EP1623497A1

EP1623497A1 - Method for modulating a carrier signal, and method for demodulating a modulated carrier signal

Info

Publication number: EP1623497A1
Application number: EP04709199A
Authority: EP
Inventors: Klaus University of Illinois HUBER
Original assignee: Deutsche Telekom AG
Current assignee: Deutsche Telekom AG
Priority date: 2003-05-02
Filing date: 2004-02-09
Publication date: 2006-02-08
Also published as: US7580473B2; US20060126756A1; US20090310689A1; DE10319636A1; WO2004098041A1

Abstract

The invention relates to a novel method for modulating a carrier signal used for transmitting analog or digital message signals. The module k of elliptic functions is used as a modulation parameter instead of the amplitude or the frequency as in conventional amplitude and angle modulation methods. The carrier signal modulated according to the novel modulation method is thus provided with a constant amplitude and a fixed frequency while the signal form is chronologically modified at the rhythm of the message that is to be transmitted.

Description

Method for modulating a carrier signal and method for demodulating a modulated one

carrier signal

The invention relates to a method for modulating a carrier signal for the transmission of message signals and a method for demodulating carrier signals modulated in this way. The invention further relates to an analog circuit arrangement for modulating a carrier signal, which can be represented by an elliptical function.

In communications technology, high-frequency sinusoidal or cosine-shaped carrier signals are generally used to transmit messages, e.g. Voice, music, images or data. For this, the one to be transferred

Message modulated onto a carrier signal. Known and widely used modulation methods are angle and amplitude modulation. As is well known, in amplitude modulation, the information contained in the message signal m (t) is substantially in accordance with the equation

s (t) = (ao + c-m (t)) • sin (2ττfot)

modulated onto the carrier signal. The carrier frequency is denoted by f ₀ , where a ₀ and c are constants which are selected in accordance with practical requirements. A characteristic property of the amplitude modulation is that the amplitude of the signal s (t) is modulated in the rhythm of the message m (t) to be transmitted, the frequency f _{0 of} the modulated carrier signal not changing over time.

With the known angle modulation, however, the frequency or phase is in the rhythm of the one to be transmitted Message signal m (t) changed over time. The frequency-modulated signal transmitted over a transmission channel is

s (t) = a ₀ -sin (2? f (m (t)))

the frequency f (m (t)) is mostly defined by the expression (f ₀ + cm (t)). With frequency modulation, the amplitude a _{0 is} constant.

The invention is based on the object of adding a new modulation and demodulation method to the known modulation and demodulation methods. Another task is to provide an analog modulator circuit for the new modulation method.

A core idea of the invention is that a so-called waveform modulation method is used, in which, in contrast to the amplitude and angle modulation, neither the amplitude ao nor the frequency f _{0 is changed over time} in the rhythm of the message signal to be transmitted.

Rather, the signal form of the carrier signal itself is changed.

The technical problem mentioned above is caused on the one hand by the

Process steps of claim 1 solved.

Then a method for modulating a carrier signal for the transmission of message signals is described.

According to the invention, the signal shape of the carrier signal is changed in time by a message signal to be transmitted, the amplitude and the frequency of the carrier signal remaining constant. This new modulation method is referred to as a waveform modulation method in order to differentiate it from classic amplitude and frequency modulation.

The waveform modulation method is preferably based on the modulation of carrier signals, the temporal course of which is defined by an elliptical function. Jacobi preferably uses elliptical functions, which are described, for example, in the book A. Hurwitz, Lectures on General Function Theory and Elliptical Functions ", Springer, 5th Edition, Springer Berlin Heidelberg New York, 2000.

As already mentioned, neither the amplitude nor the frequency is used as the modulation parameter, but rather the module k determining the shape of an elliptic function. The module k is changed in 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 transmitted.

The temporal course of the modulated carrier signal can by the elliptical function

s (t) = a ₀ sx (2; rfot, k (t)), where a _{0 is} the

Amplitude and fo is the frequency. π and the module k are related via the complete elliptic integral of the first kind.

According to an advantageous development, the function sx (2; rfot, k (t)) for 0 <k (t) <l is determined by the Jacobi elliptic function sn (2τrf ₀ t, k (t)) and for -l≤k ( t) <0 by the Jacobi elliptic function cn (2 f ₀ (tT / 4), | & (t) |) Are defined .

