Apparatus and method for synchronizing nonlinear systems using filtered signals

Abstract

A filtered cascaded synchronized nonlinear system includes a nonlinear transmitter having stable first and second subsystems. The first subsystem produces a first transmitter signal for driving the second subsystem, and the second subsystem produces a second transmitter signal for driving the first subsystem. A first filter filters the second transmitter signal to produce a filter output signal. A subtractor subtracts the filter output signal from the second transmitter signal to produce a transmitter output signal which is transmitted to a nonlinear cascaded receiver. The receiver includes an adder for summing the received transmitter output signal with a receiver filter output signal to restore frequencies that were subtracted from the second transmitter signal in order to produce a first receiver drive signal. The receiver includes cascaded third and fourth subsystems that are respective duplicates of the first and second subsystems. The third subsystem is driven by the first receiver drive signal to produce a first receiver signal in synchronization with the first transmitter signal. The fourth subsystem is driven by the first receiver signal to produce a second receiver signal in synchronization with the second transmitter signal. A second filter filters the second receiver signal to produce the receiver filter output signal.

Claims

What is claimed and desired to be secured by Letter Patent of the United States is:

1. A filtered cascaded synchronized nonlinear system comprising:

a nonlinear transmitter comprising:

stable first and second subsystems, said first subsystem producing a first transmitter signal for driving said second subsystem, and said second subsystem producing a second transmitter signal for driving said first subsystem;

first means for filtering the second transmitter signal to produce a filter output signal; and

second means responsive to the second transmitter signal and the filter output signal for producing a transmitter output signal; and

a nonlinear cascaded receiver comprising:

third and fourth cascaded subsystems being respective duplicates of said first and second subsystems;

third means responsive to the transmitter output signal from said second means and to a receiver filter output signal for producing a first receiver drive signal to drive said third subsystem to produce a first receiver signal in synchronization with the first transmitter signal to drive said fourth subsystem;

said fourth subsystem being responsive to the first receiver signal for producing a second receiver signal in synchronization with the second transmitter signal; and

fourth means responsive to said second receiver signal for producing the receiver filter output signal.

2. The filtered cascaded synchronized nonlinear system of claim 1 wherein:

at least part of said first subsystem is external to said second subsystem, and at least part of said second subsystem is external to said first subsystem.

3. The filtered cascaded synchronized nonlinear system of claim 1 wherein:

said second means is a subtractor for subtracting the filter output signal from the second transmitter signal to produce the transmitter output signal;

said third means is an adder for summing the receiver filter output signal with the transmitter output signal to produce the first receiver drive signal; and

each of said first and fourth means is a filter.

4. The filtered cascaded synchronized nonlinear system of claim 1 wherein:

said nonlinear transmitter further includes:

first periodic forcing means for respectively providing phase coherent, periodic first and second forcing signals to said first and second subsystems to force said first and second subsystems to oscillate, at least one of the first and second forcing signals having a non-zero value; and

said nonlinear receiver further includes:

second periodic forcing means for respectively providing phase coherent, periodic third and fourth forcing signals to said third and fourth subsystems to force said third and fourth subsystems to oscillate, all of said first, second, third and fourth forcing signals being phase coherent with each other; and

phase control means being responsive to the first receiver drive signal and the second receiver signal for producing and applying an error signal to said second periodic forcing means, the error signal being proportional to the phase difference between said second periodic forcing means and said first periodic forcing means, said second periodic forcing means being responsive to the error signal for matching the phase of the second periodic forcing means to the phase of said first periodic forcing means, said nonlinear receiver becoming synchronized to the nonlinear transmitter when the phase of the second periodic forcing means in said nonlinear receiver matches the phase of the first periodic forcing means in said nonlinear transmitter.

5. The filtered cascaded synchronized nonlinear system of claim 4 wherein:

said nonlinear receiver is synchronized to said nonlinear transmitter when the second receiver signal matches the first receiver drive signal.

