U.S. patent application number 12/547453 was filed with the patent office on 2009-12-17 for method for modulating a carrier signal and method for demodulating a modulated carrier signal.
Invention is credited to Klaus HUBER.
Application Number | 20090310689 12/547453 |
Document ID | / |
Family ID | 33305105 |
Filed Date | 2009-12-17 |
United States Patent
Application |
20090310689 |
Kind Code |
A1 |
HUBER; Klaus |
December 17, 2009 |
METHOD FOR MODULATING A CARRIER SIGNAL AND METHOD FOR DEMODULATING
A MODULATED CARRIER SIGNAL
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.
Inventors: |
HUBER; Klaus; (Darmstadt,
DE) |
Correspondence
Address: |
KENYON & KENYON LLP
ONE BROADWAY
NEW YORK
NY
10004
US
|
Family ID: |
33305105 |
Appl. No.: |
12/547453 |
Filed: |
August 25, 2009 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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10555527 |
Feb 6, 2006 |
7580473 |
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PCT/DE04/00222 |
Feb 9, 2004 |
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12547453 |
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Current U.S.
Class: |
375/259 |
Current CPC
Class: |
H03C 1/02 20130101; H03C
3/02 20130101; H04L 27/00 20130101 |
Class at
Publication: |
375/259 |
International
Class: |
H04L 27/00 20060101
H04L027/00 |
Foreign Application Data
Date |
Code |
Application Number |
May 2, 2003 |
DE |
103 19 636.6 |
Claims
1-18. (canceled)
19. A method for modulating a carrier signal for the transmission
of message signals, comprising: varying a signal shape of the
carrier signal over time by a message signal to be transmitted, an
amplitude and a frequency of the carrier signal remaining
constant.
20. The method as recited in claim 19, wherein a time
characteristic of the carrier signal is defined by an elliptic
function.
21. The method as recited in claim 20, wherein the elliptic
function is a Jacobian elliptic function.
22. The method as recited in claim 20, wherein modulus k of the
elliptic function is varied over time by the message signal to be
transmitted so as to modulate the signal shape of the carrier
signal in a rhythm of the message signal to be transmitted.
23. The method as recited in claim 20, wherein the time
characteristic of the modulated carrier signal is defined by the
elliptic function s(t)=a.sub.0 sx(2{circumflex over
(.pi.)}f.sub.0t,k(t)), a.sub.0 being the amplitude and f.sub.0 the
frequency, and {circumflex over (.pi.)} and modulus k being linked
via a complete elliptic integral of a first kind.
24. The method as recited in claim 23, wherein the function
sx(2{circumflex over (.pi.)}f.sub.0t,k(t)) for
0.ltoreq.k(t).ltoreq.1 is defined by Jacobian elliptic function
sn(2{circumflex over (.pi.)}f.sub.0t,k(t)), and for
-1.ltoreq.k(t).ltoreq.0 by Jacobian elliptic function
cn(2{circumflex over (.pi.)}f.sub.0(t-T/4),k(t)).
25. The method as recited in claim 20, wherein an orthogonal
transmission method is used, which is based on orthogonal elliptic
basic functions (sn(2{circumflex over (.pi.)}f.sub.0t,k(t)),
sd(2{circumflex over (.pi.)}f.sub.0t,k(t)), cd(2{circumflex over
(.pi.)}f.sub.0t,k(t)) and cn(2{circumflex over
(.pi.)}f.sub.0t,k(t))).
26. The method as recited in claim 20, wherein the carrier signal
defined by the elliptic function is generated using an analog
circuit configuration which supplies at least one modulated carrier
signal (s(t)) whose curve shape at least one of sectionally
corresponds to and is approximated to an elliptic function.
27. An apparatus for modulating a carrier signal for the
transmission of message signals, comprising: a modulator, wherein
the modulation by the modulator of the carrier signal (s(t)) is
implemented so that a signal shape of the carrier signal is able to
be varied over time by a message signal (m(t)) to be transmitted,
the amplitude (a.sub.0) and the frequency (f.sub.0) of the carrier
signal (s(t)) remaining constant.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] The present application is a continuation of U.S. patent
application Ser. No. 10/555,527, issuing as U.S. Pat. No.
7,580,473, which was the national stage of PCT/DE2004/000222 filed
on Feb. 9, 2004, which claimed priority to German Patent
Application No. DE 10319636.6 filed on May 2, 2003, each of which
is expressly incorporated herein in its entirety by reference
thereto.
FIELD OF THE INVENTION
[0002] 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
[0003] 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.sub.0+cm(t))sin(2.pi.f.sub.0t),
where f.sub.0 denotes the carrier frequency, and a.sub.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.sub.0 of the modulated
carrier signal not being able to be varied over time.
