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Title ARMA Digital Lattice Filter Based on New Criterion

Author(s) Miyanaga, Yoshikazu; Nagai, Nobuo; Miki, Nobuhiro

Citation IEEE Transactions on Circuits and Systems, 34(6), 617-628

Issue Date 1987-06

Doc URL http://hdl.handle.net/2115/6047

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Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

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j;,''''AN,,,cno,,, ')N '''''''''''''N' SYSTEMS, VOL. CAs·3 4, NO.6, JUNE 1987 617

,-

ARMA Digital Lattice Filter Based on New Criterion

YOSHIKAZU MIYANAGA, MEMBER, IEEE, NOBUO NAGAI, MEMBER, JEEE, AND NOBUHIRO MIKI , MEMBER, IEEE

. "",,'" ··1. this paper, II. new ARMA digitllliattice filter Is proposed. II. new criterion is defined for designing the ARMA digital lattice from given stochastic data. Based on this criterion, an ARMA

... _"" estimation algorithm is de,'eIoPfd. This algorithm ean esrimat~ ARMA parameters lIith little calculation «lSI. Then, from the fasr

.,.ri

61' IE££ TRANSACTIONS ON CI RCUITS AND SYSTE.>.IS, VOL. 0.5·)4 , NO.6, JUNE 1987

u Ck) y(k) A(d H,(z-o) -+ v (k ) •

x (Ie)

B

ii

620

These matrices are used in the recursive formulas intro-duced later.

A. AR·Type Recursive Formula

Let us show the AR-type recursive formula. This recur-sive formula estimates the parameters of an ARMA esti-mation model-as the AR order of this model increases by one. In other words, the AR-type recursive formula as-sumes that the parameters of the ARMA estimation model with s AR order and t MA order, abbreviated as the (s, I) ARMA parameters, have already been calculated, and then calculates the (s + 1, t) ARMA parameters. Since the AR order increases by one aher this calculation, this formula is regarded as the AR order update algorithm which allows ARMA parameter estimation.

By using (24), we get

[ y" E,. ,

where

[ V" 0 , ~~ , ~ l " ' 6s ,JtRs'H,, - ~~' • 0 I V"Y " , Tl I . J, /

1'1 [0 v', , 0 v,:, 1 1'3 l

I I R~+ I , f "" 0 " ,

V}, ,

0 V " 1'4 i , ". ~s ,II1R sH . 1 = [1"1 0 V' ' 1.1 : '1"\ 0]

[" ,, ] - E[E",(k - J)" ,,(k)]

" = - E[Y(k - , - l )Y,,,(k)]

'. = E[y(k - , -ljE,,,(k )],

(27)

(28)

(29)

Equations (27)~(29) are derived in Appendix II. As al-ready defined, the AR-type recursive formula calculates the parameter matrix and vectors 8, +1,,' 1 1+\,/ ' and E, +I,1 by using (27)-(29). From (19) and (20), 81+ 1". )'1-+1". and El+ l" satisfy

[ V" 0 : Vi" .. l" ~l 9,+1" R HI,I , ,->-1,,- V,. 0 ' V" 1+ 1,/ , , + 1.1 [Y"" j [0 V 1( i 0 V,h'l El+ t" R J+ l. ,· 0 1+ 1.1 I (30) , ' 0 ' ' V.+l.f : V;~" I . 1

Comparing the right-hand sides of (27)- (29) with those of (30), we can obtain (30) if the (5 + 2)th elements in the first and second rows of (27) become zero and the (s + l)th elements in the first and second rows of (28) become zero. In order to make these elements zero, this recursive for-mula employs (29). The AR-type recursive formula is given

'" 6,+ I . I = 6s.,11 + ""iE" ,12

'Y,+l. , = ['Y"I + 1-' 2E •• , l l l t +1.1 = E,,,l2 + [1l 36", + 1-' 4'Y., I + I-' sE..t1 /1

~,~ -(v,~,r 'h ,,] '" = - V Y(j V{ 2 I,r '.1

(31)

IEEE TRANSACTIONS ON CIR CUITS AN D SYSTE.\{S. VOL. CAs.34. 1'0. 6 . JU I' E 198'; '-~~

}l4 '" (V,~~1'4 - ~~, 'f) ) / [ (~~~ ) 2 - V,~ ,~~' 1 "'5 = 1-'4112

Y. +l, ,= Y. ,, + lln'fl 1'2]

v.~l,t = 'f) + 1-' 21'4

V'\ I" ""' V,~, + ['f\ 'T21 1l~ + J!4'f) + P- S'T4

J.j~l " "" J.j~ , + P-2V.~~ · The above equations are derived in Appendix. III.

