Patent Publication Number: US-10769900-B2

Title: Touch sensitive device

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This present application is a divisional of, and claims priority to, U.S. patent application Ser. No. 13/811,904, filed Jun. 11, 2013, entitled “TOUCH SENSITIVE DEVICE,” now pending, which is a United States National Stage Application under 35 U.S.C. § 371 of International Patent Application No. PCT/GB2011/051417, filed Jul. 25, 2011, which claims the benefit of and right of priority to British Application No. 1012387.5, filed Jul. 23, 2010, each of which are incorporated by reference in their entirety as if fully set forth herein. 
    
    
     TECHNICAL FIELD 
     The invention relates to touch sensitive devices including touch sensitive screens or panels. 
     BACKGROUND ART 
     U.S. Pat. Nos. 4,885,565, 5,638,060, 5,977,867, US2002/0075135 describe touch-operated apparatus. 
     DISCLOSURE OF INVENTION 
     According to a first aspect of the invention, there is provided a method of generating a set of filters for a touch sensitive device comprising a touch-sensitive member and a plurality of transducers mounted to the member, the method comprising: 
     a) determining an initial estimate of a filter to be applied to a respective signal associated with each transducer; 
     b) defining a model of the system whereby the relationship of vibration of the member to the respective signals can be calculated, the model having a plurality of parameters; 
     c) calculating the output of the model of the system; 
     d) calculating a reference error value for the output of the model by comparing the output of the model with a measured value; 
     e) determining changed parameter values of the parameters of the model; 
     f) recalculating the error value for the output of the model by comparing the output of the model with the changed parameter values with the measured value; 
     g) comparing the recalculated error value with the reference error value; 
     h) if the compared recalculated error value is less than the reference error value, setting the recalculated error value as the reference error value, setting the changed parameter values as the model parameters, and repeating the steps c) to h), or 
     if the compared recalculated error value is greater than the reference error value, outputting the model parameters; 
     generating a set of new filters each using respective output model parameters. 
     According to a second aspect of the invention, there is provided a method of generating a desired touch sensitivity in a touch sensitive device comprising a touch-sensitive member and a plurality of transducers mounted to the member, the method comprising: 
     generating a set of filters by carrying out the method of the first aspect; 
     applying the set of filters to an output signal from each transducer to generate filtered output signals; and 
     using the filtered output signals to provide the desired touch sensitivity. 
     The filter may have a plurality of coefficients, the number of filter coefficients being equal to the number of model parameters. 
     The initial estimate of the filter to be applied to a signal output from each transducer, may be an estimate of the required filter. The initial estimate of the filter to be applied to a signal output from each transducer, may be any initial estimate of the filter because the reference error minimisation procedure of the above method will determine the required output model parameters. In some examples the initial estimate of the filter may be a standard, or default, filter, or even a random filter. In some examples the initial estimate of the filter may be a close estimate of the required filter. This may provide the advantage of reducing the number of iterations of the method required to arrive at a filter giving acceptable performance. 
     The above method models the filter in the time domain and applies an iterative refinement algorithm (repeating changing, recalculating and comparing steps), to improve the performance of the filter. The initial filter may be a time-reversal (TR) filter, a simultaneous multi-region (SMR) filter or an infinite impulse response filter. SMR filters obtained analytically are exact in the frequency domain but seldom achieve a good separation in the time domain. In some examples the application of the refinement algorithm may provide double the separation. The model may comprise an inverse of the filter. 
     The initial filter may be very complex in the temporal domain. However, in the frequency domain there may be one or more key or pole frequencies that are more important than other frequencies across the frequency range of interest. These one or more key or pole frequencies may be transformed into separate initial filter components in the temporal domain. A less complex initial filter may be represented by the combination of these separate initial filter components. 
     According to a third aspect of the invention, there is provided a method of generating a set of filters for a touch sensitive device comprising a touch-sensitive member and a plurality of transducers mounted to the member, the method comprising: 
     choosing a set of frequencies for use in the filters; 
     determining an impulse response of a filter for each respective transducer to be applied to a signal associated with each transducer; 
     calculating the transfer function of each filter, wherein each filter has a transfer function with at least one pole and at least one zero, and calculating the transfer function of each said filter comprises, 
     determining at least one pole coefficient which determines at least one pole; 
     determining, using said at least one pole coefficient, a pole representation of the transfer function which filters said input signal using said at least one pole; 
     using said respective impulse response, calculating at least one zero coefficient which determines at least one zero; and 
     combining said pole representation of the transfer function with said at least one zero coefficient to calculate said transfer function of said filter; and 
     generating a set of filters comprising said calculated filters. 
     According to a fourth aspect of the invention, there is provided a method of generating a desired touch sensitivity in a touch sensitive device comprising a touch-sensitive member and a plurality of transducers mounted to the member, the method comprising: 
     generating a set of filters by carrying out the method of the third aspect; 
     applying the set of filters to an output signal from each transducer to generate filtered output signals; and 
     using the filtered output signals to provide the desired touch sensitivity. 
     The impulse response of each filter may be derived from a desired value of touch sensitivity of the member. 
     The filter may be an infinite impulse response filter and may have a transfer function of the form: 
     
       
         
           
             
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     Where d k,0  d k,1  d k,2  are zero coefficients which determine the zeros, a k,0  a k,1  a k,2  are pole coefficients which determine the poles and k is the number of poles. 
     The pole coefficients may be expressed as
 
 a   k,0 =−2Re( p   k ) and  a   k,1   =|p   k | 2  
 
     and the transfer function may be written as 
     
       
         
           
             
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                 ( 
                 
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                   d 
                 
                 ) 
               
             
             := 
             
               
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     Determining said pole representation of the transfer function may comprise determining:
 
∪ 0,k =1 ∪ 1,k =−( a   k,0 −∪ 0,k ) ∪ 1+2,k =−( a   k,0 −∪ i+1,k   +a   k,1 −∪ 1,k )
 
     The pole coefficients may be determined from the chosen set of frequencies. 
     If the output signal for each transducer is known, the zero coefficients may be determined from 
     
       
         
           
             
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     The method may further comprise calculating zero coefficients for zeros which are not paired with said determined pole coefficients. 
     According to a fifth aspect of the invention, there is provided a method of generating a set of filters for a touch sensitive device comprising a touch-sensitive member and a plurality of transducers mounted to the member, the method comprising: 
     choosing a set of frequencies for use in the filters; 
     calculating a set of transfer functions of respective filters for each transducer to be applied to a signal associated with each transducer; 
     wherein each filter has a transfer function with at least one pole and at least one zero and calculating the transfer function of each said filter comprises, 
     determining at least one pole coefficient which determines said at least one pole; 
     determining, using said at least one pole coefficient, a pole representation of the transfer function which filters said input signal using said at least one pole; 
     using an eigenvector method to determine at least one zero coefficient which determines said at least one zero; and 
     combining said pole representation of the transfer function with said at least one zero coefficient to determine said transfer function for said filter; and 
     generating a set of filters comprising said calculated filters. 
     According to a sixth aspect of the invention, there is provided a method of generating a desired touch sensitivity in a touch sensitive device comprising a touch-sensitive member and a plurality of transducers mounted to the member, the method comprising: 
     generating a set of filters by carrying out the method of the sixth aspect; 
     applying the set of filters to an output signal from each transducer to generate filtered output signals; and 
     using the filtered output signals to provide the desired touch sensitivity. 
     The filter may be an infinite impulse response filter. 
     The method may further comprise calculating zero coefficients for zeros which are not paired with said determined pole coefficients. 
     The following features may apply to all aspects. 
     The initial reference error value may be calculated using a sum squared error, for example from: 
     
       
         
           
             
               SSE 
               ⁡ 
               
                 ( 
                 
                   a 
                   , 
                   b 
                   , 
                   … 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 n 
               
               ⁢ 
               
                 
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                       Y 
                       ⁡ 
                       
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                         ) 
                       
                     
                     - 
                     
                       X 
                       ⁡ 
                       
                         ( 
                         n 
                         ) 
                       
                     
                   
                    
                 
                 2 
               
             
           
         
       
     
     where X(n) is the measured value which is measured in response to a touch at n test points and Y(n) is the output of a system model where a, b are the model parameter values. The model parameter values may correspond to the filter coefficients. 
     The SSE is used as an exemplar method for determining a measure of error between a measured value or values and modelled value or values. Other methods such as the variance, standard deviation, mean squared error, root mean square error or other techniques as would be appreciated by the person skilled in the art. The SSE is an example of a suitable method only, the present claimed Method and Device employ any or all of the methods described above, including those appreciated by the person skilled in the art but not explicitly mentioned herein. 
     The SSE may be minimised by any suitable minimisation routine. A number of known minimisation routines may be used such as Gradient Search or Gradient Decent, Synthetic Annealing, Newton and Quasi-Newton methods, Interior point methods, linear least squares methods (a regression model comprising a linear combination of the parameters), functional analysis methods (approximating to a sum of other functions), non-linear least squares methods (approximate to a linear model and refine the parameters by successive iterations) and other methods that would be appreciated by the skilled person in the art. 
     A Gradient Search method is described below for the minimisation of the SSE. This is an example of a suitable method only, the present claimed Method and Device may employ any or all of the methods described above, including those appreciated by the person skilled in the art but not explicitly mentioned herein. 
     Changing the values of the parameters of the model may comprise selecting parameters to reduce the value of SSE by finding the value of t that minimises F(t) where 
     
       
         
           
             
               F 
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             = 
             
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     This means that F(t) can be written as 
     
       
         
           
             
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                       X 
                       ⁡ 
                       
                         ( 
                         n 
                         ) 
                       
                     
                   
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                 2 
               
             
           
         
       
     
     Finding the value of t may comprise estimating the vector gradient of error (grad (SSE)), changing all the model parameters proportionally to the estimated vector gradient of error, estimating the first and second derivatives F′(t=0) and F″(t=0) of F(t=0) for the value of t equal to zero and determining t from the estimated derivatives of F(t) where 
     
       
         
           
             
               grad 
               ⁡ 
               
                 ( 
                 SSE 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       
                         ∂ 
                         SSE 
                       
                       
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                         a 
                       
                     
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                         ∂ 
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                         b 
                       
                     
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                   ) 
                 
                 T 
               
               ⁢ 
               
                   
               
               ⁢ 
               and 
             
           
         
       
       
         
           
             
               
                 F 
                 ′ 
               
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                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
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                   F 
                 
                 
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                   t 
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               and 
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   F 
                   ″ 
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               ⁢ 
               
                 
                   
                     
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                       2 
                     
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                     F 
                   
                   
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                       t 
                       2 
                     
                   
                 
                 . 
               
