Improvement of Cumulant-based Parameter estimation

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Improvement of Cumulant-based Parameter estimation

Denis Kouamé, Jean-Marc Girault

To cite this version:

Denis Kouamé, Jean-Marc Girault. Improvement of Cumulant-based Parameter estimation. IEEE

International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2001), May 2001, Salt

Lake City, United States. pp.3989–3992, �10.1109/ICASSP.2001.940718�. �hal-03149804�

Improvement of Cumulant-based

Parameter estimation

Denis Kouamk, Jean-Marc Girault LUSSI, GIP Ultrasons Tours, France

Abstmct- This paper presents a improvement of high or- der cumulant-based parameter estimation using delta oper- ator applied t o instrumental variable algorithm. It is based

gate a d-operator high order statistics recursive instrumen- tal variable method.

on a modification of the classical least squares estimation

11. PRELIMINARIES

and the utilization of the delta operator and the introduc- tion of an additional term in the parameter estimates. Com- puter simulation results are given t o illustrate the behavior of this method.

The following notations will be used.

3:

denotes a vec- tor, ~ ( n ) denotes the nth(time) element of

3:.

Uppercase ( X ) denotes a matrix. The finite difference S-operator is

I. INTRODUCTION

defined by: S 5 ( n ) =

z ( n ) - z ( n - l )

-

^1-q

T

-

-5;-3:(n),

where T is the (normalized) sampling period of the process

^2. q is

the ECOND order statistics (SOS) parameter estima- delay operator that is, for an integer i, q i z ( n ) = z ( n

-

i).

1+ ...+( - 1)'q'

S tion, using of a 6-operator has shown advantages Thus, we define Siz(n) = w 3 : ( n ) =

T'

4 . 1 .

over the classical q-operator, in terms of numerical abil- m-order cumulant of

z

is denoted by Cmz(tl, ...tm-l). Con- ity and ill-conditions processing. Moreover, it has been sider a real discrete time process,

shown that the least squares estimation convergence rate Concern-

P

.(n) = a(IC)z(n

-

k ) + u ( n ) , y(n)

=

z ( n ) + w ( n ) (1) can be improved [1]- [3] using a d-operator .

k= 1

ing high order statistics (HOS), the major problems are where

U(.)

is an independent input excitation, y(n) the the convergence rate and the fluctuations of parameter es- observed output, and w(n) a zero-mean gaussian colored timation when the algorithms, particularly the Recursive noise (which can be an MA process). Here, we consider the Instrumental Variable (RIV), work on short-length data, problem of estimating the AR parameter, a ( k ) , IC = 1, . . . , p , [4]-[6]. To this day, no &operator type algorithm applica-

ble t o HOS is available. Such an algorithm, by using the properties of &operator, could lead to an improvement of classical algorithms. The aim of this paper is to investi-

using recursive instrumental variable transversal structure.

This problem has been thoroughly covered in many papers,

e.g.[?].

The Instrumental Variable (IV) estimation of the

AR. part of (1) uses a process z ( n ) referred to as instru-

mental varianie, wnicn will ne aiscussea neiow. we aenne

111.

r w w s w

H L ~ ~ O H I ' I H M

4 T ( n ) = [y(n-l),"*?l(n-p)l,$T(a) = [z(n-l),..'z(n-p)l Now consider the extension of motje] (1) using the 6- operator, which can be written as:

and BT

=

[ a ( l ) ... u ( p ) ] . The process z ( n ) , assumed to be un- correlated to the additive noise t u ( . ) , asymptotically yields the ( p + 1) rank matrix

P

A ( S ) z ( n )

=

~ ( n ) ; w i t h A ( @

==

c ~ ( i ) B ~ ( 5 )

z=o

n

F ( n ) = 'c A"-"(k)@(k) (2) Define:

k = O

Equation ( 2 ) gives the classical correlation-based estimate eT ⁼ [ - a ( p

-

1) ... - 4 0 ) ] ( 6 )

$F(n) ⁼ [ S ( P - l ) z ( n ) ... 4 4 . ( 7 ) for z ( n ) = y(n). In this case, the parameter estimates

can be obtained via the Recursive Least Squares (RLS) y(n)

=

S ( P ) z ( n ) + w(n) method. To overcome R.LS limitations, classically z ( n ) is

chosen so that P ( n ) is an m-order cumulant matrix ( m >

2). Typical choices e.g. [4] are:

z ( n ) = y(n)y(n

f

no); this leads t o estimates based on 1D slice C3y(k, k + no). Generally no is set t o zero and z ( n ) = yz(n) and then estimates are based on the diagonal slice of the 3 T d - ~ r d e r cumulants of y.

t+V(n) = [ b ( p - - 1 ) z ( n ) ... z ( n ) ] (9)

Remark 1

:

