Abstract:
An analog circuit system for generating output signals whose curve shape, at least sectionally, corresponds or is approximate to an elliptic function. Standard analog components such as adders, integrators, multipliers and differential amplifiers can be interconnected in order to simulate elliptic time functions from the standpoint of circuit engineering.

Description:
FIELD OF THE INVENTION  
       [0001]     The present invention relates to an analog circuit system having a plurality of analog computing circuits for generating elliptic functions.  
       BACKGROUND TECHNOLOGY  
       [0002]     Elliptic functions and integrals are used in numerous applications in engineering practice. The elliptic functions occurring frequently are the so-called Jacobi elliptic functions sn(x,k), cn(x,k), dn(x,k). The characteristic of the function sn(x,k) is similar to the sine function, while the function cn(x,k) is similar to the cosine function. For k=0, the functions sn(x,0) and cn(x,0) change into the sine function and cosine function, respectively. The value of k lies mostly in the interval [0, 1].  
         [0003]     Elliptic functions play a role in information and communication technology, e.g., in the design of Cauer filters, because some parameters of the Cauer filter are linked by elliptic functions. German patent reference 102 49 050.3 apparently describes a method and an arrangement for 20- adjusting an analog filter with the aid of elliptic functions.  
         [0004]     Elliptic functions are likewise used in the two-dimensional representation, interpolation or compression of data, for example, see German patent reference 102 48 543.7.  
       SUMMARY OF INVENTION  
       [0005]     The present invention provides for analog circuit systems that are able to electrically simulate elliptic functions.  
         [0006]     For example, an analog circuit system has a plurality of analog computing circuits such as analog multipliers, adders, integrators, differential amplifiers and dividers, which generate at least one output signal whose curve shape, at least sectionally, corresponds or is approximate to an elliptic function.  
         [0007]     In embodiments of the present invention, Jacobi elliptic functions are electrically simulated by the analog circuit system.  
         [0008]     In embodiments of the present invention, an analog circuit system includes analog multipliers and integrators which are able to deliver three output signals whose curve shapes, at least sectionally, correspond or are approximate to the Jacobi elliptic time functions  
               sn   (           2   ⁢           ⁢     π   ^       T     ·   t     ,   k     )     ,             cn   (           2   ⁢     π   ^       T     ·   t     ,   k     )     ⁢           ⁢   and   ⁢           ⁢       dn   (           2   ⁢           ⁢     π   ^       T     ·   t     ,   k     )     .               
 
 In these time functions, k is the module of the elliptic functions, f=1/T is the frequency of the elliptic time functions, and  
           π   ^     =     π     M   (     1   ,       1   -     k   2           )         ,       
 
 where M(1, √{square root over (1−k 2 )}) represents the so-called arithmetic-geometric mean of 1 and √{square root over (1−k 2 )}. The value k lies mostly in the interval [0, 1]. 
 
         [0009]     An application case can frequently occur in which a specific output signal is assigned to an input signal. Therefore, in embodiments of the present invention, a plurality of analog computing circuits are interconnected in such a way that, given an input variable x, output variable y is an elliptic function of x.  
         [0010]     If a triangle function is applied as input signal to a circuit system, which, for example, realizes sn(x), an elliptic time function is obtained at the output.  
         [0011]     A circuit system able to generate this functional relationship has a first multiplier, at whose one input an input signal having the quantity x, for example, a triangular input signal, is applied, and at whose other input the factor (1−k 2 )/2 is applied. A second multiplier can be provided, at whose one input the triangular input signal is applied, and at whose other input the factor (1+k 2 )/2 is applied. A differential amplifier is connected to the output of the second multiplier, a further input of the differential amplifier being connected to ground. An adder is also provided which is connected to the output of the first multiplier and the output of the differential amplifier. Present at the output of the adder is an output signal U a  which is combined or linked with the input signal by the Jacobi elliptic function sn(U e ).  
         [0012]     Further elliptic functions may be realized with the aid of an analog division device. To generate an output signal according to the elliptic function  
         sd   (           2   ⁢           ⁢     π   ^       T     ·   t     ,   k     )     ,       
 
 output signals  
         sn   (           2   ⁢           ⁢     π   ^       T     ·   t     ,   k     )     ⁢           ⁢   and   ⁢           ⁢     dn   (           2   ⁢           ⁢     π   ^       T     ·   t     ,   k     )         
 
 are applied to the analog division device. To generate an output signal according to the elliptic function  
         cd   (           2   ⁢           ⁢     π   ^       T     ·   t     ,   k     )     ,       
 
 output signals  
         cn   (           2   ⁢     π   ^       T     ·   t     ,   k     )     ⁢           ⁢   and   ⁢           ⁢     dn   (           2   ⁢           ⁢     π   ^       T     ·   t     ,   k     )         
 
 are applied to the inputs of the analog division device. 
 