[0004] 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.sub.0sin(2{circumflex over (.pi.)}f(m(t))),
where frequency f(m(t)) in most cases being defined by the
expression (f.sub.0+c m(t)). In a frequency modulation amplitude
a.sub.0 is constant.
SUMMARY OF THE INVENTION
[0005] Embodiments of the present invention may involve adding a
new modulation and demodulation method to available modulation and
demodulation methods.
[0006] Additional embodiments of the present invention may involve
providing an analog modulator circuit for the new modulation
method.
[0007] 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.sub.0 nor frequency f.sub.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.
[0008] 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.
[0009] 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.
[0010] 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 uber
allgemeine Funktionentheorie und elliptische Funktionen" [i.e.,
"Lectures on general function theory and elliptic functions"],
5.sup.th edition, Springer Berlin Heidelberg New York, 2000,
incorporated in its entirety by reference herein, may be
utilized.
[0011] 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.
[0012] The time characteristic of the modulated carrier signal may
be defined by the elliptic function s(t)=a.sub.0sx(2{circumflex
over (.pi.)}f.sub.0t,k(t)), a.sub.0 being the amplitude and f.sub.0
the frequency. {circumflex over (.pi.)} and modulus k may be linked
via the complete elliptic integral of the first kind.
[0013] In embodiments of the present invention, the function
sx(2{circumflex over (.pi.)}f.sub.0t,k(t)) for
0.ltoreq.k(t).ltoreq.1 may be defined by the Jacobian elliptic
function sn(2{circumflex over (.pi.)}f.sub.0t,k(t)), and for
-1.ltoreq.k(t).ltoreq.0 by the Jacobian elliptic function
cn(2{circumflex over (.pi.)}f.sub.0(t-T/4), |k(t)|).
[0014] 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 (.pi.)}f.sub.0t,k(t)) and sd(2{circumflex over
(.pi.)}f.sub.0t,k(t)), or by the two orthogonal elliptic functions
cd(2{circumflex over (.pi.)}f.sub.0t,k(t)) and cn(2{circumflex over
(.pi.)}f.sub.0t,k(t)), may be utilized toward this end.
[0015] 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.
[0016] 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.sub.0sx(2{circumflex over (.pi.)}f.sub.0t, k(t)). a.sub.0 is
the amplitude and f.sub.0 is the frequency of the carrier signal,
{circumflex over (.pi.)} and modulus k being linked via the
complete elliptic integral of the first kind.
[0017] 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.sub.0. Modulus k(t)--and hence
transmitted message signal m(t)--may be obtained from the sampling
values.
[0018] In alternative embodiments, i.e., an alternative
demodulation method, received modulated carrier signal
s(t)=a.sub.0sx(2{circumflex over (.pi.)}f.sub.0t, k(t)) may be
integrated in order to obtain modulus k(t).
[0019] In alternative embodiments, i.e., another alternative
demodulation method, received modulated carrier signal
s(t)=a.sub.0sx(2{circumflex over (.pi.)}f.sub.0t,k(t)) may be
squared and then integrated.
[0020] 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.
[0021] 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.
[0022] In embodiments, the elliptic functions may be Jacobian
elliptic functions.
[0023] 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.
[0024] 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.sub.0sx(2{circumflex over (.pi.)}f.sub.0t,k(t)),
a.sub.0 being the amplitude and f.sub.0 the frequency of the
carrier signal, {circumflex over (.pi.)} and modulus k being linked
via the complete elliptic integral of the first kind.
[0025] 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 (.pi.)}f.sub.0t,k(t)); cn(2{circumflex over
(.pi.)}f.sub.0t,k(t)); and dn(2{circumflex over
(.pi.)}f.sub.0t,k(t))
[0026] In embodiments, an analog division device for forming
quotient sn(2{circumflex over (.pi.)}f.sub.0t,k(t))/dn(2{circumflex
over (.pi.)}f.sub.0t,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.sup.2)}. For 0=k(t)=1, output signal sn(2{circumflex over
(.pi.)}f.sub.0t,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
[0027] 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.
[0028] 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.
[0029] FIG. 3 shows an exemplary modulator according to the present
invention.
[0030] FIG. 4 shows an exemplary circuit configuration for
generating the elliptic function sn(2{circumflex over
(.pi.)}f.sub.0t).
[0031] FIG. 5 shows a circuit configuration for calculating the
arithmetic-geometric mean M.
[0032] FIG. 6 shows an alternative circuit configuration for
calculating the arithmetic-geometric mean M.
[0033] FIG. 7 shows a circuit configuration for calculating r
[0034] FIG. 8 shows section of the curve shape of a carrier signal
modulated according to a binary shape jump method.