B. MA-Type Recursiue Formula

(32)

(33)

Nex.t let us show the recursive formula which calculates ARMA parameters as the MA order of an estimation model increases by one. This formula assumes that the (s , 1) ARMA parameters have already been calculated, and then calculates the (s , I + 1) ARMA parameters. Since the MA order increases by one after this calculation, this fo rmula is regarded as the MA order update algorithm which allows ARMA parameter estimation .

According to (15)-(20), the MA-type recursive for mula is the complementary form of the AR- type recursive for-mula. In addition, the MA-type recursive formula is equal to the AR-type recursive formula if the variables associ-ated with the input signal x(k) and the variables associ-ated with the output signal y(k) are interchanged. Thus, if we replace y(k ), 'Y,.I and - 'Y •• ,(k) with - x(k), E •. I and ( ,.t(k), and vice versa, then the MA-lype recursive for-mula is derived. The MA-type recursive formula is given as

where

6'.1+1 = 6" ,1J + TJi Y •. ,14 E"I+1= [E, .r+ 1h Y, ,,] / 3

'Y •• I + 1 = Y" ,14 + [ TJ J6 •. 1 + 114E, .r + "I 5'Y I,,] 13

TJ I - - (~~t rl [ 'T{ 1';{ l 11 = - vYf/ V '

2 '.' ' , r

TJ 3= - ['f{ -r{1V,)

"_(V1f-r' _ VY1") / [(V,,)2 _VYVC I " 4 I.t 4 ' .r 1 I.t J. ' , ,'

V,+l,, =- V,., + TJf( 'f{ 'f;{J •

V " ~ ,' + ' ,,1 +1 J '121'4 v.~ r+ \ = v,: I + [ -r { -r:£ ] TJ~ + "141') + "I ~ 'T; V , - V' + V'' ., , +1 '.' '1 2 1, 1

[ ,; ,; ] - - ElY",(k-l)",.(k)]

,; - - E[x (k - t-ljE, ,(k) ]

" ~ E [x(k - t - l)Y", (k)].

(34)

(35)

(36)

By using (31) and (34), the ARMA parameters with arbitrary AR and MA orders can be recursively calculated fro m the correlation data of input and output sequences. These formulas provide lower calculation cost than the

MIVA NAOA el 01,: ARMA DIGITAL LATTICE ALTER BASED ON NEW CRITERION

.inverse matrix algorithm which estimates ARMA parame-~ ters in (5) by using an inverse matrix. Thus, let us call · these formulas the fast recursive method, abbreviated :FRM. , If the two proposed recursive algorithms join together, :we get the ARMA-type recursive formula which calculates ~-ARMA parameters as the AR and MA orders of an ~ estimation model increase by one simultaneously. This • ''formula is not essential but is useful in some cases. This ~ARMA-type recursive formula has already been proposed ;in [5J and [81.

v. ELEMENTARY SECTIONS USED IN ARMA DIGITAL LATTICE FILTER ,

, Levinson's algorithm is the recursive formula which ! solves the normal equation fo r the AR model with the ; lowest calculation cost. The AR digital lattice inverse filter r can be designed by using the prediction errors defined in i this recursive formula. The term" inverse filter" has been ~-coined to describe the whitening filter. The observed sto-~chastic signals are inputted to the whitening filter and the ~ whitened stochastic signals are obtained from this filter. In ;.addition, an AR lattice filter can be designed from the ~ obtained AR lattice inverse filter easily. This filter synthe-f sizes the observed stochastic signals with white Gaussian , input signals.