6. A filtered cascaded synchronized nonlinear system comprising:

a nonlinear transmitter comprising:

stable first and second subsystems, said first subsystem producing a first transmitter signal for driving said second subsystem, and said second subsystem producing a second transmitter signal for driving said first subsystem;

first means for removing undesired frequencies from the second transmitter signal to produce a transmitter output signal;

a nonlinear cascaded receiver comprising:

third and fourth cascaded subsystems being respective duplicates of said first and second subsystems;

second means responsive to the transmitter output signal from said first means and to a receiver filter output signal for producing a first receiver drive signal to drive said third subsystem to produce a first receiver signal in synchronization with the first transmitter signal to drive said fourth subsystem;

said fourth subsystem being responsive to the first receiver signal for producing a second receiver signal in synchronization with the second transmitter signal; and

third means responsive to said second receiver signal for producing the receiver filter output signal, the receiver filter output signal corresponding to the undesired frequencies removed from the second transmitter signal.

7. The filtered cascaded synchronized nonlinear system of claim 6 wherein said first means comprises:

fourth means for filtering the second transmitter signal to produce a filter output signal; and

fifth means responsive to the second transmitter signal and the filter output signal for producing the transmitter output signal.

8. The filtered cascaded synchronized nonlinear system of claim 6 wherein said first means comprises:

a communications channel for removing undesired frequencies from the second transmitter signal to produce and transmit the transmitter output signal to said second means.

9. The filtered cascaded synchronized nonlinear system of claim 6 wherein:

at least part of said first subsystem is external to said second subsystem, and at least part of said second subsystem is external to said first subsystem.

10. The filtered cascaded synchronized nonlinear system of claim 7 wherein:

said fifth means is a subtractor for subtracting the filter output signal from the second transmitter signal to produce the transmitter output signal;

said second means is an adder for summing the receiver filter output signal with the transmitter output signal to produce the first receiver drive signal; and

each of said third and fourth means is a filter.

Description

CROSS REFERENCE TO RELATED PATENTS AND APPLICATIONS

This application is related to commonly assigned U.S. Pat. Nos. 5,245,660 (having Navy Case No. 72,593); 5,379,346 (having Navy Case No. 74,222); and 5,402,334 (having Navy Case No. 73,912). This application is also related to commonly assigned U.S. patent application Ser. No. 08/267,696 filed Jun. 29, 1994 (having Navy Case No. 75,496). U.S. Pat. Nos. 5,245,660 and 5,379,346 and U.S. patent application Ser. No. 08/267,696 are all incorporated herein by reference.

FIELD OF THE INVENTION

The present invention relates generally to physical systems with dynamical characteristics which involve synchronization of a transmitter system and a receiver system and, more particularly, to a system which allows the synchronization of a nonlinear system when the driving signal has been filtered.

DESCRIPTION OF THE RELATED ART

The design of most man-made mechanical and electrical systems assumes that the systems exhibit linear behavior (stationary) or simple nonlinear behavior (cyclic). In recent years there has been an increasing understanding of a more complex form of behavior, known as chaos, which is now recognized to be generic to most nonlinear systems. Systems evolving chaotically (chaotic systems) display a sensitivity to initial conditions, such that two substantially identical chaotic systems started with slightly different initial conditions (state variable values) will quickly evolve to values which are vastly different and become totally uncorrelated, even though the overall patterns of behavior will remain the same. This makes chaotic systems nonperiodic (there are no cycles of repetition whatsoever), unpredictable over long times, and thus such systems are impossible to synchronize by conventional methods. Y. S. Tang et al., "Synchronization and Chaos," IEEE Transactions of Circuits and Systems, Vol. CAS-30, No. 9, pp. 620-626 (September 1983) discusses the relationship between synchronization and chaotic systems in which selected parameters are outside some range required for synchronization.

SUMMARY OF THE INVENTION

It is therefore an object of the invention to provide systems for producing synchronized signals, and particularly nonlinear dynamical systems.

Another object of the invention is provide communication systems for encryption utilizing synchronized nonlinear sending and receiving circuits.

Another object of the invention is to provide improved control devices which rely on wide-frequency band synchronized signals.

Another object of the invention is to provide physical systems with dynamical characteristics which involve synchronization of a transmitter system and a receiver system and, more particularly, to a system which allows the synchronization of a nonlinear system when the driving signal has been filtered.