DETAILED DESCRIPTION
[0035] 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 (.pi.)}f.sub.0t,k), cn(2{circumflex over
(.pi.)}f.sub.0t,k) and dn(2{circumflex over (.pi.)}f.sub.0t,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.sub.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.sub.0sx(2{circumflex over (.pi.)}f.sub.0t,k(t)) (1)
[0036] The role of .pi. in the classic sine or cosine carrier
signals is assumed by {circumflex over (.pi.)} in elliptic
functions. {circumflex over (.pi.)} is a function of modulus k, the
correlation between {circumflex over (.pi.)} and k being given by
the so-called complete elliptic integral of the first kind as
follows:
.pi. ^ 2 = K ( k ) = .intg. 0 .pi. / 2 .PHI. 1 - k 2 sin 2 ( .phi.
) ( 2 ) ##EQU00001##
{circumflex over (.pi.)} may easily be calculated with the aid of
the equation
.pi. ^ = .pi. M ( 1 , 1 - k 2 ) , ( 3 ) ##EQU00002##
M(1, {square root over (1-k.sup.2)} being the arithmetic-geometric
mean of 1 and {square root over (1-k.sup.2)}.
[0037] Analog circuit configurations for calculating the
arithmetic-geometric mean are shown in FIGS. 5 and 6. To be able to
generate {circumflex over (.pi.)} in terms of circuit engineering,
first of all, the arithmetic-geometric mean M(1, {square root over
(1-k.sup.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.sup.2)} is applied to its
other input. On condition that the factor {square root over
(1-k.sup.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.sup.2)} applied to the inputs of analog
computing circuit 210.
[0038] 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.sup.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.sup.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.sup.2)}.
[0039] At this point, {circumflex over (.pi.)} may be calculated
via a division device 290, shown in FIG. 7, at whose inputs are
applied the number {circumflex over (.pi.)} and the
arithmetic-geometric mean M(1, {square root over (1-k.sup.2)})
which is generated, for instance, by the circuit shown in FIG. 5 or
in FIG. 6.
[0040] 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 (.pi.)}f.sub.0t,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 (.pi.)}. The utilization of the
elliptic function sn(2{circumflex over (.pi.)}f.sub.0t,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 (.pi.)}f.sub.0t,k)
may be utilized. In order to obtain this function in the same phase
position as the Jacobian elliptic function sn(2{circumflex over
(.pi.)}f.sub.0t,k), function cn, shifted by T/4, is considered,
which may be expressed as follows:
cn ( 2 .pi. ^ ( t - T / 4 ) f 0 , k ( t ) ) = 1 - k 2 sn ( 2 .pi. ^
f 0 t , k ( t ) ) dn ( 2 .pi. ^ f 0 t , k ( t ) ) = 1 - k 2 sd ( 2
.pi. ^ f 0 t , k ( t ) ) ( 4 ) ##EQU00003##
[0041] FIG. 2 illustrates the function cn(2{circumflex over
(.pi.)}(t-T/4)f.sub.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.
[0042] 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
(.pi.)}f.sub.0t,k(t)), defined in equation 1, may be defined as
follows:
sx ( 2 .pi. ^ f 0 t , k ( t ) ) = { sn ( 2 .pi. ^ f 0 t , k ( t ) )
for 0 .ltoreq. k .ltoreq. 1 1 - k 2 sd ( s .pi. ^ f 0 t , k ) for -
1 .ltoreq. k .ltoreq. 0 ( 5 ) ##EQU00004##
[0043] In this equation, k is the modulation parameter carrying the
message. The values of k lie within the interval [-1.1].
[0044] FIG. 3 shows an exemplary modulator, which is composed of
analog computing circuits and electrically simulates the function
sx(2{circumflex over (.pi.)}f.sub.0t,k(t)).
[0045] 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 (.pi.)}/T. Multiplier 50 multiplies the output
signal of multiplier 40 by the factor
- 2 .pi. ^ T . ##EQU00005##
Multiplier 80 multiplies the output signal of multiplier 70 by the
factor
- k 2 2 .pi. ^ T . ##EQU00006##
[0046] 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.
[0047] It should be noted that measures, available in circuit
engineering, for taking into account predefined initial states
during initial operation are not marked in the circuit. Such an
analog circuit configuration, shown in FIG. 3, delivers the
Jacobian elliptic time function sn(2{circumflex over
(.pi.)}f.sub.0t) at the output of integrator 30, the Jacobian
elliptic function cn(2{circumflex over (.pi.)}f.sub.0t) at the
output of integrator 60, and the Jacobian elliptic function
dn(2{circumflex over (.pi.)}f.sub.0t) at the output of integrator
90. It should be noted that the multiplication by
.+-. 2 .pi. ^ T ##EQU00007##
in multipliers 20 and 50, respectively, and the multiplication
by
- k 2 2 .pi. ^ T ##EQU00008##
in multiplier 80 may also be carried out in integrators 30, 60 and
90. The multiplication by k.sup.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.
[0048] All three Jacobian elliptic time functions sn(2{circumflex
over (.pi.)}f.sub.0t), cn(2{circumflex over (.pi.)}f.sub.0t) and
dn(2{circumflex over (.pi.)}f.sub.0t) 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.
