NONLINEAR SYSTEMS IDENTIFICATION USING THE VOLTERRA MODEL. Georgeta Budura

Transcription

1 NONLINEAR SYSTEMS IDENTIFICATION USING THE VOLTERRA MODEL Georgeta Budura Politenica University of Timisoara, Faculty of Electronics and Telecommunications, Comm. Dep., Abstract: Nonlinear adaptive filtering tecniques are widely used for te nonlinearities identification in manny applications. Tis paper investigates te performances of te Volterra estimator by considering a nonlinear system identification application. Te Volterra estimator parameters are compared wit tose of a linear estimator. For te nonlinear estimator, based on a second order RLS Volterra filter, a new implementation is proposed. Te experimental results sow tat te proposed Volterra identifier as better performances tan te linear one. Different degrees of nonlinearity for te nonlinear system are considered. Keywords: Recursive least squares algoritm, nonlinear estimator, linear estimator, Volterra filter.. INTRODUCTION Detection, representation and identification of nonlinearities in different telecommunication systems represent important tasks in many applications and ad a major contribution to te development of te main nonlinear modeling tecniques. Te current trend in te telecommunication systems design is te identification and compensation of unwanted nonlinearities. It was demonstrated tat unwanted nonlinearities in te system will ave a determinant effect on is performance (Tsimbinos, 995). Tere are various ways of reducing te effects of undesired nonlinearities (Stenger, 000 a,b; Küc, 00 ). Te use of nonlinear models considered in tis paper to caracterize and compensate armful nonlinearities offer a possible solution. Te Volterra series ave been widely applied as nonlinear system modeling tecnique wit considerable success. However, at present, none general metod exists to calculate te Volterra kernels for nonlinear systems, altoug tey can be calculated for systems wose order is known and finite. Wen te nonlinear system order is unknown, adaptive metods and algoritms are widely used for te Volterra kernel estimation. Te accuracy of te Volterra kernels will determine te accuracy of te system model and te accuracy of te inverse system used for compensation. Te speed of kernel estimation process is also important. A fast kernel estimation metod may allow te user to construct a iger order model tat give an even better system representation. Tis paper investigates te performances of te Volterra estimator by considering a nonlinear system identification application. Te Volterra estimator parameters are compared wit tose of a linear estimator. Te nonlinear estimator is based on a second order RLS Volterra filter. A new implementation of te second order RLS Volterra filter based on te extended input vector and on te extended filter coefficients vector is also proposed. Due to te linearity of te input-output relation of te Volterra model wit respect to filter coefficients, te implementation of te RLS algoritm was realized as an extension of te RLS algoritm for linear filters.. THE VOLTERRA MODEL Tis section will discuss some important aspects of te second order Volterra model. For a discrete-time and causal nonlinear system wit memory, wit input x[n] and output y[n], te Volterra series expansion is given by (Scetzen, 980) :

2 N M y [ n] =... r[ k,..., kr ] x[ n k]... x[ n kr ] () r = k = 0 M kr = 0 were N represent te model nonlinearity degree. Coosing N =, te input-output relationsip of te second order Volterra filter (FV ) can be expressed as: y M [ n] = 0 [ k] x[ n k] k 0 M M k = 0 k = 0 = () [ k, k ] x[ n k ] x[ n k ] Te input-output relation can also be written in terms of nonlinear operators as indicated in relation (3). [ n] H [ x[ n ] H [ x[ n ] H [ x[ n ] y 0 = (3) In te above representations, te functions i, i = 0, represent te kernels associated to te nonlinear operators [ x[ n ] H i. Te nonlinear model described by te relations () and (3) is called a second order Volterra model. Note tat te above representations as te same memory for all nonlinearity orders. In te most general case te relation () may used different memory for eac nonlinearity order. A furter simplification can be made to relation () by considering symmetric Volterra kernels. Te kernel i [ k, Λ k i ] is symmetric if te indices can be intercanged witout affecting its value. Te second order Volterra kernel is a (M M) matrix. If we consider symmetric kernels of memory M, te second order Volterra kernel requires te determination of M(M)/ coefficients as indicated in relation (4). M M = k = 0 k = k [ k, k ] x[ n k ] x[ n ] y [ n] k (4) Te kernel estimation accuracy becomes te major problem in te practical applications. It was sown tat te Volterra operators are omogeneous and generally not ortogonal. As a consequence of tis last caracteristic te Volterra kernels can not be measured using te cross correlation tecniques and te values of te Volterra kernels will depend on te order of te Volterra representation being used. If te order of te Volterra model is canged te Volterra kernels will cange and tey must be recalculated (Scetzen, 980; Budura and Nafornita 00; Budura and Botoca, 00a). However, for an input aving a symmetric amplitude density function, suc as te Gaussian noise, te odd order Volterra functionals are ortogonal to te even order Volterra functionals. It follows tat for tis type of input, a nd order Volterra model, wit zero DC component, is an ortogonal model (Budura and Botoca, 005). Tis leads to direct Volterra kernel measurement by te cross-correlation metods. In practical situations wen confronted wit system wic parameters are time varying, adaptive metods for kernels estimation are widely used. Due to te linearity of te input-output relation according to te kernels, or filter coefficients, te application of adaptive algoritms for te Volterra filters implementation is quite simple. Te nonlinearity is reflected only by multiple products between te delayed versions of te input signal. Te nonlinear estimator proposed in te article is based on a second order Volterra filter (FV ) descriebed by relation ()-(4) and considering 0 = 0. Te filter structure is indicated in Fig.. x[n] Fig.. Te second order Volterra filter structure Next we will introduce te input vectors corresponding to different orders kernels. Te first order input vector is defined as follows: [ x[ n] x[ n... ] x[ n ] X (5) = M Te second order input vector can be expressed by: X * X X = (6) Wen we deal wit symmetric Volterra kernels only M(M)/ elements of te MxM matrix indicated in relation (6) are used in te input output relation (). Based on tis observation we introduced te extended input vector for te second order Volterra filter as indicated in relation (7): [ n]... x[ n M ] x [ n] x[ n] x[ n M ] X = x... x [ n M ] [k ] [k, k ] (7) and te extended filter coefficients vector according to (8). [ ] H (8) = 0... M M M M M 3. RLS VOLTERRA ESTIMATOR Adaptive metods and algoritms are widely used for te purpose of kernel estimation. A Volterra filter of FV y V [n]