Another object of the invention is to provide synchronizable systems in which only a single drive signal is transmitted between systems, multiple synchronized signals can be produced and in which the drive signal can be reproduced to confirm synchronization.

Another object of the invention is to provide synchronizable systems in which the total dimension of the synchronized systems or the number of elements can be the same.

Another object of the invention is to provide synchronizable chaotic systems when the drive signal is filtered.

Another object of the invention is to transmit information using cascaded synchronizable chaotic systems with a filtered drive signal.

A further object of the invention is to transmit information using cascaded synchronizable chaotic systems using parameter changes to transmit information when the drive signal is filtered.

A cascaded synchronized nonlinear system includes a nonlinear drive system having stable first and second subparts. The first subpart produces a first drive signal for driving the second subpart and the second subpart produces a second drive signal for driving the first subpart. The nonlinear transmitter transmits the second drive signal to a nonlinear cascaded response system. The response system, being for producing an output signal in synchronization with the second drive signal, comprises a first stage (a duplicate of the first subpart) responsive to the second drive signal for producing a first response signal. The response system further comprises a second stage (a duplicate of the second subpart) responsive to the first response signal for producing the output signal.

A cascaded synchronized nonlinear system with a filtered drive signal includes a nonlinear drive system having stable first and second subparts. The first subpart produces a first drive signal for driving the second subpart and the second subpart produces a second drive signal for driving the first subpart. The second drive signal is passed through the transmitter filter and the transmitter filter output is subtracted from the second drive signal to produce the broadcast signal. The broadcast signal is transmitted to a nonlinear receiver with a cascaded section.

The receiver, being for producing an output signal in synchronization with the second drive signal, consists of a cascaded nonlinear response system and a filter section. The cascaded system comprises a first stage (a duplicate of the first subpart) responsive to the receiver driving signal produced by the filter section. The first stage produces a first response signal. The cascaded section further comprises a second stage (a duplicate of the second subpart) responsive to the first response signal for producing the output signal.

The filter section is comprised of a receiver filter identical to the transmitter filter and an adding circuit. The receiver filter is responsive to the receiver output signal, and produces the filter output signal. The adding circuit adds the filter output signal to the broadcast signal to produce the receiver driving signal.

The filtered cascaded synchronized nonlinear system can be used in an information transfer system. The transmitter responsive to an information signal produces a broadcast signal for transmission to the receiver. An error detector compares the receiver drive signal described above and the output signal produced by the receiver to produce an error signal indicative of the information contained in the information signal. These and other objects, features and advantages of the present invention are described in or apparent from the following detailed description of preferred embodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

The preferred embodiments will be described with reference to the drawings, in which like elements have been denoted throughout by like reference numerals, and wherein:

FIG. 1 is a general block diagram of a nonlinear dynamical physical system of the prior art;

FIG. 2 is a general block diagram of an synchronized chaotic system of the prior art;

FIG. 3 is a schematic circuit diagram of a synchronized chaotic circuit system of the prior art;

FIG. 4 illustrates a cascaded synchronized system of the prior art with two stages;

FIG. 5 shows an embodiment of a two stage cascaded synchronization system of the prior art.

FIGS. 6-9 illustrate details of the prior art system of FIG. 5.

FIG. 10 is a block diagram of the present invention as applied to an autonomous nonlinear dynamical system;

FIG. 11 is a block diagram of a second embodiment of the present invention as applied to a non-autonomous nonlinear dynamical system;

FIG. 12-18 show details of the second embodiment of FIG. 11;

FIG. 19 shows a modification of FIG. 14 in which FIG. 19 replaces FIG. 16 to produce a third embodiment of the invention comprised of FIGS. 12-15 and 17-19;

FIG. 20(a) illustrates the power spectrum of the x signal from the drive system circuit 3000 (FIG. 12) described by equations 26-28;

FIG. 20(b) illustrates the power spectrum of the broadcast signal s.sub.t from the transmitter filter circuit (FIG. 19) described by equations 37-39;