A similar procedure is applied to the recursive formulas ;. given in the previous section. In other words, from the kproposed formulas, we can easily design an ARMA digital • ~Iattice inverse filter. Then, the ARMA digital lattice filter r.is also designed. Associated with the recursive formulas f'(i.e., the AR-type recursive formula and the MA-type ~re

622

x ", ,/- (k l ----_~-'.'

'Ix t+l {k l , .

Ys,t+l (k)

~.t+l (k ) \;2 - r'\ ~ +

-n~ I ts,t+l (k )

,., " ~ x. '" C > -0 ;:> C ._ ,,- , -EO :;; " " ) -'" W

y{k )

) ,., f-~

L '" c ~ -0 c __ ,,-EO I

~"'N'G' et csl. : AJO.,Mo. DIGITAL I.ATIICE FILTER 8ASED ON NEW CRITERION

model with colored input signals. However, the cost of this algorithm is the smallest among

FRA's including our algorithm. The algorilhm of (VI-4) can real ize the ARMA model

r ;I'h arbitrary AR and MA orders (i.e., s "" I). In addition, can be applied to the model with colored input signals.

it needs weak restriction. In other words, this ;o,Uhm consists of

Type 1:

Type 2:

Type 3:

Evaluation of the (s + 1, I + 1) error set as a function of the (s. t) error sel. Eval uation of the (s + 1,0) error set as a func-tion of the (s,O) error set. Evaluation of the (0, t + 1) error sel as a func-tion o( the (0, I) error set.

H~:~',~the error set denotes the four estimation errors de in (14). Thus, the (VT-4) algorithm is an expanded

of (VI-I ) and (VI-2). In other words, the recursion of forward estimation errors in (VT-4) is the same as that the (VI-I ) algorithm. If s = t , the (VI-4) algorithm

~;':~~~'~~ to the (VI-2) algorithm. Although this al-~c can realize the ARMA model with arbitrary '~~~:.orders, the arrangement for each filter section is ~ According to Type 2 and Type 3, this algorithm

to realize either the AR part or the MA part at the first if s *" t. In other words, it cannot realize the ARMA

with arbitrary arrangement. For example, this al-cannot realize the ARMA model with the arrange-

of (0. 1). (1 . 1), (2. 1). and (3. 1) ARMA o,d", In 'ms it has to realize this model with only the arrange-

of (1,0), (2,0), and (3,1). The new algorithm pro-in this paper can realize this model without any

for the arrangement. 1f the orders of the esti-model are already known, it is enough to employ However, if any information of its orders has not

given, the new algorithm may be better than (VI-4). The algorithm (VI-5) can realize any ARMA model with

arrangement for the orders of the estimation model.

~~:'~::~.the errors given in this algorithm, if they can be ~c do not satisfy orthogonal conditions even for

forward estimation model. The proposed algorithm compensates for the disad-

,>", ' a!le of the above algorithms. It has been shown that of the algori thms above shows excellent ability for a

.. Cl a;n kind of signal processing. However, the proposed together with its lattice filler also presents the same since it is an expanded fo rm of the others.

VII. ExPERIMENTAL REsULTS

carried out some experiments for the proposed digital lallice filter. In this section, let us show the

resu lts. The experiments were performed to a model reduction problem. The model reduction

has been established in system identification [9], . This is used to yield a reduced model from a given

~r,,,ence model with either high order or infin itc order. In sectio n, we show three different ways of approaching

~ 0',-...... -2 fJ

~ ::. - 40 ~ -- - 60 1!-.

624

Fig. 11.

~ 0,--,~ ___ -----,

'-' - 2 121 o

-U ::l - 40 ~

-- -6121 C-,

-< -8121\-__ .,, __ --;---"

121 Fregu;ncy (kHi)

"J

L

- L

'bJ (a) Spectrum and (b) pole- zero locations of the designed

ARMA lattice filter based on v,, / of (40)

were the AR-type lattice elementary sections. In this fiiter , the four ARMA-type lattice sections were designed first. As discussed in the previous section, any other algorithms cannot design this filter with this arrangement. The AR and the MA order were 8 and 4. The spectrum and the pole-zero locations of this filter are shown in Fig. 11. This filter is unstable according to the pole-zero locations.

