthogonal digital filter structure$

Gang Li a,�, Chaogeng Huang a, Jingyu Hua a, Chunru Wan b

a College of Information Engineering, Zhejiang University of Technology, Hangzhou, Zhejiang, PR Chinab School of EEE, Nanyang Technological University, Republic of Singapore

a r t i c l e i n f o

Article history:

Received 22 July 2008

Received in revised form

13 April 2009

Accepted 7 July 2009Available online 16 July 2009

Keywords:

Digital filters

Orthogonal structures

Finite word length effects

Roundoff noise gain

84/$ - see front matter & 2009 Elsevier B.V. A

016/j.sigpro.2009.07.006

s work was supported by NSFC-Grant 608721

responding author.

ail addresses: egli@ntu.edu.sg (G. Li),

@ntu.edu.sg (C. Wan).

a b s t r a c t

A novel structure is derived for digital filter implementation. This structure is actually

an improved version of an existing one in terms of implementation efficiency and

reducing finite word length (FWL) effects. Expression of roundoff noise gain is obtained

for the proposed structure. Design examples are given to demonstrate the performance

of this structure and to compare it with the existing one and the classical minimum

roundoff state-space realizations. Numerical examples show that the proposed structure

outperforms the others in terms of minimizing roundoff noise as well as implementa-

tion efficiency.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

The classical digital filter design methodology usuallyassumes that the designed filters are to be realized oninfinite precision devices. This is the case for computersimulations, in which the the processor is of very highprecision and the computation efficiency is not an issue.In many real-time applications such as hand-phoneproducts in wireless communications, however, the well-designed filters have ultimately to be implemented with adigital processor of a few bits/digits in order to reducethe cost of the products. It has been noted that theperformance of the FWL implemented filters may deviategreatly from their desired one [1,2]. How to minimize theFWL effects has been one of the important research topicsin digital filter design and implementation [3–14].

Consider an Nth order digital filter of transfer functionHðzÞ. This filter can be implemented with many different

ll rights reserved.

11.

structures1 such as its state-space equations

xðnþ 1Þ ¼ AxðnÞ þ BuðnÞ

yðnÞ ¼ CxðnÞ þ duðnÞ

((1)

where uðnÞ and yðnÞ are the input and output of the filter,respectively. ðA;B;C; dÞ is called a state-space realization ofHðzÞ with A 2 RN�N ;B 2 RN�1;C 2 R1�N and d 2 R, satis-fying

HðzÞ ¼ dþ CðzI � AÞ�1B (2)

For real-time applications, a digital filter has to beimplemented using a digital device of limited capacity andfinite precision. Therefore, it is desired to realize the filterwith a simple structure that has a nice performanceagainst the FWL effects. The classical minimum roundoffnoise realizations by Mullis and Roberts [3] and Hwang[4], denoted as Smrh, can reduce the roundoff noisesignificantly and have a minimum noise gain given by

Gmrh ¼ 1þ1

N

XN

k¼1

sk

!224

35ðN þ 1Þ (3)

1 Here, a structure of a digital filter means a way how the filter

output is computed for a given signal.

ARTICLE IN PRESS

G. Li et al. / Signal Processing 90 (2010) 679–683680

with sk9lkðWcWoÞ; 8k the singular values of the filter,where lkðMÞ denotes the kth eigenvalue of matrix M,while Wc , Wo are the controllability and observabilityGramians of the realization ðA;B;C; dÞ, which are thesolutions to the following equations:

Wc ¼ AWcATþ BBT

Wo ¼ ATWoAþ CTC

((4)

These structures generally require ðN þ 1Þ2 multiplicationsand NðN þ 1Þ additions for computing one output sample,leading to a very complicated implementation. To solvethis problem, a lot of effort has been made to achievesparse optimal structures [5–9].