             
           
         
       
     
     The derivative D′(x) of a function D(x) may be calculated from 
     
       
         
           
             
               
                 D 
                 ′ 
               
               ⁡ 
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               
                 lim 
                 
                   x 
                   = 
                   0 
                 
               
               ⁢ 
               
                 
                   
                     D 
                     ⁡ 
                     
                       ( 
                       
                         x 
                         + 
                         z 
                       
                       ) 
                     
                   
                   - 
                   
                     D 
                     ⁡ 
                     
                       ( 
                       x 
                       ) 
                     
                   
                 
                 z 
               
             
           
         
       
     
     Estimating the vector gradient of error with respect to each of the parameters may therefore comprise changing each parameter by a small amount, recalculating SSE and estimating the vector gradient of error using the difference between the reference value for SSE and the recalculated value of SSE. 
     Estimating F′(t)=0 and F″(t)=0 may therefore comprise determining two new parameter sets, the first parameter set being changed proportionally to the estimated vector gradient of error by addition and the second parameter set being changed proportionally to the estimated vector gradient of error by subtraction, calculating SSE values for each of the two new parameter sets and calculating estimates for F′(t)=0 and F″(t)=0 from the new SSE values. 
     The desired touch sensitivity may be a maximum at a first test point and a minimum at a second test point. Alternatively, the desired touch sensitivity may be a response which is between the minimum or maximum at a given test position, for example, where the responses at multiple test positions are to be taken into account. 
     The desired touch sensitivity may provide the sensation of a button click to a user. Alternatively, a touch sensitivity (in terms of produced displacement and/or acceleration of the touch sensitive member) may be generated to provide additional information to the user. The filtered output signal may be associated with a user action or gesture etc. Alternatively, or additionally, the filtered output may be associated with the response of the touch-sensitive surface in terms of display action or reaction. 
     The vibration may include any type of vibration, including bending wave vibration, more specifically resonant bending wave vibration. 
     The transducer may be an electromagnetic transducer. Such transducer are well known in the art. Alternatively, the exciter may be a piezoelectric transducer or a bender or torsional transducer. A plurality of transducer (perhaps of different types) may be selected to operate in a co-ordinated fashion. 
     The touch surface may be a panel-form member which is a bending wave device, for example, a resonant bending wave device. The touch screen may also be a loudspeaker wherein a second vibration exciter excites vibration which produces an acoustic output. For example, the touch screen may be a resonant bending wave mode loudspeaker as described in International Patent Application WO97/09842 which is incorporated by reference. 
     Contact on the surface may be detected and/or tracked as described in International patent applications WO 01/48684, WO 03/005292 and/or WO 04/053781 to the present applicant. These International patent applications are here incorporated by reference. Alternatively, other known methods may be used to receive and record or sense such contacts. 
     According to a seventh aspect of the invention, there is provided a touch sensitive device comprising 
     a touch-sensitive member, 
     a plurality of transducers mounted to the member, and 
     a processor configured to carry out the method of any one of the preceding aspects. 
     According to an eighth aspect of the invention, there is provided a computer program comprising program code which, when executed on a processor of a touch sensitive device, will cause the touch sensitive device to carry out the method of any one of the first to sixth aspects. 
     The invention further provides processor control code to implement the above-described methods, in particular on a data carrier such as a disk, CD- or DVD-ROM, programmed memory such as read-only memory (firmware), or on a data carrier such as an optical or electrical signal carrier. Code (and/or data) to implement embodiments of the invention may comprise source, object or executable code in a conventional programming language (interpreted or compiled) such as C, or assembly code, code for setting up or controlling an ASIC (Application Specific Integrated Circuit) or FPGA (Field Programmable Gate Array), or code for a hardware description language such as Verilog (Trade Mark) or VHDL (Very High speed integrated circuit Hardware Description Language). As the skilled person will appreciate such code and/or data may be distributed between a plurality of coupled components in communication with one another. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       The invention is diagrammatically illustrated, by way of example, in the accompanying drawings in which: — 
         FIG. 1 a    is a flowchart showing the steps for improving an estimate for a filter; 
         FIG. 1 b    is a flowchart showing the steps for an alternative method of improving an estimate for a filter; 
         FIG. 2 a    is a graph showing the initial filter set varying against time; 
         FIG. 2 b    is a graph of time variation of the filter set of  FIG. 2 a    after iteration according to the method shown in  FIG. 1 ; 
         FIG. 2 c    is a graph of signal level (dB) against time for the filtered response of  FIG. 2   a;    
         FIG. 2 d    is a graph of signal level (dB) against time for the filtered response of  FIG. 2   a;    
         FIG. 3 a    is a graph of signal level (dB) against frequency for a second initial filter set; 
         FIG. 3 b    is a graph of signal level (dB) against frequency for the filter set of  FIG. 3 a    after iteration according to the method shown in  FIG. 1 ; 
         FIG. 4  is a flowchart showing the steps for creating an initial filter set; 
         FIG. 5 a    shows a log-log plot of | force | vs frequency for each channel F and the arithmetic sum FA; 
         FIG. 5 b    shows a log-lin plot of each channel of  FIG. 5 a    divided by FA (i.e. normalised); 
         FIGS. 6 a  to 6 d    show the impulse response as it varies with time for each of the signal of  FIG. 5   a;    
         FIG. 7 a    shows the time reversal filters for each of  FIGS. 6 a    to  6   d;    
         FIG. 7 b    shows the time reversal filters of  FIG. 7 a    convolved with each respective signal of  FIGS. 6 a    to  6   d;    
         FIG. 8  is a schematic illustration of a touch sensitive device; 
         FIG. 9 a    is a flow chart of an alternative method for creating an initial filter set; 
         FIG. 9 b    is a graph plotting the imaginary part against the real part for p k  and −sin(θ k ) against cos(θ k ); 
         FIGS. 9 c  and 9 d    plot the eight variations of u with time; 
         FIG. 9 e    shows the variation with time of the transfer functions for each of the four exciters, and 
         FIGS. 10 a  and 10 b    plot the variation in time of the quiet and loud signals. 
     
    
    
     DETAILED DESCRIPTION OF DRAWINGS 
       FIG. 1 . shows a flow diagram of a first example of a method used to create an improved filter. This method involves minimising a Summed Squared Error (SSE) value, with the SSE being used as in example only.  FIG. 1 . Shows use of the gradient Search method for the minimisation of SSE, but this is used as an exemplar method only. 
       FIG. 1  shows the steps of a first method used to create an improved filter. The filter may be a simultaneous multi-region filter (SMR) filter or a time-reversed filter (TR) filter created as described in co-pending International application PCT/GB2010/050540 (the entire contents of which are incorporated herein by reference). The first step S 100  is to create an initial estimate of the filter to be applied to a signal output from each transducer whereby the filtered output signal is derived from vibration of the member in response to a touch, so that the filtered output signals from all of the transducers provide a touch sensitivity. This signal from a transducer will be referred to as an output signal, although it is of course an input signal from the point of view of the filter. The output time response for the filter is the convolution of the output signal input to the filter and the time impulse response of the filter. 
     The output signal from the transducer is considered as the output of a system model Y(n, a, b, . . . ) where a, b are the model parameters (there may be m parameters) and n is the number of test points at which the real output X(n) produced in response to a touch of known magnitude is measured. Thus, as recorded at step S 102 , the model is created using the model parameters, corresponding to the taps, i.e. the coefficients on the digital filter. As discussed above the coefficients of the filter may correspond to the model parameters. The system model may comprise an inverse of the filter. The initial estimate of the filter may be used to create the model. 
     The next step is to measure the effectiveness of the model by using the sum-squared error (SSE) which is defined as the energy of the difference between the model and the real system. This is calculated from 
     
       
         
           
             
               SSE 
               ⁡ 
               
                 ( 
                 
                   a 
                   , 
                   b 
                   , 
                   … 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 n 
               
               ⁢ 
               
                 
                    
                   
                     
                       Y 
                       ⁡ 
                       
                         ( 
                         
                           n 
                           , 
                           a 
                           , 
                           b 
                           , 
                           … 
                         
                         ) 
                       
                     
                     - 
                     
                       X 
                       ⁡ 
                       
                         ( 
                         n 
                         ) 
                       
                     
                   
                    
                 
                 2 
               
             
           
         
       
     
     In other words the difference between the model output and the real output at each test point is calculated, each difference is squared and the squared differences are summed together. The real output may be determined by measurement. For example, using contact methods such as using a stylus in contact with the touch sensitive member to cause vibration of the member by applying a touch force of known magnitude and/or other contact methods as would be appreciated by a person skilled in the art. 
     At step S 104 , a value for SSE is calculated that is termed the reference SSE value (or reference value for SSE). The reference value for SSE is calculated making use of the initial filter estimate. This means that the reference value for SSE may be calculated for the output of the model by comparing the output of the model with a measured value of output signal. 
     At step S 106 , the parameters of the model are changed in order to obtain a modelled output that tends towards the real output. This means that the parameters of the model are changed in order that the calculated output signals generated in response to the touch tends towards the measured output signals generated in response to the touch. Accordingly, each model parameter (that may correspond to each tap or coefficient of the filter) is adjusted by a small amount (i.e. by less than 10%, preferably less than 1%) and a new value for SSE is calculated. 
     In order to obtain a model output that tends towards the real output the aim or requirement is to minimise the SSE value. The SSE may be minimised by any suitable minimisation routine. Using a Gradient Search method, minimisation of SSE is achieved by setting the gradient to zero or substantially close to zero. 
     For a model containing m parameters, there are m gradient equations. Thus at step S 108 , the vector gradient of error is calculated with respect to each of the parameters using 
     
       
         
           
             
               grad 
               ⁡ 
               
                 ( 
                 SSE 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     
                       ∂ 
                       SSE 
                     
                     
                       ∂ 
                       a 
                     
                   
                   , 
                   
                     
                       ∂ 
                       SSE 
                     
                     
                       ∂ 
                       b 
                     
                   
                   , 
                   … 
                 
                 ) 
               
               T 
             
           
         
       
         
         
           
             The vector gradient represents the magnitude and direction of the slope of the error function, which in this example is the SSE. 
           