FIG. 21 shows the receiver forcing signal F" (2240) vs. the transmitter forcing signal F (2130) from the circuit implementation of the present invention when the band-stop filter arrangement of FIG. 19 is used;

FIG. 22 illustrates a dotted line showing the value of a parameter A in the drive system circuit 3000 of FIG. 12 that was varied and the solid line shows the value of the error signal .DELTA. from the phase control circuit 5200 when a set of nonautonomous circuits were synchronized according to the block diagram of FIG. 11;

FIG. 23 shows a solid line representing a time series of the numerical output of the y signal from a 4-th order Runge-Kutta integration routine executed on a digital computer to simulate the system of equations 43, and a dotted line representing the signal broadcast signal y.sub.t from the filter of equations 44 and 45;

FIG. 24 shows a power spectrum of the numerical y signal from equations 43;

FIG. 25 shows a power spectrum of the broadcast signal y.sub.t from equation 45;

FIG. 26 shows the numerical response system output signal y" from equation 48 vs the drive system output signal y from equation 43; and

FIG. 27 illustrates a block diagram of a fourth embodiment of the present invention to correct for channel filtering effects.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

All physical systems can be described by state variables. For example, a billiard game can be described by the position and the velocity of a ball at any instant of time; and an electronic circuit can be described by all of its currents and voltages at a particular time. This invention is a tangible system which can be of any form. The state variables and associated signals can be, as further examples, pressure or other force, temperature, concentration, population, or electro-magnetic field components. The evolution of a physical system depends on the dynamical relations between the state variables, which are usually expressed as functional relations between the rates of change of the variables. Thus, most but not all physical systems are describable in terms of ordinary differential equations (ODEs). Mathematical models of chaotic systems often involve two types of systems: flows and iterative maps. The former evolve as solutions of differential equations, and the latter evolve in discrete steps, such as by difference equations. For example, seasonal measurements of populations can be modeled as iterative maps. Cf., Eckmann et al., Rev. Mod. Phys., Vol. 57, pp.617-618, 619 (1985). Some iterative maps could be considered as numerical solutions to differential equations. Solution or approximate solution of these equations, such as approximate, numerical, or analytical solution, provides information about the qualitative and quantitative behavior of the system defined by the equations.

As used herein, the synchronization of two or more evolving state variables of a physical system means the process by which the variables converge toward the same or linearly related but changing set of values. Thus, if one synchronized variable changes by a certain amount, the change of the other synchronized variable will also approach a linear function of the same amount. Graphically, the plot of the synchronized variables against each other as they evolve over time would approach a straight line.

Referring to FIG. 1, an n-dimensional autonomous nonlinear dynamical drive system 9 can be arbitrarily divided, as shown, into first and second parts or subsystems 12 and 14, each of which subsystems is also a nonlinear dynamical system. Drive system 9 and, more specifically, subsystem 14, has output signal S.sub.o.

The following discussion involves mathematical modeling of the system 10 in terms of solutions to differential equations and provides theoretical support for the invention. However, it is not necessary in practicing this invention that system 10 be susceptible to such modeling. For example, as stated earlier, some iterated maps cannot be modeled as solutions to differential equations, and yet this invention encompasses Systems evolving according to iterated maps. As a further example, it is impractical to accurately model an ideal gas by individually considering the position and momentum of every molecule because of the vast number of molecules and variables involved.

This discussion about mathematical modeling is in two parts to correspond to two sources of difficulty in synchronizing signals: instability within a single system (chaos) and instability between two systems (structural instability). It is understood that both discussions apply to this invention and neither part should be read separately as limiting the practice of this invention.

