Abstract:
A system for modulating and demodulating signals that provides a new class of signal modulators and their corresponding demodulators. The modulation scheme embeds in an information signal a carrier signal by modulating the oscillatory rate of the carrier signal. The invention generalizes the possible carrier signals to any signal which can be generated by a dynamical system that have a known exponentially convergent observer, as in certain chaotic systems.

Description:
PRIORITY INFORMATION 
     This application claims priority from provisional application Ser. No. 60/269,052 filed Feb. 15, 2001. 
    
    
     This invention was made with government support under Grant No. F49620-96-1-0072 awarded by the United States Air Force and Cooperative Agreement No. DAAL01-96-2-0001 awarded by the United States Army. The government has certain rights in the invention. 
    
    
     BACKGROUND OF THE INVENTION 
     Many approaches have been taken to embed continuous-time information bearing waveforms onto continuous-time carrier waveforms for the purpose of communication. Amplitude modulation (AM) and frequency modulation (FM) are common examples of such approaches to communication. In both AM and FM, the carrier wave is a sinusoidal wave. There is a need in the art for a systematic procedure for constructing continuous-time modulators and their corresponding demodulators using carrier waves that are generated by nonlinear systems that have periodic, almost-periodic, quasi-periodic or chaotic attractors. 
     SUMMARY OF THE INVENTION 
     Accordingly, the invention provides a new class of signal modulators and their corresponding demodulators. The modulator embeds into a carrier signal an information signal by modulating the oscillatory rate of the carrier signal in a manner proportional to the information signal. The permissible carrier signals are any signals that can be generated by a nonlinear dynamical system that has a known exponentially convergent observer. modulating the rate at which the non-linear system evolves in a manner proportional to the information signal. The method includes multiplying the information signal by a constant to produce a first signal value. The method further includes adding the first signal value and a nominal rate of evolution of the dynamical system to generate a second signal value. The method further includes providing a feedback path that includes a first and second path, wherein the input to the first path is the integration of a multiplication of the second signal value and output of the second path, and the input of the second path is output of the first path, such that the second path is a first function that defines the non-linear dynamical system, and providing the output of the first path as input to a second function that produces a transmitted signal. 
     According to another aspect of the invention, a system for demodulating a transmitted modulated carrier signal is provided. The demodulator system includes an observer component that receives as input the transmitted signal and a rate estimate and produces an estimate of the current state of the transmitter. The demodulator system also includes a rate estimator that receives as input the transmitted signal and the estimate of the current state of the transmitter to produce the rate estimate. The observer component and rate estimator are interconnected in a feedback loop arrangement, and as shown below, this arrangement recovers the information signal assuming that the dynamical system possesses a local exponential observability property. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of the modulator system; 
         FIG. 2  is a block diagram of the demodulator  20 ; 
         FIG. 3  is a block diagram of the demodulator  30 ; 
         FIG. 4  is a schematic of an exemplary modulator circuit  40 ; 
         FIG. 5  is a schematic of the observer component  50  of the exemplary demodulator circuit  52 ; 
         FIG. 6  is a schematic of the rate estimator component  60  of the exemplary demodulator circuit  52 ; and 
         FIG. 7  is a table illustrating exemplary components of the exemplary modulator and exemplary demodulator circuits  40 ,  50 , and  60 . 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       FIG. 1  is a block diagram of the modulator system  10 . An information signal m(t) is supplied to a modulating system  10 . The modulating system  10  consists of a non-linear system that has a periodic, quasi-periodic, almost periodic, or chaotic attractor. The information signal, m(t), is embedded in the non-linear system  10  by modulating the rate at which the dynamical system evolves. The non-linear dynamically system used to construct the system in  10  is denoted using the notation
   {dot over (x)}=ω   c   f ( x ),  (1) 
where x is an N-dimensional vector and ω c  is a constant. Modulation is achieved by applying the information signal, m(t), to the non-linear dynamical system as
   {dot over (x)}= (ω c   +βm ( t )) f ( x ),  (2) 
where ω c  is the nominal rate of evolution and β is a parameter that characterizes the degree of modulation. The effects of modulation on the non-linear system are readily seen by denoting the solution to the system without modulation giving in (1) as x o (t). Thus, the addition of modulation as described by (2) results in the modulated signal that can be expressed in term of x o (t) as
   x ( t ) =x   0 ( t+β∫   0   t   m (τ) dτ ).  (3) 
Modulating in this fashion does not modify the attractor of the non-linear system. It only modulates the rate at which the dynamical system evolves along the attractor.
 