         
       
    
     At step S 110 , two new models are created by changing the model parameters by a value which is proportional to the calculated vector gradient. One model output is determined by adding the value to each of the model parameters and calculating a model output and the second model output is determined by subtracting this value from each of the model parameters and calculating a model output. Using these two model outputs, two new SSEs are calculated; one called SSE_p (from the new model where the value is added to model parameters) and one called SSE_m (from the new model where the value is subtracted from model parameters). 
     The next step S 112  is to obtain a set of parameters which aims to reduce the value of SSE. Using a Gradient Search method, the value of SSE may be minimised by attempting to find the value of t that minimises F(t), where F(t) is defined as: 
     
       
         
           
             
               F 
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               SSE 
               ⁡ 
               
                 ( 
                 
                   
                     a 
                     + 
                     
                       
                         t 
                         · 
                         
                           ∂ 
                           SSE 
                         
                       
                       
                         ∂ 
                         a 
                       
                     
                   
                   , 
                   
                     b 
                     + 
                     
                       
                         t 
                         · 
                         
                           ∂ 
                           SSE 
                         
                       
                       
                         ∂ 
                         b 
                       
                     
                   
                   , 
                   … 
                 
                 ) 
               
             
           
         
       
     
     At the value of t where F(t) is minimised, the gradient of F is zero, i.e. for F(t) the derivative, F′(t)=0. The Newton-Raphson method is used to solve for t, starting with t 0 =0. Other equivalent known methods may be used to solve for t. For the Newton Raphson method a new value for t (t k+1 ) can be calculated from a present or start value for t (t k ) from 
     
       
         
           
             
               
                 t 
                 
                   k 
                   + 
                   1 
                 
               
               = 
               
                 
                   t 
                   k 
                 
                 - 
                 
                   
                     
                       F 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         t 
                         k 
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     / 
                   
                   ⁢ 
                   
                     
                       F 
                       ″ 
                     
                     ⁡ 
                     
                       ( 
                       
                         t 
                         k 
                       
                       ) 
                     
                   
                 
               
             
             , 
             where 
           
         
       
       
         
           
             
               
                 F 
                 ′ 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   ∂ 
                   F 
                 
                 
                   ∂ 
                   t 
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               and 
             
           
         
       
       
         
           
             
               
                 F 
                 ″ 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             ⁢ 
             
               
                 
                   ∂ 
                   2 
                 
                 ⁢ 
                 F 
               
               
                 ∂ 
                 
                   t 
                   2 
                 
               
             
           
         
       
     
     Starting with t=0, a first solution is 
     
       
         
           
             t 
             = 
             
               
                 
                   F 
                   ′ 
                 
                 ⁡ 
                 
                   ( 
                   0 
                   ) 
                 
               
               
                 
                   F 
                   ″ 
                 
                 ⁡ 
                 
                   ( 
                   0 
                   ) 
                 
               
             
           
         
       
     
     The first and second derivatives of F(t) may be found using finite difference methods. Two-sided estimates of F′(0) and F″(0) are used to determine t, using the standard equations: 
     
       
         
           
             
               
                 df 
                 ⁡ 
                 
                   ( 
                   x 
                   ) 
                 
               
               dx 
             
             ∼ 
             
               
                 
                   
                     f 
                     ⁡ 
                     
                       ( 
                       
                         x 
                         + 
                         dx 
                       
                       ) 
                     
                   
                   - 
                   
                     f 
                     ⁡ 
                     
                       ( 
                       
                         x 
                         - 
                         dx 
                       
                       ) 
                     
                   
                 
                 
                   2 
                   ⁢ 
                   dx 
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               and 
             
           
         
       
       
         
           
             
               
                 
                   d 
                   2 
                 
                 ⁢ 
                 
                   f 
                   ⁡ 
                   
                     ( 
                     x 
                     ) 
                   
                 
               
               
                 dx 
                 2 
               
             
             ∼ 
             
               
                 
                   f 
                   ⁡ 
                   
                     ( 
                     
                       x 
                       + 
                       dx 
                     
                     ) 
                   
                 
                 + 
                 
                   f 
                   ⁡ 
                   
                     ( 
                     
                       x 
                       - 
                       dx 
                     
                     ) 
                   
                 
                 - 
                 
                   2 
                   ⁢ 
                   
                     f 
                     ⁡ 
                     
                       ( 
                       x 
                       ) 
                     
                   
                 
               
               
                 dx 
                 2 
               
             
           
         
       
     
     Once a value for t is calculated, this is used to generate a new set of model parameters and hence a new SSE value. If this value is less than the reference value calculated in step S 104 , the new SSE value is set as the reference value (step S 116 ) and steps S 106  to S 112  are repeated. If this new value is greater than the reference value (either the value calculated in step S 104  or a previous value of SSE calculated using steps S 106  to S 110 ), the model parameters for the improved filter are output. If the new value is greater than the reference value, then no further improvement can be achieved by this methodology and an optimum solution has been achieved by the previous set of model parameter values. 
       FIG. 1 b    shows an alternative iterative method for calculating an improved filter. The filter may be a simultaneous multi-region filter (SMR) filter or a time-reversed filter (TR) filter. The method may show computational efficiencies as only one parameter of the model need be varied within a calculation step. Consider the general situation where we have the multi-channel impulse responses measured at two target locations, Q is desired to be “insensitive” and L is desired to be “sensitive”. 
     At step S 200 , we define a set of filter impulse responses, h, which represent an estimate of the filters needed to achieve the separation of the Q and L target sensitivities. These may be an estimate of the simultaneous multi-region filter (SMR) filter or time-reversed filter (TR) filter. 
     At step S 202 , we calculate a reference value to be used in calculating an improved filter, using the sum over all channels of the convolution product of the filter impulse response at each channel and the response measured for that channel for a touch at the quiet location and the sum over all channels of the convolution product of the filter impulse response at each channel and the response measured for that channel for a touch at at the loud location. The reference value is termed SSE. Thus the reference value for SSE is 
     
       
         
           
             SSE 
             = 
             
               
                 
                    
                   
                     
                       ∑ 
                       chan 
                     
                     ⁢ 
                     
                       
                         Q 
                         chan 
                       
                       * 
                       
                         h 
                         chan 
                       
                     
                   
                    
                 
                 2 
               
               
                 
                    
                   
                     
                       ∑ 
                       chan 
                     
                     ⁢ 
                     
                       
                         L 
                         chan 
                       
                       * 
                       
                         h 
                         chan 
                       
                     
                   
                    
                 
                 2 
               
             
           
         
       
     
     where the star signifies the convolution product, Q chan  is the measured value for the quiet location, and L chan  is the measured value for the loud location. 
     At step S 204 , we “perturb” the set of filter impulse responses h by adjusting a tap (i.e. a digital coefficient) by a small amount α in only one filter channel j. So
 
 h 1 =h ,chan≠ j, h 1 =h+α·δ ( t−n,T ),chan= j, T =sample period
 
     where δ is the delta (sampling) function and where
 
δ( x )=1if  x= 0,0 otherwise
 
     At step S 206 , we calculate a new value for SSE for the perturbed value from: 
     
       
         
           
             
               SSE 
               ⁡ 
               
                 ( 
                 α 
                 ) 
               
             
             = 
             
               
                 
                    
                   
                     
                       ∑ 
                       chan 
                     
                     ⁢ 
                     
                       
                         Q 
                         chan 
                       
                       * 
                       
                         ( 
                         
                           
                             h 
                             chan 
                           
                           + 
                           
                             α 
                             · 
                             
                               δ 
                               
                                 chan 
                                 , 
                                 j 
                               
                             
                             · 
                             
                               z 
                               
                                 - 
                                 n 
                               
                             
                             · 
                             
                               δ 
                               ⁡ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                    
                 
                 2 
               
               
                 
                    
                   
                     
                       ∑ 
                       chan 
                     
                     ⁢ 
                     
                       
                         L 
                         chan 
                       
                       * 
                       
                         ( 
                         
                           
                             h 
                             chan 
                           
                           + 
                           
                             α 
                             · 
                             
                               δ 
                               
                                 chan 
                                 , 
                                 j 
                               
                             
                             · 
                             
                               z 
                               
                                 - 
                                 n 
                               
                             
                             · 
                             
                               δ 
                               ⁡ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                    
                 
                 2 
               
             
           
         
       
     
     Where z is the unit delay operator 
     δ i,j  is the Kronecker delta which is 1 if i=j, and 0 otherwise (i.e a discrete version of the delta function) 
     At step S 206 , we define the gradient vector exactly as follows; 
     Now as the numerator and denominator for SSE(α) are identical in form, lets consider just one term for now. 
     
       
         
           
             
               term 
               ⁡ 
               
                 ( 
                 α 
                 ) 
               
             
             = 
             
               
                  
                 
                   
                     ∑ 
                     chan 
                   
                   ⁢ 
                   
                     
                       R 
                       chan 
                     
                     * 
                     
                       ( 
                       
                         
                           h 
                           chan 
                         
                         + 
                         
                           α 
                           · 
                           
                             δ 
                             
                               chan 
                               , 
                               j 
                             
                           
                           · 
                           
                             z 
                             
                               - 
                               n 
                             
                           
                           · 
                           
                             δ 
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                       ) 
                     
                   
                 
                  
               
               2 
             
           
         
       
     
     Where R represents either Q or L. 
     Expanding this explicitly, assuming that the second order term is vanishingly small will give: 
     
       
         
           
             
               term 
               ⁡ 
               
                 ( 
                 α 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   chan 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
               ⁢ 
               
                 
                   R 
                   
                     chan 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
                 * 
                 
                   
                     ( 
                     
                       
                         h 
                         
                           chan 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                       + 
                       
                         α 
                         · 
                         
                           δ 
                           
                             
                               chan 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                             , 
                             j 
                           
                         
                         · 
                         
                           z 
                           
                             - 
                             n 
                           
                         
                         · 
                         
                           δ 
                           ⁡ 
                           
                             ( 
                             0 
                             ) 
                           
                         
                       
                     
                     ) 
                   
                   · 
                   
                     
                       ∑ 
                       
                         chan 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         R 
                         
                           chan 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                       * 
                       
                         ( 
                         
                           
                             h 
                             
                               chan 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                           + 
                           
                             
                               α 
                               · 
                               
                                 δ 
                                 
                                   
                                     chan 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                   , 
                                   j 
                                 
                               
                               · 
                               
                                 z 
                                 
                                   - 
                                   n 
                                 
                               
                               · 
                               δ 
                             
                             ⁢ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               
                 term 
                 ⁡ 
                 
                   ( 
                   α 
                   ) 
                 
               
               = 
               
                 
                   term 
                   ⁡ 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 + 
                 
                   2 
                   · 
                   α 
                   · 
                   
                     ( 
                     
                       
                         z 
                         
                           - 
                           n 
                         
                       
                       , 
                       
                         R 
                         j 
                       
                     
                     ) 
                   