An orthogonal structure is the one whose equivalentstate-space realization has its Wc equal to the identitymatrix I. This class of structures has some nice propertiessuch as yielding a small roundoff noise gain and makingoverflow oscillation impossible [1,12–14]. The orthogonalstate-space realizations, like Smrh, are usually very com-plicated. In [6], a sparse orthogonal structure wasderived by Li et al., denoted as Slgs. Such a structurerequires 7N � 3 multiplications and 6N � 3 additions forcomputing one output sample and yields a noise gainsmaller than Gmrh for a certain class of filters.

The main objective of this paper is to derive novelorthogonal filter structure which is much simpler than theSlgs structure, requiring only 4N � 1 multiplications and4N � 1 additions for computing one filter output. Theproposed structure also outperforms Slgs in terms ofreducing roundoff noise and even beats Smrh for a muchwider range of filters.

2. The novel orthogonal structure

In [6], the following orthogonal realization, denoted asSorth, is discussed

A ¼ ðI þFÞðI �FÞ�1

B ¼

ffiffiffi2p

2ðI þ AÞJ

C ¼

ffiffiffi2p

2LðI þ AÞ

8>>>>><>>>>>:

(5)

where

F ¼

0 a1 0 � � � 0 0

�a1 0 a2 � � � 0 0

0 �a2 0 � � � 0 0

..

. ... ..

.� � � ..

. ...

0 0 0 � � � 0 aN�1

0 0 0 � � � �aN�1 �aN

26666666664

37777777775; J ¼

0

0

0

..

.

0ffiffiffiffiffiffiffiffiffi2aN

p

26666666664

37777777775

(6)

with ak40;8k and L 2 R1�N having no special structure.Like Smrh, Sorth is usually fully parametrized, the round

noise gain is given by

Gorth ¼ 1þXN

k¼1

s2k

!ðN þ 1Þ (7)

with fskg defined before. It is easy to see that Gorth � Gmrh

and the equality holds if and only if sj ¼ sk; 8j; k.To reduce the implementation complexity of Sorth, we

now derive an equivalent structure of this realization. Thekey for that is based on the following factorization ofðI �FÞ�1.

Define N092ðN � 1Þ and

bkþ19� akþ1gk; gk91

1� akbk

(8)

for k ¼ 1;2;3; . . . ;N � 2 with b1 ¼ �a1 and gN�1 ¼ 1=ð1þ aN � aN�1bN�1Þ.

Let Uði; j; xÞ be the identity matrix of dimension N

except that its ði; jÞth element is x, 8ði; jÞ, andAN0

9Uð1;2;a1ÞUð2;2; g1Þ. Then

ðI �FÞAN0¼

1 0 0 � � � 0 0

�b1 1 �a2 � � � 0 0

0 �b2 1 � � � 0 0

0 0 a3 � � � 0 0

..

. ... ..

.� � � ..

.0

0 0 0 � � � 1 �aN�1

0 0 0 � � � aN�1 1þ aN

26666666666664

37777777777775

Denote AN0þ1�k9Uðk; kþ 1;akÞUðkþ 1; kþ 1; gkÞ; k ¼

1;2; . . . ;N � 1, which is in fact the identity matrix except

AN0þ1�kðk : kþ 1; k : kþ 1Þ ¼1 ak

0 gk

" #(9)

where ak9akgk, k ¼ 1;2; . . . ;N � 1.It can be shown that ðI �FÞAN0

� � �AN0þ1�k � � �AN

is equal to G, where G is the identity matrix exceptGðkþ 1; kÞ ¼ �bk for k ¼ 1;2; . . . ;N � 1.

Noting that G�1 can be factorized as G�1¼ AN�1 � � �

Ak � � �A1, where

Ak9Uðkþ 1; k;bkÞ; k ¼ 1;2; . . . ;N � 1 (10)

one finally has

ðI �FÞ�1¼ AN0

� � �Am � � �A19YN0

m¼1

Am (11)

with Am specified by (10) for 1 � m � N � 1 and (9) forN � m � N0.