     The signal that is transmitted to the receiver is a scalar function of the state variables of the non-linear dynamical system in the transmitter. As shown in  FIG. 2 , y(t) is denoted as
 
 y ( t )= h ( x ),  (4)
 
where h(x) is a function that maps the N-dimensional vector, x, to a one-dimensional signal, y. The complete modulator system is represented mathematically as
 
 {dot over (x)}= (ω c   +βm ( t ))ƒ( x ),
 
 y=h ( x ),  (5)
 
where x is a N-dimensional signal and y is a one-dimensional signal.
 
       FIG. 2  is a block diagram of the demodulator  20 . The demodulator  20  must track changes in the rate of evolution of a dynamical system. The demodulator  20  consists of two fundamental components—an observer component  22  and a rate estimator component  24 . The observer component  22  reconstructs the state of the demodulator  20 , z, given y(t) and the rate estimate, {circumflex over (m)}(t) . It is assumed assume that the dynamical system without modulation as given in (1) has a known exponentially convergent observer. We define an exponentially convergent observer to be a dynamical system
   ż={circumflex over (f)} ( z,h ( x   0 ))  (6) 
that has the property that ∥z−x 0 ∥≦e −λt  for some λ&gt;0, where x 0  is the solution to the unmodulated dynamical system given in equation (1). The rate estimator estimates m(t) given the state estimate, z, and the transmitted signal, y. The interconnection of these components comprises the complete demodulator system  20 . Given the dynamical system is used in modulator system  10 , the observer component  24  is given by
   ż= (ω c   +β{circumflex over (m)} ) {circumflex over (f)} ( z,y ),  (6) 
where {circumflex over (m)} is the rate estimate. The rate estimator  24  takes as input the reconstructed state z from the observer component  24  and the transmitted signal y(t) and tracks m(t). The low-pass filter  26  removes any spectral energy known to be absent from the original modulating signal, m(t). The low-pass filter  26  is an optional component to the demodulator  20 .
 
     Assuming that the unmodulated dynamical system used in modulator  10  has a known exponentially convergent local observer function, the observer component  24  can be modified so that it is an exponentially convergent observer of the modulator  10  when m(t)=m 0 , where m 0  is an unknown constant. If the rate estimator  24  converges to a value of m 0 , then the augmented observer is assumed to be able to track a time-varying m(t) provided that m(t) varies sufficiently slow. 
     The design of the rate estimator  24  is based on a technique that is referred to as a backwards perturbation expansion. The essential step in this perturbation expansion is to express the modulator state, x, as a perturbation expansion about the demodulator state, z, in terms of the rate estimate error, e m ={circumflex over (m)}−m 0 . By expanding the modulator state x as perturbation about the demodulator state z, the resulting expansion terms depend only on variables local to the demodulator  20 . The perturbation variables can then be combined in such a way that they force the demodulator rate estimate error to zero. If a dynamical system used in a modulator is of the form
 