With the help of elliptical functions, known orthogonal transmission methods based on sine and cosine carriers can be generalized. In this way it is possible to use new orthogonal modulation methods. For this purpose, orthogonal carrier signals are used, which are represented by the two orthogonal elliptical functions sn (2; fot, k (t)) and sd (2 ^ f ₀ t, k (t)) or the two orhogonal elliptical functions cd (2πfnt, k ( t)) and cn (2 ^ fot, k (t)) can be defined.

The carrier signals defined by an elliptical function are expediently generated using an analog circuit arrangement. Analog circuit arrangements are composed of operational amplifiers, integrators, multipliers, differential amplifiers and dividers known per se. Analog circuit arrangements for generating elliptical functions are in the patent application filed on the same day with the internal

File number P03025 described in detail. The content of this German patent application is hereby incorporated in full.

The above-mentioned technical problem is also solved by the method steps of claim 9. A method for demodulating a modulated carrier signal is then made available, the temporal course of which is described by the elliptic function s (t) = a ₀ "sx (2 ^ fo ^f t, k (t)), an is the amplitude and fπ is the frequency of the carrier signal, where π and the module k are related via the complete elliptical integral of the first kind. For the demodulation, the received modulated carrier signal is sampled at times which correspond to the odd multiples of T / 8, with T = l / f ₀ . The module k (t) and thus the transmitted message signal m (t) is obtained from the sampled values.

According to an alternative demodulation method, the received modulated carrier signal s (t) = ao 'sx (2 fo ^# t, k (t)) is integrated in order to obtain the module k (t).

According to a further alternative demodulation method, the received modulated carrier signal s (t) = ao * sx (2 ^ f _0-1 , k (t)) is squared and then integrated.

The above technical problem is also solved by a modulator according to claim 13.

The modulator is characterized in that the carrier signal is modulated in that the signal shape of the carrier signal can be changed over time by a message signal to be transmitted, the amplitude and the frequency of the carrier signal remaining constant.

A special embodiment of the modulator provides an analog circuit arrangement that delivers at least one modulated carrier signal, the curve shape of which corresponds at least in sections to an elliptical function or is approximated.

The elliptical functions are preferably Jacobi elliptical functions. Since the novel modulator neither modulates the amplitude nor the frequency of the carrier signal, devices are provided which temporally change the module k of an elliptical function by the message signal to be transmitted in order to modulate the signal form of the carrier signal in the rhythm of the message signal to be transmitted.

According to an alternative embodiment, the analog circuit arrangement of the modulator generates a modulated carrier signal, the temporal course of which is due to the elliptical function

s (t) = a ₀ * sx (2? fo-t, k (t))

is defined, where ao is the amplitude and f _{0 is} the frequency of the carrier signal and where π and the module k are related via the complete elliptical integral of the first type.

The circuit arrangement expediently has first analog multipliers and analog integrators which are connected together in such a way that the circuit arrangement has the three output functions

sn (2 ^ f ₀ t, k (t)) cn (2 f ₀ t, k (t)) dn (2τrf ₀ t, k (t))

supplies.

Furthermore, an analog division device is provided for forming the quotient sn (2πf ₀ t, k (t)) / dn (2 f ₀ t, k (t)) and a second analog multiplier assigned to the division device, which multiplier outputs the output signal of the division device multiplied by the factor -yl - k ² . For 0 = k (t) = l the output signal sn (2 fot, k (t)) forms the modulated carrier signal, whereas for -l = k (t) = 0 the output signal of the second analog multiplier forms the modulated carrier signal.

The invention is explained in more detail below using several exemplary embodiments in conjunction with the accompanying drawings. Show it:

1 shows a quarter period of the curve profiles of a carrier signal modulated by means of the module k, where 0 = k (t) = 1, FIG. 2 shows a quarter period of the curve profiles of a carrier signal modulated by means of the module k, where -l = k (t) = 0 .

Fig. 3 shows an exemplary modulator according to the

Invention, Fig. 4 shows an exemplary circuit arrangement for

Generation of the elliptic function sn (2 ^ f ₀ t), FIG. 5 shows a circuit arrangement for calculating the arithmetic-geometric mean M, FIG. 6 shows an alternative circuit arrangement for

Calculation of the arithmetic-geometric mean M, and FIG. 7 shows a circuit arrangement for calculating π,

8 shows a section of the curve shape of a carrier signal modulated according to a binary shape jump method.

