Abstract:
A method for compensating non-linearities of a read signal generated by a variable-capacitance inertial sensor including a first fixed electrode and a second fixed electrode and a mobile electrode, which is spatially arranged between the first and second fixed electrodes and is capacitively coupled to the first and second fixed electrodes, said method comprising the steps of: acquiring the read signal; identifying a first linear component and at least one first nonlinear component of the read signal; a generating a compensated output signal by subtracting the first nonlinear component from the read signal.

Description:
BACKGROUND 
       [0001]    1. Technical Field 
         [0002]    The present disclosure relates to a method and system for compensating systematic non-linearities in a signal supplied by a capacitive inertial sensor, in particular an inertial micro-electromechanical (MEMS) sensor, such as for example an accelerometer. 
         [0003]    2. Description of the Related Art 
         [0004]    Known in the prior art are inertial-measurement units or systems typically comprising an acceleration sensor (accelerometer) having one or more (e.g., three) sensing axes X, Y, Z, designed to measure movements (accelerations) to which the accelerometer is subject during use with respect to the Earth&#39;s reference system. Other inertial measurement systems, such as gyroscopes, are available in the prior art. 
         [0005]    MEMS technology has favored miniaturization of accelerometers. Schematically and by way of example, an inertial sensor of a known type includes one or more fixed parts (also referred to as fixed masses, or stators) and a mobile mass (rotor). The rotor is capacitively coupled to the stators so that it forms one or more capacitors with each stator. In other words, the stators and the rotor form the respective plates of one or more capacitors. The signal of variation of capacitance of said capacitors indicates a displacement of the rotor with respect to the stator and generates the output signal of the inertial sensor, which indicates the acceleration to which the rotor is subject during use of the accelerometer. 
         [0006]    At the end of the manufacturing steps, the inertial sensor is calibrated so for making up, at least in part, for systematic errors generated by the manufacturing process. In particular, a factor that affects the output signal of the inertial sensor is the misalignment of the mobile mass (rotor) with respect to the ideal position that it should occupy with respect to the fixed parts (stators). In particular, according to a known embodiment, stator electrodes (e.g., two stator electrodes) constitute as many plates of respective capacitors, whereas a rotor electrode constitutes a common plate of said capacitors; in this case, the rotor electrode is spatially arranged between the stator electrodes. 
         [0007]    In ideal manufacturing conditions, the plate that forms the rotor electrode is spaced at equal distances apart from the plates that form the stator electrodes so that the respective capacitors show, in conditions of rest, a same value of capacitance. However, in real cases, there may exist an undesirable misalignment on account of which said capacitors show, in conditions of rest, a different value of capacitance. When the inertial sensor operates as differential capacitive sensor, the output signal is given by the difference of variation of capacitance of the two capacitors formed by the stator electrodes with the rotor electrode. It is evident that, in the case of the aforementioned manufacturing errors, an undesirable misalignment of said electrodes causes a nonzero output signal also in conditions of rest and further introduces a deterioration of the performance of nonlinearity of the output signal. In particular, in the presence of a marked initial misalignment of the position of the rotor towards the stators, for high values of acceleration there is a markedly nonlinear behavior of the output signal (of a parabolic type). Said behavior is undesirable in the majority of applications in which accelerometers are used. 
       BRIEF SUMMARY 
       [0008]    Some embodiments of the present disclosure are a method and a system for compensating non-linearities in a signal supplied by a capacitive inertial sensor that will be able to overcome the drawbacks of the known art. 
     
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
         [0009]    According to the present disclosure, a method and a system are provided for compensating non-linearities in a signal supplied by a capacitive inertial sensor. 
           [0010]    For a better understanding of the present disclosure, preferred embodiments thereof are now described, purely by way of non-limiting example and with reference to the attached drawings, wherein: 
           [0011]      FIG. 1  shows, in schematic form, a sensor module of an inertial sensor of a known type including electrodes of a fixed mass and an electrode of a mobile mass; 
           [0012]      FIG. 2  shows a stage for reading a differential capacitive signal generated by the sensor module of  FIG. 1  and including a system for compensating non-linearities according to the present disclosure; 
           [0013]      FIG. 3  shows the variation of a differential-capacitance signal generated by an inertial sensor including one or more sensor modules according to  FIG. 1  as the acceleration to which the mobile-mass electrode is subject during use varies; 
           [0014]      FIG. 4  shows, according to one embodiment of the present disclosure, a digital circuit for compensating non-linearities of a differential capacitive signal generated by the sensor module of  FIG. 1 ; and 
           [0015]      FIGS. 5 and 6  show respective electronic devices including an inertial sensor provided with a sensor module of the type illustrated in  FIG. 1  and a digital circuit for compensating non-linearities of the type illustrated in  FIG. 4 . 
       