Next, let us consider the following criterion:

v,:, ~ a!;/(2wj)¢I( B( ,-' )/A(,-'))

A(,-') - ii(,-') I' d,/, This criterion is the same as the modified least square criterion introduced by Mullis and Roberts. To calculate the parameters which minimize v,.~ I' we needed the serond~order information and the first~order information (i.e., autocorrelation data and impulse responses of the reference model). In this experiment, after we calculated the 256 points of the impulse response of B(Z - l)/A(z - l), both data were calculated. The estimated ARMA lattice filter was the same as that in Fig. 10. Although the criterion is the same as that of Mullis and Roberts, we could design the ARMA latlice filter with a different ARMA order (i.e., s = 8 and t = 4). The spectrum and the pole- zero locations of the designed ARMA lattice filter are shown in Fig. 12. Comparing this filter with the reference model, the spectrum envelope of this estimated filter is quite similar to that of the reference model. In addition, all the poles and zeros lie inside the unit circle (i.e., this filter is stable). Thus, this filter is considered as a fine reduced ARMA model.

lEU TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAs,34, No.6, JUN[ 1987

Fig. 12. (a)

0r--rr------,

("

• -1. 1. •

- 1.

(bl

Spectrum and (b) pole- zero locations of ARMA lattice filter based on )//, of (41).

the designed

Let us show the other example. The criterion is defined

V':', ~ a!;/(2wj)¢IA( ,-')/ A(, - ')

- ii(, - ')/B(, - ') I' d,/, We needed both the impulse response of l /A(z - l) and the impulse response of 1/ B( z - I) to calculate the estimation ARMA parameters. We calculated the 256 points of these impulse responses. The designed ARMA lattice filter was the same as in Fig. 10. The spectrum and the pole- zero locations of this filter are shown in Fig. 13. This spectrum and the pole- zero locations are similar to those of the previous lattice filter. Thus, this filter is also the fine reduced ARMA lattice filter.

Now, let us consider why the estimated lattice filter has different characteristics for different criteria. The ARMA parameters of each filter are estimated to minimize each criterion. The estimation model .4(z-') and .8(z- ' ) have to identify the resonances and the antiresonances of the reference model. Thus, in v.,/ of (40), A(Z-I) and B(Z - I ) have to identify the zero points of A(Z - l) and 8(Z -I ) in the unit circle, respectively. However, even if an estimated zero point is slightly shifted from the given zero point, the value of v:..1 is not greatly changed. In other words, the sensitivity of the estimation is poor because the estimation error around the resonances and the antiresonances does not influence the criterion greatly. On the other hand, for V.:" A(Z-I) and .8(Z -I ) have to identify the pole points of A(Z - l) and the zero points of B(Z-I) in the unit circle. Thus, if an estimated pole point of .,i(z-') is slightly shifted from the given pole point, the value of V/ 1 is

,,.,,,,, el al.: "RMA OiOITAL LAmCE fILTER BASE D ON NEW CRITERION

"'TO -60 ~

626

Using (17) and (18) again, P I' /32' /1), and f34 in (A6) are

p, ~ - E[ r,,,(k-I)y(k -s -1)1 ~ - E[r,,,( k - I){, ,,(k - 1)1 ~ V,:~

fi, ~ EIr,., (k - I) x(k - s - l)1

~E[( r,,,(k- I)) ' I ~V/, p,~ E[{,,,(k- I )y(k-s-I)I

~ E[(",,(k-I» ' I ~vi, p, - - E[{,,(k - I )x(k - I - I )]

- - E[{,., (k - I)y.,,( k- I)I - v,:~. (A7)

[n addition, using (At) and (AS). the left-hand side of (A6) is rewritten as

E[[ - r,. ,(k-I)]h (k-I)] { (k - I) '.' '. ,

-E m::} ,.,(k -I) Th,,(k - I) 1 [ Y,., 1

- ~ R •. ,. ,. , Thus, we get (20) from (A6)- (A8).