It follows from A ¼ ðI þFÞðI �FÞ�1¼ 2ðI �FÞ�1

� I

and hence B ¼ffiffiffi2p

=2ðI þ AÞJ ¼ffiffiffi2pðI �FÞ�1J that the ortho-

gonal realization Sorth can be represented as

xðnþ 1Þ ¼ ðI �FÞ�1½2xðnÞ þ JuðnÞ� � xðnÞ

yðnÞ ¼ CxðnÞ þ duðnÞ

((12)

based on which, the following (equivalent) structure,referred as Slhhw, is obtained:

xð0ÞðnÞ92xðnÞ þ JuðnÞ

xðmÞðnÞ ¼ Amxðm�1ÞðnÞ; 8m

xðnþ 1Þ ¼ xðN0ÞðnÞ � xðnÞ

yðnÞ ¼ CxðnÞ þ duðnÞ

8>>>><>>>>:

(13)

where J9ffiffiffi2p

J has the same sparse structure as J, havingonly one non-zero parameter.

ARTICLE IN PRESS

G. Li et al. / Signal Processing 90 (2010) 679–683 681

Noting the sparse structure of Am specified by (10)and (9), one can see that to compute each output sample,Slhhw requires at most 4N � 1 multiplications2 and 4N � 1additions.

An alternative equivalent structure of Sorth was given in[6], which is denoted as Slgs and presented below:

xð0ÞðnÞ9xðnÞ

xðmÞðnÞ ¼ Amxðm�1ÞðnÞ; 8m

xðnþ 1Þ ¼ xðN0ÞðnÞ þ BuðnÞ

yðnÞ ¼ CxðnÞ þ duðnÞ

8>>>><>>>>:

(14)

where N093ðN � 1Þ þ 1 and Am is given by

Am ¼

Am; 1 � m � N � 1

Uðkþ 1; kþ 1; gkÞ; m ¼ N0 � 2k; 8k

Uðk; kþ 1;akÞ; m ¼ N0 � 2kþ 1; 8k

I þF; m ¼ N0

8>>><>>>:

(15)

Simple calculation shows that the Slgs needs 7N � 3multiplications and 6N � 3 additions for computing onefilter output. Therefore, the proposed Slhhw is moreefficient. With less multiplications involved, it is alsoexpected to have a smaller roundoff gain. These areactually the main motivations of this new structure.

3. Performance analysis

As any digital signal processor used for filter imple-mentation has a fixed signal dynamical range, all the statevariables in xðnÞ, which are needed for updating xðnþ 1Þ,have to be stored. The magnitude of these variables isstructure dependent. To keep these signals within acertain dynamical range, the actually implemented struc-ture should be properly scaled in order to avoid signaloverflow.

There exist several scaling schemes, among which thel2-scaling is most popularly used. The l2-scaling, whichmeans that each state variable should have a unit variancewhen the input is a white noise with a unit variance, canbe achieved if [1]

Wcðk; kÞ ¼ 1; 8k (16)

As the controllability Gramian of the orthogonal filterstructures is the identity matrix, (16) automatically holds.

One of the serious FWL related issues is the roundofferrors due to multiplications between signals and non-trivial parameters in the filter structure used for imple-mentation. Let t be a non-trivial parameter (multiplier) ina filter structure. For an implementation of less-than-double precision with rounding after multiplication, theproduct tsðnÞ has to be rounded by a quantizer q½:�. Denote�tðnÞ9q½tsðnÞ� � tsðnÞ as the roundoff noise due to t. Theroundoff noises from different quantizers are usuallymodelled as statistically independent white processes[3] and s2

09E½�2t ðnÞ� is constant, uniquely determined by

the word length used for representing the states.

2 Note the factor 2 in the 1st equation of (13) is implemented as a

shifting operation.