    
    
     DETAILED DESCRIPTION 
       [0016]    An inertial sensor, for example an accelerometer, is a micro-electromechanical structure comprising one or more sensor modules of the type illustrated by way of example in  FIG. 1 . With reference to  FIG. 1 , the sensor module comprises at least one mobile mass (also referred to as “rotor”)  2 , and a fixed structure (also referred to as “stator”)  3 . Typically, the mobile mass  2  is mechanically connected to the fixed structure  3  by springs and is mobile with respect to the fixed structure  3  according to pre-set degrees of freedom. The mobile mass  2  is further electrically coupled to the fixed structure  3  via capacitive structures (capacitors C 1  and C 2 ). 
         [0017]    The mobile mass  2  includes an electrode  2   a , and the fixed structure  3  includes a first electrode  3   a  and a second electrode  3   b . The electrode  2   a  is arranged between the electrodes  3   a  and  3   b , respectively, for forming a capacitive structure with planar parallel plates. In this example, the capacitive coupling is of a differential type, obtained by parallel-plate electrodes perpendicular to the sensing direction (here the sensing direction shown is the direction X). The movement in the direction X of the mobile mass  2  with respect to the fixed body  3 , for example on account of an external stress, modifies the capacitance of the capacitors C 1  and C 2 . By detecting the variation of differential capacitance of the capacitors C 1  and C 2  it is possible to trace back to the relative displacement of the mobile mass  2  with respect to the fixed structure  3  and thus to the acceleration to which the inertial sensor, which integrates the mobile mass  2  and the fixed body  3 , is subject during use. Instead, by supplying appropriate biasing voltages, it is possible to apply an electrostatic force to the mobile mass  2  to arrange it in motion, in particular at a certain resonance frequency ω. In this case, the inertial sensor comprises a driving device (not illustrated), which has the task of keeping the mobile mass  2  in oscillation. For instance, in a per se known manner, it is possible to supply, in open loop, periodic stresses at the resonance frequency ω of the mobile mass  2 . Alternatively, it is possible to use feedback driving circuits, based upon the use of sigma-delta modulators. Other solutions are further possible. 
         [0018]      FIG. 2  shows schematically a reading system  1  including a chain for processing the signal supplied by the inertial sensor, in particular for analog-to-digital conversion of said signal and for compensation of non-linearities, according to one embodiment of the present disclosure. For instance, the reading system  1  is integrated in an application-specific integrated circuit (ASIC) (here not illustrated). The reading system  1  comprises a charge amplifier AMP_C  4 , a lowpass filter LPF  6  for filtering possible noise components and for limiting the band of the signal supplied by the inertial sensor, and an analog-to-digital conversion stage ADC  7 , cascaded together. The charge amplifier AMP_C  4  is, for example, of a fully differential switched-capacitor type. The charge amplifier AMP_C  4  has inputs  4   a ,  4   b  connected to the terminals of the mobile mass  2 . According to the operation of the charge amplifier AMP_C  4 , present on its outputs are read voltages indicating displacement of the mobile mass  2 . 
         [0019]    The output of the charge amplifier AMP_C  4  is supplied to the filter LPF  6  and then to the analog-to-digital conversion stage ADC  7 , which makes a conversion of the signal received at input into a digital word, in a known way, for example on a number of bits comprised between 8 and 16. The charge amplifier AMP_C  4 , the filter LPF  6  and the analog-to-digital conversion stage ADC  7  are known and already used in the chain for reading and processing the signal supplied by an inertial sensor, such as an accelerometer. These elements are consequently not described in detail. 
         [0020]    According to one aspect of the present disclosure, the reading stage  1  further comprises a linearization block  10  operatively coupled to the output of the analog-to-digital conversion stage ADC  7 . 
         [0021]    At output from the inertial sensor, in the case provided by way of example of a triaxial inertial sensor, three signals are generated, one for each sensing axis X, Y, Z. In this case, the processing performed by the blocks of  FIG. 2  is executed for each of the signals supplied at output by the inertial sensor, selected by a multiplexer (e.g., with time multiplexing, not illustrated in  FIG. 2 ) present upstream of the charge amplifier AMP_C  4 . In the case of an inertial sensor having just one sensing axis, the multiplexer is not necessary. 
         [0022]    Alternatively, once again in the case of multiaxial (e.g., triaxial) inertial sensor, it is possible to envisage three read systems of the type illustrated in  FIG. 2 , one for each signal generated for a respective axis. 
         [0023]    According to what is illustrated in  FIG. 2 , the linearization block  10  receives at input the signal S int  converted by the analog-to-digital conversion stage  7  (e.g., a digital word) and performs an operation of linearization of said signal to generate at output a signal S out . 
         [0024]    The differential capacitive signal supplied at output from the inertial sensor is affected by non-linearities. This effect is all the more evident, the more the mobile mass  2  of the inertial sensor is subject to misalignments (offsets) along X with respect to the ideal position that it should assume, i.e., evenly spaced apart, along X, from the electrodes of the fixed structure  3 . This may happen on account of imperfections introduced during the manufacturing process. On account of these imperfections, the signal supplied at output from the inertial sensor presents a parabolic shape. 
         [0025]    Reference may be made, for example, to  FIG. 3 , which shows, designated by the reference number  12 , an ideal curve (desired linear plot), and by the reference number  14  a real curve that illustrates a signal at output from the inertial sensor, regarding a sensing axis (e.g., axis X). The reference system of  FIG. 3  shows, on the axis of the abscissae, values of acceleration to which the mobile mass of the inertial sensor is subject (the value 0 means no acceleration), whereas present on the axis of the ordinates are the values, expressed in femtofarads, of variation of differential capacitance between the stator electrodes  3   a ,  3   b  and the rotor electrode  2   a . The scale of the axis of the abscissae is made in units g of acceleration of gravity from −8 g to +8 g. 
         [0026]    In this example, both of the curves are normalized in such a way that corresponding to a zero value of acceleration is a zero value of differential capacitance. 
         [0027]    As may be noted, the real curve  14  is deviates from the ideal curve  12 , in particular for high values (in module) of acceleration, presenting a plot of a parabolic type, in particular between 4 g and 8 g. 
         [0028]    The variation of differential capacitance ΔC between the electrode  2   a  of the mobile mass  2  and the electrodes  3   a ,  3   b  of the fixed structure  3  is given, in a known way, by the following formula (1): 
         [0000]    
       
