Abstract:
The invention relates to an arrangement and method for estimating the linear and nonlinear parameters of a model  11  describing a transducer  1  which converts input signals x(t) into output signals y(t) (e.g., electrical, mechanical or acoustical signals). Transducers of this kind are primarily actuators (loudspeakers) and sensors (microphones), but also electrical systems for storing, transmitting and converting signals. The model describes the internal states of the transducer and the transfer behavior between input and output both in the small- and large-signal domain. This information is the basis for measurement applications, quality assessment, failure diagnostics and for controlling the transducer actively. The identification of linear and nonlinear parameters P l  and P n  of the model without systematic error (bias) is the objective of the current invention. This is achieved by using a transformation system  55  to estimate the linear parameters P l  and the nonlinear parameters P n  with separate cost functions.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    1. Field of the Invention 
         [0002]    The invention relates generally to an arrangement and a method for estimating the linear and nonlinear parameters of a model describing a transducer which converts input signals (e.g., electrical, mechanical or acoustical signals) into output signals (e.g., electrical, mechanical or acoustical signals). Transducers of this kind are primarily actuators (such as loudspeakers) and sensors (such as microphones), but also electrical systems for storing, transmitting and converting signals. The model is nonlinear and describes the internal states of the transducer and the transfer behavior between input and output at small and high amplitudes. The model has free parameters which have to be identified for the particular transducer at high precision while avoiding any systematic error (bias). The identification of nonlinear systems is the basis for measurement applications, quality assessment and failure diagnostics and for controlling the transducer actively. 
         [0003]    2. Description of the Related Art 
         [0004]    Most of the nonlinear system identification techniques known in prior art are based on generic structures such as polynomial filters using the Volterra-Wiener-series as described by V. J. Mathews, Adaptive Polynomial Filters, IEEE SP MAGAZINE, July 1991, pages 10-26. Those methods use structures with sufficient complexity and a large number of free parameters to model the real system with sufficient accuracy. This approach is not applicable to an electro-acoustical transducer as the computational load can not be processed by available digital signal processors (DSPs). However, by exploiting a priori information on physical relationships it is possible to develop special models dedicated to a particular transducer as disclosed in U.S. Pat. No. 5,438,625 and by J. Suykens, et al., “Feedback linearization of Nonlinear Distortion in Electro-dynamic Loudspeakers,” J. Audio Eng. Soc., 43, pp 690-694). Those models have a relatively low complexity and use a minimal number of states (displacement, current, voltage, etc.) and free parameters (mass, stiffness, resistance, inductance, etc.). Static and dynamic methods have been developed for measuring the parameters of those transducer-oriented models. The technique disclosed by W. Klippel, “The Mirror Filter—a New Basis for Reducing Nonlinear Distortion Reduction and Equalizing Response in Woofer Systems”,  J. Audio Eng. Society  32 (1992), pp. 675-691, is based on a traditional method for measuring nonlinear distortion. The excitation signal is a two-tone signal generating sparse distortion components which can be identified as harmonic, summed-tone or difference tone components of a certain order. This method is time consuming and can not be extended to a multi-tone stimulus because the distortion components interfere if the number of fundamental tones is high. In order to estimate the nonlinear parameters with an audio-like signal (e.g., music), adaptive methods have been disclosed in DE 4332804A1 or W. Klippel, “Adaptive Nonlinear Control of Loudspeaker Systems,”  J Audio Eng. Society  46, pp. 939-954 (1998). 
         [0005]    Patents DE 4334040, WO 97/25833, US 2003/0118193, U.S. Pat. No. 6,269,318 and U.S. Pat. No. 5,523,715 disclose control systems based on the measurement of current and voltage at the loudspeaker terminals while dispensing with an additional acoustical or mechanical sensor. 
         [0006]    Other identification methods, such as those disclosed in U.S. Pat. No. 4,196,418, U.S. Pat. No. 4,862,160, U.S. Pat. No. 5,539,482, EP1466289, U.S. Pat. No. 5,268,834, U.S. Pat. No. 5,266,875, U.S. Pat. No. 4,291,277, EP1423664, U.S. Pat. No. 6,611,823, WO 02/02974, WO 02/095650, provide only optimal estimates for the model parameters if the model describes the behavior of the transducer completely. However, there are always differences between the theoretical model and the real transducer which causes significant errors in the estimated nonlinear parameters (bias). This shall be described in the following section in greater detail: 