Assume that wðnÞ is the error signal due to thequantizer q½MvðnÞ� rounding all products that occur inthe multiplication of a matrix M and a vector vðnÞ ofproper dimension. Let DyðnÞ be the corresponding outputdeviation of the filter and FðzÞ, the transfer functionbetween wðnÞ and DyðnÞ. It is well known that

E½ðDyðnÞÞ2� ¼1

j2p

Ijzj¼1

FðzÞRwFHðzÞz�1 dz

where E½:�, H denote the statistical average operation andthe conjugate-transpose, respectively, whileRw9E½wðnÞwTðnÞ�. The roundoff noise gain due to M isdefined as the ratio between E½ðDyðnÞÞ2� and s2

0.Let ðAF ;BF ;CF ;DF Þ be a realization of FðzÞ and WF

o , thecorresponding observability Gramian. According to theresidue theory, it can be shown that

E½ðDyðnÞÞ2� ¼ tr½ðBTF WF

oBF þ DTF DF ÞRw� (17)

where tr½:� is the trace operator.In the proposed Slhhw, the structure parameters are

fak; gk;bkg, the N elements in C, d, and the Nth element of Jthat is denoted as aN .

First of all, let us consider the quantization errorcaused by aN. In this case, (13) becomes

xð0ÞðnÞ ¼ 2xðnÞ þ q½JuðnÞ�

xðmÞðnÞ ¼ Amx

ðm�1ÞðnÞ; 8m

xðnþ 1Þ ¼ xðN0ÞðnÞ � xðnÞ

yðnÞ ¼ CxðnÞ þ duðnÞ

8>>>>><>>>>>:

Denote e0ðnÞ9q½JuðnÞ� � JuðnÞ. So, e0ðnÞ ¼ vN�aNðnÞ,

where vk denotes the elementary (column) vector ofproper dimension, whose elements are all zero except thekth one which is equal to 1, and �aN

ðnÞ is the unit roundoffnoise produced by aN. Denote DxðnÞ9xðnÞ � xðnÞ,DxðmÞðnÞ9x

ðmÞðnÞ � xðmÞðnÞ and DyðnÞ9yðnÞ � yðnÞ. It follows

from (13) and the above equation that

Dxð0ÞðnÞ ¼ 2DxðnÞ þ vN�aNðnÞ

DxðmÞðnÞ ¼ AmDxðm�1ÞðnÞ; 8m

Dxðnþ 1Þ ¼ DxðN0ÞðnÞ �DxðnÞ

DyðnÞ ¼ CDxðnÞ

8>>>><>>>>:

Comparing with (13), one realizes that DyðnÞ is actuallythe output of the following system having the realization½A; ðI �FÞ�1vN ;C;0�, when excited by �aN

ðnÞ. The corre-sponding roundoff noise gain, according to (17), is

G0 ¼ tr½vTN ðI �FÞ�TWoðI �FÞ�1vN� (18)

where Wo is the observability Gramian of the state-spacerealization Sorth.

Similarly, denote ekðnÞ9q½Akxðk�1ÞðnÞ� � Akxðk�1ÞðnÞ.With the same notations set before, one has

Dxð0ÞðnÞ ¼ 2DxðnÞ

DxðmÞðnÞ ¼ AmDxðm�1ÞðnÞ; 8mak

DxðkÞðnÞ ¼ AkDxðk�1ÞðnÞ þ ekðnÞ

Dxðnþ 1Þ ¼ DxðN0ÞðnÞ �DxðnÞ

DyðnÞ ¼ CDxðnÞ

8>>>>>><>>>>>>:

ARTICLE IN PRESS

2 3 4 5 6 7 8 9 10 11 12 130

10

20

30

40

50

60

70

80

Fig. 1. Roundoff noise gain G (vertical axis in linear scale) versus filter

order N (horizontal axis) with BW ¼ 0:125: solid line—Slhhw , dotted

line—Slgs , dashed line—Sorth , and dashdot line—Smrh.

Table 1Statistics for each of the four structures.

