RU2366076C1 - Method to demodulate phase-modulated rf-frequency signals and device to this end - Google Patents

Abstract

FIELD: radio engineering.

SUBSTANCE: invention relates to radio communication. It can be used to demodulate (DM) phase-manipulated and phase-modulated signals by converting phase-modulated (PM) signal into amplitude-phase-modulated (APM) signal, via shaping the slope of APM-demodulator with preset ratio of absolute magnitudes m₂₁ of transmission factors at two preset frequencies f₁, f₂. The latter correspond to appropriate extreme values of input PM-signal frequency variation and subsequent ADM. In compliance with proposed invention, PM-signal is sent to DM connected in between RF PM-signal source and LF-load. Proposed DM consists of linear reactive quadripole (QP), two-electrode nonlinear element (NLE), LF-filter (LFF). PM-signal is converted into APM-signal by feeding this signal to the RH or LH slope of APM-module of complex resistance. NLE helps decompose APM-signal spectrum into HF- and LF-components. LFF is used to isolate LF data signal with its amplitude by law of LF data input signal phase variation. Two electrode NLE is connected in between the FM-signal source and FC into cross circuit. FC consists of reactive dipoles, their number making at least two with their parametres selected from required percentage of AM-APM-signal.

EFFECT: demodulation of PM-signals without using master oscillator and parallel oscillating circuit, along with conversion of PM-signal into APM-signal using HF-part of demodulation at preset amplitude modulation percentage to add to receiver noise immunity.

2 cl, 4 dwg

Description

The invention relates to radio communications and can be used to demodulate phase-shifted as well as phase-modulated signals.

A known method of demodulating phase-modulated signals (PMS), consisting in the fact that two non-linear elements simultaneously fed in phase out of phase high-frequency PMS and in phase high-frequency reference oscillation with a frequency equal to the carrier frequency of the PMS. As a result, the FMS phase changes in time and the constant phase of the reference oscillation is compared, as a result of which the FMS is converted into an amplitude-modulated and phase-modulated signal (AFMS). In this case, the amplitude changes according to the law of phase change. This signal then undergoes the same transformations as in the amplitude demodulator [S. Baskakov Radio circuits and signals. M.: Higher School, 1988, pp. 286-292]. This means that on nonlinear elements the AFMS spectrum is destroyed (decomposed) into low-frequency and high-frequency components. Then, using the low-pass filter, a low-frequency component is extracted, the amplitude of which changes according to the law of phase change of the input FMS. Then, using the separation capacitance included in the longitudinal chain (sequentially), the constant component that occurs on the nonlinear elements as a result of interaction with the AFMS is eliminated. After that, low-frequency vibrations containing useful information are allocated to the low-frequency load.

The disadvantage of this method and device for its implementation is that in order to isolate a low-frequency signal, the amplitude of which changes in accordance with the law of phase change of the high-frequency PMS, it is necessary to have a reference oscillator. Another disadvantage is the inability to correct the depth of amplitude modulation of the AFMS, which when passing through the resonant circuits leads to a decrease in this characteristic, that is, to the well-known phenomenon of partial demodulation of the AFMS or to a decrease in noise immunity.

The closest in technical essence and the achieved result (prototype) is a method of demodulating phase-modulated signals, which consists in the fact that to demodulate the FMS, a frequency detector is used, consisting of a cascade-connected amplitude limiter, a frequency-modulated signal (HMS) converter in the amplitude-frequency modulated signal (AFMC) in the form of a parallel oscillatory circuit and a conventional amplitude demodulator. Further, the process of isolating the low-frequency component is carried out in the same way as described above. The peculiarity of using a frequency detector for FMS demodulation is that if the frequency of the FMS carrier signal is located on the right slope of the amplitude-frequency characteristic (AFC) of the circuit, then the low-frequency component is fed to the differentiating circuit. If the frequency of the carrier signal of the FMS is located on the left slope of the frequency response of the circuit, then the low-frequency component is fed to the integrating circuit [S. Baskakov Radio circuits and signals. M.: Higher School, 1988, pp. 286-292]. If necessary, between the source of modulated signals and the nonlinear element or between the nonlinear element and the load include a reactive or resistive four-terminal network for matching and additional selection of signal and interference. As a result, at the output of the device, we have a low-frequency oscillation, the amplitude of which changes according to the law of variation of the envelope of the input high-frequency phase-modulated oscillation.