         [0007]    The output signal y(t) of the transducer: 
         [0000]        y ( t )= y   nlin ( t )+ y   lin ( t )  (1) 
         [0000]    consists of a nonlinear signal part: 
         [0000]        y   nlin ( t )= P   sn   G   n ( t )  (2) 
         [0000]    and a linear signal part: 
         [0000]        y   lin ( t )= P   sl   G   l ( t )+ e   r ( t ).  (3) 
         [0008]    The linear signal part y lin (t) comprises a scalar product P sl G l (t) of a linear parameter vector P sl , a gradient vector G l (t) and a residual signal e r (t) due to measurement noise and imperfections of the model. 
         [0009]    The nonlinear signal part y nlin (t) can be interpreted as nonlinear distortion and can be described as a scalar product of the parameter vector: 
         [0000]      P sn =[p s,1  p s,2  . . . p s,N ]  (4) 
         [0000]    and the gradient vector: 
         [0000]        G   n   T ( t )=[ g   1 ( t ) g   2 ( t ) . . .  g   N ( t )]  (5) 
         [0000]    which may contain, for example: 
         [0000]        G   n   T ( t )=[ i ( t ) x ( t ) 2   i ( t ) x ( t ) 4   i ( t ) x   6 ]  (6) 
         [0000]    products of input signal x(t) and the input current: 
         [0000]        i ( t )= h   i ( t )* x ( t ).  (7) 
         [0000]    The model generates an output signal: 
         [0000]        y ′( t )= P   n   G   n ( t )+ P   l   G   l ( t ),  (8) 
         [0000]    which comprises scalar products of the nonlinear parameter vector: 
         [0000]      P n =[P n,1  P n,2  . . . P n,N ]  (9) 
         [0000]    and the linear parameter vector: 
         [0000]      P l =[P l,1  P l,2  . . . P l,L ]  (10) 
         [0000]    with the corresponding linear and nonlinear gradient vector G n (t) and G l (t), respectively. It is the target of the optimal system identification that the parameters of the model coincide with the true parameters of the transducer (P n →P s,n , P l →P s,l ). 
         [0010]    A suitable criterion for the agreement between model and reality is the error time signal: 
         [0000]        e ( t )= y ( t )− y ′( t ),  (11) 
         [0000]    which can be represented as a sum: 
         [0000]        e ( t )= e   n ( t )+ e   l ( t )+ e   r ( t )  (12) 
         [0000]    comprising a nonlinear error part: 
         [0000]        e   n ( t )=Δ P   n   G   n =( P   sn   −P   n ) G   n ,  (13) 
         [0000]    a linear error part: 
         [0000]        e   l ( t )=Δ P   l   G   l ( P   sl   −P   l ) G   l   (14) 
         [0000]    and the residual signal e r (t). 
         [0011]    System identification techniques known in the prior art determine the linear and nonlinear parameters of the model by minimizing the total error e(t) in a cost function: 
         [0000]        C=E{e ( t ) 2 }→Minimum.  (15) 
         [0012]    The linear parameters P l  are estimated by inserting Eqs. (1) and (8) into Eq. (11), multiplying with the transposed gradient vector G l   T (t) and calculating the expectation value 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0013]    Considering that the residual error e r (t) is not correlated with the linear gradient signals in G l (t), this results in the Wiener-Hopf-equation: 
         [0000]        P   l   E{G   l ( t ) G   l   T ( t )}= E{y   lin ( t ) G   l   T ( t )}− E{e   n ( t ) G   l   T ( t )} P   l   S   GGl   =S   yGl   +S   rGl   (17) 
         [0000]    which can be solved directly by multiplying this equation with the inverted matrix S GGl : 
         [0000]        P   l =( S   yGl   +S   rGl ) S   GGl   −1   P   l   =P   sl   +ΔP   l   (18) 
         [0000]    or determined iteratively by using the LMS algorithm: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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         [0000]    with parameter μ changing the speed of convergence. The linear parameters P l  are estimated with a systematic bias ΔP l  if there is a correlation between the nonlinear error e n (t) and the linear gradient vector G l (t). 
         [0014]    Minimizing the total error in the cost function in Eq. (15) may also cause a systematic bias in the estimation of the nonlinear parameters P n . Inserting Eq. (1) and (8) into Eq. (11) and multiplying with the transposed gradient vector G n   T (t) results in the Wiener-Hopf-equation for the nonlinear parameters: 
         [0000]        P   n   E{G   n ( t ) G   n   T ( t )}= E{y   nlin ( t ) G   n   T ( t )}− E{[e   l ( t )+ e   r ( t )] G   n   T ( t )},  P   n   S   GGn   =S   yGn   =S   yGn   +S   rGn   (20) 
         [0000]    where S GGn  is the autocorrelation of the gradient signals, S yGn  is the cross-correlation between the gradients and the signal y nlin (t) and S rgn  is the cross-correlation of the residual error e r  with the gradient signals. The nonlinear parameters of the model can directly be calculated by inverting the matrix S GGn : 
