Journal of Applied Research and Technology

ISSN: 1665-6423

jart@aleph.cinstrum.unam.mx

Centro de Ciencias Aplicadas y Desarrollo

Tecnológico

México

Jovanovic-Dolecek, G.; M-Alvarez, M.; Martinez, M.

ONE SIMPLE METHOD FOR THE DESIGN OF MULTIPLIERLESS FIR FILTERS

Journal of Applied Research and Technology, vol. 3, núm. 2, agosto, 2005, pp. 125-138

Centro de Ciencias Aplicadas y Desarrollo Tecnológico

Distrito Federal, México

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o

G. jovanovlc-Dolecekl , M. M-Alvarez2 & M. MartlnezJ

lINAOE, Puebla, México

gordana@inaoep.mx

21T, Mérida, México

JUniversidad Politecnica de Valencia, Spain

Received: July 1ih, 2004. Accepted: May 30th, 2005

ABSTRAG

Thls paper presents one simple method for the design of multlplierless flnlte Impulse response (FIR) fllters by the

repeated use of the same fllter. The prototype fllter Is a cascade of a second order recurslve running sum (RRS)

fllter, known as a coslne fllter, and its corresponding expanded verslons. As a result, no multlpliers are requlred to

implement this fllter.

RESUMEN

Este artículo presenta un método simple para el diseño de filtros digitales con respuesta al impulso finita (FIR) sin

multlplicadores, usando el mismo filtro varias veces. El filtro prototipo es una cascada de los filtros recursivos

corrientes (RRS) de segundo orden conocidos como filtros cosenos. Así mismo se usan versiones expandidas de

estos filtros. Como resultado no se necesitan ningunos multiplicadores para implementar este filtro.

KEY'vVORDS: FIR Fllter5, Multlpllerless Filter5, SharpenlnQ, Co5ine Filter5.

1.INTRODUGION

Dlp;ltal slp;nal processinp; (DSP)is an area of science and enp;ineerlnp; that has been rapldly developed over the past

years [1 & 2]. ThlS rapid development is the result of sip;nlflcant advances in dip;ital computers technolop;y and

Intep;rated circuits fabrlcatlon [3-5].

A typlcal operatlon In DSP 15 filterlnQ. The fundlon of a dlQltal fllter i5 to prOCe55 a Qlven Input 5equence ,\{n) and

Qenerate the output 5equence }.(n) wlth the de5ired charaderi5tfc5. If the Input 5iQnal ,\{n) 15 the unit samp/e

sequence, the output 5iQnal would be the charaderi5tfc of the filter, called the unit samp/e response /;< n).

Dependlng on the length of fj,n), the fllters can be elther FIR (Finite impulse response filters), or I IR (/nfinite impulse

response filters).

FIR fllters are often preferred over IIR fllters because they have several very deslrable propertles such as linear phase,

stablllty, and absence of Ilmit cycle. The main dlsadvantage of FIR filters is that they involve a hlgher degree of

computational complexlty compared to IIR filters with equlvalent magnitude response [6-8].

125 Imlrnal nF Annllpd Ilp4;parrh and Tprhnnlnov

FIR fllters of length Nrequlre, (M 1 )/2 multlpllers if Nis odd, N/2 multlpllers if Nis even, and Nl adders and Nl delays.

The complexity of the Implementatlon increases wlth the Increase In the number of multlpliers.

Over the past years there has been a number of attempts to reduce the number of multlpllers [9-12] etc.

Another approach Is a true multiplier-free (also referred as multipllerless) desl~n where the coefflclents are reduced to

simple Inte~ers or to simple comblnations of powers of two, for example [13-16], etc.

One alternatlve approach 1s based on combinlng simple sub-fjlters which do not require any multiplierS. Ramakrishnan

and Gopinathan [11] proposed to sharpen a cascade of recursive running sum (RRS) fjlters which are multiplier-freefjlters. .

