Abstract:
A method for modulating a carrier signal used for transmitting analog or digital message signals is provided. The module k of elliptic functions is used as a modulation parameter instead of the amplitude or the frequency. The carrier signal modulated according to this modulation method is provided with a constant amplitude and a fixed frequency while the signal form is chronologically modified at the rhythm of the message that is to be transmitted.

Description:
FIELD OF THE INVENTION  
       [0001]     The present invention relates to a method for modulating a carrier signal for the transmission of message signals. The present invention also relates to a method for demodulating such modulated carrier signals. The present invention also relates to an analog circuit configuration for modulating a carrier signal that may be represented by an elliptic function.  
       BACKGROUND TECHNOLOGY  
       [0002]     In information technology, high-frequency, sine-shaped or cosine-shaped carrier signals are generally utilized so as to be able to transmit information such as language, music, images or data. To this end, the message to be transmitted is modulated onto a carrier signal. Available modulation methods are the angle and amplitude modulation. In amplitude modulation the information contained in the message signal m(t) is modulated onto the carrier signal essentially according to the equation s(t)=(a 0 +c·m(t))·sin (2πf 0 t), where f 0  denotes the carrier frequency, and a 0  and c are constants that are selected according to the practical requirements. A characteristic property of amplitude modulation is that the amplitude of the signal s(t) is modulated in the rhythm of message m(t) to be transmitted, frequency f 0  of the modulated carrier signal not being able to be varied over time.  
         [0003]     In the available angle modulation, the frequency or the phase is varied over time in the rhythm of the message signal m(t) to be transmitted. The frequency-modulated signal transmitted via a transmission channel is s(t)=a 0 ·sin (2{circumflex over (π)}f(m(t))), where frequency f(m(t)) in most cases being defined by the expression (f 0 +c m(t)). In a frequency modulation amplitude a 0  is constant.  
       SUMMARY OF THE INVENTION  
       [0004]     Embodiments of the present invention may involve adding a new modulation and demodulation method to available modulation and demodulation methods.  
         [0005]     Additional embodiments of the present invention may involve providing an analog modulator circuit for the new modulation method.  
         [0006]     Additional embodiments of the present invention may involve applying a so-called signal shape modulation method in which—in contrast to the amplitude and angle modulation—neither amplitude a 0  nor frequency f 0  is varied over time in the rhythm of the message signal to be transmitted. Instead, the signal shape of the carrier signal itself is varied.  
         [0007]     A method for modulating a carrier signal for the transmission of message signals is described herein. In embodiments of the present invention, the signal shape of the carrier signal may be varied over time by a message signal to be transmitted, the amplitude and the frequency of the carrier signal remaining constant.  
         [0008]     For the purpose of delimiting it from the classic amplitude and frequency modulation, the new modulation method also will be referred to as the signal shape modulation method.  
         [0009]     The signal shape modulation method may be based on the modulation of carrier signals whose time characteristic is defined by an elliptic function. Jacobian elliptic functions, which, for example, are described in the book by A. Hurwitz, “Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen” [i.e., “Lectures on general function theory and elliptic functions”], 5 th  edition, Springer Berlin Heidelberg New York, 2000, incorporated in its entirety by reference herein, may be utilized.  
         [0010]     In embodiments of the present invention, neither amplitude nor frequency but modulus k, which determines the form of an elliptic function, may be used as modulation parameters. Modulus k may be varied over time by the message signal to be transmitted so as to modulate the signal shape of the carrier signal in the rhythm of the message signal to be transmitted.  
         [0011]     The time characteristic of the modulated carrier signal may be defined by the elliptic function s(t)=a 0 sx(2{circumflex over (π)}f 0 t,k(t)) a 0  being the amplitude and f 0  the frequency. {circumflex over (π)} and modulus k may be linked via the complete elliptic integral of the first kind.  
         [0012]     In embodiments of the present invention, the function sx(2{circumflex over (π)}f 0 t,k(t)) for 0≦k(t)≦1 may be defined by the Jacobian elliptic function sn(2{circumflex over (π)}f 0 t,k(t)), and for −1≦k(t)≦0 by the Jacobian elliptic function cn(2{circumflex over (π)}f 0 (t−T/4), |k(t)|).  
         [0013]     In embodiments of the present invention, using elliptic functions, available orthogonal transmission methods based on sine and cosine carriers may be generalized, thus making it possible to use new orthogonal modulation methods. Orthogonal carrier signals which are defined by the two orthogonal elliptic functions sn(2{circumflex over (π)}f 0 t,k(t)) and sd(2{circumflex over (π)}f 0 t,k(t)), or by the two orthogonal elliptic functions cd(2{circumflex over (π)}f 0 t,k(t)) and cn(2{circumflex over (π)}f 0 t,k(t)), may be utilized toward this end.  