 {dot over (x)}=ƒ ( x ),
 
 y=h ( x ),  (7)
 
and has an exponentially observer of the form
 
 ż={circumflex over (ƒ)} ( z,y ),  (8)
 
then a modulator as can be constructed as 
                 x   .     =       (       ω   c     +     β   ⁢           ⁢     m   ⁡     (   t   )           )     ⁢     f   ⁡     (   x   )           ,     
     ⁢     y   =     h   ⁡     (   x   )         ,           (   9   )             
 
which is similar to the modulator  10  defined in (5) above and a demodulator will be constructed as 
                 z   .     =       (       ω   c     +     β   ⁢           ⁢     m   ^         )     ⁢       f   ^     ⁡     (     z   ,   y     )           ,     
     ⁢       y   ^     =     h   ⁡     (   z   )         ,     
     ⁢         m   ^     .     =       K   ⁡     (       y   ^     -   y     )       ⁢     g   ⁡     (         ∂   h       ∂   z       ⁢       (   z   )     ·     ξ   1         )           ,     
     ⁢         ξ   .     1     =         (         (       ω   c     +     β   ⁢           ⁢     m   ^         )     ⁢       ∂     f   ^         ∂   z       ⁢     (     z   ,   y     )       +     K   ⁢           ⁢     r   ⁡     (     z   ,     ξ   1       )           )     ⁢     ξ   1       -     β   ⁢           ⁢     f   ⁡     (     z   ,   y     )             ,           (   10   )             
 
which is the same as demodulator  20  as defined in (6) where 0&lt;K&lt;K* for some K*&gt;0, and r(•,•) is a scalar valued function defined by 
               r   ⁡     (     z   ,     ξ   1       )       =       (         ∂   h       ∂   z       ⁢       (   z   )     ·     ξ   1         )     ·     g   ⁡     (         ∂   h       ∂   z       ⁢       (   z   )     ·     ξ   1         )                 (   11   )             
 
where g(•) is scalar function such that sgn(g(α))=sgn(α), and 
                        r   ⁡     (     z   ,     e   z       )                   e   z            &lt;   γ     ,           (   12   )             
 
for some γ&gt;0. The observer  22  and rate estimator  24  are interconnected in a feedback loop as shown in  FIG. 3 .
 
     Many systems have been shown to possess an exponentially convergent observer, including some chaotic systems. One such system is a chaotic system that is described by the Lorenz equations that are defined as 
                   x   .     1     =     (     σ   ⁡     (       x   2     -     x   1       )       )       ⁢     
     ⁢         x   .     2     =     (       r   ⁢           ⁢     x   1       -       x   1     ⁢     x   3       -     x   2       )       ⁢     
     ⁢         x   .     3     =     (         x   1     ⁢     x   2       -     b   ⁢           ⁢     x   3         )               (   13   )             
 
where σ, r, and b are constant parameters. An exponentially convergent observer of the system given in (13) when y(t)=x 1 (t), 
                   z   .     1     =     σ   ⁡     (       z   2     -     z   1       )         ⁢     
     ⁢         z   .     2     =     (       r   ⁢           ⁢   y     -     y   ⁢           ⁢     z   3       -     z   2       )       ⁢     
     ⁢         z   .     3     =       (       y   ⁢           ⁢     z   2       -     b   ⁢           ⁢     z   3         )     .               (   14   )             
 
       FIG. 3  is block diagram of a demodulator  30 . Three enhancements have been made to the demodulator  30 . First, a low-pass filter  36  is added between the rate estimator  32  and observer  34  to remove spectral energy in {circumflex over (m)}(t) that is known to be outside the bandwidth of m(t). Second, K is increased beyond the value for which the perturbation expansion analysis guarantees stability. Although stability is no longer guaranteed, the demodulator  30  may remain stable over a range of K&gt;K*. A larger value for K allows {circumflex over (m)}(t) to track faster signals. Finally, many of the nonlinearities in the demodulator  30  are removed by approximating the rate estimator  32  with a linear system. The system that results from the approximation is equivalent to a least-squares approach to designing a demodulator. 
     To remove the spectral energy in {circumflex over (m)}(t) that is outside the bandwidth of m(t), a filter  36  is added to the feedback path. The filtering operation, denoted as &lt;•&gt;, is given by
 