         
           
             
               
                 
                   
                     Δ 
                      
                     
                         
                     
                      
                     C 
                   
                   = 
                   
                     
                       
                         
                           ɛ 
                           0 
                         
                          
                         
                           NA 
                           i 
                         
                       
                       
                         ( 
                         
                           
                             x 
                             0 
                           
                           - 
                           x 
                         
                         ) 
                       
                     
                     - 
                     
                       
                         
                           ɛ 
                           0 
                         
                          
                         
                           NA 
                           i 
                         
                       
                       
                         ( 
                         
                           
                             x 
                             0 
                           
                           + 
                           x 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where: ∈ 0  is the dielectric constant, or electrical permittivity, of vacuum; A i  is the value in square meters of the area of the stator electrode  3   a  (or electrode  3   b , which are assumed as having identical areas) directly facing the rotor electrode  2   a ; N is the number of plane-plate electrodes belonging to the fixed structure  3  (with reference to  FIG. 1 , N=2); x 0  is the ideal (desired) distance (see  FIG. 1 ), considered along the sensing axis X, between one electrode  3   a ,  3   b  and the electrode  2   a ; and x is the displacement, measured in meters, of the electrode  2   a  with respect to the condition of rest during use. 
         [0029]    When possible manufacturing imperfections are considered whereby the electrode  2   a  of the mobile mass  2  does not occupy an ideal position perfectly symmetrical between two respective electrodes  3   a ,  3   b  of the fixed structure  3 , but is shifted by an amount x offset  approaching one of the two electrodes  3   a ,  3   b  (and moving away from the other between the electrodes  3   a ,  3   b ), then Eq. (1) assumes the following form (2): 
         [0000]    
       
         
           
             
               
                 
                   
                     Δ 
                      
                     
                         