         [0000]        P   n =( S   yGn   +S   rGn ) S   GGn   −1   P   n   =P   sn   +ΔP   n   (20) 
         [0000]    or iteratively by using the LMS-algorithm: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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         [0015]    These techniques known in prior art generate a systematic deviation ΔP n  from the true parameter values if either the linear error e l  or the residual error e r  correlates with the nonlinear gradient G n : 
         [0000]        E{G   n ( t )[ e   l ( t )+ e   r ( t )]}≠0  (23) 
         [0016]    The bias ΔP n  in the estimation of P n  is significant (&gt;50%) if the nonlinear distortion y nlin  is small in comparison to the residual signal e r (t), which is mainly caused by imperfections in the linear modeling. 
         [0017]    To cope with this problem, the prior art increases the complexity of the linear model (e.g. the number of taps in an FIR-filter) to describe the real impulse response h m (t) more completely. This demand can not be realized in many practical applications. For example, the suspension in a loudspeaker has a visco-elastic behavior which can hardly be modeled by a linear filter of reasonable order. The eddy currents induced in the pole plate of a loudspeaker also generate a high complexity of the electrical input impedance. In addition, loudspeakers also behave as time varying systems where aging and changing ambient conditions (temperature, humidity) cause a mismatch between reality and model which increases the residual error signal e r (t). 
       OBJECTS OF THE INVENTION 
       [0018]    There is thus a need for an identification system which estimates the nonlinear parameters P n  and the linear parameters P l  of the model without a systematic error (bias) if the measured signals are disturbed by noise or there are imperfections in the modeling of the transducer. The free parameters of the model should be identified by exciting the transducer with a normal audio signal (e.g. music), a synthetic test signal (e.g. noise) or a control signal as used in active noise cancellation having sufficient amplitude and bandwidth to provide persistent excitation. The transferred signal shall not or only minimally be changed by the identification system to avoid any degradation of the subjectively perceived sound quality. A further object is to realize an identification system for transducers comprising a minimum of elements and requiring minimal processing capacity in a digital signal processor (DSP) to keep the cost of the system low. 
       SUMMARY OF THE INVENTION 
       [0019]    According to the invention, the nonlinear parameters P n  are estimated by minimizing the cost function: 
         [0000]        C   n   =E{e   n ( t ) 2 }→Minimum  (24) 
         [0000]    which considers the nonlinear error part e n  only. In this case, the correlation: 
         [0000]        E{G   n ( t ) e   n ( t )}=0  (25) 
         [0000]    between nonlinear error part e n (t) and nonlinear gradient signal G n (t) vanishes. Thus, a systematic error (bias) in the estimated value of the nonlinear parameter P n  can be avoided. 
         [0020]    The nonlinear cost function C n  is not suitable for an error-free estimation of the linear parameters P l . Using different cost functions for the estimation of the linear and nonlinear parameters is a feature of the current invention not found in prior art. This requirement can be realized theoretically by splitting the total error e(t) into error components according to Eq. (12) and using only the nonlinear error part e n (t) for the estimation of the nonlinear parameters P n . However, the practical realization is difficult and it is more advantageous to apply an appropriate transformation T g  to the gradient signal G n (t) and to generate a modified gradient signal: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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         [0000]    and/or to transform the error signal e(t) by an appropriate transformation T e  into a modified error signal: 
         [0000]        e′   j ( t )= T   e,j   {e ( t )}= e′   n,j ( t )+ e′   res,j ( t ), j=1, . . . , N.  (27) 
         [0021]    The transformations T g  and T e  have to be chosen to ensure that the correlation: 
         [0000]        E{g′   j ( t ) e′   res,j ( t )}=0, j=1, . . . , N  (28) 
         [0000]    between the transformed residual error e′ res,j (t) and the transformed gradient signals g′ j (t) will vanish, and a positive correlation: 
         [0000]        E{g   j ( t ) g′   j ( t ) T }&gt;0, j=1, . . . , N  (29) 
         [0000]    between original and transformed gradient signals and a positive correlation: 
         [0000]        E{e   n ( t ) e′   n,j ( t )}&gt;0, j=1, . . . , N  (30) 
