An RML algorithm for retrieval of sinusoids with cascaded notch filters

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An RML algorithm for retrieval of sinusoids with cascaded notch filters

Nacer K M’Sirdi, Ioan Doré Landau

To cite this version:

Nacer K M’Sirdi, Ioan Doré Landau. An RML algorithm for retrieval of sinusoids with cascaded notch

filters. ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing, New York,

NY, USA, 1988, pp. 2484-2487 vol.4., Sep 1988, Grenoble, France. �hal-02482009�

E1.15

AN BML ALGORITHM FOB RETRIEVAL OF SINUSOIDS WITH CASCADED NOTCH FILTERS

' LABORATOIRE D'AUTOMATIQUE ET ROBOTIQUE WY. P. M. CURIE ( P m 6)

*

LABORATOIRE DAUTOMATIQUE DE GRENOBLE (1H.P.G.) et GRECO SYEITEMES ADAPTATES (CH.RB]

W.NB.1.E.O.) B.P. 46 -38402 ST MhRTM D'XBRES FRANCE

ABSTRACT:

To p v i d e the frequencies estimation , Adaptive Notch Film [ 1-41 can be used by implementation in a cascad blocks of second ordercells. Samestmtegiestotracktimevaryingparametm canbeused [6]. Then, inasecondsteptheestimatedfrequenciesare used to provide the estimation of amplitudes and phases in a recursive manner. The proposed recursive algorithm consists

in

two steps. The first step involves a M a x i " Likelihood algorithm to adapt the cascaded fim parameters [ 1][4], which will provide the frequencies estimates. The second step uses the last estimates, and then the estimations of amplitudes and phases are given by a Recursive

Least

Squares algorlthm[6]. The proposed algorithm is asymptotically consistent and robust faced with the neglected dymmcs. In case of time varying signals, its tracking capabilities insure the goodness of the estimations. The accuracy of estimation is better with the RML than with the ELS method far which an upper limit of the debiasing parameter is crucial in order to have a convergencewithoutlocalinstabilities.

I- IN TRODUC TIOM:

Our main interesr in this contribution, is the retrieval of sinusoidal signals in noise. AR aad

ARMA

modelling techniques have powerfull potentialities and wellsuited for signal spectrum analysis. However, when the noise level is high and I or when the

signal

model bas

poles

near or on the unit circle, serious problems arise[ 11. In thenarrowband case, successfull applications have been realized with AR m o d e m [2-31. The high order aumgressve FIR firer is shown to be asympwtically equivalent to an Adaptive

Notch

Filtm(ANF). The noise effect is reduced by inmasing the order of of the AR

filter

at the expense of computational cost.Thus the a

priori

knowlege of a narrowband leads to the

Notch

Filter

Structure (NFS) with constrained poles and zeros. This has recedybeenproposedin extenSivesimulations [4-71. Atheoretical analysis have been presented in [5][7] and the applicable estimation methodsaretwiewedin[8].

These last structures introduce in ParameterAdaptationAlgorithms

(€'AA)

some nolinearity which may give some problems in the transient period of the estimation. The cascade implementation of second order have been claimed possible.

Fast

Least Squares

(FLS)

estimation algorithm case have been considemi and studied in simdations[9]. The main interest of the ANF Cascade form is to simplify computation of the estimated frequencies and in case of independly time varymgfreqnencies, a second order filter appears faster to trackvariation than ahigher order one.

In section

II

of this paper we develop an implementation of cascaded

ANF.

The Predictionk Method Qmr) is applied lea- to a Recursive Maximum Likelihood (RML) algorithm.

This

PAA

allows the computation ofthe estimated frequencies. In a second step, using these estimates, a Recursive

Least

Squares

(RLS)

algorithm is employed for the estimation of amplitudes and phases. The analysis and performance evaluation of these algorithms ispresented insectionII1.

11-Recursive estimation algorithm and filter structure.

2.1 Frequency estimation whit cascaded ANF

The recursive estimation procedure consists of two stages.

The first one involves an

ANF

in cascade form for the estimation of the frequencies. The other one uses an adaptive algorithm to p v i d e the amplitudes and phases estimations. The involved signalsmay be modeled asfollowswhere vkisthenoise disturbance:

9

yk = t+&, Sh(fdi. k+ fii)

+

vk (1)

AdaptiveNotchFiltersareverywell suitedforestimationof the sinusoidal component frequencies.

Let

us consider the notch filtertransfertfunctionincascadeform:

With no loss of generality we assume identical bandwidth

2484

for the notches.

if

the frequencies are all independant we can write

(3) and the prediction emrris:

%

= H,(q'l)

72

(4)

The last equation presents a second order notch filter. Thus independence between frequencies yields independence between parametem

9

of each second order cell. Then each cellcan be adaptedindependentlyoftheothersafterptefilteriagthesignal bythe others. If we suppose that the filters Hi@') with

8

and i=l ...p have converged then the filter Hj(2-l) will remove the

remaining

component. Thus there exits a unique global minimum for the quadratic criterion. Followingthe

Prediction

k t e c h n i q u e forthe estimationalgorithm, weobtainthegradient:

The RML equations:

algorithm may be summarized by the following

An approximation must be done to reduce computational complexity. It yields an Appruximate RML algorithm. The computational complexity is essentally du to the computation of the gradient. Wecanthenusethe formula:

(7)

2.2 Amplitude and phase estimation with WRLS:

In this second step we assume the frequencies known and use a Weighted Recursive

Least

Squares algorithm to estimate the amplitudes and phases as in [lo]. The frequency are provided bythe firststep.

Model (1) of the signal may be w r i m :

wheretheamplitudes~andphases $(P=l ...p) aregivenby:

In equation (12) the filtered version of the signal produced by the first stage, can beused.