         [0013]     In many cases, one wants to selectively control or influence the frequency  
         f   =     1   T       ,       
 
 as well as the value k of an elliptic function. An exemplary application case is, for example, the voltage-controlled change of frequency f, oscillation period T or module k. For this purpose, one should specifically select the value of, frequency f and the value of {circumflex over (π)}. As mentioned above, the variables {circumflex over (π)} and π can have the following relationship:  
         π   ^     =     π     M   (     1   ,       1   -     k   2           )           
 
         [0014]     For this reason, the arithmetic-geometric mean M(1, √{square root over (1−k 2 )}) can be simulated with the aid of analog computing circuits.  
         [0015]     In embodiments of the present invention, at least one analog computing circuit is provided, at whose first input, the value 1 is applied, and at whose second input, the factor √{square root over (1−k 2 )} is applied. The arithmetic mean of the two input signals is present at the first output of the analog computing circuit, whereas the geometric mean of the two input signals is present at the second output of the analog computing circuit. Moreover, an analog computing circuit, connected to the outputs of the analog computing devices or circuits, is provided for calculating the arithmetic mean, which corresponds approximately to the arithmetic-geometric mean 
 
M(1, √{square root over (1−k 2 )}) of 1 and √{square root over (1−k 2 )}. 
 
         [0016]     An alternative analog circuit system for generating the arithmetic-geometric mean M(1, √{square root over (1−k 2 )}) has one analog computing circuit for calculating the minimum from two input signals, one analog computing circuit for calculating the maximum from two input signals, one analog computing circuit for calculating the arithmetic mean from two input signals, and one analog computing circuit for calculating the geometric mean from two input signals. The output of the analog computing circuit for calculating the minimum is connected to an input of the analog computing circuit for calculating the arithmetic mean and an input of the analog computing circuit for calculating the geometric mean. The output of the analog computing circuit for calculating the maximum is connected to another input of the analog computing circuit for calculating the arithmetic mean and another input of the analog computing circuit for calculating the geometric mean. One input of the analog computing circuit for calculating the minimum is connected to the output of the analog computing circuit for calculating the arithmetic mean, the value 1 being applied to the other input. One input of the analog computing circuit for calculating the maximum is connected to the output of the analog computing circuit for calculating the geometric mean, the value √{square root over (1−k 2 )} being applied to the other input.  
         [0017]     Consequently, the arithmetic-geometric mean M o f 1 and √{square root over (1−k 2 )} is present at the output of the analog computing circuit for calculating the geometric mean and at the output of the analog computing circuit for calculating the arithmetic mean.  
         [0018]     To be able to provide the value {circumflex over (π)} in terms of circuit engineering, a device, for example, a divider, is provided, at whose inputs, the arithmetic-geometric mean M(1, √{square root over (1−k 2 )}) and the number π are applied. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0019]      FIG. 1  shows an analog circuit system for generating three output signals, each corresponding to a Jacobi elliptic time function.  
         [0020]      FIG. 2  shows an analog circuit system for generating an output signal which corresponds to the Jacobi elliptic time function  
         sn   (         2   ⁢           ⁢     π   ^       T     ·   t     )     .         
         [0021]      FIG. 3  shows an analog circuit system for generating an output signal which is combined with a triangular input signal by the Jacobi elliptic time function sn(U e ).  
         [0022]      FIG. 4  shows an analog circuit system which, from two input signals, supplies an estimate for the arithmetic-geometric mean M.  
         [0023]      FIG. 5  shows an alternative analog circuit system for calculating the arithmetic-geometric mean M from two input signals.  
         [0024]      FIG. 6  shows a divider for generating the value {circumflex over (π)}. 
     