                     
                      
                     C 
                   
                   = 
                   
                     
                       
                         
                           ɛ 
                           0 
                         
                          
                         A 
                       
                       
                         ( 
                         
                           
                             x 
                             0 
                           
                           - 
                           
                             x 
                             offset 
                           
                           - 
                           x 
                         
                         ) 
                       
                     
                     - 
                     
                       
                         
                           ɛ 
                           0 
                         
                          
                         A 
                       
                       
                         ( 
                         
                           
                             x 
                             0 
                           
                           + 
                           
                             x 
                             offset 
                           
                           + 
                           x 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the error x offset  has been introduced, and, for simplicity, A is the numeric value of N·A i  identified in Eq. (1). 
         [0030]    The value x offset  is an error and varies, obviously, on the basis of the manufacturing process. However, once a certain process of production of the inertial sensor is set, it is possible to estimate (for example, by simulation or tests) a mean value of x offset , which is thus known (or estimated) at the end of the manufacturing process. 
         [0031]    Thus, once a value x offset  is fixed, it is possible to calculate numerically the value of ΔC according to Eq. (2). The value of x may for example be the zero value (zero acceleration), or else a value calculated considering a value of acceleration a provided by way of example, according to Eq. (3): 
         [0000]    
       
         
           
             
               
                 
                   x 
                   = 
                   
                     
                       9.81 
                       
                         ω 
                         2 
                       
                     
                      
                     a 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where ω is the resonance frequency (which is known) chosen for the mobile mass, 9.81 is the acceleration of gravity, and a is a value of acceleration to which the inertial sensor is subjected along the sensing axis considered (in this example, X). 
         [0032]    Eq. (2) may be approximated by a polynomial expansion of partial derivatives up to the third order, of the type illustrated in the following Eq. (4): 
         [0000]    
       
         
           
             
               
                 
                   
                     Δ 
                      
                     
                         
                     
                      
                     
                       C 
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           
                             ∂ 
                             Δ 
                           
                            
                           
                               
                           
                            
                           C 
                         
                         
                           ∂ 
                           x 
                         
                       
                        
                       x 
                     
                     + 
                     
                       
                         1 
                         2 
                       
                        
                       
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             Δ 
                           
                            
                           
                               
                           
                            
                           C 
                         
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                        
                       
                         x 
                         2 
                       
                     
                     + 
                     
                       
                         1 
                         6 
                       
                        
                       
                         
                           
                             
                               ∂ 
                               3 
                             
                              
                             Δ 
                           
                            
                           
                               
                           
                            
                           C 
                         
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                        
                       
                         x 
                         3 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0033]    Eq. (4) approximates the continuous and x-differentiatable function ΔC(x) according to Eq. (2), and may be represented generically with a polynomial of an arbitrary degree n (i.e., a degree other than the third degree, for example the second degree, or a degree higher than the third). In particular, Eq. (4) represents a development in Taylor series or, more precisely, a McLaurin development, where the partial derivatives are calculated in a pre-set point x (for example, as has been said, x=0). 
         [0034]    We have that Eq. (4) may be expressed in the following Eq. (5): 
         [0000]      Δ C ( x )=α x+βx   2   +γx   3   (5)
 
         [0000]    where α is the first derivative of ΔC(x); β is the second derivative, divided by the factor 2, of ΔC(x); and γ is the third derivative, divided by the factor 6, of ΔC(x). All the terms α, β, and γ have a value that is a function of the ideal value  x0  and of the value of the error x offset . Eq. (5) represents a third-order equation, where αβ and γ are the coefficients of the equation. 
         [0035]    It is desirable to compensate, or annul, the nonlinear terms of Eq. (5) (i.e., the terms β·x 2  and γ·x 3 ), in such a way as to obtain a value of variation of capacitance ΔC(x) that is a function exclusively of the linear term α·x. The desired variation of capacitance is consequently the following ΔC corr  expressed by Eq. (6): 
         [0000]      Δ C   corr ( x )=α x   (6)
 
         [0000]    wherein we obtain x from Eq. (7): 
         [0000]    
       
         
           
             
               
                 
                   x 
                   = 
                   
                     
                       Δ 
                        
                       
                           
                       
                        
                       
                         
                           C 
                           corr 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                     
                     α 
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0036]    Substituting the expression of x according to Eq. (7) in Eq. (5), the following Eq. (8) is obtained: 
         [0000]    
       