Structure Smrh Sorth Slgs Slhhw

G 23.3947 32.3886 24.8863 16.0523

Nmul 81 81 53 31

Nadd 72 72 45 31

G. Li et al. / Signal Processing 90 (2010) 679–683682

which leads to

Dxðnþ 1Þ ¼ ADxðnÞ þ PkekðnÞ

DyðnÞ ¼ CDxðnÞ

(

where

Pk9

I; k ¼ N0QN0

m¼kþ1

Am; kaN0

8><>: (19)

and with Ak specified by (10) for 1 � m � N � 1 and (9) forN � m � N0,

ekðnÞ ¼

vkþ1�bkðnÞ; 1 � k � N � 1

vm�amðnÞ þ vmþ1�gm

ðnÞ; k ¼ N0 þ 1�m

1 � m � N � 1

8><>:

According to (17), the roundoff noise gain due to Ak is

Gk ¼ trðPTk WoPkRkÞ; 8k (20)

where

Rk9E½ekðnÞeTk ðnÞ� ¼

vkþ1vTkþ1; 1 � k � N � 1

vmvTm þ vmþ1vT

mþ1; k ¼ N0 þ 1�m

1 � m � N � 1

8><>:

As to C and d, it is easy to show that the correspondingroundoff noise gain is

Gcd ¼ Ncd

where Ncd is the number of non-trivial elements in thevector ½C d�.

The overall roundoff noise gain of our proposedstructure is then given by

Glhhw ¼XN0

k¼0

Gk þ Ncd (21)

For the Slgs specified by (14), it can be shown in the sameway that the corresponding roundoff noise gain is givenby

Glgs ¼XN0

m¼1

trðPTm WoPmRmÞ þ trðWoÞ þ Ncd (22)

where

Pm9

I; m ¼ N0QN0

k¼mþ1

Ak; maN0

8>><>>: (23)

and

Rm ¼

vmþ1vTmþ1; 1 � m � N � 1

vkþ1vTkþ1; m ¼ N0 � 2k; 8k

vkvTk ; m ¼ N0 � 2kþ 1; 8k

diagð1;2; . . . ;2Þ; m ¼ N0

8>>>>><>>>>>:

(24)

Though there is no specific relationship obtained yetbetween Gmrh;Glgs and Glhhw, the sparseness of Slhhw

implies that the proposed structure should have a verygood performance against the roundoff noises, which is

supported by the numerical examples provided in thenext section.

4. Design examples

In this section, a number of design examples arepresented. There are four structures to be examined:the classical minimum roundoff noise realization Smrh, theorthogonal state-space realization Sorth, the Slgs structure,and the proposed Slhhw.

The examples used here are generated with MATLABcommand: ellipðN;0:25;40;2BWÞ, which yields an Nthorder low-pass elliptic filter with a normalized pass-bandfrequency of BW , a pass-band peak-to-peak ripple of0.25 dB, and a stop-band attenuation of 40 dB.

For the first example, BW ¼ 0:125 and N ¼ 8 are used.Table 1 shows the statistics for each of the four structures,in which G, Nmul and Nadd are the roundoff noise gain, thenumber of multiplications and additions for computingone output sample, respectively, for each structure.

Comment 4.1. The results are self-explanatory. Besidesthe obvious advantage in structure simplicity, the round-off noise gain by the proposed Slhhw is just about 68.62%and 64.50% of that by Smrh and Slgs, respectively. Thissuperiority of the proposed Slhhw in reducing roundoffnoises, though not proved theoretically yet, is supportedby extensive numerical examples.

Fig. 1 shows how the roundoff noise gain changeswith the order of the low-pass elliptic filters generated

ARTICLE IN PRESS

0.05 0.1 0.15 0.2 0.25

15

20

25

30

35

40

Fig. 2. Roundoff noise gain G (vertical axis in linear scale) versus filter

bandwidth BW (horizontal axis) with N ¼ 8: solid line—Slhhw , dotted

line—Slgs , dashed line—Sorth , and dashdot line—Smrh .