3 fixed order and fixed memory adapts to te unknown nonlinear system using one of te various adaptive algoritms. Te use of adaptive tecniques for Volterra kernel estimation as been well publised. Most of te previous work considers nd order Volterra filters and some consider te 3 rd order case. A simple and commonly used algoritm uses an LMS adaptation criterion (Matews, 99; Budura and Botoca, 00b; Budura and Botoca 004). Adaptive Volterra filters based on te LMS adaptation algoritm are computational simple but suffer from slow and input signal dependent convergence beavior and ence are not useful in many applications. Te aim of tis section is to discuss te efficient implementation of te RLS adaptive algoritm on a second order Volterra filter. As in te linear case te adaptive nonlinear system minimizes te following cost function at eac time: J n n k T ( n) = λ y( k) H ( n) X ( k) k = 0 were H ( n) and ( n) (9) X are te coefficients and te input signal vectors, respectively, as defined in (8) and (7), is a factor tat controls te memory span of te adaptive filter and y( k) represents te desired output. Te solution of equation (8) can be obtained recursively using te RLS algoritm. Based on te second order Volterra filter presented in Section we ave implemented te RLS nonlinear estimator. Te filter coefficients are adapted according to te following steps: I. Initialization: -define te filter memory(lengt for H(n) and X(n)); ( 0 ) = [ ] H ; ( 0) δ I C = * ; II. Operations: for C n =, nr. of iterations. Create te extended input vector: X ( n) ;. Compute te error: e ( n / n ) = d( n) H ( n ) * X ( n) ; 3. Compute te scalar: ( n) = X ( n) * C ( n ) * X ( n) µ ; 4. Compute te matrix: ( n) = C ( n ) * H ( n ) /( λ µ ); G 5. Updates te filter vector: H ( n) H ( n ) e( n / n ) * G ( n) = ; 6. Updates te matrix C : ( n) = λ *( C ( n ) G( n) * X ( n) * C ( n ) ) ; In te relations above C denotes te inverse autocorrelation matrix of te extended input signal. Inversion was done according to te matrix inversion lemma (Mortensen, 987). 4. EXPERIMENTS AND RESULTS Te RLS Volterra estimator performances were studied and compared wit te linear estimator performances in a typical nonlinear system identification application presented in figure. were is a small positive constant; FV y V [n] _ Noise generator z[n] FIR x [n] Nonlinear system y[n] Adaptive linear filter y l [n] _ Fig.. Te nonlinear system identification

4 Te nonlinear system wit memory being identified is represented in figure 3 and consists of a linear filter(fir) wit impulse response given by: ( ) [ n] = 0 n M n (0) followed by a nonlinear system witout memory wic input-output relation is: y [ n] = u[ n] bu [ n] () Te linear filter memory in relation (0) is M=0. Te coeficient b permits to cange te nonlinearity degree. Te input signal x[n] is generated by colouring te wite Gaussian sequence z[n], wit te autoregresive filter (FIR) described by: [ n] x[ n ] 0,9x[ n] 0, z[ n] x = 5 () Fig.4. Te error of te RLS Volterra identifier b = 0.0 Te error using te linear adaptive filter as identifier is presented in Fig.5. Experiments ave been done regarding te identification of a second order nonlinear system wit different degrees of nonlinearity. x[n] FIR u[n] Nonlinear system witout memory y[n] Nonlinear system Fig. 3. Te nonlinear system structure Te performance of te RLS adaptive Volterra filter was appreciated by comparing te error of te nonlinear identifier wit te error of a linear identifier. Te same memory M=0 as been cosen for te linear and for te second order Volterra kernel according to te relation (). For te linear identifier we fixed te same memory. Te factor was cosen equal to 0,995. Te simulations ave been done in te MATLAB software. Te following degrees of nonlinearity are considered: -Nonlinear system identified: b=0,0; Te error using te RLS Volterra identifier is indicated in Fig.4. Fig.5. Te error of te linear identifier b = 0, 0 -Nonlinear system identified: b=0,; Te error using te RLS Volterra identifier is indicated in Fig.6 and te error using te linear identifier is depicted in Fig.7. Fig.6. Te error of te RLS Volterra identifier b = 0.