    
     DETAILED DESCRIPTION  
       [0025]     Herein, analog circuit systems are discussed which generate at least one output signal whose curve shape corresponds or is approximate to a Jacobi elliptic time function. The so-called Jacobi elliptic functions sn(x,k), cn(x,k) and dn(x,k) are used in the following embodiment. In considering time functions, the variable x is replaced by t in the above functions, and, to simplify matters, the value of k is omitted in the following formulas.  
         [0026]     Under these conditions, the following well-known equations may be indicated with respect to the Jacobi elliptic functions:  
                 ⅆ     ⅆ   t       ⁢     sn   ⁡     (   t   )         =       cn   ⁡     (   t   )       ·     dn   ⁡     (   t   )                 (   1   )                   ⅆ     ⅆ   t       ⁢     cn   ⁡     (   t   )         =       -     sn   ⁡     (   t   )         ·     dn   ⁡     (   t   )                 (   2   )                   ⅆ     ⅆ   t       ⁢     dn   ⁡     (   t   )         =       -     k   2       ⁢       sn   ⁡     (   t   )       ·       cn   ⁡     (   t   )       .                 (   3   )             
 
         [0027]     Further, descriptions regarding elliptic functions may be found, inter alia, in the reference “Vorlesungen über allgemeine Funktionentheorie und elliptischen Funktionen,” A. Hurwitz, Springer Verlag, 2000, page 204.  
         [0028]     To permit electrical simulation of elliptic functions in which frequency f can be changed, it is necessary, similarly as in the case of the circular functions, to take into account corresponding multiplicative constants which appear in conjunction with variable t. Instead of circular constant π, constant {circumflex over (π)} is used. Variable {circumflex over (π)} has the following relation with variable π:  
               π   ^     =     π     M   ⁡     (     1   ,       1   -     k   2           )                 (   4   )             
 
         [0029]     The function M(1, √{square root over (1−k 2 )}) forms the so-called arithmetic-geometric mean of 1 and (√{square root over (1−k 2 )}).  
         [0030]     With period duration T and the insertion of {circumflex over (π)}, the following differential equations result:  
                 ⅆ     ⅆ   t       ⁢   s   ⁢           ⁢     n   ⁡     (         2   ⁢           ⁢     π   ^       T     ·   t     )         =           2   ⁢           ⁢     π   ^       T     ·   c     ⁢           ⁢       n   ⁡     (         2   ⁢           ⁢     π   ^       T     ·   t     )       ·   d     ⁢           ⁢     n   ⁡     (         2   ⁢           ⁢     π   ^       T     ·   t     )                 (   5   )                   ⅆ     ⅆ   t       ⁢   c   ⁢           ⁢     n   ⁡     (         2   ⁢           ⁢     π   ^       T     ·   t     )         =         -       2   ⁢           ⁢     π   ^       T       ·   s     ⁢           ⁢       n   ⁡     (         2   ⁢           ⁢     π   ^       T     ·   t     )       ·   d     ⁢           ⁢     n   ⁡     (         2   ⁢           ⁢     π   ^       T     ·   t     )                 (   6   )                   ⅆ     ⅆ   t       ⁢   d   ⁢           ⁢     n   ⁡     (         2   ⁢           ⁢     π   ^       T     ·   t     )         =       -     k   2       ⁢         2   ⁢           ⁢     π   ^       T     ·   s     ⁢           ⁢       n   ⁡     (         2   ⁢           ⁢     π   ^       T     ·   t     )       ·   c     ⁢           ⁢     n   ⁡     (         2   ⁢           ⁢     π   ^       T     ·   t     )                 (   7   )             
 
 where f=1/T is the frequency of the elliptic functions. 
 
         [0031]      FIG. 1  shows an analog circuit system which generates three output signals whose curve shapes correspond to the Jacobi elliptic functions.  
         [0032]     In  FIG. 1 , a multiplier  10 , a multiplier  20 , and an analog integrator  30 , are connected in series. Moreover, an analog multiplier  40 , an analog multiplier  50 , and a further analog integrator  60 , are connected in series. A third series connection includes a further analog multiplier  70 , an analog multiplier  80 , and an analog integrator  90 . Analog multiplier  20  multiplies the output signal of multiplier  10  by the factor 2 {circumflex over (π)}/T. Multiplier  50  multiplies the output signal of multiplier  40  by the factor  
       -         2   ⁢           ⁢     π   ^       T     .         
 
 Multiplier  80  multiplies the output signal of multiplier  70  by the factor  
         -     k   2       ⁢         2   ⁢           ⁢     π   ^       T     .         
 