The disadvantage of the method and device for its implementation is that after converting the FMS to AFMS, the depth of amplitude modulation of the AFMS is not controlled and, as a rule, is insignificant in size, which impairs noise immunity [S. Baskakov. Radio circuits and signals. M.: Higher School, 1988, pp. 286-292. Gonorovsky I.S. Radio circuits and signals. M .: Radio and communications, 1986, p. 247-252]. Another disadvantage is the additional presence of an oscillatory circuit for converting FMS to AFMS. This drawback is due to the fact that the classical theory of radio circuits assumes that the nonlinear element is purely resistive and inertia-free, and therefore does not react to a change in the frequency and phase of the input signal, but only responds to a change in amplitude. Meanwhile, everyday experience shows that nonlinear elements have internal capacitances and inductances, which have a significant impact on the formation of the dependence of their conductivity (resistance or elements of the matrix of conductivities or resistances) on frequency and phase. This is especially significant with an increase in frequency, which is currently mainly sought by designers of new systems and means of radio communications.

The technical result of the invention is to provide demodulation of the FMS without using the reference oscillator and parallel oscillatory circuit with the conversion of the FMS to AFMS using the high-frequency part of the demodulator at a given depth of amplitude modulation of the AFMS at high frequency load, which increases the noise immunity of the receiver.

,1. The specified result is achieved by the fact that in the method of demodulating phase-modulated signals, consisting in the fact that the demodulator, consisting of a linear reactive four-terminal, two-electrode nonlinear element and a selective load, is supplied with a phase-modulated signal, a phase-modulated signal is converted into an amplitude-phase-modulated signal, phase-modulated conversion signal in the amplitude-phase-modulated signal is carried out by applying this signal to the right or left slope of the frequency response, low-frequency the nth component of the amplitude-phase-modulated signal is fed to a differentiating or integrating circuit, respectively, using a nonlinear element, the spectrum of the amplitude-phase-modulated signal is destroyed into high-frequency and low-frequency components, an information low-frequency signal is extracted with a low-pass filter, the amplitude of which changes according to the law of phase change of the phase-modulated input signal, additionally non-linear element include between the phase-modulated signal source and four lyusnikom the transverse chain, converting the phase-modulated signal in the amplitude-phase-modulated signal is performed by forming a frequency response of the demodulator slope with a predetermined ratio of moduli m ₂₁ transfer coefficients at two predetermined frequencies f _1, f _2, corresponding to the extreme values of the changes of the input phase modulated signal frequency quadripole operate from among reactive two-terminal, at least two, the resistance values of which are selected from the condition of providing a given amplitude modulation depth M ₂₁ am plateau-phase modulated signal

,

for m ₂₁ > 1 or

,

,

moreover, for m ₂₁ > 1 and f ₁ > f ₂ , the left slope of the frequency response is formed, for f ₁ <f _{2 the} right is formed, and for m ₂₁ <1 and f ₁ > f ₂ the right slope of the frequency response is formed, for f ₁ > f ₂ - left, while this condition is implemented by providing the following relationships between the elements x ₁₁ , x ₁₂ = -x ₁₂ , x _{22 of the} quadrupole resistance matrix:

;

;

;

;Where

;

;

;

;

r _{n1, n2} , x _{n1, n2} and r _01.02 , x _01.02 are the specified real and imaginary components of the resistances of the high-frequency load and the signal source at the two extreme frequencies of the phase-modulated or amplitude-phase-modulated signal;

g _1,2 , b _1,2 - specified real and imaginary components of the conductivity of a two-electrode nonlinear element at two specified known frequencies; φ ₂₁ - the set double value of the phase deviation of the input phase-modulated signal.