A system with extreme sensitivity to initial conditions is considered chaotic. The same chaotic system started at infinitesimally different initial conditions may reach significantly different states after a period of time. As known to persons skilled in the art and discussed further, for example, in Wolf et al., Determining Lyapunov Exponents from a Time Series, Physica, Vol. 16D, p. 285 et seq. (1985), Lyapunov exponents (also known in the art as "characteristic exponents") measure this divergence. A system will have a complete set (or spectrum) of Lyapunov exponents, each of which is the average exponential rate of convergence (if negative) or divergence (if positive) of nearby orbits in phase space as expressed in terms of appropriate variables and components. If all the Lyapunov exponents are negative, then the same system started with slightly different initial conditions will converge (exponentially) over time to the same values, which values may vary over time. On the other hand, if at least one of the Lyapunov exponents is positive, then the same system started with slightly different initial conditions will not converge, and the system behaves chaotically. It is also known by persons skilled in the art that "in almost all real systems there exist ranges of parameters or initial conditions for which the system turns out to be a system with chaos . . . ." Chernikov et al., Chaos: How Regular Can It Be?, 27 Phys. Today 27, 29 (Nov. 1988).

Drive system 9 can be described by the ODE

du(t)/dt=f{u(t)} or u=f(u) (1)

where u(t) are the n-dimensional state variables.

Defined in terms of the state variables v and w for subsystems 12 and 14, respectively, where u=(v,w), the ODEs for subsystems 12 and 14 are, respectively:

v=g(v,w)

w=h(v,w) (2)

where v and w are m and n-m dimensional, respectively, that is,

where v=(u.sub.1, . . . ,u.sub.m), g=(f.sub.1 (u) . . . , f.sub.m (u)), w=(u.sub.m+1, . . . , u.sub.n) and h=(f.sub.m+1 (u), . . . , f.sub.n (u)).

The division of drive system 9 into subsystems 12 and 14 is truly arbitrary since the reordering of the u.sub.i variables before assigning them to v, w, g and h is allowed.

If a new subsystem 16 identical to subsystem 14 is added to drive system 9, thereby forming system 10, then substituting the variables v for the corresponding variables in the function h augments equations (2) for the new three-subsystem system 10 as follows:

v=g(v,w)

w=h(v,w)

w'=h(v,w'). (3)

Subsystem 16 has output signal S.sub.o '.

The w and w' subsystems (subsystems 14 and 16) will only synchronize if .DELTA.w.fwdarw.0 as T.fwdarw..infin., where .DELTA.w=w'-w.

The rate of change of .DELTA.w (for small .DELTA.w) is:

.DELTA.w=d.DELTA.w/dt=h(v,w')-h(v,w)=D.sub.w h(v,w).DELTA.w+W;(4)

where D.sub.w h(v,w) is the Jacobian of the w subsystem vector field with respect to w only, that is: an (n-m).times.(n-m) linear operator (matrix)

(D.sub.w h).sub.ij =.differential.h.sub.i /.differential.w.sub.j(5)

for (m+1).ltoreq.i.ltoreq.n and 1.ltoreq.j.ltoreq.(n-m), and where W is a nonlinear operator. When Equation 4 is divided by .vertline..DELTA.w(0).vertline., and .xi.=.DELTA.w(t)/.DELTA.w(0), an equation for the rate of change (the growth or shrinkage) of the unit displacement (n-m) dimensional vector, .xi., is obtained. In the infinitesimal limit, the nonlinear operator vanishes and this leads to the variational equation for the subsystem

d.xi./dt=D.sub.w h(v(t),w(t)).xi.. (6)

The behavior of this equation or its matrix version, using the usual fundamental matrix approach, depends on the Lyapunov exponents of the w subsystem. These are hereinafter referred to as sub-Lyapunov exponents to distinguish them from the full Lyapunov spectrum of the (v,w)=(u) system. Since the w subsystem 14 is driven by the v subsystem 12, the sub-Lyapunov exponents of the w subsystem 14 are dependent on the m dimensional v variable. If at least one of the sub-Lyapunov exponents is positive, the unit displacement vector .xi. will grow without bounds and synchronization will not take place. Accordingly, the subsystems 14 and 16 (w and w') will synchronize only if the sub-Lyapunov exponents are all negative. This principle provides a criterion in terms of computable quantities (the sub-Lyapunov exponents) that is used to design synchronizing systems in accordance with the present invention.