Tai and Lin [13] prapased a deslgn af multlpller-free fllters based an sharpening technlque where the pratatype fllter

is a cascade af the caslne fllters whlch requires no multlpllers and anly same adders. Hawever, ta satlsfy the deslred

speclflcatlan the arder af the sharpenlng palynamlal must be hlgh, thereby resultlng In hlgh camplexlty.

In this paper we propose a modification of thls method which results in lower overall complexlty. The paper is

or~anlzed in the followin~ way. In Section 2 we describe cosine fjlters, while in Section 3 we present the sharpenin~

technique. The proposed desi~n, alon~ with one example is ~iven in Sectlon 4.

2. COSINE FILTERS

The slmplest low pass finlte impulse response (FIR) filter is the M-point moving-average(MA) filter, also known as the

comb fllter, wlth an Impulse response

g(n) = {1/M,O,

for O~n~M-

otherwise,11\

where Mis an Integer. Its system functlon 15 glven by

M-l

-LzM k=O

+ Z-(M-l)) = -k (2)1

G(Z) = -(1 + zM

+

The scaling factor l/M is needed to provide a dc gain of O dB. This filter does not reQuire any multiplications or

coefficient storages. A more convenient form of the previous system fundion for realization purposes is given by

I~\G(z) = ~I1

which Is also known as a recurslve runnlng-sum flIter(RRS) [9].

For example, for M = 2, we have

(4)l r l-z-2 1 1 -1 G(z)=- =-(l+z )

2 l-z-1 2

The ma~nitude re5pOn5e of the fllter for M = 2 15

126

Vol. 1 NO.2 August 2005

(5)

Because of this cosine form this filter is called a coslne fllter.

The Impulse res pon se and magnltude response of thls fllter are plotted in Figure 1

The Nexpanded fllter 15 obtalned by insertlng Nl zeros between each sample of the Impulse response. In z-domainthat means that each delay is replaced by N delays

G(zN) = .!.(1 + z-N ) .(6)2

The corresponding magnitude response is

The sequences in Figure 1 are expanded in time domain by a fador of N, and in the frequency domain are

compressed by the same fador N As a consequence, the frequency of the first zero of the magnitude response at 7r

in Figure lb would be compressed by N For example for N=l it would be at 7r/2, (Flg. la), for N=3 at 7r/3, (Figure lb),

for N=4 at 7r/4, (Figure lc), etc.

127

journalof Applled Research and Technology

Figure 2 shows the Impulse responses and the magnltude responses of the expanded cosine fllters for dlfferent

values of N

We can also notlce that the cascade of different expanded cosine filters will result in one low pass magnitude

charaderistic because the zeros of one filter will cancel the side lobes of the adJacent filter .

As llIustrated in Figure 3, the zero of the filter in FIQure 2( at frequency á) =0.75 trdecreases the side lobe of the filter

in Figure 2b around that frequency.

The transfer fundlon of the cascade of K expanded coslne fllters iS given by

K K 1H(z)= n G(ZN)= n -(l+z-N

N=l N=l 2(8)

=I IÍ COS(NúJ/2)

1N=}

The correspondinQ maQnitude response is then

IH(ej{1J)1 = I ñ G(ej{1JNN=l

In\

129

journalof App!led Research and Technology

We can notice that

The cascade of cosine fllters is multiplier-free. (5ee Equation 8).

The ma~nitude characterlstlc is low pass but it has a bad stop b~nd attenuatlon and si~niflcant pass band

drooD.

.

In arder ta imprave the magnltude charaderistic af this cascade, we use the sharpenlng technique, as explalned in

the fallawin~ sedlan.