                   · 
                   
                     ( 
                     
                       
                         ∑ 
                         chan 
                       
                       ⁢ 
                       
                         
                           R 
                           chan 
                         
                         * 
                         
                           h 
                           chan 
                         
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     𝒪 
                     ⁡ 
                     
                       ( 
                       α 
                       ) 
                     
                   
                   2 
                 
               
             
           
         
       
     
     Using this result, we may define the exact gradient of SSE as follows; 
     
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   grad 
                   ⁡ 
                   
                     ( 
                     SSE 
                     ) 
                   
                 
                 
                   n 
                   , 
                   j 
                 
               
               = 
               
                 
                   lim 
                   
                     n 
                     → 
                     0 
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       
                         SSE 
                         ⁡ 
                         
                           ( 
                           α 
                           ) 
                         
                       
                       - 
                       SSE 
                     
                     α 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 grad 
                 ⁡ 
                 
                   ( 
                   SSE 
                   ) 
                 
               
               
                 n 
                 , 
                 j 
               
             
             = 
             
               
                 2 
                 
                   
                      
                     
                       
                         ∑ 
                         chan 
                       
                       ⁢ 
                       
                         
                           L 
                           chan 
                         
                         * 
                         
                           h 
                           chan 
                         
                       
                     
                      
                   
                   2 
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     
                       ( 
                       
                         
                           z 
                           
                             - 
                             n 
                           
                         
                         · 
                         
                           Q 
                           j 
                         
                       
                       ) 
                     
                     · 
                     
                       ( 
                       
                         
                           ∑ 
                           chan 
                         
                         ⁢ 
                         
                           
                             Q 
                             chan 
                           
                           * 
                           
                             h 
                             chan 
                           
                         
                       
                       ) 
                     
                   
                   - 
                   
                     SSE 
                     · 
                     
                       ( 
                       
                         
                           z 
                           
                             - 
                             n 
                           
                         
                         · 
                         
                           L 
                           j 
                         
                       
                       ) 
                     
                     · 
                     
                       ( 
                       
                         
                           ∑ 
                           chan 
                         
                         ⁢ 
                         
                           
                             L 
                             chan 
                           
                           * 
                           
                             h 
                             chan 
                           
                         
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     Now we have only 2 convolutions and some shifts and dot products, and the result is exact. Thus at Step S 210  we calculate the value at which the gradient vector and follow the same final steps as in the previously described method. 
       FIGS. 2 a  to 2 d    show the method of  FIG. 1  applied to a specific example, namely a TR filter (see  FIGS. 4 to 8 ). The TR filter produces a simultaneous minimum and maximum, the said Quiet and Loud signals produced by touches at two separate points having respective lower and higher relative sensitivities for a touch sensitive device having four transducers in this specific example, however as appreciated by the person skilled in the art the claimed method and device may operate with other numbers of transducers, such as 2 or more transducers.  FIG. 2 a    shows the filter for each of the four transducers which is essentially a time-reversed impulse response.  FIG. 2 c    shows the output time response of the filters of  FIG. 2 a    (i.e. shows the convolution of the impulse responses of the filter with the input signals). The maximum signal is shown as testABL k  and the minimum signal as testABQ k . The SSE for this filter set is −2.6 dB which is better than creating a filter at random but not significantly so. 
       FIG. 2 b    shows the filter for each of the four transducers after application of ten iterations of the optimisation method of  FIG. 1 .  FIG. 2 d    shows the output time response of the filters of  FIG. 2 c   . The SSE for this filter set is −17.9 dB. This translates in to a clearly visible difference between the maximum output of the maximum signal (testABL k ) and the maximum output of the minimum signal (testABQ k ). The difference is considerably greater than the same difference for the original filters and renders the optimised filters effective whereas the original filters were not very effective. 
       FIGS. 3 a  and 3 b    show the application of the optimisation method of  FIG. 1  to an SMR filter which has been obtained analytically using the eigenvector method. These show two traces for output signals produced by touches at the loud position and two traces for output signals produced by touches at the quiet position, both before and after application of the optimisation method. 
     One set of filters is created for a net-book portable computer with a touch-screen by dividing the touch screen into a grid having three rows and five columns. Measurements are taken for touches at the “2,2” target (i.e. second row, second column) and the “3,1” target to generate two sets of SMR filters with the goal of minimising the response at one target and maximising it at the other, that is, minimising the sensitivity at one target and maximising the sensitivity at the other. The filtered output signals resulting from applying the calculated filters to an output signal were measured and are shown in  FIG. 3 a   . For both filter sets, the intendedly loud signals ( 50 ,  52 ) are stronger than the intendedly quiet signals ( 60 ,  62 ). However, at some frequencies the difference is less than desired. Moreover, such SMR filters are exact in the frequency domain but seldom achieve a theoretical separation better than 20 dB in the time-domain. This can be improved somewhat by improving the frequency resolution of the measurement data and by adjusting the delay of the filters. However, there still seems to be a fundamental limit to the effectiveness of the filters in the time-domain. One explanation may be the Heisenberg uncertainty relationship between frequency and time means that without an infinitely long filter and an infinitely fine frequency resolution, the time-domain filters obtained this way will never be exact. 
       FIG. 3 b    shows the result of iteratively refining the initial filter set according to the method of  FIG. 1 . In real use, the filters still have to contend with system non-linearity and time variation but the measured responses still show a very useful improvement in performance. The output filtered signals resulting from applying the optimised filters to an input signal were measured and are shown in  FIG. 3 b   . The separation between responses to touches at the loud and quiet locations is more marked with the refined filter set than with the original set. The refinement process has effectively doubled the dB separation of the analytically derived filters. In real use, the filters still have to contend with system non-linearity and time variation but the measured responses still show a very useful improvement in performance. 
     Moreover, it can be shown that the process makes empirically derived SMR filters effective. 
     The “exact” solutions, i.e. the analytically calculated SMR filters, result in filter delays of about 1/10 th  of the delays from the time-reversal solution and may be preferable. Time reversal filters may intrinsically incur temporal delays, as shown in  FIG. 2   a.    
       FIGS. 4 to 7   b  illustrate one method for creating an initial filter set, in this case using a time-reversed impulse response. As explained in more detail below, time-reversed impulse response (TR) shows how to create a single maximum sensitivity. With a minimum amount of effort, it can also give a single minimum sensitivity. To get simultaneously min and max sensitivities at different locations would probably use the minimising set, and could either rely on there being some sensitivity at the other location, or require extra “empirical” combining of the minimising and maximising filters. Thus, it is possible to empirically derive an SMR using the TR process. The method of improving the filter set described in relation to  FIGS. 1 to 3   b  could be used in relation to any filter set, including conceptually a blind guess. However, the better the initial estimate, the better the result. 
     As shown in  FIG. 4 , the first step S 200  in creating a filter set is to physically touch the screen with a known force at test positions and to measure the resulting output signal at this plurality of locations (S 202 ). As explained with reference to  FIG. 5 a   , each measured response is optionally whitened (S 204 ) and then transformed into the time domain (S 206 ). As explained in  FIGS. 6 a  to 6 d   , the filter is formed by taking a snapshot of each impulse response (S 208 ) and reversing this snapshot (S 210 ). 
     The spectrum of the time-reversed signal is the complex conjugate of the original 
     original: x(t)−&gt;X(f) 
     filter: y(t)=x(−t); Y(f)=conj(X(f)) 
     This is approximated by adding a fixed delay, so
 
 z ( t )= x ( T−t ) if &lt;= T , or  z ( t )=0 if  t&gt;T  
 
     When the filter is applied to the signal (ignoring the approximation for now), the phase information is removed, but the amplitude information is reinforced.
 
 y ( t )* x ( t )−&gt; X ( f )× Y ( f )=| X ( f )|{circumflex over ( )}2
 
     (In fact, the resulting time response is the autocorrelation function). 
       FIG. 5 a    is a log-log plot of power vs frequency for each channel (F) (i.e. for signals from four vibration transducers) and the arithmetic sum (FA) of these four power signals.  FIG. 5 b    is a log-lin plot of each of the 4 channel responses divided by FA (FN), i.e. normalised. Dividing each response by FA renders them more spectrally white, which improves the effectiveness of the method. This is because the response is squared in the filtering process and thus it is beneficial if the signal is spectrally “white”, or flat. 
       FIGS. 6 a  to 6 d    show each of the four normalised responses transformed into the time domain to give the individual impulse responses (GYtime &lt;i&gt; ). These impulse responses have also been normalised by dividing by the peak for each response. 
     The time reversal filters, TR, are formed by taking a finite snapshot of the impulse responses of  FIGS. 6 a  to 6 d    and then reversing them in time. The TR have built into them a delay equal to the length of the sample, as shown in  FIG. 2 a   .  FIG. 7 a    shows the time reveral filter for each channel (0, 1, 2 and 3) 
     
       
         
           
             
               TR 
               
                 k 
                 , 
                 j 
               
             
             := 
             
               
                 if 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       k 
                       ≥ 
                       kmax 
                     
                     , 
                     0 
                     , 
                     
                       
                         GYtime 
                         
                           
                             kmax 
                             - 
                             k 
                             - 
                             1 
                           
                           , 
                           
                             j 
                             + 
                             1 
                           
                         
                       
                       · 
                       
                         win 
                         
                           kmax 
                           - 
                           k 
                           - 
                           1 
                         
                       
                     
                   
                   ) 
                 
               
               
                 peak 
                 
                   j 
                   + 
                   1 
                 
               
             
           
         
       
     
     where kmax=samples (length) 
       FIG. 7 b    shows the results of convolved the filters with the appropriate impulse responses in the time domain (GYtime) to give the filtered response GYresp. The convolution is expressed as: 
     
       
         
           
             
               GY 
               
                 resp 
                 
                   k 
                   , 
                   j 
                 
               
             
             := 
             
               
                 ∑ 
                 
                   j 
                   = 
                   0 
                 
                 k 
               
               ⁢ 
               
                 ( 
                 
                   
                     GYtime 
                     
                       
                         k 
                         - 
                         i 
                       
                       , 
                       
                         j 
                         + 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     TR 
                     
                       i 
                       , 
                       j 
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               amp 
               j 
             
             := 
             
               max 
               ⁡ 
               
                 ( 
                 
                   GYresp 
                   
                     〈 
                     j 
                     〉 
                   
                 
                 ) 
               
             
           
         
       