         
           
             
               
                 
                   
                     Δ 
                      
                     
                         
                     
                      
                     
                       
                         C 
                         corr 
                       
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       Δ 
                        
                       
                           
                       
                        
                       
                         C 
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                     
                     - 
                     
                       
                         β 
                         
                           α 
                           2 
                         
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       
                         
                           
                             C 
                             corr 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                         2 
                       
                     
                     - 
                     
                       
                         γ 
                         
                           α 
                           3 
                         
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       
                         
                           
                             C 
                             corr 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                         3 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0037]    From Eq. (8) it may be noted that: 
         [0038]    (i) the values of α, β, and γ may be calculated numerically by computing the partial derivative with respect to x of the formula of ΔC according to Eq. (2), where the value x offset  is estimated or measured experimentally, and the value x is set at a predetermined value, in particular the zero value; the other values of Eq. (2) are known in so far as they are design parameters of the inertial sensor; and
       (ii) the value of ΔC corr  is unknown.       
 
         [0040]    Consequently, since ΔC corr  is unknown, the following simplification of Eq. (8) is made: 
         [0000]      Δ C   corr ( X )=Δ C ( x )− BΔC ( x ) 2   −CΔC ( x ) 3   (9)
 
         [0000]    where ΔC(x) is the variation of instantaneous capacitance of the microstructure formed by the rotor electrode and by the stator electrodes; in other words, ΔC(x) is a differential-voltage signal indicating the displacement of the mobile mass  2  along the corresponding sensing axis (here, X) in the instant considered. 
         [0041]    The values of B and C of Eq. (8) are given by B=β/α 2  and C=γ/α 3 . Since, as has been said, α, β, and γ may be calculated numerically, the values of B and C may be determined. 
         [0042]    A numeric non-limiting example of the present disclosure is now provided for calculation of the coefficients α, β and γ, and thus of B and C. On the basis of what has been set forth previously, we have that the expressions of α, β and γ are expressed by the following Eqs. (10a-10c): 
         [0000]    
       
         
           
             
               
                 
                   
                     α 
                     = 
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     x 
                                     0 
                                   
                                   + 
                                   
                                     x 
                                     offset 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     x 
                                     0 
                                   
                                   - 
                                   
                                     x 
                                     offset 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   
                                     x 
                                     0 
                                   
                                   + 
                                   
                                     x 
                                     offset 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             · 
                             
                               
                                 ( 
                                 
                                   
                                     x 
                                     0 
                                   
                                   - 
                                   
                                     x 
                                     offset 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                         · 
                         
                           ɛ 
                           0 
                         
                       
                        
                       A 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     β 
                     = 
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     x 
                                     0 
                                   
                                   + 
                                   
                                     x 
                                     offset 
                                   
                                 
                                 ) 
                               
                               3 
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     x 
                                     0 
                                   
                                   - 
                                   
                                     x 
                                     offset 
                                   
                                 
                                 ) 
                               
                               3 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   
                                     x 
                                     0 
                                   
                                   + 
                                   
                                     x 
                                     offset 
                                   
                                 
                                 ) 
                               
                               3 
                             
                             · 
                             
                               
                                 ( 
                                 
                                   
                                     x 
                                     0 
                                   
                                   - 
                                   
                                     x 
                                     offset 
                                   
                                 
                                 ) 
                               
                               3 
                             
                           
                         
                         · 
                         
                           ɛ 
                           0 
                         
                       
                        
                       A 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     γ 
                     = 
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     x 
                                     0 
                                   
                                   + 
                                   
                                     x 
                                     offset 
                                   
                                 
                                 ) 
                               
                               4 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     x 
                                     0 
                                   
                                   - 
                                   
                                     x 
                                     offset 
                                   
                                 
                                 ) 
                               
                               4 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   
                                     x 
                                     0 
                                   
                                   + 
                                   
                                     x 
                                     offset 
                                   
                                 
                                 ) 
                               
                               4 
                             
                             · 
                             
                               
                                 ( 
                                 
                                   
                                     x 
                                     0 
                                   
                                   - 
                                   
                                     x 
                                     offset 
                                   
                                 
                                 ) 
                               
                               4 
                             
                           
                         
                         · 
                         
                           ɛ 
                           0 
                         
                       
                        
                       A 
                     
                   
                 