3. SHARPENING TECHNIQUE

The filter sharpenlng technlque Introduced by Kaiser and Hammlng. [17J. can be used for slmultaneous Improvement

of both pass band and stop band charaderlstlcs of a Ilnear-phase FIR dlgital filter. The technlque uses amplltudechange fundlon (ACF). An ACF 15 a polynomlal relatlonshlp of the form Ho=f(HJ between the amplitudes of the overall

and the prototype filters. Ho and H, respedlvely. The Improvement in the pass band, near H=1. or In the stop band.

near H=O. depends on the order oftangenCies mand nofthe ACF at H=1 or at H=O.

The expressions for ACF, proposed by Kaiser and HamminQ [17], for the mth and nth order tanQencies of the ACF at H =

1 and H= O, respedively, are Qiven as,

Ho =Hn+li~(l-H)k =Hn+liC(n+k,k)(l-H)k .k=O n.k. k=O

where C (n+k, k) 15 the blnomlal coefflcient.

(10)

The values of the ACF for some typlcal values of m and n are ~iven in Table

~

i"~~

¡g

~~.

,~~

f' , 0010.2 03 O. 05 08 0

0/1

c)K=5 d)K=6Figure 5. Sharpening of the cascade of cosine filters

130

Vol. ] No.2 Augusl 2005

Figure 5 shows the results of applying the sharpening technlQue to cosine filters of Figure 4. Three dlfferent ACF'sfrom Table I are used: m = n = 1; m = 1, n = 2; and m = 2, n = 1. Notice that the pass band is Improved by Increasing m,

whlle the stop band 15 Improved by increasing n.

In order to obtain more flexibility in the desi~n, instead of usin~ a sin~le filter h(z) in sharpenin~ polynomials of the

Table I, we can use a cascade of k fjlters H.z). For example, the sharpeñin~ polynomial for m=l, n=l and k=1 is 3H2-

2H3. If we use k=2 filters Hin the cascade, the correspondinQ polynomial is 3~ -2¡f .

JABLE I. ACF po/ynomla/s for m = 1, ¿ 3 and n=l, ¿ 3

110 2H-H2-

3H2_2~

4~-3Jt

5Jt -4H5

6H5_51f

H3_3H2+3H

3Jt -8H3+6H2~

-1

1

1

1

2-¡

2

6H5-15Jt+l0HJ

w~s:¡:¡s¡¡;15H~ EI'+2Iii:i

:¿ j2 43 O3 13 23 3

-Jt+4H3_6H2+4H

-4H5+ 15Jt -20~+ 1 OH2

-I OJt+ 36H5_45Jt+ 2oF

-20H7+70Jt -84H'+35Jt

Figure 6a compares the results for the cascade of K=3 cosine filters (8) using the polynomial with m=l, n=l, k=2,

polynomial wlth m=l, n=3, k=l, and the polynomial with m=l, n=4, k=l.

The correspondin~ pass bands are plotted In Fi~ure 6b

K=3

nr : : : : : : : :: ~ , : I -m=1n=1k=2 I; ; '~ m=1 n=1 k=2 : : :

: .m=1'n=3'k=1 : : :

¡ -:- m=1 ~n=4:k=,1 ! ~1,r!=3,k=1i

.~ ., ,, ,, ,!g -50

,,-~og-e!

~~

'j: -100

K=3

, , ,, , ,

:l: j f : -_.; f : 1 f 1 .!_~ c

_..:-:: ! , ::::::::j:::::::::::::-l:- ~:::c

\-\-~" :m=1,1)=4,k=1

11.,...,

..~j t -_.r r o.

-0002

-0004

a¡-O(XE"O

..."' -0008co

~!! -001"

{ -O012

~ -0014

-O.016;

.0.018 --;-; ; ~ ¡. ...¡

.0-02n --1&! w _.-~

a) Overa// magnitude response b) Pass band zoom

Figure 6. Sharpening of the cosine fi/ters; K = J

1.11

Journalof Applled Research and Technology

We can notlce that when the parameter k=2 in the sharpenin~ polynomial m=1 and n=1 iS used. a new polynomial

which fjlls the characteristlcs between the polynomial m=1, n= 3. and the polynomial m=1. n=4 results.

we obtain new polynomials which flt theSlmllarly, uslng dlfferent values of k In the polynomlals of Tablecharaderlstlcs of the ortginal polynomlals of the Table I.