Relations between general model (5) (say a ( n ) ) and (l),(saY a(.)) come immediately by expanding (5), for example

8(1) ⁼ 2 + TB(1);8(2) = -1 - $1) + T G ( 2 )

. ^{~ ( n )}

^{= y3(n) -} 3a2y(n), with u 2 = ~ ~ ~ ( 0 ) ;

this leads It is well known that the methodology in developing IV es- elements of p to be sample estimates of c4y(k, ^{k, k)} the

diagonal slice of the 4th-oder cumulant of y.

timates is based

on

modification of least squares covariance matrix so that the observations are not correlated t o the Here, we consider the case of estimation using m-order noise, through introduction of an instrumental process [ 7 ] . cumulant(m > 2 ) . The classical q-operator Recursive In-

strumental Variable algorithm is then given by

:

Following this methodology, the 6-operator RIV estimates are dervived. Using g(n) = e'(n)

-

e'(n - 1) and initial cu- mulant matrix Po, the orthogonality condition [7] leads t o

with 0 < A 5 1 This algorithm suffers from high fluctua- tions related to the use of High Order Cumulants (HOC) when applied to short-length data.

3990

\---

J n \ K )

uses tne Iorgetting factor

^A

to account

Ior

slow van- compute o q n ) eq.

^ti~),

ana r t n ) using tne aennition ations of the parameters. Parameter estimates are then

given by

:

of 6P(n) as specified in section 11.

compute 6 ( n ) eq. (12) and i ( n ) using the definition of

n

&(n) as specified in section 11.

;(n) = ~ ( n ) fn(/c)q(/c) [ ~ ( k ) - 4~(r~)i?(n

^-

111

the algorithm ends here if the model used is (5);

k=O

r

^{n 1 - 1}

Unlike the classical algorithm, here, the case without for- getting factor corresponds t o x ^{= 0.}

^1.5

For convenience, in the following parts we assume, without lost of generality, that T = 1. From these definitions, using similar derivations as in [ 8 ] , the &operator RIV estimates

1

0.5

g o

E g

^-0.5

-..-.-.-.-.-.-.-

are obtained by:

^-1

&n) = ~(n){~;ldi(n

-

1) + t,i(n)[jj(n)

-

$T(n>5(n - I)])

-1.5

-2

0 50 100 150 200 250 300 350 400 450

(12)

^-2.5

1 P ( n ) d ( n ) 4 T ( 4 m } (13)

^samples

SP(n) = -(XP(n)

-

1 - X 1 + p ( n ) P ( n ) q ( n )

Fig. 1.

In

practice,

Po

initial cumulant

marix

is

chosen by setting

0

Parameter estimate with classical RIV algorithm. Theo- retical values (dash dot) and estimated values (solid lines) with a single realization

Po = y I , where y is a scalar and I the identity matrix.

Here, one can notice that equations (12) and (13), by using an additional term P(n)PG1S8(n - 1) and by estimating the finite difference of parameters and cumulant matrix, differ from equations (3) and (4). This algorithm, first results in the reduction of parameter estimates fluctuations

and thus accelerates convergence speed. Secondly, it takes The counterpart of this algorithm is the computation of advantage of good numerical properties of the &operator the additional term and finite differences. But this is [1],[3]. The algorithm can be summarized by

:

choose initial values of P (e.g. Po = 71 with I the iden- tity matrix and y a positive constant) and e ( 0 )

For n=1,2, ^...

compute @(n) &n);

y(n);

GT(n) eqs. (6)-(9);

a small price t o pay compared t o the improvement in

terms of fluctuations reduction. To illustrate this, we con-

sider for the purpose of a computer simulation the pro-

cess

:

s ( n )

^-

0.6z(n - 1)

-

0.27z(n - 2)

=

u(n); with

w(n) = -0.49w(n

-

2) + ^e(n);

^y(n)

^{= z(n)} + ^{w ( n ) where}

111

^{estimate :} lhearelical

I1 I

a(2) : 0.5

p o E

I - 0 . 5

-1

aistinguisn Detween tne two parameter estimates Decause of the high fluctuations.

IV. CONCLUSJON

In this paper we have presented a new parameter estima- tion algorithm which has been applied t o HOC . This algo-

a(1) :

rithm is based on the modification of the classical approach

by i using a &operator. It allows reduction of parameter

-2

as shown by computer simulation.

Fig. 2.

Parameter estimate with new R I V algorithm. Theoretical REFERENCES

values [dash dot) and estimated values [solid lines) with a

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a

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mates using

the 3 r d - ~ r d e r

C m n k ”

with

C l a S S i C d

RIV and the new algorithm for SNR=10 dB. For this simulation,

we

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the new algorithm yields less fluctuations (fig.(2))

ⁱ ‘Om-

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