         [0014]     In embodiments of the present invention, the carrier signals defined by an elliptic function may be generated using an analog circuit configuration. Analog circuit configurations may be made up of operational amplifiers, integrators, multipliers, differential amplifiers and dividers known per se. Analog circuit configurations for generating elliptic functions are described in the patent application bearing Attorney Docket No. 2345/217, having title “Analog Circuit System for Generating Elliptic Functions,” filed as International Application No. PCT/DE2004/000223, and being filed as a U.S. patent application on Nov. 2, 2005, which is hereby incorporated in its entirety by reference.  
         [0015]     Embodiments of the present invention may involve a method for demodulating a modulated carrier signal is provided whose time characteristic is described by elliptic function s(t)=a 0 ·sx(2{circumflex over (π)}f 0 ·t, k(t)). a 0  is the amplitude and f 0  is the frequency of the carrier signal, {circumflex over (π)} and modulus k being linked via the complete elliptic integral of the first kind.  
         [0016]     In embodiments, for demodulation, the received modulated carrier signal may be sampled at instants that correspond to the odd multiples of T/8, with T=1/f 0 . Modulus k(t)—and hence transmitted message signal m(t)—may be obtained from the sampling values.  
         [0017]     In alternative embodiments, i.e., an alternative demodulation method, received modulated carrier signal s(t)=a 0 ·sx(2{circumflex over (π)}f 0 ·t, k(t)) may be integrated in order to obtain modulus k(t).  
         [0018]     In alternative embodiments, i.e., another alternative demodulation method, received modulated carrier signal s(t)=a 0 ·sx(2{circumflex over (π)}f 0 ·t,k(t)) may be squared and then integrated.  
         [0019]     In embodiments, the modulator may be distinguished by the fact that the modulation of the carrier signal is implemented in such a way that the signal shape of the carrier signal is able to be varied over time by a message signal to be transmitted, the amplitude and the frequency of the carrier signal remaining constant.  
         [0020]     In embodiments, a special development of the modulator may have an analog circuit configuration which provides at least one modulated carrier signal whose curve profile corresponds to or approximates an elliptic function at least in sections.  
         [0021]     In embodiments, the elliptic functions may be Jacobian elliptic functions.  
         [0022]     In embodiments, since the modulator modulates neither the amplitude nor the frequency of the carrier signal, devices may be provided that vary modulus k of an elliptic function over time by the message signal to be transmitted in order to modulate the signal shape of the carrier signal in the rhythm of the message signal to be modulated.  
         [0023]     In embodiments, the analog circuit configuration of the modulator may generate a modulated carrier signal whose time characteristic is defined by the elliptic function s(t)=a 0 ·sx(2{circumflex over (π)}f 0 ·t, k(t)), a 0  being the amplitude and f 0  the frequency of the carrier signal, {circumflex over (π)} and modulus k being linked via the complete elliptic integral of the first kind.  
         [0024]     In embodiments, the circuit configuration may have first analog multipliers as well as analog integrators which are interconnected in such a way that the circuit configuration provides the three output functions sn(2{circumflex over (π)}f 0 t,k(t) ); cn(2{circumflex over (π)}f 0 t,k(t)); and dn(2{circumflex over (π)}f 0 t,k(t)).  
         [0025]     In embodiments, an analog division device for forming quotient sn(2{circumflex over (π)}f 0 t,k(t))/dn(2{circumflex over (π)}f 0 t,k(t)), and a second analog multiplier, assigned to the division device, may be provided, which multiplies the output signal of the division device by factor √{square root over (1−k 2 )}. For 0=k(t)=1, output signal sn(2{circumflex over (π)}f 0 t,k(t)) forms the modulated carrier signal, whereas for −1=k(t)=0, the output signal of the second analog multiplier forms the modulated carrier signal. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0026]      FIG. 1  shows a quarter period of the curve shapes of a carrier signal modulated with the aid of modulus k, 0=k(t)=1.  
         [0027]      FIG. 2  shows a quarter period of the curve shapes of a carrier signal modulated with the aid of modulus k, −1=k(t)=0.  
         [0028]      FIG. 3  shows an exemplary modulator according to the present invention.  
         [0029]      FIG. 4  shows an exemplary circuit configuration for generating the elliptic function sn(2{circumflex over (π)}f 0 t).  
         [0030]      FIG. 5  shows a circuit configuration for calculating the arithmetic-geometric mean M.  
         [0031]      FIG. 6  shows an alternative circuit configuration for calculating the arithmetic-geometric mean M.  
         [0032]      FIG. 7  shows a circuit configuration for calculating {circumflex over (π)}.  
         [0033]      FIG. 8  shows section of the curve shape of a carrier signal modulated according to a binary shape jump method.  
     