In the following, a new modulation method for data transmission is described in detail, which does not use the amplitude or the frequency of a carrier signal as the modulation parameter, but the signal form. The New modulation method is preferably based on elliptical functions and is characterized in that, in contrast to amplitude modulation, the amplitude of the carrier signal remains unchanged and, in contrast to frequency modulation, the frequency of the carrier signal also remains unchanged. As mentioned, the new modulation method is preferably based on the Jacobi elliptic functions sn (2; rf ₀ t, k), cn (2 ^ f ₀ t, k) and dn (2 ^ f ₀ t, k). The second argument of the Jacobi elliptic functions, the value k, is called the module of the elliptic functions and, as detailed below, is used as a new modulation parameter. In other words, the module of the Jacobi elliptic functions is modulated according to a message m (t) to be transmitted. The module k is thus a function of time and described by k (t). We assume that the frequency of the message to be transmitted and thus the frequency of the change in k (t) is small compared to the frequency f ₀ = l / T of the change in the carrier signal. The modulated carrier signal transmitted via a message channel can be through

s (t) = a ₀ -sx (2 ^ fo-t), k (t)) (1)

can be specified. The role of π in the classical sine or cosine carrier signals is assumed by π in the case of elliptical functions, π is a function of the module k, the relationship between π and k being given by the so-called complete elliptic integral of the first kind as follows:

π <> π / 2 dφ

_Λ = - K-vk) = I, ^ψ - (2)

2 ^Jo ^ \ - k ² sm ² (φ) The calculation of π can easily be done using the equation

7t π = done, (3)

where M (l, Λjl - k ² ) is the arithmetic-geometric mean of 1

Analog circuit arrangements for calculating the arithmetic-geometric mean are shown in FIGS. 5 and 6.

In order to be able to generate π in terms of circuitry, the arithmetic-geometric mean M (1, For example, can be realized with an analog circuit arrangement, which is shown in Fig. 5. The circuit arrangement shown in FIG. 5 consists of a plurality of analog arithmetic circuits 210, 220, 230, designated AG, and an analog arithmetic circuit 240 for calculating the arithmetic mean of two input signals. The analog computing circuits 210 to 230 are designed such 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 becomes 1 at the first input of the analog computing circuit 210 and the value becomes at its other input created. Provided that the factor \ - k ^{2 is} between 0 and 1, the output signal of the analog circuit device 240 corresponds approximately to the arithmetic-geometric mean M of the values 1 and sjl - k ² applied to the inputs of the analog computing circuit 210 6 shows an alternative analog circuit arrangement for calculating the arithmetic-geometric mean M of the two values 1 and 4l ~ k ² . The circuit arrangement shown in FIG. 6 has an analog arithmetic circuit 250 for calculating the minimum of two input signals, one

Analog arithmetic circuit 260 for calculating the maximum from two input signals, an analog arithmetic circuit 270 for calculating the arithmetic mean from two input signals, and an analog arithmetic circuit 280 for calculating a geometric mean from two input signals. At an entrance to the

Analog arithmetic circuit 250 has the value 1, whereas an input of the analog arithmetic circuit 260 has the value 1- & ² . The output of the analog arithmetic circuit 250 for calculating the minimum of two input signals is connected to the input of the analog arithmetic circuit 270 and the analog arithmetic circuit 280. The output of the analog arithmetic circuit 260 for calculating the maximum of two input signals is with an input of the analog arithmetic circuit 270 and an input of

Analog computing circuit 280 connected. The output of the analog arithmetic circuit 270 is connected to an input of the analog arithmetic circuit 250, whereas the output of the analog arithmetic circuit 280 is connected to an input of the analog arithmetic circuit 260. In the analog circuit arrangement shown in FIG. 6, the outputs of the analog arithmetic circuit 270 and 280 each provide the arithmetic-geometric mean M of 1 and \ - k ² .

The calculation of π can now take place via a division device 290 shown in FIG. 7, at the inputs of which the number π and the arithmetic-geometric mean

M (1, 4l - k ² ), which for example of that in Fig. 5 or 6 shown in FIG.