         [0033]     The output signal of integrator  30  is coupled back to multiplier  40  and to the input of multiplier  70 . The output signal of integrator  60  is coupled back to the input of multiplier  10  and to the input of multiplier  70 . The output of integrator  90  is coupled back to the input of multiplier  40  and to the input of multiplier  10 . Measures, available in circuit engineering, for taking into account predefined initial states during initial operation are not marked in the circuit. Such an analog circuit system, shown in  FIG. 1 , delivers the Jacobi elliptic time function sn(2 {circumflex over (π)} ft) at the output of integrator 30, the Jacobi elliptic function cn(2 {circumflex over (π)} ft) at the output of integrator  60 , and the Jacobi elliptic function dn(2 {circumflex over (π)} ft) at the output of integrator  90 . The multiplication by  
       ±       2   ⁢           ⁢     π   ^       T         
 
 in multipliers  20 ,  50 , respectively, and the multiplication by  
         -     k   2       ⁢       2   ⁢           ⁢     π   ^       T         
 
 in multiplier  80  may also be carried out in integrators  30 ,  60 ,  90 . The multiplication by k 2  may also be put at the output of integrator  90 . Moreover, in further embodiments, it is possible to add familiar stabilization circuits to the circuit system shown in  FIG. 1 . See, for example, reference “ Halbleiter Schaltungstechnik,”  Tietze, Schenk, Springer Verlag, 5 th  edition, 1980, Berlin, pages 435-438. 
 
         [0034]     All three Jacobi elliptic time functions sn(2 {circumflex over (π)} ft), cn(2 {circumflex over (π)} ft) and dn(2 {circumflex over (π)} ft) may be realized simultaneously using the analog circuit system shown in  FIG. 1 . In addition, the derivatives of the Jacobi elliptic time functions sn, cn and dn are obtained at the output of the multipliers  10 ,  40 ,  70 , respectively.  
         [0035]     If, for example, only the Jacobi elliptic time function sn((2 {circumflex over (π)} ft)) is to be realized using an analog circuit system, it is possible to get along with fewer multipliers by considering the differential equation of the second degree, valid for sn(2 {circumflex over (π)} ft), which may be derived from the differential equations indicated above. The differential equation of the second degree valid for sn(2 {circumflex over (π)} ft) reads:  
                   ⅆ   2       ⅆ     t   2         ⁢   s   ⁢           ⁢     n   ⁡     (         2   ⁢           ⁢     π   ^       T     ·   t     )         =         -       (       2   ⁢           ⁢     π   ^       T     )     2       ·   s     ⁢           ⁢       n   ⁡     (         2   ⁢           ⁢     π   ^       T     ·   t     )       ·     (     1   +     k     2   -       -     2   ⁢           ⁢     k   2     ⁢   s   ⁢           ⁢       n   2     ⁡     (         2   ⁢           ⁢     π   ^       T     ·   t     )           )                 (   8   )             
 
         [0036]     An exemplary analog circuit system which simulates this differential equation (8) is shown in  FIG. 2 .  
         [0037]     The analog circuit system has a multiplier  100  whose output is connected to a series-connected multiplier  110 . Moreover, the factor −2k 2  is applied to the input of multiplier  110 . The output of multiplier  110  is connected to an input of an adder  120 . The factor 1+k 2  is applied to a second input of adder  120 . The output of adder  120  is connected to the input of a multiplier  130 . The factor  
       -       (       2   ⁢           ⁢     π   ^       T     )     2         
 
 is applied to a further input of multiplier  130 . The output of multiplier  130  is connected to an input of a multiplier  140 . The output of multiplier  140  is connected to an input of an integrator  150 . The output of integrator  150  is connected to the input of an integrator  160 . The output of integrator  160  is coupled back to the input of multiplier  140  and to two inputs of multiplier  100 . In this way, an output signal whose curve shape corresponds to the Jacobi elliptic time function  
       s   ⁢           ⁢     n   ⁡     (         2   ⁢           ⁢     π   ^       T     ·   t     )           
 
 appears at the output of integrator  160 . 
 
         [0038]     The multiplication by the factor  
         (       2   ⁢           ⁢     π   ^       T     )     2       
 
 may expediently be carried out again in integrators  150  and  160 . 
 
         [0039]     In  FIG. 3 , an exemplary embodiment is described in which a functional relationship corresponding to the Jacobi elliptic function sn(2 {circumflex over (π)} ft) approximatively exists between an input signal and an output signal.  
         [0040]     The analog circuit system shown in  FIG. 3  includes a differential amplifier  170 , a multiplier  180 , a multiplier  190  and an adder  200 . An input signal having a triangular voltage curve is applied, for example, at each input of the multipliers  180 ,  190 . Moreover, the factor (1−k 2 )/2 is applied to multiplier  180 , whereas the factor (1+k 2 )/2 is applied to multiplier  190 . The output signal of multiplier  190  is fed to differential amplifier  170 . The second input of the differential amplifier is connected to ground. The output of multiplier  180  and the output of differential amplifier  170  are connected to the inputs of adder  200 .  
         [0041]     Because of the fact that differential-amplifier circuit  70  has a relation between input signal U e  and output signal U a  according to the equation  
                 U   a     =     R   ·   I   ·     tanh   ⁡     (       U   e       2   ⁢           ⁢     U   T         )           ,           (   9   )             
 
 given suitably selected parameters of the differential amplifier, the circuit system shown in  FIG. 3  generates at the output, a signal U a , which is approximatively combined with input signal U e  via the Jacobi elliptic function sn. Notably, combining or linking an output signal and an input signal via the Jacobi elliptic function cn or dn in a circuit system is available knowledge in the art. 
 