         [0000]    between the original and transformed error is maintained. 
         [0022]    The transformations T g  and T e  suppress the linear signal parts y nlin  primarily, but preserve most of the information of the nonlinear signal part y nlin  required for the estimation of the nonlinear parameters. 
         [0023]    Using the transformed gradient signal g′(t) and the transformed error signal e′(t) in the LMS algorithm: 
         [0000]        P   n,j ( t )= p   n,j ( t− 1)+μ g′   j ( t ) e′   j ( t ), j=1, . . . , N P n →P sn   (31) 
         [0000]    results in an error-free estimation of the nonlinear parameters (P n =P sn ). Suitable transformations can be realized by different methods: 
         [0024]    The first method developed here is a new decorrelation technique which has the benefit that a modification of the input signal x(t) is not required. A signal with arbitrary temporal and spectral properties ensuring persistent excitation of the transducer is supplied to the transducer input  7 . Although the decorrelation technique can be applied to the output signal y(t), it is beneficial to calculate the decorrelated error signal: 
         [0000]        e′   j ( t )= T   e,j   {e ( t )}= e ( t )+ C   j B j , j=1, . . . , N  (32) 
         [0000]    which is the sum of the original error signal and the jth compensation vector: 
         [0000]      B j   T =[b j,l  b j,k  . . . b j,K ],  (33) 
         [0000]    weighted by the jth decorrelation parameter vector: 
         [0000]      C j =[c j,l  c j,k  . . . c j,K ].  (34) 
         [0025]    All compensation vectors B j  with j=1, . . . , N comprise only decorrelation signals b j,i  with i=1, . . . , K, which have a linear relationship with the input signal x(t). Those decorrelation signals b j,i  have to be derived from the transducer model and correspond with the gradient signals g′ j . The expectation value: 
         [0000]      E{e′ j g j }=ΣΠ{η k η l }  (35) 
         [0000]    which is the product of the error signal e′ j  and the gradient signal g j  can be decomposed into a sum of products in which each product comprises only expectation values of two basic signals κ k  and κ l  (as described, for example, in “Average of the Product of Gaussian Variables,” in M. Schertzen, “The Volterra and Wiener Theories of Nonlinear Systems”, Robert E. Krieger Publishing Company, Malabar, Fla., 1989.) 
         [0026]    Applying Eq. (35) to the first gradient signal g l (t)=ix 2  presented as an example in Eq. (6) results in: 
         [0000]        E{ix   2   e′   l ( t )}=2 E{ix}E{xe   l ′( t )}+ E{ie   l ′( t )} E{x   2 }.  (36) 
         [0027]    The correlation between the nonlinear gradient signal ix 2  and an arbitrary (linear) error part e′ res,l (t) in e′ l (t) vanishes if the following conditions: 
         [0000]        E{xe   l ′( t )}=0 E{ie   l ′( t )}=0  (37) 
         [0000]    hold. 
         [0028]    The transformation T e,j  of the error signals e(t) has to remove the correlation between e′ l (t) and x and the correlation between e′ l (t) and i as well. The compensation vector: 
         [0000]      B l   T =[xi]  (38) 
         [0000]    for j=1 comprises only displacement x(t) and current i(t) which are weighted by C j  and added to the original error signal e(t) as decorrelation signals according to Eq. (32). The optimal decorrelation parameter C j  can be determined adaptively by the following iterative relationship: 
         [0000]        C   j   T ( t )= C   j   T ( t− 1)+μ B   j   e′   j ( t ), j=1, . . . , N.  (39) 
         [0029]    Using the additive decorrelation method the transformed gradient signal G′(t)=T g {G n }=G n  is equal to the gradient signal G n . The nonlinear information in the error part e n  which is required for the estimation of the nonlinear parameters P n  is preserved in the transformed error signal e′ j (t). If the error signal e′ j (t) contains the nonlinear gradient signal e′ n,l (t)=ix 2 , the expectation value: 
         [0000]        E{ix   2   ix   2 }=2 E{ii}E{xx}   2 +4 E{ix}   2   E{xx}≠ 0,  (40) 
         [0000]    will not vanish and the condition in Eq. (29) is fulfilled. 
         [0030]    The transformed error signal e′ i (t) can not be used for the estimation of the linear parameters P l ; the original error signal e(t) according to Eq. (19) should be used instead. 
         [0031]    An alternative transformation which fulfills the requirements of Eqs. (28)-(30) can be realized by performing a filtering: 
         [0000]        x ( t )= h   g ( t )* u ( t )  (41) 
         [0000]    of the excitation signal with the filter function: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                   42 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where FT{ } is Fourier transformation and the function δ(f) is defined as: 
         [0000]    
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     
                       ( 
                       f 
                       ) 
                     