The v=(v.sub.1, . . . ,v.sub.m) components (subsystem 12) can be viewed more broadly as driving variables and the w'=(w'.sub.m+1, . . . , w'.sub.n) components (subsystem 16) as responding variables. The drive system 9 (v,w) can be viewed as generating at least one drive signal S.sub.d, in the formula v(t), which is applied to the response systems w and w' (subsystems 14 and 16, respectively) to synchronize the drive system and the response system outputs. This is the approach taken in accordance with the present invention to provide synchronized nonlinear dynamical systems.

In practicing this invention, the above discussion applies to identical subsystems 14 and 16. This might be achievable, for example, in digital systems. In such systems 10, the signals S.sub.o and S.sub.o ' may each be chaotic because the system 9 might be chaotic. They may differ because of different initial conditions in subsystems 14 and 16. However, they will approach each other (.DELTA.w.fwdarw.0) because systems 14 and 16 are stable (that is, with all negative sub-Lyapunov exponents) when considered as driven by the same at least one drive signal S.sub.d.

In most physical systems, subsystems 14 and 16 are not identical. For example, two electrical components with the same specifications typically do not have identical characteristics. The following explanation based on mathematical modeling shows that the signals S.sub.o and S.sub.o ' will nevertheless be synchronized if both subsystems 14 and 16 have negative sub-Lyapunov exponents. According to this mathematical model, the synchronization is affected by differences in parameters between the w and w' systems which are found in real-life applications. Let .mu. be a vector of the parameters of the w subsystem (subsystem 14) and .mu.' of the w' subsystem (subsystem 16), so that h=h(v,w,.mu.), for example. If the w subsystem were one-dimensional, then for small .DELTA.w and small .DELTA..mu.=.mu.'-.mu.:

.DELTA.w.apprxeq.h.sub.w .DELTA.w+h.sub..mu. .DELTA..mu. (7)

where h.sub.w and h.sub..mu. are the partial derivatives of h with respect to w and .mu., respectively. Roughly, if h.sub.w and h.sub..mu. are nearly constant in time, the solution of this equation will follow the formula ##EQU1##

If h.sub.w <0, the difference between w and w' will level off at some constant value and the systems will be synchronized. Although this is a simple one dimensional approximation, it turns out to be the case for all systems that have been investigated numerically, even when the differences in parameters are rather large (.about.10-20%). This is also the case in the exemplary electronic synchronizing circuit described in more detail hereinbelow. Furthermore, it can be established on a mathematical basis that the small changes in parameters only lead to proportionally small degradations of synchronization, which approach a constant value. See Pecora et al., "Driving systems with chaotic signals", Physical Review A, Vol. 44, No. 4, Aug. 15, 1991, pages 2374-2383.

Since m-dimensional variable v may be dependent on (n-m)-dimensional variable w, there may be feedback from subsystem 14 to subsystem 12. As shown in FIG. 2, a response part 15 of subsystem 14 may produce a feedback signal S.sub.r responsive to the m-dimensional driving variable v, and a drive part 17 of subsystem 12 may respond to the feedback signal S.sub.F to produce the at least one drive signal S.sub.d.

As shown in FIG. 2, subsystems 14 and 16 need not be driven by the same at least one drive signal S.sub.d but could be driven by at least one input signals S.sub.I and S.sub.I responsive to the at least one drive signal S.sub.d. System 10 could have primary and secondary means 18 and 19, respectively, coupled to subsystem 12 and responsive to the at least one drive signal S.sub.d for generating input signals S.sub.I and S.sub.I ', respectively. If these primary and secondary means 18 and 19, respectively are linearly responsive to the at least one drive signal S.sub.d, then the above mathematical analysis would continue to apply since linear transformations do not affect the signs of the sub-Lyapunov exponents.

In accordance with the prior art, in order to develop electrical circuits, for example, which have chaotic dynamics, but which will synchronize, a nonlinear dynamical circuit (the drive subsystem) is duplicated (to form a response subsystem). A selected portion of the response circuit is removed, and all broken connections are connected to voltages produced at their counterparts in the drive circuit. These driving voltages constitute the at least one drive signal S.sub.d shown in FIG. 1, and advantageously are connected to the response circuit via a buffer amplifier to ensure that the drive circuit is not affected by the connection to the response circuit, i.e., it remains autonomous.