 &lt;{circumflex over (m)} ( t )&gt;=∫ 0   t ∫ψ( t,τ ) {circumflex over (m)} (τ) dτ ,  (15)
 
where ψ(t,τ) is the filter kernel. If the support of this kernel is sufficiently small compared to the rate at which m and {circumflex over (m)} vary, then 
                     〈       m   ^     .     〉     =       ⁢       -   K     ⁢     〈         (         ∂   h       ∂   z       ⁢       (   z   )     ·     ξ   1         )     ·     g   ⁡     (         ∂   h       ∂   z       ⁢       (   z   )     ·     ξ   1         )         ⁢     (       m   ^     -   m     )       〉                   ≈       ⁢       -   K     ⁢     〈       (         ∂   h       ∂   z       ⁢       (   z   )     ·     ξ   1         )     ·     g   ⁡     (         ∂   h       ∂   z       ⁢       (   z   )     ·     ξ   1         )         〉     ⁢       (       m   ^     -   m     )     .                     (   16   )             
 
     As mentioned above, the gain parameter K had to be smaller than some K* to guarantee that the perturbation expansion was bounded. For K&lt;K*, the rate estimate converges exponentially and monotonically. Choosing K slightly larger than K* affects the demodulator  30  in two ways. First, since K scales the derivate of {circumflex over (m)}, increasing K also increases the rate which {circumflex over (m)} can vary, allowing {circumflex over (m)} to track signals that vary more rapidly. Second, the perturbation term may not remain bounded. Returning to the equation for the perturbation variable, 
                   ξ   .     1     =         (       ω   c     +     β   ⁢           ⁢     m   ^         )     ⁢     (           ∂     f   ^         ∂   z       ⁢     (     z   ,   y     )       +     K   ⁢           ⁢     r   ⁡     (     z   ,     ξ   1       )           )     ⁢     ξ   1       -     β   ⁢           ⁢     f   ⁡     (     z   ,   y     )             ,           (   17   )             
 
where 
               r   ⁡     (     z   ,     ξ   1       )       =       (         ∂   h       ∂   z       ⁢       (   z   )     ·     ξ   1         )     ·       g   ⁡     (         ∂   h       ∂   z       ⁢       (   z   )     ·     ξ   1         )       .               (   18   )             
 
     If K is set to zero in (18), ξ 1  remains bounded. However, the dynamics of demodulator  30  is changed and convergence is no longer guaranteed, demodulator  30  may be stable for a range of K&gt;K*.
 
An example, consider the Lorenz based modulation/demodulation system based on the systems described in (13) and (14) with K&gt;K*. The term Kr(z,ξ 1 ) is dropped from (17) and the demodulator equation for the Lorenz equation becomes 
                   z   .     1     =       (       ω   c     +     β   ⁢           ⁢     m   ^         )     ⁢     (     σ   ⁡     (       z   2     -     z   1       )       )         ⁢     
     ⁢         z   .     2     =       (       ω   c     +     β   ⁢           ⁢     m   ^         )     ⁢     (       r   ⁢           ⁢   y     -     y   ⁢           ⁢     z   3       -     z   2       )         ⁢     
     ⁢           z   .     3     =         (       ω   c     +     β   ⁢           ⁢     m   ^         )     ⁢       (       y   ⁢           ⁢     z   2       -     b   ⁢           ⁢     z   3         )     .     
     ⁢       m   ^     .         =       K   ⁡     (       z   1     -   y     )       ⁢     sgn   ⁡     (     ψ   1     )             ,     
     ⁢       [             ψ   .     1                 ψ   .     2                 ψ   .     3           ]     =           (       ω   c     +     β   ⁢           ⁢     m   ^         )     ⁡     [         σ       σ       0           0         -   1           -   y             0       y         -   b           ]       ⁢           [           ψ   1               ψ   2               ψ   3           ]     -       β   ⁡     [           σ   ⁡     (       z   2     -     z   1       )                   r   ⁢           ⁢   y     -     y   ⁢           ⁢     z   3       -     z   2                   y   ⁢           ⁢     z   2       -     b   ⁢           ⁢     z   3               ]       .                   (   19   )             
 