               
               
                 
                   ( 
                   
                     10 
                      
                     a 
                      
                     
                       - 
                     
                      
                     10 
                      
                     c 
                   
                   ) 
                 
               
             
           
         
       
     
         [0043]    Considering that ∈ 0 =8.85·10 −12  and assuming the following values: A=9.6·10 −8  m 2 , x offset =100·10 −9  m,  x0 =2·10 −6  m, and zero acceleration (x=0), we have: α=4.28·10 −7  F/m, β=0.0327 F/m 2 , γ=1.0888·105 F/m 3 . Thus, we obtain the values of B and C, i.e., B=1.7538·10 11  F −1  and C=1.3887·10 24  F −2 . 
         [0044]    According to a further embodiment of the present disclosure, in order to improve the effect of linearization (e.g., of the curve  14  of  FIG. 3 ) it is possible to act also on the parameter x 0 . In fact, even though x 0  is known from the design of the inertial sensor, its effective value may vary on account of process spread. Thus, estimating that the effective value of x 0  varies, for a given manufacturing process, in a range x 0 ±x 0 ′ (with x 0 ′ equal to a fraction of x 0 ), it is possible to calculate the coefficients B and C for a plurality of values of x 0  included in the range x 0 ±x 0 ′ considered. The value of x 0  to be used for calculation of the coefficients B and C will be that value such that the best linearization of the curve considered is obtained (e.g., of the curve  14  of  FIG. 3 ). For this purpose, according to one embodiment of the disclosure, to find the optimal coefficients B and C as x 0  varies, it is possible to use a method based upon the Monte Carlo algorithm. In this way, it is possible to obtain a solution to the problem of linearization of the curve of the signal ΔC. 
         [0045]    Furthermore, according to a further embodiment, the values of B and C may be obtained by tests and simulations, choosing those values that, substituted in Eq. (9), enable a signal ΔC corr  to be obtained that approximates a straight line (e.g., the straight line  12  of  FIG. 3 ). 
         [0046]    To return to  FIG. 2 , we have that the linearization block  10  (illustrated in  FIG. 4  according to one embodiment) digitally implements Eq. (9) in such a way as to process the signal S int  that it receives at input, to generate at output a signal S out  equal to 
         [0000]        S   out   =S   int −( B′·S   int   2   +C′·S   int   3 )  (11)
 