    
     DETAILED DESCRIPTION  
       [0034]     In the following, a new modulation method for data transmission is described, which uses as modulation parameters not the amplitude or frequency of a carrier signal, but the signal shape. The new modulation method may be based on elliptic functions and is distinguished in that, in contrast to the amplitude modulation, the amplitude of the carrier signal remains unchanged and that, in contrast to the frequency modulation, the frequency of the carrier signal remains unchanged as well. As mentioned, the new modulation method may be based on the Jacobian elliptic functions sn(2{circumflex over (π)}f 0 t,k), cn(2{circumflex over (π)}f 0 t,k) and dn(2{circumflex over (π)}f 0 t,k). The second argument of Jacobian elliptic functions, value k, is called the modulus of the elliptic functions and—as described in more detail herein—is used as a new modulation parameter. In other words, for example, the modulus of Jacobian elliptic functions is modulated in accordance with a message m(t) to be transmitted. Modulus k thus becomes a function of time and is described by k(t). It is assumed here that the frequency of the message to be transmitted and thus the frequency of the change of k(t) is small with respect to frequency f 0 =1/T of the variation of the carrier signal. The modulated carrier signal transmitted via a message channel may be indicated by 
 
 s ( t )= a   0   ·sx (2 {circumflex over (π)}f   0   ·t, k ( t ))   (1) 
 
         [0035]     The role of π in the classic sine or cosine carrier signals is assumed by {circumflex over (π)} in elliptic functions. {circumflex over (π)} is a function of modulus k, the correlation between {circumflex over (π)} and k being given by the so-called complete elliptic integral of the first kind as follows:  
                 π   ^     2     =       K   ⁡     (   k   )       =       ∫   0     π   /   2       ⁢       d   ⁢           ⁢   φ         1   -       k   2     ⁢       sin   2     ⁡     (   ϕ   )                           (   2   )             
 
 {circumflex over (π)} may easily be calculated with the aid of the equation  
                 π   ^     =     π     M   ⁡     (     1   ,       1   -     k   2           )           ,           (   3   )             
 
 M( 1, √{square root over (1−k     2     )}  being the arithmetic-geometric mean of 1 and √{square root over (1−K 2 )}. 
 