The carrier signal s (t) is modulated in accordance with the time-changing value of k, but the zero crossings and the amplitude of the carrier signal remain unchanged. 1 shows various waveforms of a waveform-modulated carrier signal over a quarter period of the function sn (2 ^ f ₀ t, k) for k = 0, k = 0.8, k = 0.95 and k = 0.99 , Note that for k = 0 the elliptic function represents the sine function and for k-1 the tangent hyperbolic. The period of tangent hyperbolic is infinite, but leads to an impulse due to the scaling with π. With the elliptic function sn (2 fot, k) one obtains signal forms which are above the sine function for 0 = t = T / 4. The Jacobi elliptic function cn (2 τrf ₀ t, k) can also be used to generate waveforms below the sine function. In order to obtain this function in the same phase position as the Jacobi elliptic function sn (2 f ₀ t, k), the function cn shifted by T / 4 is considered, which can be expressed as follows:

cn (2 £ (tT / 4) f ₀ , k (t)) = T ^M ^ '»ώι (2πf ₀ t, k (t))

= 4l - k ² sd (2 f ₀ t, k (t)) (4)

2 shows the function cn (2π (tT / 4) f ₀ , k (t)) for k = 0, k = 0.8, k = 0.95 and k = 0.99. For k = 0, the sine function is obtained again.

It can be seen that by using the Jacobi elliptic functions sn and cn a wide variety of waveforms can be covered. Accordingly, the function sx (2? Fot, k (t)) defined in equation 1 can be defined as follows:

In this equation, k is the modulation parameter that carries the message. The values of k are in the interval

[-1,1].

An exemplary modulator constructed from analog arithmetic circuits is shown in FIG. 3, which electrically simulates the function sx (2 f ₀ t, k (t)).

As shown in FIG. 3, a multiplier 10 is a

Multiplier 20 and an analog integrator 30 connected in series. Furthermore, an analog multiplier 40, an analog multiplier 50 and another analog integrator 60 are connected in series. A third series circuit comprises a further analog multiplier 70, an analog multiplier 80 and an analog integrator 90. The analog multiplier 20 multiplies the output signal of the multiplier 10 by the factor 2π / Υ. The multiplier 50 multiplies that

2π output signal of the multiplier 40 with the factor ~ ~ f •

The multiplier 80 multiplies the output signal of the

_? 2π multiplier 70 by the factor -k -.

The output signal of the integrator 30 is fed back to the multiplier 40 and to the input of the multiplier 70. The output signal of the integrator 60 becomes fed back to the input of multiplier 10 and to the input of multiplier 70. The exit of the

Integrator 90 is fed back to the input of multiplier 40 and to the input of multiplier 10. It should be noted that measures known in terms of circuit technology take account of predefined initial states

Commissioning are not shown in the circuit.

Such an analog circuit arrangement shown in FIG. 3 supplies the Jacobi elliptical time function sn (2 ^ f ₀ t) at the output of the integrator 30 and the at the output of the integrator 60

Jacobi elliptic function cn (2 ^ f ₀ t) and at the exit of the

Integrator 90 the Jacobi elliptic function dn (2 ^ f ₀ t).

2π It should be noted that the multiplication by ± ^ r in the

Multipliers 20 and 50 and the multiplication by -k - in the multiplier 80 can also take place in the integrators 30, 60 and 90. The multiplication by k ² can also be applied to the output of the integrator 90. Furthermore, it is possible to add known stabilization circuits to the circuit arrangement shown in FIG. 3, such as those found in the technical literature “semiconductors

Circuit technology ", Tietze, Schenk, Springer Verlag, 5th edition, 1980, Berlin Heidelberg New York, pages 435-438.

With the analog circuit arrangement shown in FIG. 3, all three Jacobi elliptical time functions sn (2; rfot), cn (2 fot) and dn (2 ^ f ₀ t) can be implemented simultaneously. In addition, the derivatives of the Jacobi elliptical time functions sn, cn and dn are obtained at the output of the multipliers 10, 40 and 70. Furthermore, a division device 96 is connected to the outputs of the integrators 30 and 90 to, in conjunction with a multiplier 97, perform the elliptical function τrf ₀ t, k (t)) which, as explained above, corresponds to the elliptic function cn (2 f ₀ t, k (t)) shifted by T / 4.

The modulator can thus deliver a waveform-modulated carrier signal according to the Jacobi elliptic function sn (2 f ₀ t, (t)) at the output of the integrator 30, specifically for 0 <k (t) <1. At the output of the multiplier 97, the modulator generates a waveform-modulated carrier signal according to the

Jacobi supply elliptic function 4l - k ² sd (2 f ₀ t, k (t)),

The waveform modulation takes place via k or π in the multipliers 20, 50 and 80. As mentioned, the module k and π are related via the completely elliptical integral of the first type. An exemplary analog circuit for calculating π depending on the one to be transmitted

Message signal m (t) that modulates module k is shown in FIG. 7.