                   
                   = 
                   
                     
                       { 
                       
                         
                           
                             
                               1 
                               , 
                               
                                 
                                   for 
                                    
                                   
                                       
                                   
                                    
                                   f 
                                 
                                 = 
                                 0 
                               
                             
                           
                         
                         
                           
                             
                               0 
                               , 
                               
                                 
                                   for 
                                    
                                   
                                       
                                   
                                    
                                   f 
                                 
                                 ≠ 
                                 0 
                               
                             
                           
                         
                       
                       } 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
     
         [0032]    A few selected spectral components at frequencies f i  with i=1, . . . I do not pass the filter, but the remaining signal components are transferred without attenuation. 
         [0033]    A second filter with a transfer function: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       H 
                       a 
                     
                      
                     
                       ( 
                       f 
                       ) 
                     
                   
                   = 
                   
                     
                       FT 
                        
                       
                         { 
                         
                           
                             h 
                             a 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                         } 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         I 
                       
                        
                       
                           
                       
                        
                       
                         δ 
                          
                         
                           ( 
                           
                             f 
                             - 
                             
                               f 
                               i 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
         [0000]    is used for the transformation T e  of the error signal: 
         [0000]        e′   j ( t )= h   a ( t )* e   j ( t ), j=1, . . . , N.  (45) 
         [0034]    Since the filter H a (f) lets pass only spectral components which are not in the input signal x(t), the transformed error signal e′ j (t) will not be correlated with the linear error signal e′ res (t) fulfilling the first condition in Eq. (28). However, the error signal e′ j (t) contains sufficient nonlinear spectral components from e n,j , to ensure a correlation between both error signals according to the second condition in Eq. (30). 
         [0035]    The LMS algorithm applied to the filtered error signal: 
         [0000]        p   n,j ( t )= p   n,j ( t− 1)+μ g   j ( t ) e′   j ( t ), j=1, . . . , N P n →P sn   (46) 
         [0000]    results in an error-free estimation of the nonlinear parameters as long as measurement noise is not correlated with the gradient signal G n (t). If the error signal is filtered, the transformed gradient signal G′(t)=T g {G n }=G n  is identical with original gradient signal. 
         [0036]    A third alternative to realize the conditions in Eq. (28)-(30) is the filtering of the gradient signals: 
         [0000]        G ′( t )= h   a ( t )* G   n ( t )  (47) 
         [0000]    by using the filter function H a (f) defined in Eq. (44) while filtering the input signal with the filter function H g (f) according Eq. (42). 
         [0037]    This transformation ensures that the filtered gradient signal g′ j (t) is neither correlated with the linear error e r (t) nor with the residual error eat): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               E 
                                
                               
                                 { 
                                 
                                   
                                     
                                       g 
                                       j 
                                       ′ 
                                     
                                      
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       e 
                                       r 
                                     
                                      
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                 
                                 } 
                               
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             
                               E 
                                
                               
                                 { 
                                 
                                   
                                     
                                       g 
                                       j 
                                       ′ 
                                     
                                      
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       e 
                                       l 
                                     
                                      
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                 
                                 } 
                               
                             
                             = 
                             0 
                           
                         
                       
                     
                     } 
                   
                   , 
                   
                     j 
                     = 
                     1 
                   
                   , 
                   
                     … 
                      
                     
                         