The modulator equations are 
                         x   .     1     =       (       ω   c     +     β   ⁢           ⁢   m       )     ⁢     (     σ   ⁡     (       x   2     -     x   1       )       )         ⁢     
     ⁢         x   .     2     =       (       ω   c     +     β   ⁢           ⁢   m       )     ⁢     (       r   ⁢           ⁢     x   1       -       x   1     ⁢     x   3       -     x   2       )         ⁢     
     ⁢         x   .     3     =       (       ω   c     +     β   ⁢           ⁢   m       )     ⁢     (         x   1     ⁢     x   2       -     b   ⁢           ⁢     x   3         )                 (   20   )               
 
     The demodulator  30  has additional nonlinearities added beyond those already present in the dynamical system. These additional nonlinearities appear in the equation for ξ 1  as given in (17). Even when K&gt;K* and Kr(z,ξ 1 ) is removed, a nonlinear equation remains. 
     The last term, βf(z,y), also appears in the observer component  34  of the demodulator  30 . Since {circumflex over (f)}(z,y) is required by the observer component  34 , removing it from the rate estimator  32  does not reduce the total number of nonlinearities present in the demodulator  30 . This term is left as it is. The first term, 
           (       ω   c     +     β   ⁢           ⁢     m   ^         )     ⁢       ∂     f   ^         ∂   z       ⁢     ξ   1       ,       
 
is generally nonlinear and does not appear else where in system  30 . Approximating this term with a linear, time-invariant system simplifies the hardware implementation of the demodulator  30 .
 
     First, if ω c &gt;&gt;β{circumflex over (m)} then 
                 ξ   .     1     ≈         ω   c     ⁢       ∂     f   ^         ∂   z       ⁢     (     z   ,   y     )     ⁢     ξ   1       -     β   ⁢           ⁢         f   ^     ⁡     (     z   ,   y     )       .                 (   20   )             
 
The varying gain matrix, 
             ∂     f   ^         ∂   z       ⁢     (     z   ,   y     )       ,       
 
is generally nonlinear. However, the differential equation in (20) is linear with to ξ 1  and is a time varying linear filter with β{circumflex over (f)}(z,y) as its input. Using the notation &lt;•&gt; to denote the filtering operation,
 
 ξ   1   =−β&lt;{circumflex over (f)} ( z,y )&gt;.  (21)
 
Replacing this linear time-varying filter with a linear time-invariant filter makes the equation for ξ 1  consists of only linear components and {circumflex over (f)}(x,y), the latter of which is already present in the demodulator  30 .
 
     The difference between the derivatives of ŷ and y can be approximated as, 
                   y   ^     .     -     y   .       =           (       ω   c     +     β   ⁢           ⁢     m   ^         )     ⁢       ∂   h       ∂   z       ⁢     (   z   )     ⁢       f   ^     ⁡     (     z   ,   y     )         -       (       ω   c     +     β   ⁡     (       m   ^     -     e   m       )         )     ⁢       ∂   h       ∂   z       ⁢     (     z   +       ξ   1     ⁢     e   m       +   …     )     ⁢       f   ^     ⁡     (       z   +       ξ   1     ⁢     e   m       +   …     ⁢           ,   y     )           ≈       e   m     ⁢   β   ⁢       ∂   h       ∂   z       ⁢     (   z   )     ⁢         f   ^     ⁡     (     z   ,   y     )       .                 (   22   )             
 