The waveform modulation of the carrier signal s (t) takes place in the multiplier 80 by the expression -k ² 2;? / T, in the multiplier 50 by the factor -2 π / Υ and in the multiplier 20 by the factor 2 π / Υ.

With the aid of the waveform modulation method, not only analog messages but also digital messages can be modulated onto a carrier signal. A simple binary so-called form jump method (FSV) can be defined, for example, by the agreement to send a carrier signal s (t) according to the elliptic function aosn (2 ^ f ₀ t) if a "1" is to be transmitted, and a carrier signal Function a _O 4l - k ² sd (2 f ₀ t) to be sent if a "0" is to be transmitted. The modulation parameter k is set to 0.9 in both cases, for example. Under the simplified assumption that one bit is transmitted per period, the bit sequence "10" is transmitted with the two successive signals. The corresponding curve profile is shown in FIG. 8.

In the following, three exemplary demodulation methods are specified in order to recover the transmitted message signal m (t) from the received modulated carrier signal s (t).

The first demodulation method is based on the fact that the frequency fo = 1 / T of the carrier signal is fixed and the modulated carrier signal s (t) goes through zero twice every T seconds. The function s (t) has the value zero at the times zero and T / 2, the value a ₀ at the times T / 4 and the value -a ₀ at the time 3T / 4. At times T / 8 and 3T / 8, the function value anSx (T / 8) results. At the times

T

5T / 8 and 7T / 8 the function value is -a ₀ sx (-).

⁸ ^ The value of sx (-) is 1 / l + k 'for waveforms

8 above the sine function and 4k ^x / 4l + k 'for waveforms below the sine function. The expression k 'is equal to VI- & ² . Therefore, the modulation parameter k (t), which slowly changes with respect to the frequency f _{0 of} the carrier signal, and thus the message m (t) can be transmitted Sampling can be retrieved at the odd multiples of T / 8.

In the second demodulation method, the message signal is obtained by integrating the received modulated carrier signal s (t) over a quarter period T / 4 or half a period T / 2. With the integrals

J * Kx, *) _Ä - -l "W *) + ^ (*)) _f i _{cn (χ k) dχ} arcsin (k. Sn (x) k rC

for example in I.S. Gradshteyn, I.M. Ryzhik, "Table of Integrals, Series, and Products", Corrected and Enlarged Edition, Academic Press, 1980, pages 630, 5,133), we receive

An integration over a quarter period results in half of the values.

According to the third demodulation method, the modulated carrier signal s (t) is first squared and then according to the equation

integrated, E (k) is the so-called complete elliptic integral of the second kind and k 'is 4l - k ² . An integration over half (a quarter) period gives half (a quarter) of the value.

With the help of elliptic functions, well-known orthogonal modulation methods based on sine and cosine carriers can also be generalized. Instead of the sine function we use the function sx (x) from equation (5) and instead of the cosine function we use the

Function sy (x) with x = 2 ^ f ₀ t, which is defined as follows.

The function cd (x) is the sn (x) function shifted by K, i.e. cd (x) = sn (x + K). It can be expressed by cd (x) = cn (x) / dn (x). Then the orthogonality property applies:

p4K sx (x) • sy (x) dt

As a result, elliptic functions can be used for orthogonal modulation. Given values for ao, fo, and k, you have two basis functions per dimension (sn and k 'sd in the x direction and cd and cn in the y direction) compared to only one basis function for classical ones

Sinus carriers. The orthogonality can be used in the base and / or in the transmission band.

Claims

claims

1. A method for modulating a carrier signal for the transmission of message signals, characterized in that the shape of the carrier signal is changed in time by a message signal to be transmitted (m (t)), the amplitude (a ₀ ) and the frequency (f ₀ ) of the carrier signal (s (t)) remain constant.

2. The method according to claim 1, characterized in that the temporal course of the carrier signal is defined by an elliptical function.

3. The method according to claim 2, characterized in that Jacobi elliptical functions are used.

4. The method according to claim 2 or 3, characterized in that the module k of an elliptical function is changed in time by the message signal to be transmitted (m (t)) in order to determine the signal shape of the carrier signal in rhythm with the message signal to be transmitted (m (t) ) to modulate.

5. The method according to any one of claims 2 to 4, characterized in that the temporal profile of the modulated carrier signal is defined by the elliptic function s (t) = ao sx (2 f ₀ t, k (t)), where ao is the amplitude and fo is the frequency and where π and the module k are related via the complete elliptic integral of the first kind.