[0049] 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.sup.2)}sd(2{circumflex over
(.pi.)}f.sub.0t,k(t)) in conjunction with a multiplier 97,
which--as explained herein--corresponds to the elliptic function
cn(2{circumflex over (.pi.)}f.sub.0t,k(t)) shifted by T/4.
[0050] 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
(.pi.)}f.sub.0t,k(t)), namely for 0.ltoreq.k(t).ltoreq.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.sup.2)}sd(2{circumflex
over (.pi.)}f.sub.0t,k(t)), namely for
-1.ltoreq.|k(t)|.ltoreq.1.
[0051] The signal-shape modulation is implemented via k or
{circumflex over (.pi.)} in multipliers 20, 50 and 80. As
mentioned, modulus k and {circumflex over (.pi.)} are linked via
the complete elliptic integral of the first kind.
[0052] FIG. 7 illustrates an exemplary analog circuit for
calculating {circumflex over (.pi.)} as a function of message
signal m(t) to be transmitted, which modulates modulus k.
[0053] The signal-form modulation of carrier signal s(t) takes
place in multiplier 80 via the expression -k.sup.22{circumflex over
(.pi.)}/T, in multiplier 50 via factor -2{circumflex over
(.pi.)}/T, and in multiplier 20 by factor 2{circumflex over
(.pi.)}/T.
[0054] 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.
[0055] 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.sub.0sn(2{circumflex over (.pi.)}f.sub.0t) if a "1" is
to be transmitted, and to transmit a carrier signal of the function
a.sub.0 {square root over (1-k.sup.2)}sd(2{circumflex over
(.pi.)}f.sub.0t) 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.
[0056] Hereinafter, three exemplary demodulation methods are
indicated to recover transmitted message signal m(t) from received
modulated carrier signal s(t).
[0057] The first demodulation method is based on the fact that
frequency f.sub.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.sub.0; and at instant 3T/4 it has
the value -a.sub.0. At instants T/8 and 3T/8, function value
a.sub.0sx(T/8) results. At instants 5T/8 and 7T/8, the function
value is
- a 0 sx T 8 . ##EQU00009##
[0058] The value of sxT/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.sup.2)}.
Modulation parameter k(t), which changes slowly with respect to
frequency f.sub.0 of the carrier signal, and thus message m(t), may
therefore be recovered by sampling in the odd multiples of T/8.
[0059] 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
.intg. sn ( x , k ) x = - ln ( dn ( x ) + kcn ( x ) ) k , .intg. cn
( x , k ) x = arcsin ( k sn ( x ) k , ##EQU00010##
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
.intg. 0 T / 2 s ( t ) t = { .intg. 0 T / 2 a 0 sn ( 2 .pi. ^ t / T
) t = a 0 T 2 .pi. ^ ( k ) k ln 1 + k 1 - k .intg. 0 T / 2 a 0 cn (
2 .pi. ^ ( t - T / 4 ) / T ) t = a 0 T .pi. ^ ( k ) k arcsin k
##EQU00011##
[0060] An integration over a quarter period in each case results in
one half of the values.
[0061] According to the third demodulation method, modulated
carrier signal s(t) is first squared and then integrated according
to the equation
.intg. 0 T s ( t ) 2 = { .intg. 0 T ( a 0 sn ( 2 .pi. ^ t / T ) ) 2
t = a 0 2 T K ( k ) - E ( k ) k 2 K ( k ) .intg. 0 T ( a 0 cn ( 2
.pi. t ^ / T ) ) 2 t = a 0 2 T E ( k ) - k '2 K ( k ) k 2 K ( k )
##EQU00012##
[0062] E(k) is the so-called complete elliptic integral of the
second kind, and k' is {square root over (1-k.sup.2)}). An
integration over half (a quarter of) a period in each case results
in half (a quarter o)f the value.
[0063] 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 (.pi.)}f.sub.0t may be
used, which is defined as follows:
sy ( x , k ( t ) ) = { cd ( x , k ( t ) ) for 0 .ltoreq. k .ltoreq.
1 cn ( x , k ) for - 1 .ltoreq. k .ltoreq. 0 ##EQU00013##
[0064] 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
.intg..sub.0.sup.4Ksx(x)sy(x)dt=0
applies.
[0065] As a result, elliptic functions may be used for the
orthogonal modulation. When values are given for a.sub.0, f.sub.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.
* * * * *