         [0042]     To be able to generate further elliptic functions, a division device (not shown) may be connected in series to the circuit system shown in  FIG. 1 . For instance, to generate the elliptic function sd(x)=sn(x)/dn(x), the output signals of the integrators  30 ,  60  may be fed (or added) to the division device. Furthermore, the output signals of the integrators  60 ,  90  may be fed to the division device, in order to generate the elliptic function cd(x)=cn(x)/dn(x).  
         [0043]     In embodiments, it may be desirable to selectively control frequency f or the value of k.  
         [0044]     According to equation (4), it is possible to change the value {circumflex over (π)} by changing the value k. That is to say, {circumflex over (π)} and therefore k may be calculated by calculating the arithmetic-geometric mean M(1, √{square root over (1−k 2 )}). One possibility for altering the frequency of the Jacobi elliptic functions generated using the circuit system according to  FIG. 1  is to feed a selectively altered value for {circumflex over (π)} to the multipliers  20 ,  50 ,  80 .  
         [0045]     To be able to generate {circumflex over (π)} in terms of circuit engineering, the arithmetic-geometric mean M(1, √{square root over (1−k 2 )}) may be realized, for example, using an analog circuit system which is shown in  FIG. 4 . The circuit system shown in  FIG. 4  is made up of a plurality of analog computing circuits  210 ,  220 ,  230 , denoted by AG, as well as an analog computing circuit  240  for calculating the arithmetic mean from two input signals. Some analog computing circuits  210 ,  220 ,  230  are adapted in such a way that they generate the arithmetic mean of the two input signals at one output, and the geometric mean of the two input signals at the other output. As shown in  FIG. 4 , the factor 1 is applied to the first input of analog computing circuit  210 , and the factor √{square root over (1−k 2 )} is applied to its other input. On condition that the factor √{square root over (1−k 2 )} lies between 0 and 1, the output signal of analog computing circuit  240  corresponds approximately to the arithmetic-geometric mean M of the factors 1 and √{square root over (1−k 2 )} applied to the inputs of analog computing circuit  210 .  
         [0046]      FIG. 5  shows an alternative analog circuit system for calculating the arithmetic-geometric mean M of the two factors 1 and √{square root over (1−k 2 )}. The circuit system shown in  FIG. 5  has an analog computing circuit  250  for calculating the minimum from two input signals, an analog computing circuit  260  for calculating the maximum from two input signals, an analog computing circuit  270  for calculating the arithmetic mean from two input signals and an analog computing circuit  280  for calculating a geometric mean from two input signals. The factor 1 is applied to an input of analog computing circuit  250 , whereas the factor √{square root over (1−k 2 )} is applied to an input of analog computing circuit  260 . The output of analog computing circuit  250  for calculating the minimum from two input signals is connected to the input of analog computing circuit  270  and analog computing circuit  280 . The output of analog computing circuit  260  for calculating the maximum from two input signals is connected to an input of analog computing circuit  270  and an input of analog computing circuit  280 . The output of analog computing circuit  270  is connected to an input of analog computing circuit  250 , whereas the output of analog computing circuit  280  is connected to an input of analog computing circuit  260 . In the analog circuit system shown in  FIG. 5 , the outputs of analog computing circuits  270  and  280  in each case supply the arithmetic-geometric mean M of 1 and √{square root over (1−k 2 )}.  
         [0047]     Transit-time effects, which can be handled with methods (e.g., sample-and-hold elements) generally used in circuit engineering, are not taken into account in the technical implementation of the circuit system according to  FIG. 5 .  
         [0048]     At this point, {circumflex over (π)} i may be calculated via a division device  290 , shown in  FIG. 6 , at whose inputs are applied the number π and the arithmetic-geometric mean M(1, √{square root over (1−k 2 )}), which is generated, for example, by the circuit shown in  FIG. 4  or in  FIG. 5 .  
         [0049]     In this way, selectively altered values for {circumflex over (π)} may be fed to multipliers  20 ,  50 ,  80  of the circuit system according to  FIG. 1 , which means the frequency response of the output functions may be selectively influenced.