As a specific example, FIG. 3 shows an electrical circuit system 20 constructed in accordance with the prior art which has two synchronized nonlinear dynamical subsystems, a drive circuit 22 and a response circuit 16. Circuits 22 and 16 correspond to the u and w' systems, respectively, discussed above.

Drive circuit 22 comprises a hysteretic circuit formed by a differential amplifier 30, resistors 42, 44, 46, 48, 50 and 52; potentiometer 74; capacitor 76; and diodes 82 and 84 connected as shown; and an unstable oscillator circuit formed by differential amplifiers 32, 34, 36, 38 and 40; resistors 58, 60, 62, 64, 66, 68, 70 and 72; and capacitors 78 and 80 connected as shown. In an experimental implementation of circuit system 20 which has been successfully tested, amplifiers 30-40 were MM741 operational amplifiers, and diodes 82 and 84 were 1N4739A diodes. Component values for the resistors and capacitors which were used are set forth in the following table:

    ______________________________________
    Resistor 42 = 10K.OMEGA.
                      Resistor 62 = 220K.OMEGA.
    Resistor 44 = 10K.OMEGA.
                      Resistor 64 = 150K.OMEGA.
    Resistor 46 = 10K.OMEGA.
                      Resistor 66 = 150K.OMEGA.
    Resistor 48 = 20K.OMEGA.
                      Resistor 68 = 330K.OMEGA.
    Resistor 50 = 100K.OMEGA.
                      Resistor 70 = 100K.OMEGA.
    Resistor 52 = 50K.OMEGA.
                      Resistor 72 = 100K.OMEGA.
    Resistor 54 = 3K.OMEGA.
                      Potentiometer 74 = 10K.OMEGA.
    Resistor 56 = 20K.OMEGA.
                      Capacitor 76 = 0.01 .mu.F
    Resistor 58 = 100K.OMEGA.
                      Capacitor 78 = 0.01 .mu.F
    Resistor 60 = 100K.OMEGA.
                      Capacitor 80 = 0.001 .mu.F
    ______________________________________

    ______________________________________
    Resistor R1 = 100k.OMEGA.
                       Resistor R11 = 221k.OMEGA.
    Resistor R2 = 100k.OMEGA.
                       Resistor R12 = R.sub.v
    Resistor R3 = 100k.OMEGA.
                       Resistor R13 = 100k.OMEGA.
    Resistor R4 = 100k.OMEGA.
                       Resistor R14 = 200k.OMEGA.
    Resistor R5 = 68.2k.OMEGA.
                       Resistor R15 = 100k.OMEGA.
    Resistor R6 = 100k.OMEGA.
                       Resistor R16 = 100k.OMEGA.
    Resistor R7 = 100k.OMEGA.
                       Resistor R17 = 100k.OMEGA.
    Resistor R8 = 68.2k.OMEGA.
                       Resistor R18 = 100.OMEGA.
    Resistor R9 = 1m.OMEGA.
    Resistor R10 = 100k.OMEGA.
    Capacitor C1 = 910pf
                       Capacitor C3 = 910pf
    Capacitor C2 = 910pf
                       Capacitor C4 = 910pf.
    ______________________________________

    ______________________________________
    Resistors R21 = 27.4k.OMEGA.
                       Resistors R29 = 50.1.OMEGA.
    Resistors R22 = 27.4k.OMEGA.
                       Resistors R30 = 50.1.OMEGA.
    Resistors R23 = 49.9k.OMEGA.
                       Resistors R31 = 50.1.OMEGA.
    Resistors R24 = 49.9k.OMEGA.
                       Resistors R32 = 50.1.OMEGA.
    Resistors R25 = 200k.OMEGA.
                       Resistors R33 = 20k.OMEGA.
    Resistors R26 = 200k.OMEGA.
                       Resistors R34 = 178k.OMEGA.
    Resistors R27 = 825k.OMEGA.
                       Resistors R35 = 156.2k.OMEGA.
    Resistors R28 = 825k.OMEGA.
                       Resistors R36 = 100k.OMEGA.
    ______________________________________