                     
                      
                     
                       N 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
         [0038]    Assuming that the measurement noise is not correlated with g′ j (t), the nonlinear parameters: 
         [0000]        p   n,j ( t )= p   n,j ( t− 1)+μ g′   j ( t ) e   j ( t ), j=1, . . . , N P n →P sn   (49) 
         [0000]    can be estimated without bias using the LSM algorithm. 
         [0039]    The total number of frequencies I and the values f i  with i=1, . . . , I have to be selected in such a way to provide persistent excitation of the transducer and to get sufficient information from the nonlinear system. If the number I of frequencies is too large, the filtering of the input signal impairs the quality of transferred audio signal (music, speech). 
         [0040]    The number I of the frequencies f i  can be significantly reduced (e.g., I=1) if the values of the frequencies are not constant but rather vary with a function f i =f(t) of time. This extends the learning time, but causes only minimal changes in the transferred audio signal. When a relatively small number of frequencies I is used, it is not possible to identify the order and contribution of each distortion component. This is a difference with respect to traditional methods used in prior art for distortion measurements and nonlinear system identification. 
         [0041]    If the nonlinear parameters P n  have been estimated without bias and the nonlinear error e n (t) disappears, the linear parameters P l  can be estimated without bias by minimizing the cost function C in Eq. (15). 
         [0042]    The current invention has the benefit that the linear and nonlinear parameters can be determined without a systematic error (bias), even if the modeling of the linear properties of the transducer is not perfect. This reduces the effort in modeling complicated mechanisms (e.g., creep of the suspension) and makes it possible to use models with lower complexity and a minimal number of free parameters. This is beneficial for speeding up the identification process, improving the robustness and reducing implementation cost. 
         [0043]    These and other features, aspects, and advantages of the present invention will become better understood with reference to the following drawings, description, and claims. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0044]      FIG. 1  is a general block diagram showing a parameter identification system for a measurement, diagnostics and control application known in the prior art. 
           [0045]      FIG. 2  is a general block diagram showing a parameter identification system for a measurement, diagnostics and control application in accordance with the present invention. 
           [0046]      FIG. 3  shows a first embodiment of a transformation system using an additive decorrelation technique. 
           [0047]      FIG. 4  shows an alternative embodiment of the present invention which uses a filter technique. 
           [0048]      FIG. 5  shows an embodiment of a transformation system using an error filter in accordance with the present invention. 
           [0049]      FIG. 6  shows an embodiment of a transformation system using a gradient filter in accordance with the present invention. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0050]      FIG. 1  is a general block diagram showing a parameter identification system for measurement, diagnostics and control application in the prior art. The real transducer system  1  consists of a loudspeaker  3  (actuator) converting an electrical input signal x(t) (e.g., voltage at the terminals) at input  7  into a acoustical signal and a microphone  5  (sensor) converting an acoustical signal into an electrical signal y(t) at output  9  which is supplied to the non-inverting input of an amplifier  51 . The transfer behavior of transducer system  1  is represented by Eq. (1). The input signal x(t) is also supplied via input  13  to a model  11 . The model  11  describes the linear and nonlinear transfer behavior of transducer system  1  and generates an output signal y′(t) at output  15  which is supplied to the inverting input of the amplifier  51 . The nonlinear Eq. (8) describes the transfer behavior of the model  11 , whereas the product of the linear parameter P l  and linear gradient vector G l (t) is realized by using an FIR-filter. The nonlinear term P n G n (t) is realized by using linear filter, multiplier, adder and scaling elements according to Eqs. (6), (7) and (9). The error signal e(t) generated at output  49  of the amplifier  51  according Eq. (11) is supplied to input  35  of the nonlinear parameter estimator  23  and to input  31  of a linear parameter estimator  21 . According to prior art designs, both parameter estimators  21  and  23  minimize the error signal e(t) using the same cost function given in Eq. (15). The linear parameter estimator  21  is provided with the gradient vector G l  which is generated in a linear gradient system  47  by using delay units and supplied from output  37  to the input  29 . Similarly the nonlinear gradient system  41  generates the nonlinear gradient vector G n  according to Eq. (6) which is supplied via output  45  