Figure 7 demonstrates polynomials with K = 6, m=1, n=1, and k=1, 2, and 3

K=6.m=IF1 K=6.m=IF1............ , :~'l~

.i \.!g"..~

[..!!

i§,~

:~:-o¡ ;J, :

¡ k=3 \..T, ¡

'.'.-'-'!":

-0.002

-0004

m-O(D¡-a

-o.0C8

~ -001.,-a--0012

.i -0.014

-0.0161I

-0018 ...'..."'"""i '. '."¡ "" :¡"""_." -0.02

0.025:.: -0'3- 0'4 -0'5- 0.6 0'7 -0'8 0'9..~

a)Overall magnitude response

-'-0.1 o.,

b) Pass band zoom

Figure 7 Sharpening using a cascade of cosine fllters, K =6

As explained in the next sectlon, we can also combine different ACF's for a better improvement in both the pass band

and the stop band.

4. DESIGN PROCEDURE

We conslder a typlcallow pass filter with the pass band edQe aJp and the stop band edQe {iJs The stop band frequency

determines the number K as follows [13]

K = int{ 1/ OJs (11)

where int{)(} means the closest InteQer of X

We propose to apply dlfferent ACF's to different groups of fllters of the cascade ( 8 ).

The fllters in the cascade (8) wlth hip;her value N have more droop In the pass band and more side lobes, and

therefore require an ACF with hlp;her values of m and n. At the other extreme, the fjlters wlth smaller values of N have

wlder pass band and fewer slde lobes and consequently requlre lower values of m and n in the ACF .

We divide the cascade of fllters Into the subgroups and apply dlfferent ACF to each subgroup. The number of fllters In

a subgroup Is at most 3. As a result. the hlghest values for m and n are typlcally 3. The procedure Is Implemented in

MATLAB.

The followinQ example 1Iiustrates the method.

Examp/e ,;

132

Vol. J No.2 AuIJusf 2005

We desiQn a filter with the followinQ specifications: The pass band and stop band frequencies are COp=.O2 and {j)s=.ll ,

respectively. The pass band ripple is Rp=O.15 dB and the stop band attenuation iS As= -80dB. A desiQn usinQ the Parks

McCIellan alQorithm results in a filter of order 78 and requires 39 multipliers.

From (11) it follows that K=9.

The prototype filter has the magnitude response given by

COS(úJ /2) cos(úJ ) cos(3úJ /2)... cos(9úJ /2)

This magnitude response and the pass band zoom are shown in Figure 8.

We can notlce that both the pass band and the stop band spedflcatlon are not satlsfled. Toremedy thls problem, we

flrst form the followlng groups of co~lne fllters:

Gl(OJ) = cos(9OJ/2)cos(8OJ/2)cos(7OJ/2), G2(OJ) = cos(5OJ/2)cos(6OJ/2)

G3 (OJ) = COS( 4OJ /2) cos(3OJ/ 2) , G4 (OJ) = COS(OJ) , G5 (01) = COS( OJ/ 2).

In the followin~ steps we apply varlous sharpenin~ polynomlals to ~roups In ( 13 ).

Step 7: Sharpenlng of G\((O)

We apply the ACF fundlon wlth m= 3, {1= 3, and k= 1 to GI ( m),

The corresponding magnitude response iS

1 Hl(OJ) 1=1 F{Hl(Gl,3,3,1)}G2(0J)G3(0J)G4(0J)G5(0J) I

where /{ H} means Fourler transform of H

journalof Applled Research and Technology

FI~ure 9 demonstrates the overall and the pass band magnitude response ( 15 ).

0.1!' , , , .,o, '.¡ ! l.:,.:.L i !