         [0047]    Here, S int  is a signal representing the differential-capacitance signal ΔC(x), represented in digital format. Likewise, also the values of B′ and C′ of Eq. (11) are values correlated to the values of B and C referred to previously, but expressed in digital format so that they may be appropriately processed by the linearization block  10 , which, as has been said, operates on digital signals according to one embodiment of the present disclosure. 
         [0048]      FIG. 4  is a schematic illustration of a possible implementation via logic blocks of the linearization block  10 , which implements the linearization according to Eq. (11). 
         [0049]    In detail, the linearization block  10  includes an input  10   a , which receives the signal S int , of a digital type. The signal S int  is represented on a number of bits defined as required, for example on the basis of the resolution of the ADC converter  7  of  FIG. 2 , for instance comprised between 8 bits and 16 bits (but any other value may be used). In this example, the signal S int  is represented on 14 bits. The signal S int  is supplied simultaneously to a first input  20   a  and to a second input  20   b  of a multiplier  20 ; the latter performs an operation of squaring of the signal S int , supplying on the output  20   c  the signal S int   2 , represented on 28 bits (i.e., on a number of bits twice that of the bits of the signal S int ). 
         [0050]    Furthermore, the signal S int  is supplied to the input  22   a  of a further multiplier  22 ; a second input  22   b  of the multiplier  22  receives the signal S int   2 . The multiplier  22  supplies at output a signal that is the signal S int   2  multiplied by the signal S int , i.e., the signal S int  cubed, S int   3 . The signal S int   3  is represented on a number of bits that is three times the number of bits on which the signal S int  is represented. 
         [0051]    This is followed by multiplication of the signal S int   2  by the coefficient B′ and multiplication of the signal S int   3  by the coefficient C′. 
         [0052]    For this purpose, the signal S int   2  is supplied to an input  24   a  of a multiplier  24 ; the latter receives on a further input  24   b  the coefficient B′ and supplies at output  24   c  a signal that is B′·S int   2  and may be represented on 33 bits. 
         [0053]    According to one embodiment of the present disclosure, the coefficient B′ is a power of 2 (digital word) represented on a number of bits chosen as required. For instance, 20 bits are sufficient for representing in digital format the value of B referred to previously. To be able to modify or update the value of B′, according to one embodiment of the present disclosure, a memory, or register, for example of a Flash type,  26  is present, which is accessible outside the linearization block  10 . The register  26  stores a value, for example on 5 bits, which is to be multiplied by the value of the coefficient B′ for supplying to the multiplier  24  a value of the coefficient B′ that may be updated as required. 
         [0054]    In general, the value of the coefficient B′ may be modified or updated, for example to carry out operations of re-calibration of the inertial sensor. The signal generated by the inertial sensor, in fact, may undergo variations or drift during the operating life of the sensor. To guarantee a linear output signal S out  in each stage of operating life of the sensor, it is possible to vary the value of the coefficient B′ in such a way as to restore the condition of linearity required for the output signal S out . 
         [0055]    By a further multiplier  28  the next step, as has been said, is multiplication of the signal S int   3  by the coefficient C′. For this purpose, the signal S int   3  is supplied to an input  28   a  of the multiplier  28 ; the latter receives the coefficient C′ on a further input  28   b.    
         [0056]    The multiplier  28  supplies at output  28   c  a signal that is C′·S int   3 , here represented on 45 bits. 
         [0057]    The coefficient C′ is also a power of 2 (digital word), for example represented on 35 bits, which are sufficient for representing, in digital format, the coefficient C referred to previously. 
         [0058]    An adder  30  receives at input the signals B′·S int   2  and C′·S int   3 , and supplies at output a signal that is the sum of the inputs, i.e., the signal S SUM =B′·S int   2 +C′·S int   3 . The signal S SUM  is represented on 45 bits, i.e., on the number of bits of the signal C′·S int   3 . 
         [0059]    This is followed by a step of subtraction by a subtractor  32 , to implement the operation of subtraction between the signal S int  at input to the linearization block  10  and the signal S SUM  represented by Eq. (11), mentioned previously. In order to have uniformity of representation in bits, the signal S int  (originally on 14 bits) is represented on 45 bits before being set at input to the subtractor  32 . The signal S int  represented on 45 bits is denoted in  FIG. 4  as S int     —     ex . Thus, the subtractor  32  receives at input both of the signals S int     —     ex  and S SUM , and performs the operation S int     —     ex −S SUM =S int     —     ex −(B′·S int   2 +C′·S int   3 )=S int −(B′·S int   2 +C′·S int   3 ). 
         [0060]    The signal S out     —     ex  at output from the subtractor  32  is again represented on 45 bits. However, following upon the operation of subtraction, the information carried by the signal S out     —     ex  may once again be represented on the same number of bits (14 bits) as the signal S int  at input to the linearization block  10 . This is thus followed by an operation of saturation, via the block  33 , for generating at output from the linearization block  10  an output signal S out  represented on 14 bits or, more in general, on the same number of bits as that with on which the input signal S int  is represented. 
         [0061]    The linearization block  10  of  FIG. 4  uses digital multipliers, adders, and subtractors, and performs the operation of linearization according to Eq. (11) described previously. 
         [0062]    The values of the digital signals described with reference to  FIG. 4  are preferably represented by floating-point or fixed-point numbers. This representation, however, requires a greater capacity and processing complexity than do integers. To reduce the processing complexity it is possible, according to one embodiment, to make a conversion from decimal-point values to integer values. An evaluation should, however, be made on a case-by-case basis to make sure that the loss of precision is negligible. 