5 Table Te identifiers parameters b m el σ el m ev σ ev 0, , , 0,03 0,068 6, ,353 0,675 7,50-5 0, CONCLUSIONS Fig.7. Te error of te RLS linear identifier b = 0. -Nonlinear system identified: b=; Te error signal using te RLS Volterra identifier is indicated in Fig.8 and te error using te linear identifier is depicted in Fig.9. Fig.8. Te error of te RLS Volterra identifier b = Te identification and compensation of unwanted nonlinearities are required in manny practical application in order to improve te system performances. Usualy tis are week nonlinearities and can be well represented wit te Volterra model. Tis paper investigates te performances of te Volterra estimator by considering a nonlinear system identification application. Te nonlinear identifier is based on te RLS Volterra filter. A new implementation of te second order RLS Volterra filter based on te extended input vector and on te extended filter coefficients vector is also proposed. Due to te linearity of te input-output relation of te Volterra model wit respect to filter coefficients, te implementation of te RLS algoritm was realized as an extension of te RLS algoritm for linear filters. For simplicity, tere were considered only second order nonlinearities, but te proposed tecnique can be extented to iger order nonlinearities. Te nonlinear adaptive filter performances were evaluated in a typical system identification application and compared wit te performances of a linear identifier. Te experimental results sowed tat te RLS Volterra filter performed better tan te linear filter wic performances were inacceptable wen te nonlinearity degree was increased. Te costs of tese performances are paid by te computational complexity required by te nonlinear adaptive Volterra filter implementation. REFERENCES Fig. 9. Te error of te RLS linear identifier b = Te mean and te dispersion of te error signals were calculated too, in order to caracterize te performances of bot identifiers. Te corresponding values are indicated in Table. As it can be seen te nonlinear identifier performances ( mev, σ ev ) are very good for different nonlinearity degrees, wile te linear identifier performances ( mel, σ el, ) are unsatisfactory wen te nonlinearity increases. Budura Georgeta., I. Nafornia, (00),"Kernels Measurement Tecniques for Constructing Nonlinear Models", Scientific Bulletin of te Politenica" University of Timioara, Romania, Vol.I.,Timioara, pag Budura Georgeta, C.Botoca (00a), Analyses and Modeling of Systems wit Nonlinearities, Proceedings of te Symposium on Electronics and Telecomm. `ETC 00`, Vol.I, Timioara, pp Budura Georgeta, C.Botoca (00b), "Applications of te Volterra Models in Nonlinear Systems Identification", Proceedings of te Symposium on Electronics and Telecomm. `ETC 00`, Vol.I, Timioara, pag Budura Georgeta, C. Botoca (004), "Efficient Implementation of Te Second Order Volterra Filter", Scientific Bulletin of te Politenica" University of Timioara, Romania, Transactions

6 on Electronics and Communications, 49 tom, Fascicle, pag Budura Georgeta., C.Botoca, (005) "La construction dun modele non lineaire a laide de series Volterra et Wiener", accepted to be publised in Revue de lacademie roumaine, Bucarest. Küc, F.,W. Kellermann, (00), Nonlinear Eco Cancellation using a Second Order Volterra Filter, ICCASSP 00, {kuec,wk}@lnt.de. Matews, V. J. (99), "Adaptive Polynomial Filters", IEEE, Signal Processing Magazine, July pp Mortensen R.E., (987), "Random Signals and Systems", New York, Jon Wiley & Sons, 05 pages. Scetzen M. (980), "Te Volterra and Wiener teories of nonlinear systems", New York, Jon Wiley and Sons,. Stenger A., W. Kellermann (000), RLS-Adapted Polynomial for Nonlinear Acoustic Eco Cancelling, Proceeedings of te X EUSIPCO 000, Tampere, Finland, pp Stenger A., W. Kellermann (000), Adaptation of a Memoryless Preprocesor for Nonlinear Acoustic Eco Cancelling, Signal Processing, No. 9, 000, pp Tsimbinos J. (995), "Identification And Compensation Of Nonlinear Distortion", Tesis, Institute for Telecommunications Researc, University of Sout Australia, ttp:// 37 pages.