ApPENDIX n DERIVATION OF (27)- (29)

(A8)

Let us derive (27). According to (24), 8s"I\ is given as

91.,1, _ [0 Iii'" a: 00 1, 1 ~i" '~,x ] . (A9)

1 d{ . .. a: , Obi" . b( Thus, using R H 1.,= E( hs+ I,/(k)TJr s ... 1.,(k)1 , we gel

6s .J \R . + I.t - E [9,.I I \h.+l,,( k) T h 1+ I,I( k) 1

~ E[ ". ,(k) Th,.,., (k)1

[V" 0 '. , - VII 0 • '1"\ i V/ , 00] T,l vx y I I. I (AlO)

where (14) and (A3) are used in (AlO), and '1"1 -= - £[.v:'/(k)y(k - s - 1)/ and 1'"2 = E{"!,,(k)y(k - s -I)). By using (16), the values '1"\ and 1'"2 are rewritten as

h T,j - E[y(k -s - 1»..,(k)1 - E[{,., (k -1)',.,( k)1

where ~",(k -1) is given from (14) as

{ ,.,( k - I) ~ y(k - s- I)H~y (k -s ) · ··

(All)

Hiy(k -1)- &!x(k - I)'" - &!x(k -I). Thus, from (AI D) and (All) we get (27).

Next, let us derive (28) . Using (24), we gel

[Y I 1 [ao "" aO, I,rl 0 ,~1 t ,I I "" a~·· ./i( . 0 . -1

o l ~l

(AU)

IEEE TRANSACTIO:-IS ON (: IRCt:ITS AND SYSTEMS, VOL. CAs-34, NO.6, JUN I! 1 9~7

v,:, ] V" '. ,

(AD)

where (14) and (A7) are used in (Al3) and 'T) - - El y(k - s - 1)Y,.,(k)] and T, = Ely(k -, - IH,.,(k)J.

Equation (29) is derived as follows. Using (24), we get

, bllYANAGA el at ,: ARM ... DIGITAL urnCE FILTER BASED ON NEW CRJTERION

r [ Next, consider the following equation :

[ [ Y." + JL2t,ll 11R s+l,1

I.'" [0 v.:~ + JL2VL 1') + JL 2'T4 : 0 V.:I + JL2V.:H, (A20)

!EqUation (A20) is derived from (28). Thus, if JL2 = :- v.:~/ VL, then the (s + l)tb element of the right-hand ~ide becomes zero, From (A20) and (30), we get Y,+1.t = h s,1 + JL2E.,,1 / 1 (A2l)

P- =- VY~/ V~ 2 ',I 1,1

V.!..(I,1 = 'T) + P- 2'T4 V , -V' + V" . + 1.1 - ',I P-2 ,,1 ' (A22)

Note that the (s + 2)th and (s + t + 3)th elements of the right-hand side in (A21) must be 0 and 1 since the ele-ments of YH I 1 associated with their positions are 0 and 1, • [eSpectively. · Finally, let us derive the recursion of E.+I.I' We have to notice that the (s +2)th and (s + t + 3)th elements of E'+I,I are 1 and 0, respectively. First, consider the following equation:

=[ -7"2 0 i\:

where (27) is used in (A23) and

,

V " , . , (A23)

(30). Thus, comparing (Al7) with (A25), we get

E.+ I,t - J14Y.+I,t = ~., ,12 + }L 36$" I1·

627

(A28)

From (A21 ), (A24), (A27), and (A28), we finally obtain

ES+I,t = ~" ,12 + [}LA" + }L 4Ys ,1 + P- SEs" Pl }L3 = - ['T1'T2] VI~ll

(A29)

JL 4 = ~~1.t I v.~ I., = ( 'T) + JL21"4) I( v.:, + P-2 v.:D = ( V,:~7"4 - v.~ 11'3) I[ (VI~n2 - v.:,v.~ ,1 (A30)

J1 5 = JL4P-2

Vs~ I , 1 = v.~ I + i\ + P.4VS~~1.1 .-= V.~I + ['T17"2] }L{ + J141"3 + J1 S1"4 ' (A31)

From (A1S), (A21), and (A29), the AR-type recursive formula is derived .