     
     As shown in  FIG. 7 b   , all the responses share a common maximum, but exhibit some ringing. The common maximum occurs because the phase/time information has been corrected. The ringing occurs because the amplitude information is exaggerated. 
     As shown at step S 212  in  FIG. 4 , the filter amplitudes may be adjusted to maximise or minimise the sum of the four signals produced from the touch location. The filter is then applied to each impulse response to generate a filtered output signal derived from each location (S 214 ). 
       FIG. 8  shows a touch-sensitive haptics device  10  with four haptics-input and touch sensing vibration transducers  12  mounted to a touch-sensitive screen  14  (in this specific example there are four transducers, however as appreciated by the person skilled in the art there may be any number of input transducers on the screen). The transducers  12  are each coupled to a system processor  20  via a two-way amplifier  22 . A stylus  16  is also connected to the processor  20  via a two-way amplifier  24 . 
     The touch-sensitive device shown in  FIG. 8  may be used to create an initial filter set as set out in  FIG. 4  and may then be used to apply the improved filter set created as described in  FIG. 1 . The device has two operational modes, normal use and training mode. In normal use, i.e. when a user is using the screen  14  of the touch-sensitive device  10 , the transducers  12  produce output signals in response to touches on the surface. These output signals can then be used to determine properties of the touches such as location and force of touch. The method of determining these properties is not critical to the operation of the device and may be as described in any known techniques. The output signals can be processed to control touch sensitivity of the device. The touch sensitivity may be simple, for example detecting touches only in specific locations, or may be more complex to identify complex touch activity, i.e. associated with sliding movements, increasing/decreasing intensity of touch etc. The more complex sensations may be associated with gestures such as sliding, pitching or rotating fingers on the screen. 
     The transducers  12  also produce any required localized haptic force feedback. 
     The transducers are thus reciprocal transducers able to work as both output devices to generate excitation signals which create vibration in the screen and as input devices to sense vibration in the screen and convert the vibration into signals to be analysed. It is preferable for all the transducer to be reciprocal devices but it is possible to have a device in which not all transducers are reciprocal; such a device is more complicated. 
     In training mode, the stylus  16  is used to inject vibrational signals at specified test points; thus the stylus  16  may be considered to be a “force pencil”. The system processor  20  generates the signals which are sent to the stylus  16  via the two-way amplifier  24  and receives the signals from the transducers  12 . The two-way amplifiers  22  are also connected between the system processor  20  and each transducer  12 ; one amplifier for each channel, i.e. one amplifier for each transducer. The stylus  16  is also arranged to sense haptic feedback signals in the screen originating from the transducers  12  and to feed the sensed signals to the processor  20  via the two-way amplifier  24 . 
       FIGS. 9 a  to 10 b    illustrate an alternative method for creating an initial filter set, in this case a simultaneous multi-region filter (SMR) having a maximum touch sensitivity at one location on the touch sensitive screen and a minimum touch sensitivity at another discrete location on the screen. Four transducers are used in the filter so there are four channels with j:=0 chan-1. In this specific example there are four transducers, however as appreciated by the person skilled in the art the claimed method and device may operate with 2 or more transducers. 
     The alternative method for creating an initial filter set may be carried in two ways. 
     A first way may be used when the filter impulse response is known. This means that the required theoretical filter impulse response to give the desired member touch sensitivity output may be known. It will be understood that even if the desired impulse response is known, the implementation of the required filter in practice may be difficult, as will be appreciated by the person skilled in the art. The solution may be to make use of Quiet and Loud measurements made by applying a touch to different points on the member or panel, and a generalised inverse matrix technique (also known as the Moore-Penrose technique) may be used to determine recursive filters, the output from which may be used to provide a filtered impulse response tending towards the required theoretical impulse response to give the desired touch sensitivity response. 
     The second way may be used even when the filter impulse response is not known. The solution may be to use to make use of Quiet and Loud measurements made by applying a touch to different points on the member or panel, and a solving an eigenvalue problem may be used to determine recursive filters, the output from which may be used to provide a filtered impulse response tending towards the required theoretical impulse response to give the desired vibration output. 
     The two ways of creating an initial filter set using an alternative method have a number of common steps, before the generalised inverse matrix technique or solution of an eigenvalue problem are undertaken. 
     It should be understood that although the terms quiet and loud are used to refer to locations having relatively less and more sensitivity to a touch, producing a desired touch sensitivity profile, this wording is used only to assist understanding. It is not necessary that the touch produces an audible sound. 
     As set out in  FIG. 9 a   , the first step S 300  is to choose a set of frequencies spanning the frequency range of interest. The set of frequencies may be linearly, logarithmically or otherwise nonlinearly distributed across the frequency range of interest. For example, the frequency range of the filter may be from 150 Hz (bot) to 600 Hz (top). Q (and q) is an abbreviation for quiet (i.e. the minimum touch sensitivity and response) and L (and l) is an abbreviation for loud (i.e. the maximum touch sensitivity and response). Thus the initial variables are defined as: 
     
       
         
           
             Time 
             = 
             
               Qdata 
               
                 〈 
                 0 
                 〉 
               
             
           
         
       
       
         
           
             chans 
             = 
             
               
                 cols 
                 ⁡ 
                 
                   ( 
                   Qdata 
                   ) 
                 
               
               - 
               1 
             
           
         
       
       
         
           
             
               q 
               
                 〈 
                 j 
                 〉 
               
             
             = 
             
               Ldata 
               
                 〈 
                 
                   j 
                   + 
                   1 
                 
                 〉 
               
             
           
         
       
       
         
           
             
               F 
               s 
             
             = 
             
               
                 
                   length 
                   ⁡ 
                   
                     ( 
                     time 
                     ) 
                   
                 
                 - 
                 1 
               
               
                 
                   time 
                   
                     last 
                     ⁡ 
                     
                       ( 
                       time 
                       ) 
                     
                   
                 
                 - 
                 
                   time 
                   0 
                 
               
             
           
         
       
       
         
           
             
               F 
               s 
             
             = 
             
               5512.5 
               ⁢ 
               
                   
               
               ⁢ 
               
                 ( 
                 
                   the 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   sampling 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   frequency 
                 
                 ) 
               
             
           
         
       
     
     An IIR filter is used to create the desired effect. An IIR is an infinite impulse response filter. Such filters use feedback since the output and next internal state are determined from a linear combination of the previous inputs and outputs. A second order IIR filter is often termed a biquad because its transfer function is the ratio of two quadratic functions, i.e. 
     
       
         
           
             
               Hz 
               ⁡ 
               
                 ( 
                 
                   z 
                   , 
                   d 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 k 
               
               ⁢ 
               
                 
                   
                     d 
                     
                       k 
                       , 
                       0 
                     
                   
                   + 
                   
                     
                       d 
                       
                         k 
                         , 
                         1 
                       
                     
                     · 
                     
                       z 
                       - 
                     
                   
                   + 
                   
                     
                       d 
                       
                         k 
                         , 
                         2 
                       
                     
                     · 
                     
                       z 
                       
                         - 
                         2 
                       
                     
                   
                 
                 
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           p 
                           k 
                         
                         · 
                         
                           z 
                           
                             - 
                             1 
                           
                         
                       
                     
                     ) 
                   
                   · 
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           
                             p 
                             k 
                           
                           _ 
                         
                         · 
                         
                           z 
                           
                             - 
                             1 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     The transfer function of such a filter has two poles and two zeros. A pole of a function f(z) is a point a such that f(z) approaches infinity as z approaches a and a zero is a point b such that f(z) equals zero when z equals b. Thus, the d k,0  d k,1  d k,2  co-efficients determine the zeros and the p k  coefficients determines the poles. The set of frequencies within the frequency range of interest is associated with a corresponding pole, where k may equal the number of frequencies within the set, equal to the number of poles. 
     The pole coefficients p k  may be written as:
 
 p   k   :=e   −0.5,Δθ     k     e   −j.θ     k    
 
     As shown in  FIG. 9 a   , the next step S 302  of the method is to calculate θ k  and Δθ k  which may be derived from the initial variables as follows: 
     
       
         
           
             
               K 
               := 
               
                 floor 
                 ⁡ 
                 
                   ( 
                   
                     
                       per 
                       · 
                       
                         log 
                         ⁡ 
                         
                           ( 
                           
                             
                               top 
                               bot 
                             
                             · 
                             2 
                           
                           ) 
                         
                       
                     
                     + 
                     0.5 
                   
                   ) 
                 
               
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               K 
               = 
               8 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             k 
             := 
             
               
                 0 
                 ⁢ 
                 … 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 K 
               
               - 
               1 
             
           
         
       
       
         
           
             kf 
             := 
             
               
                 1 
                 ⁢ 
                 … 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 K 
               
               - 
               2 
             
           
         
       
       
         
           
             
               f 
               k 
             
             := 
             
               
                 bot 
                 ⁡ 
                 
                   ( 
                   
                     top 
                     bot 
                   
                   ) 
                 
               
               
                 k 
                 
                   K 
                   - 
                   t 
                 
               
             
           
         
       
       
         
           
             
               f 
               
                 K 
                 - 
                 1 
               
             
             := 
             top 
           
         
       
       
         
           
             
               θ 
               k 
             
             := 
             
               
                 2 
                 · 
                 π 
                 · 
                 
                   f 
                   k 
                 
               
               
                 F 
                 s 
               
             
           
         
       
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               
                 θ 
                 
                   K 
                   - 
                   1 
                 
               
             
             := 
             
               
                 θ 
                 
                   K 
                   - 
                   1 
                 
               
               - 
               
                 θ 
                 
                   K 
                   - 
                   2 
                 
               
             
           
         
       
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               
                 θ 
                 0 
               
             
             := 
             
               
                 θ 
                 1 
               
               - 
               
                 θ 
                 0 
               
             
           
         
       
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               
                 θ 
                 
                   k 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
             
             := 
             
               
                 
                   θ 
                   
                     
                       k 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       f 
                     
                     + 
                     1 
                   
                 
                 - 
                 
                   θ 
                   
                     kf 
                     - 
                     1 
                   
                 
               
               2 
             
           
         
       
     
     Where K is the number of poles, f k  is the frequency of the k th  pole, F s  is the sampling frequency, θ k  is an angle related to the frequency of the k th  pole 
       FIG. 9 b    plots the value of the imaginary part of the pole coefficient p k  against the real part of p k .  FIG. 9 b    also plots the value of −sin(θ k ) against cos(θ k ) and shows the locus of pole positions p k  plotted along with an arc of the unit circle in the z-plane. 
     The p k  values are complex. If we wish to consider real coefficient values, the transfer function may be written as: 
     
       
         
           
             