to the input  33  of the nonlinear parameter estimator  23 . Both parameter estimators  21  and  23  use the LMS algorithm as described in Eqs. (19) and (22). Both the linear and the nonlinear gradient system  47  and  41  are supplied with the input signal x(t) via inputs  39  and  43 , respectively. The linear parameter vector P n  is generated at output  25  of the linear parameter estimator  21  and supplied to the input  19  of the model  11 . The nonlinear parameter vector P n  is generated at output  27  of the nonlinear parameter estimator  23  and is supplied via input  17  to the model  11 . The linear and nonlinear parameter vectors P l  and P a  are also supplied to the diagnostic system  53  and to a controller  58 , which is supplied with the control input z(t) and generates the input signal x(t) which is supplied to the transducer input  7 . The controller  58  performs a protection and linearization function for the transducer system  1 . If the model  11  describes the linear properties of the transducer system  1  incompletely, the minimization of the cost function in Eq. (15) causes a systematic error (bias) in the estimation of the nonlinear parameter P n  as shown in Eqs. (21) and (22). 
         [0051]      FIG. 2  is a block diagram showing a parameter identification system in accordance with the invention, which avoids the bias in the estimation of the nonlinear parameters. The transducer system  1  comprising loudspeaker  3  and microphone  5 , model  11 , the linear and nonlinear parameter estimators  21  and  23 , respectively, the controller  58  and the diagnostic system  53  are identical with the corresponding elements shown in  FIG. 1 . The main difference to the prior art is that a transformation system  55  generates a modified error signal e′(t) and/or a modified nonlinear gradient signal G′ which is supplied via outputs  67  and  69  to the inputs  33  and  35 , respectively, of the nonlinear parameter estimator  23 . The total error signal e(t) is transformed according T e  in Eq. (27) into the modified error signal e′(t). The nonlinear gradient vector G n  is transformed according to T g  in Eq. (26) into the gradient vector G′. The special cost function C n  in Eq. (24) is used for estimating the nonlinear parameters P n  and the cost function C in Eq. (15) is used for the estimation of the linear parameters P l . Using two different cost functions is a typical characteristic of the current invention. The transformation system  55  is supplied with the output signal y(t) from output  9  of the transducer system  1  via input  57  and with the output signal y′(t) from output  15  of the model  11  via input  59 . The input signal x(t) from input  7  of the transducer system  1  is also supplied to the input  61  of the transformation system  55 . 
         [0052]      FIG. 3  shows a first embodiment of the transformation system  55  using an additive decorrelation technique. The transformation system  55  comprises a linear gradient system  71 , a nonlinear gradient system  87 , an amplifier  85  and a decorrelation system  94 . The linear and nonlinear gradient systems  71  and  87  correspond with the gradient systems  47  and  41  in  FIG. 1 , respectively. The input signal x(t) at input  61  is supplied to both the input of the linear gradient system  71  and the input of the nonlinear gradient system  87 . The output of the linear gradient system  71  is connected to an output  63  of the transformation system  55  at which the gradient vector G l (t) is generated. The gradient vector G′(t)=G n (t) at the output of the nonlinear gradient system  87  is supplied to an output  69  of the transformation system  55 . The linear gradient system  71  can be realized as a FIR-filter and the nonlinear gradient system  87  can be realized by using linear filters, multipliers, adders and scaling elements according Eq. (6). The transformation system  55  also includes an amplifier  85  similar to the amplifier  51  in  FIG. 1 . The modeled output signal y′(t) and the measured signal y(t) at inputs  59  and  57  of the transformation system  55  are supplied to the inverting and non-inverting inputs of the amplifier  85 , respectively. The error signal e(t) is generated at the output  101  of the amplifier  85  according to Eq. (11) and supplied to an output  65  of the transformation system  55  and to an error input  92  of the decorrelation system  94 . The decorrelation system  94  comprises a synthesis system  81 , weighting elements  79  and  99 , adders  83  and  75 , a multiplier  77  and a storage element  73 . The error signal e(t) at input  92  is supplied to the scalar input of the adder  83 . The adder  83  also has a vector input  84  provided with input signal C j B j  and a vector output  86  providing the output signal e j ′(t) with j=1, . . . , N according Eq. (32) to an output  67  of the transformation system  55 . The signals in the jth compensation vector B j  with j=1, . . . , N are generated by using the synthesis system  81 . The input  89  of synthesis system  81  is connected to input  61  of the transformation system  55 . The synthesis system  81  contains linear filters which may be realized by digital signal processing. For each nonlinear gradient