6. The method according to claim 5, characterized in that the function sx (2 f ₀ t, k (t)) for 0 <k (t) <l by the Jacobi elliptic function sn (2 f ₀ t, k (t) ) and for -l <k (t) <0 by the Jacobi elliptic function cn (2 f ₀ (tT / 4), k (t)) is defined.

7. The method according to any one of claims 2 to 6, characterized in that an orthogonal transmission method is used, which is based on orthogonal elliptical basic functions (sn (2 f ₀ t, k (t)), sd (2 f ₀ t, k (t )), cd (2 f ₀ t, k (t)) and cn (2 f ₀ t, k (t))).

8. The method according to any one of claims 2 to 6, characterized in that the carrier signal defined by an elliptical function is generated using an analog circuit arrangement.

9. Method for demodulating a modulated carrier signal, the temporal course of which by the elliptical function

s (t) = ao sx (2 ^ f ₀ t, k (t)) is defined, where ao is the amplitude and f _{0 is} the frequency and where π and the module k are related via the complete elliptic integral of the first kind, according to which the received modulated carrier signal (s (t)) at times with the odd multiples of T / 8, with T = l / f ₀ , correspond to, is sampled, and from the sample values the module k (t) and thus the transmitted message signal (m (t)) is obtained.

10. Method for demodulating a modulated

Carrier signal, the temporal course of which is defined by the elliptic function s (t) = ao sx (2 f ₀ t, k (t)), where a _{0 is} the amplitude and fo the frequency and where π and the module k are about the complete elliptic integral of the first type are connected, according to which the received modulated carrier signal (s (t)) is integrated in order to obtain the time-dependent module k (t) and thus the transmitted message signal (m (t)).

11. The method according to claim 10, characterized in that the function sx (2; rf ₀ t, k (t)) for 0 <k (t) <l by the Jacobi elliptic function sn (2; τf ₀ t, k ( t)) and for -l <| A: (t) | <0 is defined by the Jacobi elliptic function cn (2 ^ f ₀ (tT / 4), k (t)).

12. Method for demodulating a modulated carrier signal, the temporal course of which is defined by the elliptic function s (t) = a ₀ sx (2τf ₀ t, k (t)), where ao is the amplitude and fo is the frequency and where π and the module k is related to the complete elliptic integral of the first type, according to which the received modulated carrier signal is squared and then integrated in order to obtain the module k (t) and thus the transmitted message signal (m (t)).

13. modulator for modulating a carrier signal for the transmission of message signals, characterized in that the modulation of the carrier signal (s (t)) takes place in that the signal form of the carrier signal can be changed over time by a message signal (m (t) to be transmitted, the Amplitude (a ₀ ) and the frequency (f ₀ ) of the carrier signal (s (t)) remain constant.

14. The modulator according to claim 13, characterized by an analog circuit arrangement that delivers at least one modulated carrier signal (s (t)), the curve shape of which corresponds at least in sections to an elliptical function or is approximated.

15. Modulator according to claim 14, characterized in that the elliptical function is a Jacobian elliptical function.

16. The modulator according to claim 14 or 15, characterized in that the analog circuit arrangement has means for changing the module k over time of an elliptical function by the message signal to be transmitted (m (t)), in order to determine the signal shape of the carrier signal in rhythm with the message signal to be transmitted (m (t)) to modulate.

17. Modulator according to one of claims 14 to 16, characterized in that the analog circuit arrangement generates a modulated carrier signal, the time course of which the elliptic function s (t) = a ₀ sx (2rfot, k (t)) is defined, where a _{0 is} the amplitude and f _{0 is} the frequency, and where π and the module k are related via the complete elliptic integral of the first kind.

18. Modulator according to claim 17, characterized by first analog multipliers (10, 20, 40, 50, 70, 80) and analog integrators (30, 60, 90), which are interconnected such that these the three output signals sn (2 f ₀ (tT / 4), k (t)), cn (2 ^ f ₀ (t- T / 4), k (t)), dn (2 fo (tT / 4), k (t)), an analog division device (96) for forming the quotient sn (2 ^ f ₀ (tT / 4), k (t)) / dn (2 ^ f ₀ (tT / 4), k (t)), a second one Analog multiplier (97) assigned to the division device (96), which multiplies the output signal of the division device (96) by the factor 4l-k ² , the output signal sn (2 f ₀ t, k (t for 0 <k (t) <l )) forms the modulated carrier signal and for -l <k (t) <0 the output signal of the second analog multiplier (97) forms the modulated carrier signal.