111- Analysis aud Perform" Evaluation

3.1 Convergence of the ANF

8 )

Second

order

(am)

Let

us f i consider the case of a single cell Hjjz)

(see

(4)) driven by the signal defined in (3) and assume the f i l m &(z) i=l..~-lg+l,.,nhaveconvergedtotheiroptimalvalue. Thus@is composed by a single frequency W. plus additive independant

noise.

1

The bandwidth are large enough to remow(or attenuate) the other frequencies.

T a k q for the s p l

aa

AR model with poles

on

the Unit circleandapplyingtheRMLalgorithm,thea~~errorcanbe written: A

Itiswell kaownthattheRML

algorithmneedsastabilitymonitoriag

ingeneral case.

For

t h e m this procedure canbe removedin vittu of the following Lemma.

Lemma 1:

Let

r by exponentially rime varying, from 0 to one,

Thenthereexitan fd,fO a n d r f N c h t h a t ~ ( r ~ g - l ) i s i ~ ~ stable.

proof : we can take

y

< inverse of the m a x i " modulus of the unstable poles of A(q-l).

particul~Theorem4of [ll]leadstothefollowing:

~ C c o r d i t o :

rk=

rd. fk-l+(l-rO). ff ,

Now applying

the theoret,ical bakground in [ill and

in

2485

P1: lim(Cj-wk)=O wpl p2:

lim( 9

⁾

^e

^{Dc wpl}

whhae Dc istheconwrgence domain

~ e m a 2:

b'

^$ai there exists an r such that

-

^;.I

- -

^{is SPR.}

A 8 The la# condition on r is less restrictive than the one for the stability.

Lemmas

1 and 2 allow us to assert the global nability of the ANF i f r is appqriately chosen such that A(r.q-l) is always stable duringesimation.

b)ELSprt.PPeterad.pt.tionalgocithm:

A

Ifthe ELSestimationmethodisapplied(l3) becomes [5]:

The SPR conditionis more restrictive than the stability one onr and Theorem 1 canaotbeappliedinallcases. Thereexitsanupperlimitl for the debiasiog parameter in order to have the SPR condition.

Tldslimitpdependafthesignalfrequencyandlimitstheaccuracyof

Thus the local stability m o t be etablisbed although the global stabilitycaabe ensured.

c)

Uniqueness

of the

RML

estimates:

t h k a t e s .

Intheorem 1 wehaveasymptoticallywpl (Pl):

Itfollowsfrom (17)and(18):

At9-Y

- A &-')

(19)

A64.') - A(vq-7

andthenai=al withprobabilityone(wp1) A

4

^{P # j}

.

"heorer 2: If the order is cotrect the RML estimates for

ANF

are unique. 8

i)the global convqence can be ensured with

an

appropriate choice Tbemainredbconserning ANF are :

of thedebiasing parameterwithout anystability monitormg.

ii)TheRML estimates are unique.

For

p cells in cascade adapted along (3_6),each cell w i l i converge near to a local minimum [5]. The filming (3) will make

t h a e ~ a ~ s t j n c t ~ ~ , f ~ e s c h ~ ~ , t h e w h ~ f ~ u e n ~ e s .

3.2 Amplitude .ad p h u e eaia8ter:

Exponential convergence of the amplitude and phase

estimates is garanteed by the following theorem which uses the conceptofpersisrentexcitation[ 121.

Theorem 3: The RLS estimation scheme (12) with exponential forgettingfactorisexponentixponenriallysrable.

The p o f of this theorem can be conducted in the same lines as in [ 121. The regressors are composed by p ^sinusoids having different frequencies. Then the persistent excitation condition is

Equation (20) yields by lemma 1 of [ 121:

$an+

The adaptation is bounded and will never be zero ( t r a c h g capabilities). Finallyreasoniogasin[12]weobtab.

IV- Conclusion

The main results demonstrate the good performances observed in simulation for the ANF. The exponedal convergence of the WRLS estimates of amplitudes and phases is p v e d by use of the theoretical backgmuag on persistent excitation[ 121.

These results are very impoaaat for the time varyiag systems OT ia case of neglected dynamics due to the resulting robustness of the algorirhms. The implementation form studied here is computationally attmctive and robust also when frequency, amplitudeorphaseistimevarying.

Aknowlegement

The authors wantto tank 0. Macchi forthe helpfull discussions.

2486

R E F E R E N C E S

[l] I. D. LANDAU, N. K. M'SIRDI and

M. M'SAAD.

Techniques de modelisation recursive pour l ' d y s e speurale parametaique adaptative. T r a i t " do signal, [2] B. WIDROW et d(1975). Adaptive noise cancelling:

Principlesandapplications. P m . IEEE,V63, Dec. 1976 [3] K. GLOVER. (1977). Adaptive Noise Cancelling appliedtosinusoidalinterfemnces. IEEEASSP, V25,N6 [4] A.

NEHORAI

(1985). A minimal parameter adaptive notch film with constrainedpoles and

zeros.

IEEE Trans.

[5]

N.K.

M'SJRDI et LD. LANDAU (1985).

Analyse

spectrale adaptative de signaux 8 bandes &troitfs. Rapport intane86-70. L.A.G.

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M.

T. ROMANO and

M.

BELLANGER (1987).

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least squares adaptive notcb filtering. 8th confmnce [7] D.V. BASKARRA0andS.Y. KUNG(1984). Adaptive notch filtering for retrieval of sinusoids ia noise, IEEE Trans. ASSP, Vol. 32,

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I.D.

LANDAU (1987). Adaptive Evolutionary Spectrum Analysis for Narrow band Signals. ICASSP87, Dallas,TEXAS.

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Volume V begins on page 2493

2487