         [0036]     Analog circuit configurations for calculating the arithmetic-geometric mean are shown in  FIGS. 5 and 6 . To be able to generate {circumflex over (π)} in terms of circuit engineering, first of all, the arithmetic-geometric mean M( 1, √{square root over (1−k     2     )} ) may be realized, for example, using an analog circuit configuration, which is shown in  FIG. 5 . The circuit configuration shown in  FIG. 5  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. Analog computing circuits  210  through  230  are implemented 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. 5 , the value 1 is applied to the first input of analog computing circuit  210 , and the value √{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 circuit device or analog computing circuit  240  corresponds approximately to the arithmetic-geometric mean M of the values 1 and √{square root over (1−k 2 )} applied to the inputs of analog computing circuit  210 .  
         [0037]      FIG. 6  shows an alternative analog circuit configuration for calculating the arithmetic-geometric mean M of the two values 1 and √ {square root over (1−k     2     )} . The circuit configuration shown in  FIG. 6  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 value 1 is applied to an input of analog computing circuit  250 , whereas the value √{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 configuration shown in  FIG. 6 , 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 )}.  
         [0038]     At this point, {circumflex over (π)} may be calculated via a division device  290 , shown in  FIG. 7 , 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 instance, by the circuit shown in  FIG. 5  or in  FIG. 6 .  
         [0039]     A signal shape modulation of the carrier signal s(t) is implemented in accordance with the value of k, which varies over time; the zero crossings and the amplitude of the carrier signal remain unchanged, however.  FIG. 1  shows various curve shapes of a carrier signal, modulated in its signal shape, over a quarter period of the function sn(2{circumflex over (π)}f 0 t,k) for k=0, k=0.8, k=0.95 and k=0.99. It should be noted that for k=0 the elliptic function reproduces the sine function, and for k=1 it reproduces the hyperbolic tangent. While the period of hyperbolic tangent is infinite, it leads to a pulse nevertheless by the scaling with {circumflex over (π)}. The utilization of the elliptic function sn(2{circumflex over (π)}f 0 t,k) yields signal shapes that lie above the sine function for 0=t=T/4. To generate signal shapes below the sine function as well, the Jacobian elliptic function cn(2{circumflex over (π)}f 0 t,k) may be utilized. In order to obtain this function in the same phase position as the Jacobian elliptic function sn(2{circumflex over (π)}f 0 t,k), function cn, shifted by T/4, is considered, which may be expressed as follows:  
                     cn   ⁡     (       2   ⁢       π   ^     ⁡     (     t   -     T   /   4       )       ⁢     f   0       ,     k   ⁡     (   t   )         )       =         1   -     k   2         ⁢       sn   ⁡     (       2   ⁢     π   ^     ⁢     f   0     ⁢   t     ,     k   ⁡     (   t   )         )         dn   ⁡     (       2   ⁢     π   ^     ⁢     f   0     ⁢   t     ,     k   ⁡     (   t   )         )                       =         1   -     k   2         ⁢     sd   ⁡     (       2   ⁢     π   ^     ⁢     f   0     ⁢   t     ,     k   ⁡     (   t   )         )                       (   4   )             
 
         [0040]      FIG. 2  illustrates the function cn(2{circumflex over (π)}(t−T/4)f 0 ,k(t)) for k=0, k=0.8, k=0.95 and k=0.99. For k=0, the sine function is obtained again.  
         [0041]     It can be seen that a great variety of signal shapes may be covered by utilizing the Jacobian elliptic functions sn and cn. Accordingly, the function sx(2{circumflex over (π)}f 0 t,k(t)), defined in equation 1, may be defined as follows:  
               sx   ⁡     (       2   ⁢     π   ^     ⁢     f   0     ⁢   t     ,     k   ⁡     (   t   )         )       =     {                   ⁢         sn   ⁡     (       2   ⁢     π   ^     ⁢     f   0     ⁢   t     ,     k   ⁡     (   t   )         )       ⁢           ⁢   for   ⁢             ⁢             ⁢   0     ≤   k   ≤   1                     1   -       k   2     ⁢     sd   ⁡     (       s   ⁢     π   ^     ⁢     f   0     ⁢   t     ,        k          )       ⁢           ⁢   for     ⁢           -   1     ≤   k   ≤   0                       (   5   )             
 
         [0042]     In this equation, k is the modulation parameter carrying the message. The values of k lie within the interval [−1.1].  
         [0043]      FIG. 3  shows an exemplary modulator, which is composed of analog computing circuits and electrically simulates the function sx(2{circumflex over (π)}f 0 t,k(t)).  
         [0044]     According to  FIG. 3 , 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 circuit includes an additional analog multiplier  70 , an analog multiplier  80 , as well as 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     .         
 
         [0045]     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 .  
         [0046]     It should be noted that measures, available in circuit engineering, for taking into account predefined initial states during initial operation are not marked in in the circuit. Such an analog circuit configuration, shown in  FIG. 3 , delivers the Jacobian elliptic time function sn(2{circumflex over (π)}f 0 t) at the output of integrator  30 , the Jacobian elliptic function cn(2{circumflex over (π)}f 0 t) at the output of integrator  60 , and the Jacobian elliptic function dn(2{circumflex over (π)}f 0 t) at the output of integrator  90 . It should be noted that the multiplication by  
       ±       2   ⁢     π   ^       T         
 
 in multipliers  20  and  50 , respectively, and the multiplication by  
         -     k   2       ⁢       2   ⁢     π   ^       T         
 
 in multiplier  80  may also be carried out in integrators  30 ,  60  and  90 . The multiplication by k 2  may also be put at the output of integrator  90 . Furthermore, it is possible to add to the circuit configuration shown in  FIG. 3  available stabilizing circuits as they are described, for example, in the technical literature “Halbleiter Schaltungstechnik”, [Semiconductor Circuit Technology”], Tietze, Schenk, Springer Verlag, 5th edition, 1980, Berlin Heidelberg New York, pages 435-438. 
 