               Hz 
               ⁡ 
               
                 ( 
                 
                   z 
                   , 
                   d 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 k 
               
               ⁢ 
               
                 ( 
                 
                   
                     
                       d 
                       
                         k 
                         , 
                         0 
                       
                     
                     + 
                     
                       
                         d 
                         
                           k 
                           , 
                           1 
                         
                       
                       · 
                       
                         z 
                         - 
                       
                     
                     + 
                     
                       
                         d 
                         
                           k 
                           , 
                           2 
                         
                       
                       · 
                       
                         z 
                         
                           - 
                           2 
                         
                       
                     
                   
                   
                     1 
                     + 
                     
                       
                         a 
                         
                           k 
                           , 
                           0 
                         
                       
                       · 
                       
                         z 
                         
                           - 
                           1 
                         
                       
                     
                     + 
                     
                       
                         a 
                         
                           k 
                           , 
                           1 
                         
                       
                       · 
                       
                         z 
                         
                           - 
                           1 
                         
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
     Now the a k,0  a k,1  coefficients determine the poles. These coefficients may be written: 
     
       
         
           
             
               a 
               
                 k 
                 , 
                 0 
               
             
             := 
             
               
                 - 
                 2 
               
               · 
               
                 θ 
                 
                   
                     - 
                     
                       Δθ 
                       k 
                     
                   
                   2 
                 
               
               · 
               
                 cos 
                 ⁡ 
                 
                   ( 
                   
                     θ 
                     k 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               a 
               
                 k 
                 , 
                 1 
               
             
             := 
             
               θ 
               
                 
                   - 
                   Δ 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   θ 
                   k 
                 
               
             
           
         
       
       
         
           
             
               i 
               . 
               e 
               . 
               
                   
               
               ⁢ 
               
                 a 
                 
                   k 
                   , 
                   0 
                 
               
             
             = 
             
               
                 
                   - 
                   2 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   Re 
                   ⁡ 
                   
                     ( 
                     
                       p 
                       k 
                     
                     ) 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   a 
                   
                     k 
                     , 
                     1 
                   
                 
               
               = 
               
                 
                    
                   
                     p 
                     k 
                   
                    
                 
                 2 
               
             
           
         
       
     
     At this step we also solve for the poles by considering: 
     
       
         
           
             
               Y 
               X 
             
             = 
             
               1 
               
                 1 
                 - 
                 
                   a 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     0 
                     · 
                     
                       z 
                       
                         - 
                         1 
                       
                     
                   
                 
                 + 
                 
                   a 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     1 
                     · 
                     
                       z 
                       
                         - 
                         2 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               Y 
               · 
               
                 ( 
                 
                   1 
                   - 
                   
                     a 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       0 
                       · 
                       
                         z 
                         
                           - 
                           1 
                         
                       
                     
                   
                   + 
                   
                     a 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       1 
                       · 
                       
                         z 
                         
                           - 
                           2 
                         
                       
                     
                   
                 
                 ) 
               
             
             = 
             X 
           
         
       
       
         
           
             Y 
             = 
             
               X 
               - 
               
                 
                   ( 
                   
                     
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         0 
                         · 
                         
                           z 
                           
                             - 
                             1 
                           
                         
                       
                     
                     + 
                     
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         1 
                         · 
                         
                           z 
                           
                             - 
                             2 
                           
                         
                       
                     
                   
                   ) 
                 
                 · 
                 Y 
               
             
           
         
       
     
     For input X(z), the result of filtering with only the poles is Y(z), hence the transfer function is Y(z)/X(z). This is then algebraically manipulated to get a result without division, which can be directly converted to the time-domain representation u(i,k) which are defined as follows:
 
∪ 0,k =1 ∪ 1,k =−( a   k,0 −∪ 0,k ) ∪ 1+2,k =−( a   k,0 −∪ i+1,k   +a   k,1 −∪ 1,k )
 
     The variations in these functions with time are plotted in  FIGS. 9 c    and  9   d.    
     The next step is to solve for the zeros, i.e. to determine the coefficients d k,0  d k,1  etc. The zeros may be paired with the poles or they may be a different set of zeros. The zeros are determined from: 
     
       
         
           
             
               - 
               
                 1 
                 
                   2 
                   · 
                   
                     d 
                     
                       k 
                       , 
                       0 
                     
                   
                 
               
             
             · 
             
               [ 
               
                 
                   
                     
                       
                         d 
                         
                           k 
                           , 
                           1 
                         
                       
                       - 
                       
                         
                           
                             
                               ( 
                               
                                 d 
                                 
                                   k 
                                   , 
                                   1 
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             4 
                             · 
                             
                               d 
                               
                                 k 
                                 , 
                                 0 
                               
                             
                             · 
                             
                               d 
                               
                                 k 
                                 , 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         d 
                         
                           k 
                           , 
                           1 
                         
                       
                       + 
                       
                         
                           
                             
                               ( 
                               
                                 d 
                                 
                                   k 
                                   , 
                                   1 
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             4 
                             · 
                             
                               d 
                               
                                 k 
                                 , 
                                 0 
                               
                             
                             · 
                             
                               d 
                               
                                 k 
                                 , 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     If the impulse response X(z) is known as shown in  FIG. 9 a    at step S 304 , the unique solution may be found at step S 306 , e.g. by using the generalised inverse matrix which is defined as follows:
 
 M   l+Δ,3,k =∪ l,k    M   l+Δ+1,3,k+1 =∪ l,k    M   l+Δ+2,3−k+2 =∪ l,k    M   l,3−K+m =δ( i,m )
 
 M :=submatrix( M, 0,last(time),0,cols( M )−1)  MI =geninv( M )
 
     The zero coefficients are then d &lt;j&gt; =MI·h &lt;j&gt;   
     The calculated matrix is shown below: 
     
       
         
           
               
               
               
               
               
               
               
             
               
                   
                   
               
               
                   
                   
                   
                 0 
                 1 
                 2 
                 3 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                 d= 
                 0 
                 3.405 
                 2.672 
                 42.262 
                 15.453 
               
               
                   
                   
                 1 
                 −0.537 
                 −3.023 
                 4.175 
                 1.281 
               
               
                   
                   
                 2 
                 −2.112 
                 −2.489 
                 −35.365 
                 −18.147 
               
               
                   
                   
                 3 
                 2.287 
                 7.815 
                 60.95 
                 −0.504 
               
               
                   
                   
                 4 
                 −0.437 
                 −3.615 
                 7.214 
                 −0.873 
               
               
                   
                   
                 5 
                 −0.743 
                 −8.985 
                 −48.999 
                 −5.914 
               
               
                   
                   
                 6 
                 20.899 
                 3.204 
                 49.812 
                 −12.642 
               
               
                   
                   
                 7 
                 −0.587 
                 −2.456 
                 13.298 
                 −1.674 
               
               
                   
                   
                 8 
                 −20.614 
                 −1.808 
                 −25.871 
                 5.237 
               
               
                   
                   
                 9 
                 10.879 
                 10.881 
                 −6.574 
                 −40.342 
               
               
                   
                   
                 10 
                 −1.759 
                 −6.355 
                 11.212 
                 −5.183 
               
               
                   
                   
                 11 
                 −12.454 
                 −17.547 
                 29.148 
                 27.938 
               
               
                   
                   
                 12 
                 −24.895 
                 12.807 
                 −4.562 
                 −75.415 
               
               
                   
                   
                 13 
                 −12.819 
                 −5.264 
                 −15.278 
                 0.249 
               
               
                   
                   
                 14 
                 4.693 
                 −17.369 
                 −24.445 
                 76.935 
               
               
                   
                   
                 15 
                 −9.55 
                 6.87 
                 32.191 
                 . . . 
               
               
                   
                   
               
            
           
         
       
     
     Rows(p)=8 with 
     Max (p)=09.67−0.167j and min (p)=0.729−0.594j 
     The pairs of zeros associated with the corresponding d values are thus: 
     
       
         
           
             
               z 
               ⁢ 
               
                   
               
               ⁢ 
               
                 1 
                 
                   k 
                   , 
                   j 
                 
               
             
             := 
             
               
                 
                   d 
                   
                     
                       
                         3 
                         · 
                         k 
                       
                       + 
                       1 
                     
                     , 
                     j 
                   
                 
                 - 
                 
                   
                     
                       
                         ( 
                         
                           d 
                           
                             
                               
                                 3 
                                 · 
                                 k 
                               
                               + 
                               1 
                             
                             , 
                             j 
                           
                         
                         ) 
                       
                       2 
                     
                     - 
                     
                       4 
                       · 
                       
                         d 
                         
                           
                             3 
                             · 
                             k 
                           
                           , 
                           j 
                         
                       
                       · 
                       
                         d 
                         
                           
                             
                               3 
                               · 
                               k 
                             
                             + 
                             2 
                           
                           , 
                           j 
                         
                       
                     
                   
                 
               
               
                 
                   - 
                   2 
                 
                 · 
                 
                   d 
                   
                     
                       3 
                       · 
                       k 
                     
                     , 
                     j 
                   
                 
               
             
           
         
       
       
         
           
             
               z 
               ⁢ 
               
                   
               
               ⁢ 
               
                 2 
                 
                   k 
                   , 
                   j 
                 
               
             
             := 
             
               
                 
                   d 
                   
                     
                       
                         3 
                         · 
                         k 
                       
                       + 
                       1 
                     
                     , 
                     j 
                   
                 
                 + 
                 
                   
                     
                       
                         ( 
                         
                           d 
                           
                             
                               
                                 3 
                                 · 
                                 k 
                               
                               + 
                               1 
                             
                             , 
                             j 
                           
                         
                         ) 
                       
                       2 
                     
                     - 
                     
                       4 
                       · 
                       
                         d 
                         
                           
                             3 
                             · 
                             k 
                           
                           , 
                           j 
                         
                       
                       · 
                       
                         d 
                         
                           
                             
                               3 
                               · 
                               k 
                             
                             + 
                             2 
                           
                           , 
                           j 
                         
                       
                     
                   
                 
               
               
                 
                   - 
                   2 
                 
                 · 
                 
                   d 
                   
                     
                       3 
                       · 
                       k 
                     
                     , 
                     j 
                   
                 
               
             
           
         
       
     
     The pairs of zeros relates to the zero coefficients d in the same way that the poles p relate to the pole coefficients a except that the zeros cannot be asserted to appear as complex conjugate pairs. 
     The synthesised transfer function (hs) for each filter to be applied to the output signal from each transducer to give the required sensitivity profile to touches of the member may then be determined from: 
     
       
         