Filtering {circumflex over ({dot over (y)}−{dot over (y)} with the same filter that appears in (21) results in 
                 〈         y   ^     .     -     y   .       〉     ≈     β   ⁢     〈         ∂   h       ∂   z       ⁢     (   z   )     ⁢       f   ^     ⁡     (     z   ,   y     )         〉     ⁢     e   m         ,           (   23   )             
 
assuming that e m  varies slowly with respect to the time constant of the filter so that e m  can be moved outside the filtering operation. Combining (21) and (23), the rate estimator  32  becomes 
                             m   ^     .     =       ⁢     K   ⁢           ⁢   β   ⁢     〈         y   ^     .     -     y   .       〉     ⁢     〈         ∂   h       ∂   z       ⁢     (   z   )     ⁢       f   ^     ⁡     (     z   ,   y     )         〉                   ≈       ⁢     K   ⁢           ⁢   β   ⁢       〈         ∂   h       ∂   z       ⁢     (   z   )     ⁢       f   ^     ⁡     (     z   ,   y     )         〉     2     ⁢       (       m   ^     -   m     )     .                     (   24   )               
 
From (24), the rate estimator  32  has the form
 
 {circumflex over ({dot over (m)}≈−a ( t )( {circumflex over (m)}−m ),  (25)
 
where a(t) is positive semi-definite function, which suggests {circumflex over ({dot over (m)} converge to m.
 
       FIG. 4  is a schematic of a modulator circuit  40 . The modulation/demodulation technique of the invention is applicable to any non-linear system meeting the requirements described in the previous section, a particular implementation which uses a set of equations known as the Lorenz equations is further described below. The Lorenz equations are a set of non-linear ordinary differential equations that have state trajectories that behave chaotically. A signal that is chaotic is one that exhibits long-term aperiodic behavior that has a sensitive dependence on initial conditions. 
     The Lorenz equations discussed above are repeated here for the circuit. In this circuit implementation, however, care must be given to ensure that the signal levels remain within the operating range of the circuit components. Thus, 
           X   1     =       1   2     ⁢     x   1         ,       X   2     =       1   2     ⁢     x   2         ,       and   ⁢           ⁢     X   3       =       2   9     ⁢       x   3     .             
 
The resealed Lorenz equations implemented are 
                         X   .     1     =       (       ω   c     +     β   ⁢           ⁢   m       )     ⁢     (     σ   ⁡     (       X   2     -     X   1       )       )         ⁢     
     ⁢         X   .     2     =       (       ω   c     +     β   ⁢           ⁢   m       )     ⁢     (       r   ⁢           ⁢     X   1       -       9   2     ⁢     X   1     ⁢     X   3       -     X   2       )         ⁢     
     ⁢         X   .     3     =       (       ω   c     +     β   ⁢           ⁢   m       )     ⁢       (         8   9     ⁢     X   1     ⁢     X   2       -     b   ⁢           ⁢     X   3         )     .                 (   26   )               
 
     The implementation of this modulator  40  using multipliers, operational amplifiers and capacitors, and resistors is shown in  FIG. 4 . Solving for the circuit voltage gives 
           V   .     1     =       1     C   101       ⁢     (       1     R   114       +       m   ⁢     (   t   )         10   ⁢     R   115           )     ⁢     (           R   102       R   101       ⁢     V   2       -       (       R   104         R   103     +     R   104         )     ⁢     (     1   +       R   102       R   101         )     ⁢     V   1         )           
                 V   .     2     =       ⁢       1     C   102       ⁢     (       1     R   116       +       m   ⁡     (   t   )         10   ⁢     R   117           )     ⁢       R   121       R   120       ⁢     {         -       R   106       10   ⁢     R   107           ⁢     V   1     ⁢     V   3       -         R   106       R   105       ⁢     V   2       +                         ⁢         (       R   109         R   108     +     R   109         )     ⁢     (     1   +       R   106       R   105       +       R   106       R   107         )     ⁢     V   1       }                     V   .     3     =       ⁢       1     C   103       ⁢     (       1     R   118       +       m   ⁡     (   t   )         10   ⁢     R   119           )     ⁢     {     (       R   113         R   112     +     R   113         )                           ⁢           (     1   +       R   111       R   110         )     ⁢     V   3       -         R   111       10   ⁢     R   110         ⁢     V   1     ⁢     V   2         }       .             
 