         [0047]     All three Jacobian elliptic time functions sn(2{circumflex over (π)}f 0 t), cn(2{circumflex over (π)}f 0 t) and dn(2{circumflex over (π)}f 0 t) may be realized simultaneously using the analog circuit configuration shown in  FIG. 3 . In addition, the derivatives of the Jacobian elliptic time functions sn, cn and dn may be obtained at the output of multipliers  10 ,  40  and  70 , respectively.  
         [0048]     Furthermore, a division device  96  is connected to the outputs of integrators  30  and  90  in order to generate the elliptic function √{square root over (1−k 2 )}sd(2{circumflex over (π)}f 0 t,k(t)) in conjunction with a multiplier  97 , which—as explained herein—corresponds to the elliptic function cn(2{circumflex over (π)}f 0 t,k(t)) shifted by T/4.  
         [0049]     As a result, the modulator may deliver at the output of integrator  30  a signal-shape-modulated carrier signal according to the Jacobian elliptic function sn(2{circumflex over (π)}f 0 t,k(t)), namely for 0≦k(t)≦1. At the output of multiplier  97 , the modulator is able to provide a signal-shape-modulated carrier signal according to the Jacobian elliptic function √{square root over (1−k 2 )}sd(2{circumflex over (π)}f 0 t,k(t)), namely for −1≦|k(t)|≦1.  
         [0050]     The signal-shape modulation is implemented via k or {circumflex over (π)} in multipliers  20 ,  50  and  80 . As mentioned, modulus k and {circumflex over (π)} are linked via the complete elliptic integral of the first kind.  
         [0051]      FIG. 7  illustrates an exemplary analog circuit for calculating {circumflex over (π)} as a function of message signal m(t) to be transmitted, which modulates modulus k.  
         [0052]     The signal-form modulation of carrier signal s(t) takes place in multiplier  80  via the expression −k 2 2 {circumflex over (π)}/T, in multiplier  50  via factor −2{circumflex over (π)}/T, and in multiplier  20  by factor 2{circumflex over (π)}/T.  
         [0053]     With the aid of the signal-shape modulation method, it is possible to modulate onto a carrier signal not only analog messages, but digital messages as well.  
         [0054]     A simple binary, so-called form-jump method or “Formsprungverfahren” method may be defined, for instance, by the agreement to send a carrier signal s(t) according to the elliptic function a 0 sn(2{circumflex over (π)}f 0 t) if a “1” is to be transmitted, and to transmit a carrier signal of the function a 0   √{square root over (1−k     2     )}sd( 2{circumflex over (π)}f 0 t) if a “0” is to be transmitted. In both cases modulation parameter k is set to 0.9, for instance. Under the simplified assumption that one bit is to be transmitted per period, the bit sequence “10” is transmitted by the two sequential signals. The corresponding curve shape is illustrated in  FIG. 8 .  
         [0055]     Hereinafter, three exemplary demodulation methods are indicated to recover transmitted message signal m(t) from received modulated carrier signal s(t).  
         [0056]     The first demodulation method is based on the fact that frequency f 0 =1/T of the carrier signal is fixed, and modulated carrier signal s(t) goes through zero twice every T seconds. At the instants zero and T/2, function s(t) has the zero value; at instants T/4 it has the value a 0 ; and at instant 3T/4 it has the value −a 0 . At instants T/8 and 3T/8, function value a 0 sx(T/8) results. At instants 5T/8 and 7T/8, the function value is  
         -     a   0       ⁢   sx   ⁢       T   8     .         
 
         [0057]     The value of  
       sx   ⁢     T   8         
 
 is equal to  1/√{square root over (1+k′)} for signal shapes above the sine function, and √{square root over (k′)}/√{square root over ( 1+K′)} for signal shapes below the sine function. Expression k′ is equal to √{square root over (1−k 2 )}. Modulation parameter k(t), which changes slowly with respect to frequency f 0  of the carrier signal, and thus message m(t), may therefore be recovered by sampling in the odd multiples of T/8. 
 