.:10

011;

w-~:

-40

!g -00

-00

--r f

--(\A--t1:- -' .\ ~ .

~ 005.,-

9~2! u"

"C~..-=

'f-D05

I.j..0.1

-01 0.2 0.3 04 05 0.6 0.7 OB 0.9 1 0 0.002 0004 O!JE 0.00B 001 0012 0014 0.016 0.01B 0.02

aI/a aI/a

a) Overa// magnitude re5pOn5e b) Pass band zoom

F~ílure 9. The ca5cade of co5ine fi/ter5 with 5harpening of the fir5t group ( m= J; n= J )

We notlce that now the pass band Is Improved, while the stop band is better In some frequency bands and worse In

others.

Steo 2: SharDeninQ of GI({i}) and G2({i})

In thls step we applv the sharpenln~ to the second ~roup G2(m) in (13), usin~ m=3, /7=2, and I;=

The corresDondinQ maQnltude res pon se is

I HI2(OJ) I = I F{Hl(Gl,3,3,1)}F{H2(G2,3,2,1)}G3(0J)G4(0J)G5(0J) I

The response of this cascade is shown In Figure 10.

A similar procedure Is outllned in the remalning steps.

Ste.o 3: SharpeninQ of G1({J)), G2({J)) and G3({J))

HI21 (O) I = I F{H1 (01 ,3,3,1)}F{H 2(02,3,2,1) } x F{H3(03,2,2,1)}04 (0)05(0) (19)

The result Is shown in FI~ure 11

134

1/,,1 ? Al" :1 'Sllrlllcf :1nnt;

~- 100

120

1~~

160

1~

0.2 0.3 04 05 06 07 O.B 0.9

m/2

a. Overa// magnitude response

.., o 0.002 0004 O~ 0006 001 0.012 0.014 0.016 0018 0.02

61/2

b. Pass band zoom

Figure 10. lñe cascade of cosine fllters with sharpeningofthe flrst (m=]; n=]) and the second group (m=]; n=})

Step4: Sharpenin~ of.Gl({J), G2({J), G3({J) and G4({J)

4H4(G4,0,3,1)=G4

HI234«(j)) = F{H1 (Gl ,3,3,1) }F {H 2 (G2 ,3,2,1) }F{H 3 (G3 ,2,2,1) } x F{H 4 (G4 ,0,3,1) }G5 ({J)) (21 )

0.3 04 05 0.6 0.7 08 0.9

OI!x

a. Overa// magnitude response

.

o 0.002 0.004 0.(00 0(00 001 0012 0.014 0.016 0018 002

ml2

b. Pass band zoom

Figure ". lñe cascade of coslne fl/ters wlth sharpenlngofthe flrst (m=],. n=]}, second (m=],. n=}}, and thlrd group (m=}; n=}}

The result is illustrated in Figure 12.

135

journalof Applled Research and Technology

.:¡:: : : : : : :

~.-"..!g -60

-00

~ -1~~

{-120

~ -11.:

-160

-100

_j::A--

01 0.2 03 04 05 06 07 0.8 09

lDh

a. Overall magnlfude response

Fif/ure 12. lñe cascade of cosine filters with sharpeninf/ofthe first (m=]; n=]~ second (m=]; n=2), third (m=¿. n=2~ and the fourth f/roup (m=O,. n=])

Step 5: (Last step). Sharpenlng of G¡(lV), G2(lV), G3(lV), G4(lV) and GS(lV)

(22)

H1234S({J) r = I F{H1 (G1 ,3,3,1) }F{H 2 (G2 ,3,2,1) }F{H 3 (G3 ,2,2,1) } x F{H 4 (G4 ,0,3,1) }F{H S (Gs ,0,3,1) } .(23)

Figure 13 shows the magnitude responses of the deslgned filter. We can notice that both the pass band and stop

band speclflcatlons are satlsfied.