           
             
               hs 
               
                 
                   Δ 
                   + 
                   i 
                 
                 , 
                 j 
               
             
             := 
             
               
                 ∑ 
                 k 
               
               ⁢ 
               
                 ( 
                 
                   
                     
                       uf 
                       ⁡ 
                       
                         ( 
                         
                           l 
                           , 
                           k 
                         
                         ) 
                       
                     
                     · 
                     
                       d 
                       
                         
                           3 
                           · 
                           k 
                         
                         , 
                         j 
                       
                     
                   
                   + 
                   
                     
                       uf 
                       ⁡ 
                       
                         ( 
                         
                           
                             i 
                             - 
                             1 
                           
                           , 
                           k 
                         
                         ) 
                       
                     
                     · 
                     
                       d 
                       
                         
                           
                             3 
                             · 
                             k 
                           
                           + 
                           1 
                         
                         , 
                         j 
                       
                     
                   
                   + 
                   
                     
                       uf 
                       ⁡ 
                       
                         ( 
                         
                           
                             i 
                             - 
                             2 
                           
                           , 
                           k 
                         
                         ) 
                       
                     
                     · 
                     
                       d 
                       
                         
                           
                             3 
                             · 
                             k 
                           
                           + 
                           2 
                         
                         , 
                         j 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
     As set out above, there may be zeros which are not paired with poles. In this case, the transfer function may be written as: 
     
       
         
           
             
               Hz 
               ⁡ 
               
                 ( 
                 
                   z 
                   , 
                   d 
                   , 
                   b 
                 
                 ) 
               
             
             := 
             
               
                 
                   z 
                   
                     - 
                     Δ 
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     k 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           d 
                           
                             k 
                             , 
                             0 
                           
                         
                         + 
                         
                           
                             d 
                             
                               k 
                               , 
                               1 
                             
                           
                           · 
                           
                             z 
                             
                               - 
                               1 
                             
                           
                         
                         + 
                         
                           
                             d 
                             
                               k 
                               , 
                               2 
                             
                           
                           · 
                           
                             z 
                             
                               - 
                               2 
                             
                           
                         
                       
                       
                         1 
                         + 
                         
                           
                             a 
                             
                               k 
                               , 
                               0 
                             
                           
                           · 
                           
                             z 
                             
                               - 
                               1 
                             
                           
                         
                         + 
                         
                           
                             a 
                             
                               k 
                               , 
                               1 
                             
                           
                           · 
                           
                             z 
                             
                               - 
                               2 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   ∑ 
                   m 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       b 
                       m 
                     
                     · 
                     
                       z 
                       
                         - 
                         m 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     Where the b terms are coefficients which define additional zeros with no associated poles. Such an additional term may be implemented as an additional filter which may typically be used to deal with excess-phase problems such as delay. In other words, each filter may comprise at least two separate filters which may be of different types. The individual transfer functions for each filter may be expressed as: 
     
       
         
           
             
               hs 
               
                 i 
                 , 
                 j 
               
             
             := 
             
               
                 hs 
                 
                   i 
                   , 
                   j 
                 
               
               + 
               
                 
                   ∑ 
                   m 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       d 
                       
                         
                           
                             3 
                             · 
                             K 
                           
                           + 
                           m 
                         
                         , 
                         j 
                       
                     
                     · 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           i 
                           , 
                           m 
                         
                         ) 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     Where the first term for hs sums all the contributions from the first type of filter (e.g. biquad) and the second term adds in the additional contributions from a second filter (e.g. a discrete-time filter such as a finite impulse response (FIR)). 
     Alternatively, if the impulse response is not known, a solution may be derived as follows. 
     Consider Quiet and Loud impulse responses to be determined for touches at two locations on the member. If the Quiet response is minimal or zero (this means no response to touching the member at the Quiet location) and the Loud response is the loudest possible response (this means the largest possible response to touching of the member at the Loud location), then there are a large number of possible impulse response filters for each transducer channel that can be used to obtain the Quiet response, whilst there is only one set of impulse response filters that gives the loudest possible response. Each set of filters that gives the minimal or zero quiet response to a touch at the quiet location will also give some response (which may be minimal or zero or may be non-zero) to a touch at the loud location. However, the set of filters yielding the loudest possible response to a touch at the loud position may not, and probably will not, yield minimal or zero output to a touch at the quiet location. 
     There is a non-unique set of solutions relating to sets of filters that yields a minimal or zero quite response to a touch at the quiet location, whilst responding to touching of the member at the loud location. There may be no solution relating to sets of filters that yields a minimal or zero quiet response to a touch at the quiet location whilst yielding a maximum response signal to a touch at the loud location. 
     The problem of finding an improved set of filters relates to finding the set of filters that maximises the loud signal in response to a touch at the loud location and at the same time minimises the quiet signal in response to a touch at the quiet location. The problem may be solved by considering the ratio between the quiet and loud response signals, and choosing a set of filters that yields a minimal or zero response at the quiet location, determine the response at the quiet location, determine the associated response at the loud location for the same set of filters, and determine the ratio between the signals at the quiet and loud locations. By considering one, all or any of the sets of filters that yields a minimal or zero response at the quiet location the set of filters that minimises the ratio between the quiet response at the quiet location and the loud response at the loud location may be selected. These sets of filters may be considered as the improved set of filters. 
     Further, linear combinations of the sets of filters that individually yield a minimal or zero response to a touch at the quiet location will also yield a minimal or zero response to a touch at the quiet location. Accordingly, the problem may be solved by selecting a linear combination of different ones of the sets of filters that yield a minimal or zero response to a touch at the quiet location, the linear combination being selected to maximise the response to a touch at the loud location. 
     Mathematically, a non-unique solution may be derived, for example by solving the eigenvalue problem as set out below at Step S 308 . Where we are deriving the non-unique solution, additional requirements may be imposed and only the combination of solutions that satisfies the additional requirements is generated (step S 310 ). 
     Firstly we consider the quiet response: 
     
       
         
           
             
               
                 
                   
                     MQ 
                     := 
                   
                 
               
               
                 
                   
                       
                   
                 
               
               
                 
                   
                       
                   
                 
               
               
                 
                   
                       
                   
                 
               
               
                 
                   
                       
                   
                 
               
               
                 
                   
                       
                   
                 
               
               
                 
                   
                       
                   
                 
               
               
                 
                   
                       
                   
                 
               
               
                 
                   
                       
                   
                 
               
               
                 
                   
                       
                   
                 
               
               
                 
                   
                       
                   
                 
               
             
             | 
             
               
                 
                   
                     
                       
                         M 
                         
                           
                             
                               length 
                               ⁡ 
                               
                                 ( 
                                 time 
                                 ) 
                               
                             
                             + 
                             1 
                           
                           , 
                           
                             
                               3 
                               · 
                               K 
                               · 
                               chans 
                             
                             - 
                             1 
                           
                         
                       
                       ← 
                       0 
                     
                     ⁢ 
                     
                         
                     
                   
                 
               
               
                 
                   
                     
                       
                         for 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         k 
                       
                       ∈ 
                       
                         
                           0 
                           ⁢ 
                           … 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           K 
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                         
                     
                   
                 
               
               
                 
                   
                     
                       
                         for 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         j 
                       
                       ∈ 
                       
                         
                           0 
                           ⁢ 
                           … 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           chans 
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                         
                     
                   
                 
               
               
                 
                   
                     
                       
                         for 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                       
                       ∈ 
                       
                           
                       
                       ⁢ 
                       
                         0 
                         ⁢ 
                         … 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                     ⁢ 
                     
                         
                     
                   
                 
               
               
                 
                   
                     
                       
                         for 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           k 
                           ′ 
                         
                       
                       ∈ 
                       
                         
                           0 
                           ⁢ 
                           … 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           K 
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                         
                     
                   
                 
               
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       
                         for 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           j 
                           ′ 
                         
                       
                       ∈ 
                       
                           
                       
                       ⁢ 
                       
                         
                           0 
                           ⁢ 
                           … 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           chans 
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                         
                     
                   
                 
               
               
                 
                   
                     
                       
                         for 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           n 
                           ′ 
                         
                       
                       ∈ 
                       
                         0 
                         ⁢ 
                         … 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                     ⁢ 
                     
                         
                     
                   
                 
               
               
                 
                   
                     
                       MQ 
                       
                         
                           
                             ( 
                             
                               
                                 
                                   j 
                                   ′ 
                                 
                                 · 
                                 K 
                               
                               + 
                               
                                 k 
                                 ′ 
                               
                             
                             ) 
                           
                           · 
                           3 
                           · 
                           
                             n 
                             ′ 
                           
                         
                         , 
                         
                           
                             
                               ( 
                               
                                 
                                   j 
                                   · 
                                   K 
                                 
                                 + 
                                 k 
                               
                               ) 
                             
                             · 
                             3 
                           
                           + 
                           n 
                         
                       
                     
                     ← 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         
                           last 
                           ⁡ 
                           
                             ( 
                             
                               qs 
                               kj 
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 sub 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       
                                         qs 
                                         
                                           k 
                                           , 
                                           j 
                                         
                                       
                                       , 
                                       
                                         i 
                                         - 
                                         n 
                                       
                                     
                                     ) 
                                   
                                   · 
                                 
                               
                             
                           
                           
                             
                               
                                 sub 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       qs 
                                       
                                         
                                           k 
                                           ′ 
                                         
                                         , 
                                         
                                           j 
                                           ′ 
                                         
                                       
                                     
                                     , 
                                     
                                       i 
                                       - 
                                       
                                         n 
                                         ′ 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     MQ 
                     ⁢ 
                     
                         
                     
                   
                 
               
             
           
         
       
       
         
           
             Where 
             ⁢ 
             
               : 
             
           
         
       
       
         
           
             
               qs 
               
                 k 
                 , 
                 j 
               
             
             := 
             
               
                 
                   convol 
                   ⁡ 
                   
                     ( 
                     
                       
                         u 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       , 
                       
                         q 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   sub 
                   ⁡ 
                   
                     ( 
                     
                       v 
                       , 
                       n 
                     
                     ) 
                   
                 
               
               := 
               
                 if 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       n 
                       &lt; 
                       0 
                     
                     , 
                     0 
                     , 
                     
                       v 
                       n 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     Where q &lt;j&gt;  are the determined quiet responses for the j channels. 
     Simultaneously we consider the loud response: 
     
       
         
           
             
               ML 
               
                 
                   n 
                   ′ 
                 
                 ⁢ 
                 n 
               
             
             := 
             
               
                 hls 
                 n 
               
               · 
               
                 hls 
                 
                   n 
                   ′ 
                 
               
             