         [0058]     In the second demodulation method, one obtains the message signal by integration of received modulated carrier signal s(t) over a quarter period T/4 or a half period T/2. Using the integrals  
           ∫       sn   ⁡     (     x   ,   k     )       ⁢     ⅆ   x         =         -   ln     ⁢           ⁢     (       dn   ⁡     (   x   )       +     kcn   ⁡     (   x   )         )       k       ,     
     ⁢       ∫     cn   ⁢     (     x   ,   k     )     ⁢     ⅆ   x         =       arc   ⁢           ⁢   sin   ⁢           ⁢     (     k   ·     sn   ⁡     (   x   )             k       ,       
 
 which are described, for example, in I. S. Gradshteyn, I. M. Ryzhik, “Table of Integrals, Series, and Products”, corrected and enlarged edition, Academic Press, 1980, page 630, 5.133, we obtain  
           ∫   0     T   /   2       ⁢       s   ⁡     (   t   )       ⁢           ⁢     ⅆ   t         =     {             ∫   0     T   /   2       ⁢       a   0     ⁢     sn   ⁡     (     2   ⁢     π   ^     ⁢           ⁢     t   /   T       )       ⁢           ⁢     ⅆ   t               =           a   0     ⁢   T       2   ⁢     π   ^     ⁢           ⁢     (   k   )     ⁢   k       ⁢   ln   ⁢           ⁢       1   +   k       1   -   k                       ∫   0     T   /   2       ⁢       a   0     ⁢     cn   (     2   ⁢           ⁢       π   ^     ⁡     (     t   -     T   /   4       )       ⁢           ⁢     ⅆ   t                   =           a   0     ⁢   T         π   ^     ⁢           ⁢     (   k   )     ⁢   k       ⁢   arc   ⁢           ⁢   sin   ⁢           ⁢   k                   
 
         [0059]     An integration over a quarter period in each case results in one half of the values.  
         [0060]     According to the third demodulation method, modulated carrier signal s(t) is first squared and then integrated according to the equation  
           ∫   0   T     ⁢       s   ⁡     (   t   )       2       =     {             ∫   0   T     ⁢         (       a   0     ⁢     sn   ⁡     (     2   ⁢           ⁢     π   ^     ⁢           ⁢     t   /   T       )         )     2     ⁢           ⁢     ⅆ   t               =       a   0   2     ⁢   T   ⁢         K   ⁡     (   k   )       -     E   ⁡     (   k   )             k   2     ⁢     K   ⁡     (   k   )                           ∫   0   T     ⁢         (       a   0     ⁢     cn   ⁡     (     2   ⁢           ⁢     π   ^     ⁢           ⁢     t   /   T       )         )     2     ⁢           ⁢     ⅆ   t               =       a   0   2     ⁢   T   ⁢         E   ⁡     (   k   )       -       k   ′2     ⁢     K   ⁡     (   k   )               k   2     ⁢     K   ⁡     (   k   )                           
 
         [0061]     E(k) is the so-called complete elliptic integral of the second kind, and k′ is √{square root over (1−k 2 )}). An integration over half (a quarter of) a period in each case results in half (a quarter o)f the value.  
         [0062]     Using elliptic functions, available orthogonal modulation methods based on sine and cosine carriers may be generalized as well. Instead of the sine function, the function sx(x) from equation (5) may be used, and instead of the cosine function, function sy(x) with x=2{circumflex over (π)}f 0 t may be used, which is defined as follows:  
         sy   ⁡     (     x   ,     k   ⁡     (   t   )         )       =     {               ⁢     cd   ⁡     (     x   ,     k   ⁡     (   t   )         )                   ⁢       for   ⁢           ⁢   0     ≤   k   ≤   1                     ⁢     cn   ⁡     (     x   ,        k          )                   ⁢       for   -   1     ≤   k   ≤   0                   
 
         [0063]     The function cd(x) is the sn(x) function shifted by K, i.e., cd(x)=sn(x+k). It may be expressed by cd(x)=cn(x)/dn(x). Then, the orthogonality property  
           ∫   0     4   ⁢   K       ⁢         sx   ⁡     (   x   )       ·     sy   ⁡     (   x   )         ⁢           ⁢     ⅆ   t         =   0       
 
 applies. 
 
         [0064]     As a result, elliptic functions may be used for the orthogonal modulation. When values are given for a 0 , f 0  and k, one has two basic functions per dimension (sn and k′sd in the x-direction, and cd and cn in the y-direction), compared to only one basic function in classic sine carriers. The orthogonality may be used in the basic and/or in the transmission band.