In order to compare the proposed result with the method ofTai and Lin [13J. we apply the polynomial m=2, {}=2 k=1 to

all cascades. The result In Fi~ures 14a and 14b shows that the specification is not satisfled.

, -..o 0.002 0[D4 0(0¡ 0.(0¡ 0.01 0.012 0014 0.016 0018 0.02

mIs

b. Pass band zoom

Figure 13. 7ñe deslgned fllter

136

I/nl ? i\/n 7 A,lt1ll~t 7flfl,

~

-20

~

!g-00

.!I)

--100&

"O

--120

-140

-160

-IBJ

._~-.-f---

.A-

o 0.002 0.004 o.~ 00CM3 001 0.012 0014 0.016 0018 0.02

~

b) Pass band zoom

o~j- -02 -0'30'4 0'5 06- 0'7 -0:80'9~

a) Overa// magnlfude response

J~o

Figure 14. Method ofTal and Lln [IJJ (m=n=2)

However, the polynomlal wlth m=3, n=2, k=1, satlsfles the speciflcatlon (FiQure 15). Therefore, thls example shows that

the proposed method results in a less complex fllter than the one proposed by Tai and Lln.

Figure 15. Method ofTai and Lin [1]J (m=], n=2

5. CONCLUSIONS

A simple method for the design of multiplierless FiR filters is presented. The method uses a cascade of second order

RRS filters with the corresponding expanded filters. The number of filters in the cascade depends on the val.ue of the

stop band frequency. in contrast to the method proposed in Tai and Lin. 1992. where the orders of tangencies n and

m are varled from 1 to 8, in the method proposed here the orders of tangencies are varied only from 1 to 3, thereby

resulting in a less complex filter.

A less complex design is achieved by dividing up the cascade of K cosine filters into the subgroups. The number of

fllters in a subgroup typically varles from 1 to 3. The designed examples show that it is a good choice to start wlth a

117

journalof Applled Research and Technology

5ubQroup con515tlnQ of 3 element5 for hlQher ValUe5 of N In (8). The 51ze of the 5ubQroUp5 15 decreased for lower

ValUe5 of N, and for N=1 It 15 typlcally 1.In arder to permlt more flexlbllity In the de5lQn, the prototype fllter In 5harpenlnQ polynomlal5 i5 a cascade of kfllter5

(8). The complexlty of the 5harpenlnQ polynomlal5 decrease5 with the decrease In the corre5pondlnQ ValUe5 of N

If the deslgned fllter satisfles the given specification. In the following iteration one could try to decrease complexity by

decreasin~ kand the corresponding mand nvalues ofthe sharpenlng polynomials.

The proposed method iS intended for the narrow band FIR filter deslgn. As a future work. it would be interesting to

extend the procedure to also include wide band FIR fllters as well as to analyze its implementation issues.

6. REFERENCES

[1 ]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

1]

[12]

[13]

[14]

[15]

[16]

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31, pp.260-265, 1984.Ramakrishan, K. V and Goplnathan, E., Deslgn of FIR fllters wlth reduced computatlons, Clrcults Systems Slgna/

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Tai, Y. L. and Lln, T. P. Deslgn of multlpller less FIR fllters by multlple use of the same fllter, E/ec1ronlcs /etters;

vol. 28, pp.122-123, 1992.Yll-Kaaklnen J, and saramakl T., A systematlc algorlthm for the deslgn of multlpller less FIR fllters, The 2001

Internatlona/ Symposlum on Clrcults and Systems; ISCA.S; vol. 2, pp. 185-188, 2001.Llu M. c., Chen, c. L., shln, D. Y., Lln, c. H., and Jou S. J., Low-power multlpller less FIR fllter syntheslzer based

on CsD code, 7ñe 2001lnternatlona/ Symposlum on Clrcults and Systems; ISCA.S; 4,666-669,2001.Coleman, J. O., Fadored computlng structures for multlpller less FIR fllters uslng symbol-sequence number

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