         [0063]      FIG. 5  shows a chip, designated as a whole by the reference number  50 , a die that carries an ASIC  60 , and a die that carries an inertial-measurement sensor  70 , for example an accelerometer, which for instance includes one or more sensor modules of the type illustrated in  FIG. 1 . The inertial sensor  70  is operatively coupled to the ASIC  60  to provide an acceleration signal in the form of a differential capacitive signal. The ASIC  60  is provided with a reading stage  1  of the type illustrated in  FIG. 2 , i.e., including the linearization block  10 , according to the present disclosure. To implement the steps of the linearization method described previously, the linearization block  10  includes a logic circuit of the type illustrated with reference to  FIG. 4 , or else a microprocessor configured to implement the operations of the circuit of  FIG. 4 , for example by executing instructions defined by a software program. 
         [0064]    In detail, the accelerometer  70  generates acceleration signals for each sensing axis (e.g., one, two, or three axes), in its own reference system. Each of said acceleration signals is a respective differential-voltage signal ΔC(x), of the type previously illustrated (one for each axis), where each signal S int =ΔC(x) is processed by the linearization block  10  independently of the signals regarding the other measuring axes (in a respective time interval), for example under the control of a multiplexer. 
         [0065]    Illustrated in  FIG. 6  is a portion of an electronic system  100  according to a further embodiment. The system  100  incorporates the chip  50  of  FIG. 6  and may be used in devices, such as, for example, a palmtop computer (personal digital assistant, PDA), laptop computer or portable computer, possibly with wireless capacity, a cellphone, a messaging device, a digital music player, a digital camera or other devices designed to process, store, transmit, or receive information. For instance, the chip  50  may be used in a digital camera for detecting movements and stabilizing an image. In other embodiments, the chip  50  is included in a portable computer, a PDA, or a cellphone for detecting a free-fall condition and activating a safety configuration. In a further embodiment, the chip  50  is included in a motion-activated user interface for computers or consoles for video games. In a further embodiment, the chip  50  is incorporated in a satellite-navigation device and is used for temporary tracking of position in the case of loss of the satellite positioning signal. 
         [0066]    The electronic system  100  may comprise, in addition to the chip  50 , a controller  110 , an input/output (I/O) device  120  (for example, a keyboard or a screen), a wireless interface  140 , and a memory  160 , of a volatile or nonvolatile type, coupled together through a bus  150 . In one embodiment, a battery  180  may be used for supplying the system  100 . It is to be noted that the scope of the present disclosure is not necessarily limited to embodiments having one or all of the devices listed. 
         [0067]    The controller  110  may comprise, for example, one or more microprocessors, microcontrollers, and the like. 
         [0068]    The I/O device  120  may be used for generating a message. The system  100  may use the wireless interface  140  for transmitting and receiving messages to and from a wireless communication network with a radiofrequency (RF) signal. Examples of wireless interface may comprise an antenna, a wireless transceiver, such as a dipole antenna, even though the scope of the present disclosure is not limited from this point of view. Furthermore, the I/O device  120  may supply a voltage representing what is stored either in the form of digital output (if digital information has been stored) or in the form of analog output (if analog information has been stored). 
         [0069]    Finally, it is evident that modifications and variations may be made to the resonant micro-electromechanical system described, without thereby departing from the scope of the present disclosure. 
         [0070]    For instance, the reading stage  1  of  FIG. 2  may further comprise an anti-aliasing filter arranged downstream of the ADC stage  7 . 
         [0071]    Furthermore, the disclosure may advantageously be integrated in the signal-reading stage of capacitive electromechanical oscillators of a type different from what has been described (for example, of a non-differential type). 
         [0072]    Furthermore, it is possible to use one or more clock signals, in particular for driving the mobile mass and for synchronizing the steps described for processing (linearization) of the signal S int  (see  FIG. 4 ). In this connection, it is possible to generate clock signals using just one main clock signal supplied by an asynchronous oscillator calibrated at the driving frequency. 
         [0073]    The advantages of the present disclosure and of the corresponding manufacturing method emerge clearly from the foregoing description. 
         [0074]    In particular, the present disclosure enables execution of an on-chip compensation/linearization of the output signal of the inertial sensor in a fast and inexpensive way, in particular integrating a low-cost hardware/software engine directly within the ASIC. 
         [0075]    The method according to  FIG. 4  does not require high computing capacity and expensive hardware. Furthermore, since this method is implemented in a continuous way, linearization of the output signal is obtained in real time, always guaranteeing good measuring accuracy of the inertial sensor. 
         [0076]    Finally, it is clear that modifications and variations may be made to what has been described and illustrated herein, without thereby departing from the scope of the present disclosure. 
         [0077]    The various embodiments described above can be combined to provide further embodiments. These and other changes can be made to the embodiments in light of the above-detailed description. In general, in the following claims, the terms used should not be construed to limit the claims to the specific embodiments disclosed in the specification and the claims, but should be construed to include all possible embodiments along with the full scope of equivalents to which such claims are entitled. Accordingly, the claims are not limited by the disclosure.