It is assumed that the output of the multipliers is the product of its input divided by 10, which is typical of multiplier circuits. The nominal rates at which the modulator circuit  40  modulates are governed by the capacitors.
 
       FIG. 5  is a schematic of the observer component  50  of demodulator  52 . After solving the circuit voltages in the observer portion of the demodulator gives 
                         V   ^     .     2     =       ⁢       1     C   201       ⁢     (       1     R   208       +       m   ⁡     (   t   )         10   ⁢     R   209           )     ⁢       R   207       R   206       ⁢     {         -       R   202       10   ⁢     R   203           ⁢     yV   3       -         R   202       R   201       ⁢     V   2       +                         ⁢         (       R   205         R   204     +     R   205         )     ⁢     (     1   +       R   202       R   201       +       R   202       R   203         )     ⁢   y     }                       V   ^     .     3     =       ⁢       1     C   202       ⁢     (       1     R   214       +       m   ⁡     (   t   )         10   ⁢     R   215           )     ⁢     {     (       R   213         R   212     +     R   213         )                           ⁢           (     1   +       R   211       R   210         )     ⁢     V   3       -       R   211       10   ⁢     R   210           }       ,                 (   16   )               
where y=V 1 .
 
       FIG. 6  is a schematic of the rate estimator component  60  of demodulator circuit  52 . After solving the circuit voltages in the rate estimator  60  portion of the demodulator gives 
                   m   ^     .     =       -     1     100   ⁢     C   303           ⁢     {         1     R   309       ⁢     (         R   304       R   305       +         R   304       10   ⁢     R   303         ⁢     m   ^         )     ⁢       〈   g   〉     2       -       1       ω   1     ⁢     R   308         ⁢     〈     y   .     〉     ⁢     〈   g   〉         }         ,           (   17   )               
where 
         ω   L     =     1       R   309     ⁢     C   302               
and &lt;•&gt; denotes the filtering with a first order low pass filter with a cut-off frequency of ω L . The circuit of  FIG. 7  utilizes the multiplier that is particular to the AD374. The AD374 has four inputs. It multiplies the difference between inputs one and two with the difference between inputs three and four. The multiplier M 3  multiplies y minus the output of OP 2  with the output of OP 1 .
 
       FIG. 7  is a table illustrating exemplary components of the modulator  40  and demodulator circuits  50  and  60 . The components values are illustrative of the circuits described in  FIGS. 5 ,  6 , and  7 . Based upon these component values 
                     ω   c     =     400   ,   000                 β   =     80   ,   000                   ω   L     =     80   ,   000.                   (   18   )               
However, the modulator and demodulator circuits  40 ,  50 , and  60  may utilize other component values.
 
     The goal of the invention is to describe and analyze a class of signals that can be used in the framework of the modulation technique described above, which is referred to as a generalized frequency modulation. The invention has various practical application which are low power communication system, because the modulation technique is applicable to nonlinear dynamical systems, thus the system is not constrained to operate circuit components in their linear regime. This potentially reduces the number of circuit components, simplifies the circuit, and increases the efficiency. Another is the Spread-Spectrum communication system, which the modulation technique of the invention can be applied to chaotic systems, which are naturally spread-spectrum signals. Due to sensitive dependence on initial conditions exhibited in chaotic systems, chaotic signals are difficult to track without precise knowledge of all of the parameters of the chaotic dynamical system, which suggests that chaotic carrier signals may be advantageous in the context of private communications. 
     Although the invention has been shown and described with respect to several preferred embodiments thereof, various changes, omissions and additions to the form and detail thereof, may be made therein, without departing from the spirit and scope of the invention.