    ______________________________________
    Resistor R1 = 10k.OMEGA.
                       Resistor R2 = 39.2k.OMEGA.
    Resistor R3 = 10k.OMEGA.
                       Resistor R4 = 10k.OMEGA.
    Resistor R5 = 10k.OMEGA.
                       Resistor R6 = 10k.OMEGA.
    Resistor R7 = 100k.OMEGA.
                       Resistor R8 = 1M.OMEGA.
    Resistor R9 = 1M.OMEGA.
                       Resistor R10 = 100k.OMEGA.
    Resistor R11 = 1M.OMEGA.
                       Resistor R12 = 100k.OMEGA.
    Resistor R13 = 100k.OMEGA.
                       Resistor R14 = 100k.OMEGA.
    Resistor R15 = 5.2k.OMEGA.
                       Resistor R16 = 100k.OMEGA.
    Resistor R17 = 100k.OMEGA.
                       Resistor R18 = 1M.OMEGA.
    Capacitor C1 = 1nF Capacitor C2 = 1nF
    Capacitor C3 = 1nF
    ______________________________________

    ______________________________________
    Resistor R19 = 100k.OMEGA.
                      Resistor R20 = 100k.OMEGA. Resistor
    R21 = 100k.OMEGA. Resistor R22 = 100k.OMEGA. Resistor
    R23 = 680k.OMEGA. Resistor R24 = 2M.OMEGA. Resistor
    R25 = 680k.OMEGA. Resistor R26 = 2M.OMEGA. Resistor
    R27 = 100k.OMEGA.
    Potentiometer P1 = 20k.OMEGA.
                      Potentiometer P2 = 50k.OMEGA.
    Potentiometer P3 = 20k.OMEGA.
                      Potentiometer P4 = 50k.OMEGA.
    Potentiometer P5 = 20k.OMEGA. in
                      Potentiometer P6 = 20k.OMEGA. in
    parallel with a 100.OMEGA.
                      parallel with a 100.OMEGA.
    resistor (not shown)
                      resistor (not shown)
    Potentiometer P7 = 20k.OMEGA. in
                      Potentiometer P8 = 20k.OMEGA. in
    parallel with a 100.OMEGA.
                      parallel with a 100.OMEGA.
    resistor (not shown)
                      resistor (not shown)
    ______________________________________

    ______________________________________
    Resistor R28 = 10k.OMEGA.
                       Resistor R29 = 10k.OMEGA.
    Resistor R30 = 490k.OMEGA.
                       Resistor R31 = 490k.OMEGA.
    Resistor R32 = 50k.OMEGA.
                       Resistor R33 = 50k.OMEGA.
    Resistor R34 = 20k.OMEGA.
                       Resistor R35 = 100k.OMEGA.
    Resistor R36 = 100k.OMEGA.
                       Resistor R37 = 100k.OMEGA..
    ______________________________________

    ______________________________________
    j = 1             j = 2    j = 3
    ______________________________________
    i = 1   204,000 .OMEGA.
                          408,000 .OMEGA.
                                   1026 .OMEGA.
    i = 2   102,000 .OMEGA.
                          204,000 .OMEGA.
                                   513 .OMEGA.
    i = 3   68,000 .OMEGA.
                          136,000 .OMEGA.
                                   342 .OMEGA.
    i = 4   51,000 .OMEGA.
                          102,000 .OMEGA.
                                   256 .OMEGA.
    i = 5   40,800 .OMEGA.
                           82,000 .OMEGA.
                                   205 .OMEGA.
    ______________________________________

• Inventors
  Carroll, Thomas L.;
• Assignee
  The United States of America as represented by the Secretary of the Navy (Washington, DC)
• Published
  Aug-5-1997
• Current US Classes:
  380/263
  380/28
  380/46
• Application #
  536027
• International Classes
  H04L 009/12
• Field of Search
  380/28 380/46 380/48
• Examiner
  Barron, Jr.; Gilberto
• Agent
  McDonnell; Thomas E., Jameson; George
• US Patent References:
  5245660
  5291555
  5379346
  5402334