signal g j (t), a set of decorrelation signals b j,i  in vector B j  is found by splitting the expectation value E{g j (t)e j (t)} according to Eq. (36) in a sum of products. The output  91  of the synthesis system  81  is connected to the input  95  of weighting element  79 . The weighting element  79  also has a vector input  93  provided with the decorrelation parameters C j  and an output  97  generating the weighted compensation signal C j B j  supplied to input  84  of the adder  83 . The optimal decorrelation parameters C j  are generated adaptively according to Eq. (37). The transformed error signal e′ j (t) is supplied to a first input  96  of multiplier  77 , and the compensation vector B j  is supplied to the second (vector) input  98  of multiplier  77 . The output signal B j e′ j (t) of the multiplier  77  is supplied to an input of the weighting element  99  and is weighted by the learning constant μ. The output signal μB j e′ j (t) is added to the decorrelation parameter vector C j  stored in the storage element  73  by using adder  75 , and the sum is supplied to a control input  93  of the weighting element  79 . 
         [0053]      FIG. 4  shows a further embodiment of the invention using a filter  121  which changes the spectral properties of the signal supplied to the transducer system  1 . The filter  121  has an input  119  supplied with the input signal u(t) and an output  123  connected via controller  58  to the input  7  of the transducer system  1 . The filter  121  has a linear transfer function according to Eq. (42). A few spectral components at frequencies f i =1, . . . I are suppressed while all the other components pass through filter  121  without attenuation. The filter  121  can be realized by using multiple filters with a band-stop characteristic which are connected in series between filter input  119  and filter output  123 . Alternatively, the filter  121  may be realized in a DSP by performing a complex multiplication of the filter response H g (f) with the input signal transformed into the frequency domain. The filter  121  may be equipped with an additional control input  117  connected with the output of a frequency control system  115  to vary the frequencies f i  during parameter identification. The frequency control system  115  can be realized as a simple oscillator generating a low frequency signal varying the frequency f, of the band-stop filter for I=1. The output of the frequency control system  115  is also supplied to a control input  56  of the transformation system  55 . 
         [0054]      FIG. 5  shows an embodiment of the transformation system  55  which performs filtering of the error signal. The transformation system  55  contains a linear gradient system  71  and a nonlinear gradient system  87 , having inputs connected with input  61  and having outputs providing the linear gradient vector G l  and the nonlinear gradient vector G′=G n  to the outputs  63  and  69 , respectively—similar to  FIG. 3 . The transformation system  55  in  FIG. 5  also contains an amplifier  85  having inverting and non-inverting inputs connected to inputs  57  and  59 , respectively. The total error signal e(t) at output  101  is connected in the same way as in  FIG. 3  to output  65 . An additional filter  105  having a signal input  104  supplied with the output  101  of the amplifier  85  and having a filter output  103  generating the transformed error signal e′(t) has a linear transfer response according to Eq. (44) which can be changed by the control signal provided via an input  106  from transformation system input  56 . The filter  105  may be realized in the frequency domain in a similar way as filter  121 . 
         [0055]      FIG. 6  shows an embodiment of the transformation system  55  which performs filtering of the gradient signals. The linear gradient system  71 , the nonlinear gradient system  87  and the amplifier  85  are connected in the same way as described in  FIG. 5 . The total error signal e(t) supplied via output  101  to output  65  is identical with the transformed error signal e′(t) at vector output  67 . The main difference to the previous embodiments in  FIGS. 3 and 4  is a filter  109  having a vector input  107  provided with nonlinear gradient vector G n  from the output of the nonlinear gradient systems  87 . The filter  109  has a transfer function according to Eq. (44) which can be realized by a complex multiplication in the frequency domain similar to the realization of filter  121 . However, the filtering of the gradient signals generates a higher computational load than the filtering of the error signal. The transformed gradient vector G′ at the vector output of filter  109  is supplied to output  69  of the transformation system  55 . The transfer behavior of filter  109  may be varied by a control signal which is provided via transformation system input  56  to an input  113  of filter  109 . 
         [0056]    The embodiments of the invention described herein are exemplary and numerous modifications, variations and rearrangements can be readily envisioned to achieve substantially equivalent results, all of which are intended to be embraced within the spirit and scope of the invention as defined in the appended claims.