G. Li et al. / Signal Processing 90 (2010) 679–683 683

with ellipðN;0:25;40;0:25Þ, where the order N variesfrom 2 to 13. Fig. 2 shows how the roundoff noise gainchanges with the bandwidth BW of the filters generatedwith ellipð8;0:25;40;2BWÞ, where BW varies from 0.0125to 0.25.

Comment 4.2. As noted in [6], Slgs can outperform Smrh interms of reducing roundoff noise gain just for those highorder filters of narrow bandwidth, while the proposedSlhhw seems always the best without those constraints.

5. Conclusions

In this paper, the problem of digital filter structuredesign has been investigated. Based on an orthogonalstate-space realization, a sparse structure Slhhw has beenderived, which can be considered as an improved versionof the structure Slgs proposed in [6]. Due to the improve-ment in structure sparseness, the proposed structure Slhhw

not only outperforms Slgs but also the classical minimum

roundoff noise realizations Smrh. These properties makesthe structure an interesting candidate for real-timeapplications.

An Nth order recursive filter has 2N þ 1 parameters.Note that the l2-scaling (16) introduces N constraints. It isconjectured that the sparsest l2-scaled structures possessslightly more than 3N parameters. Finding such astructure with good robustness against FWL effects isunder investigation.

References

[1] R.A. Roberts, C.T. Mullis, Digital Signal Processing, Addison Wesley,Reading, MA, 1987.

[2] M. Gevers, G. Li, Parametrizations in Control, Estimation andFiltering Problems: Accuracy Aspects, in: Communication andControl Engineering Series, Springer, London, 1993.

[3] C.T. Mullis, R.A. Roberts, Synthesis of minimum roundoff noisefixed-point digital filters, IEEE Trans. Circuits Syst. CAS-23(September 1976) 551–562.

[4] S.Y. Hwang, Minimum uncorrelated unit noise in state-spacedigital filtering, IEEE Trans. Acoust. Speech Signal Process.ASSP-25 (August 1977) 273–281.

[5] G. Amit, U. Shaked, Small roundoff realization of fixed-point digitalfilters and controllers, IEEE Trans. Acoust. Speech Signal Process.ASSP-36 (June 1988) 880–891.

[6] G. Li, M. Gevers, Y.X. Sun, Performance analysis of a new struc-ture for digital filter implementation, IEEE Trans. Circuits Syst. I 47(April 2000) 474–482.

[7] N. Wong, T.S. Ng, A generalized direct-form delta operator-based IIRfilter with minimum noise gain and sensitivity, IEEE Trans. CircuitsSyst. II 48 (April 2001) 425–431.

[8] G. Li, Z.X. Zhao, On the generalized DFIIt structure and its state-space realization in digital filter implementation, IEEE Trans.Circuits Syst. I 51 (April 2004) 769–778.

[9] G. Li, J. Chu, J. Wu, A matrix factorization-based structure for digitalfilters, IEEE Trans. Signal Process. 55 (10) (October 2007)5108–5122.

[10] D.L. Wang, A. Zilouchian, J. Zhao, Z. Huang, Modular structurerealizations of 2D-separable-in-denominator recursive digitalfilters, Signal Process. 87 (11) (November 2007) 2686–2694.

[11] T. Hilaire, P. Chevrel, J. Whidborne, A unifying framework for finitewordlength realizations, IEEE Trans. Circuits Syst. I 54 (8) (August2007) 1765–1774.

[12] R. Wirski, On the realization of 2D orthogonal state-space systems,Signal Process. 88 (11) (November 2008) 2747–2758.

[13] V. Singh, Stability analysis of a class of digital filters utilizing signalsaturation nonlinearity, Automatica 44 (2008) 282–285.

[14] X.X. He, G. Li, C.R. Wan, T. Wu, On normal realizations of digitalfilters with minimum roundoff noise gain, Signal Process. 89 (2)(February 2009) 226–231.