           
         
       
       
         
           Where 
         
       
       
         
           
             
               hls 
               n 
             
             := 
             
               
                 ∑ 
                 j 
               
               ⁢ 
               
                 convol 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         ( 
                         
                           hs 
                           n 
                         
                         ) 
                       
                       
                         ( 
                         j 
                         ) 
                       
                     
                     , 
                     
                       l 
                       
                         ( 
                         j 
                         ) 
                       
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           With 
         
       
       
         
           
             
               
                 
                   
                     
                       hs 
                       n 
                     
                     := 
                   
                 
               
               
                 
                   
                       
                   
                 
               
               
                 
                   
                       
                   
                 
               
               
                 
                   
                       
                   
                 
               
               
                 
                   
                       
                   
                 
               
               
                 
                   
                       
                   
                 
               
               
                 
                   
                       
                   
                 
               
               
                 
                   
                       
                   
                 
               
             
             | 
             
               
                 
                   
                     
                       g 
                       ← 
                       
                         E 
                         
                           〈 
                           
                             n 
                             + 
                             pick 
                           
                           〉 
                         
                       
                     
                     ⁢ 
                     
                         
                     
                   
                 
               
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       
                         for 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         i 
                       
                       ∈ 
                       
                         
                           0 
                           ⁢ 
                           … 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             rows 
                             ⁡ 
                             
                               ( 
                               u 
                               ) 
                             
                           
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                         
                     
                   
                 
               
               
                 
                   
                     
                       
                         for 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         j 
                       
                       ∈ 
                       
                         
                           0 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           … 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           chans 
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                         
                     
                   
                 
               
               
                 
                   
                     
                       hsn 
                       
                         i 
                         , 
                         j 
                       
                     
                     ← 
                     
                       
                         ∑ 
                         k 
                       
                       ⁢ 
                       
                         [ 
                         
                           
                             ∑ 
                             
                               r 
                               = 
                               0 
                             
                             2 
                           
                           ⁢ 
                           
                             [ 
                             
                               
                                 g 
                                 
                                   ( 
                                   
                                     
                                       j 
                                       · 
                                       K 
                                     
                                     + 
                                     
                                       k 
                                       · 
                                       3 
                                     
                                     + 
                                     r 
                                   
                                 
                               
                               · 
                               
                                 sub 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       u 
                                       
                                         〈 
                                         k 
                                         〉 
                                       
                                     
                                     , 
                                     
                                       i 
                                       - 
                                       r 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   
                     hsn 
                     ⁢ 
                     
                         
                     
                   
                 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             E 
             := 
             
                 
             
             ⁢ 
             
               Re 
               ⁡ 
               
                 ( 
                 
                   elgenvecs 
                   ⁡ 
                   
                     ( 
                     MQ 
                     ) 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             λ 
             := 
             
               elgenvals 
               ⁢ 
               
                   
               
               ⁢ 
               
                 ( 
                 MQ 
                 ) 
               
             
           
         
       
       
         
           
             pick 
             := 
             
               
                 
                   K 
                   · 
                   chans 
                   · 
                   3 
                 
                 2 
               
               - 
               1 
             
           
         
       
       
         
           
             n 
             := 
             
               
                 0 
                 ⁢ 
                 … 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   K 
                   · 
                   chans 
                   · 
                   3 
                 
               
               - 
               pick 
               - 
               1 
             
           
         
       
       
         
           
             
               n 
               ′ 
             
             := 
             
               
                 0 
                 ⁢ 
                 … 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   K 
                   · 
                   chans 
                   · 
                   3 
                 
               
               - 
               pick 
               - 
               1 
             
           
         
       
       
         
           
             EL 
             := 
             
               elgenvecs 
               ⁡ 
               
                 ( 
                 ML 
                 ) 
               
             
           
         
       
       
         
           
             
               λ 
               ⁢ 
               
                   
               
               ⁢ 
               L 
             
             := 
             
               eigenvals 
               ⁡ 
               
                 ( 
                 ML 
                 ) 
               
             
           
         
       
       
         
           
             c 
             := 
             
               EL 
               
                 〈 
                 n 
                 〉 
               
             
           
         
       
       
         
           
             d 
             := 
             
               
                 ∑ 
                 n 
               
               ⁢ 
               
                 ( 
                 
                   
                     c 
                     n 
                   
                   · 
                   
                     E 
                     
                       ( 
                       
                         n 
                         + 
                         pick 
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     The transfer function for each transducer is plotted in  FIG. 9 e    and is calculated from: 
     
       
         
           
             
               hs 
               
                 i 
                 , 
                 j 
               
             
             := 
             
               
                 ∑ 
                 k 
               
               ⁢ 
               
                 [ 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       0 
                     
                     2 
                   
                   ⁢ 
                   
                     [ 
                     
                       
                         d 
                         
                           
                             
                               ( 
                               
                                 
                                   j 
                                   · 
                                   K 
                                 
                                 + 
                                 k 
                               
                               ) 
                             
                             · 
                             3 
                           
                           + 
                           n 
                         
                       
                       · 
                       
                         sub 
                         ⁡ 
                         
                           ( 
                           
                             
                               u 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                             , 
                             
                               i 
                               - 
                               n 
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
                 ] 
               
             
           
         
       
     
       FIG. 10 a    plots the quiet and loud responses against time which are produced using these transfer functions. They are calculated from: 
     
       
         
           
             Q 
             := 
             
               
                 ∑ 
                 j 
               
               ⁢ 
               
                 convol 
                 ⁡ 
                 
                   ( 
                   
                     
                       hs 
                       
                         ( 
                         j 
                         ) 
                       
                     
                     , 
                     
                       q 
                       
                         ( 
                         j 
                         ) 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             L 
             := 
             
               
                 ∑ 
                 j 
               
               ⁢ 
               
                 convol 
                 ⁡ 
                 
                   ( 
                   
                     
                       hs 
                       
                         ( 
                         j 
                         ) 
                       
                     
                     , 
                     
                       q 
                       
                         ( 
                         j 
                         ) 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                  
                 Q 
                  
               
               
                  
                 L 
                  
               
             
             = 
             
               3.887 
               × 
               
                 10 
                 
                   - 
                   3 
                 
               
             
           
         
       
     
     The maximum response has a magnitude which is approximate 4000 times greater than the minimum response. 
       FIG. 10 b    shows the results showing the filtered output signal from each transducer. 
     Again the quiet and loud responses are plotted against time and they are calculated from: 
     
       
         
           
             Q 
             := 
             
               convol 
               ⁡ 
               
                 ( 
                 
                   TB 
                   , 
                   Q 
                 
                 ) 
               
             
           
         
       
       
         
           
             L 
             := 
             
               convol 
               ⁡ 
               
                 ( 
                 
                   TB 
                   , 
                   L 
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                  
                 Q 
                  
               
               
                  
                 L 
                  
               
             
             = 
             
               2.382 
               × 
               
                 10 
                 
                   - 
                   3 
                 
               
             
           
         
       
     
     The maximum response has a magnitude which is approximate 2000 times greater than the minimum response. 
     The apparatus described above may be implemented at least in part in software. Those skilled in the art will appreciate that the apparatus described above may be implemented using general purpose computer equipment or using bespoke equipment. 
     The hardware elements, operating systems and programming languages of such computers are conventional in nature, and it is presumed that those skilled in the art are adequately familiar therewith. Of course, the computing functions may be implemented in a distributed fashion on a number of similar platforms, to distribute the processing load. 
     Here, aspects of the methods and apparatuses described herein can be executed on a mobile device and on a computing device such as a server. Program aspects of the technology can be thought of as “products” or “articles of manufacture” typically in the form of executable code and/or associated data that is carried on or embodied in a type of machine readable medium. “Storage” type media include any or all of the memory of the mobile stations, computers, processors or the like, or associated modules thereof, such as various semiconductor memories, tape drives, disk drives, and the like, which may provide storage at any time for the software programming. All or portions of the software may at times be communicated through the Internet or various other telecommunications networks. Such communications, for example, may enable loading of the software from one computer or processor into another computer or processor. Thus, another type of media that may bear the software elements includes optical, electrical and electromagnetic waves, such as used across physical interfaces between local devices, through wired and optical landline networks and over various air-links. The physical elements that carry such waves, such as wired or wireless links, optical links or the like, also may be considered as media bearing the software. As used herein, unless restricted to tangible non-transitory “storage” media, terms such as computer or machine “readable medium” refer to any medium that participates in providing instructions to a processor for execution. 
     Hence, a machine readable medium may take many forms, including but not limited to, a tangible storage carrier, a carrier wave medium or physical transaction medium. Non-volatile storage media include, for example, optical or magnetic disks, such as any of the storage devices in computer(s) or the like, such as may be used to implement the encoder, the decoder, etc. shown in the drawings. Volatile storage media include dynamic memory, such as the main memory of a computer platform. Tangible transmission media include coaxial cables; copper wire and fiber optics, including the wires that comprise the bus within a computer system. Carrier-wave transmission media can take the form of electric or electromagnetic signals, or acoustic or light waves such as those generated during radio frequency (RF) and infrared (IR) data communications. Common forms of computer-readable media therefore include for example: a floppy disk, a flexible disk, hard disk, magnetic tape, any other magnetic medium, a CD-ROM, DVD or DVD-ROM, any other optical medium, punch cards, paper tape, any other physical storage medium with patterns of holes, a RAM, a PROM and EPROM, a FLASH-EPROM, any other memory chip or cartridge, a carrier wave transporting data or instructions, cables or links transporting such a carrier wave, or any other medium from which a computer can read programming code and/or data. Many of these forms of computer readable media may be involved in carrying one or more sequences of one or more instructions to a processor for execution. 
     Those skilled in the art will appreciate that while the foregoing has described what are considered to be the best mode and, where appropriate, other modes of performing the invention, the invention should not be limited to specific apparatus configurations or method steps disclosed in this description of the preferred embodiment. It is understood that various modifications may be made therein and that the subject matter disclosed herein may be implemented in various forms and examples, and that the teachings may be applied in numerous applications, only some of which have been described herein. It is intended by the following claims to claim any and all applications, modifications and variations that fall within the true scope of the present teachings. Those skilled in the art will recognize that the invention has a broad range of applications, and that the embodiments may take a wide range of modifications without departing from the inventive concept as defined in the appended claims. 
     Although the present invention has been described in terms of specific exemplary embodiments, it will be appreciated that various modifications, alterations and/or combinations of features disclosed herein will be apparent to those skilled in the art without departing from the spirit and scope of the invention as set forth in the following claims.