ApPENDIX IV

DERIVATION OF (37) AND (3S)

First, let us derive (37). From (24), 11 and 12 are rewritten by a new vector i i as

JT- [-T -T l- el··· e"" l

J T _ [-T - T 2 - e2 "'es+ 2 -T -T OT] il ,+4 '" ils + t +] (A32)

where i; is the (s + t + 3) dimensional row vector which has 1 at the jth element and zero at the others. Thus, we ,

11- 3 = - [ 7"1 1'2 J v,.~,t

i\ = [ 'TI 'T2] }L{· (A24) get

h, .. ,,(k) Jr-[y(k)"·y(k-, -I) : x(k)· ··x(k-t) ]Jr

~[y(k)···y(k-,): x(k)···x(k-t)] (A33)

h , .. ,,(k)J[-[y(k)···y(k -,- I): x(k)··x(k-t)]I[

~[y(k -I) ···y(k-'-I): x(k- I)···x(k-t) 0]. (A34)

Thus, from (29) and (A23), we get

(Es.J2+ }L 36,.JI) R .+1.1"", [0 v,.~I+i\ i 0 0]. (A25)

; However, from (A9) and (A14), the vector;. ,12 + }L 36, ,II , " . '

IS ·

. 10 a' .. · a' I : 0 b',,· b' 1 o s - 1 , 0 1- 1

; -aX ... ax , , aY . , • ay , ,

0 : 1 , o : 0 ,

:·In (A26), the ( s + t + 3)th element (Le., the rightmost " :element) does not become zero. Thus, we have to make it : !.ero in order to obtain t+ I I ' Let us consider the follow-, . : jng equation: •

, Jt+ 1,1 - JL4Ys+l.JR s+ 1,1 = [0 , ~,

" 'i

v' -" V" :, 0 01 . + 1,1 1'"'4 .+1,1 (A27)

~here P-4 = vs"!iI.tI V.~ I ,1' Equation (A27) is derived from

From (A33) and (A34), (37) is obtained. Let us derive (3S) next. From (14) and (31), we get

vs+ 1,I(k) ... h •. H .t (k) 6':1.t

= h $-+-1.1 (k) [ I t6s~, + I [E;'/}L IJ ~ " . .(k) +,,)k-l) . ,. (A35)

From (14) and (31), the other errors Y, +I, I(k) and ~ . + I,t(k) are given as follows:

Y,+I,, (k) = - h ,+l ,l(k)y;"'1.t

= - h,+ I,A k )Jt[ ytl + P- 2E;,,] ~ y,)k) - ~,',.,( k)

~$ + I , , ( k ) = h.+ 1.1 ( k )E;+1.t

= h.+ 1,, (k )J[E;" + hs-+-l.l(k) Jt

. [6s~I }LI + P-4yL + JLsE;'I]

(A36)

~ ,,) k - I) + , , )k ).j - ~,y,.,(k ) + ~"",( k )

~ ",,(k - I) H,) k ).j - ~'Y'H .,( k) (A37)

628 lEU TRANSACfl ONS ON CIRCU ITS AND SYSTEMS, VOl.. CAs-14 , NO. 6, JUNE 1987

where (A37) uses (A36) and P-5 = !l-.1l2: From (A35)- (A37), (38) is obtained.

ACKNOWLEDGMENT

The authors express their appreciation to Prof. K. Hatori of Hokkaido University, Sapporo, for his useful · advice.

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(31 Y. Miyanaga, N. Miki, N. Nagai, and K. Halori, " A speech analysIs algorithm which eliminates the influence of pitch using the model reference adaptive system," I£EE Trans. ACOWSI., Spudl. Signal Process., vol. ASSP·30, no. 1, pp. 88-96, Feb. 1982.

[4] J. D. Markel , and A, H. G ray, Linear Prediction of Speech. New York : Springer-Verlag, 1976,

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[101 R. A. E'J-attar, and M. Vidyasagar, "Order reduction by II and I-norm minimization," IEEE Trans. Au/omat. Cen/r., vol. AC-23, no. 4, pp. 731-734, Aug. 1978.

(11) A. Benveniste, and C. Chaure, " AR and ARMA identification algori thms of Levinson type: An innovations afproach," IEEE Trans. Automa/. Centr., vol. AC·26, no. 6, pp. 243- ~26 1 , Dec. 1981.

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Yoshikazu Miyanaga (S'80- M'8S) was born in Sapporo, Japan, on De