f₁, f₂, соответствующих крайним значениям изменения частоты входного фазомодулированного сигнала с помощью следующих математических выражений:2. The specified result is achieved by the fact that in the device for demodulating phase-modulated signals connected between the source of phase-modulated signals and the low-frequency load and consisting of a converter of phase-modulated signals into an amplitude-phase-modulated signal in the form of a linear reactive four-terminal device, a two-electrode nonlinear element, a low-pass filter, additionally parallel to the output a high-frequency load is introduced at the quadripole, a two-electrode nonlinear element is connected between the source m of the signal and a four-terminal in the transverse circuit, the four-terminal is made in the form of an overlapped bridge connection of four reactive two-terminal, the resistances of which are selected from the condition of the slope of the frequency response of the demodulator with a given ratio of modules m ₂₁ transmission coefficients at two given frequencies

f ₁ , f ₂ corresponding to the extreme values of the frequency change of the input phase-modulated signal using the following mathematical expressions:

x ₃ = x ₁ ; x ₄ = x ₂ ,

Where

r _n1 , _n2 , x _n1 , _n2 and r ₀₁ , ₀₂ , x ₀₁ , ₀₂ are the specified real and imaginary components of the resistances of the high-frequency load and the signal source at the two extreme frequencies of the phase-modulated or amplitude-phase-modulated signal;

g _1,2 , b _1,2 - specified real and imaginary components of the resistance of a two-electrode nonlinear element at two specified known frequencies; φ ₂₁ is the predetermined double value of the phase deviation of the input phase-modulated signal, with each of the two-terminal circuits being formed from parallel parallel L _1k , C _1k and serial L _2k , C _2k oscillatory circuits, and the parameters of the parallel oscillatory circuit are determined using the following formulas:

where k = 1,2,3,4 is the number of the two-terminal network; x _k - resistance of four reactive two-terminal networks; L _2k , С _2k - inductances and capacitances of successive oscillatory circuits, the values of which are selected from the condition for providing physically feasible values of inductances and capacitances L _1k , С _1k .

Figure 1 shows a diagram of a device for demodulating phase-modulated RF signals (prototype).

Figure 2 shows the structural diagram of the proposed device according to claim 2.

Figure 3 shows a diagram of a four-terminal device of the proposed device according to claim 2.

Figure 4 shows a diagram of each of the two-terminal circuits forming a four-terminal network (figure 3) of the proposed device according to claim 2

The prototype device contains a source 1 of phase-modulated signals, a four-terminal 2, a nonlinear element 3, a low-pass filter 4 on the elements R, C, a separation capacitance 5 on the element C _p and a low-frequency load 6 on the elements R _H , C _H.

The principle of operation of the device demodulation phase-modulated signals (prototype) is as follows.

The phase-modulated signal from source 1 is supplied to the demodulator from a series-connected semiconductor diode to a low-pass filter. The principle of operation of a device that implements this method is that, using a reactive four-terminal 2, which is a parallel oscillatory circuit and connected between the FMS source and the nonlinear element, the FMS is converted into AFMS, using the nonlinear element 3, the AFMS spectrum is destroyed into high-frequency and low-frequency components . The latter are allocated using a low-pass filter 4 and enter the low-frequency load 6. The separation capacitance 5 eliminates the constant component. As a result, at the output of the device, we have a low-frequency oscillation, the amplitude of which changes according to the law of variation of the envelope of the input high-frequency amplitude-modulated oscillation.

The disadvantage of the method and device for its implementation is that when passing the FMS through the specified circuit, after converting the FMS to AFMS, the depth of the amplitude modulation of the latter is insignificant. This is due to the large width of the FMS spectrum or to the low quality factor of the circuit. On the other hand, the narrower the bandwidth of the circuit, the greater the distortion of the received signal. In the general case, the depth of amplitude modulation of AFMS decreases and becomes, as a rule, unknown.

The high-frequency part of the structural diagram of the generalized proposed device (up to the low-pass filter) according to claim 2 (Fig. 2) consists of a cascade-connected source of FMS 1, a reactive four-terminal 2, a nonlinear element 3, and a high-frequency load 7. The low-frequency part of the structural circuit contains a lower filter frequencies 4, isolation capacitance 5 and low-frequency load 6. The reactive four-terminal 2 is made in the form of a bridge connection of four reactive two-terminal (8, 9, 10, 11), the resistances of which are selected from the conditions of formation with womb AFC demodulator to specify the ratio of m modules ₂₁ transmit the coefficients at the two predetermined frequencies f _1, f _2, corresponding to the extreme values of the changes of the input phase modulated signal with special mathematical expressions.

The principle of operation of this device is that when the FMS is supplied from source 1 with resistance z ₀ , a left or right slope of the frequency response of the demodulator with a given ratio of modules m ₂₁ transmission coefficients at two given frequencies f ₁ , f will be formed as a result of a special choice of values of bipolar elements ₂ , corresponding to the extreme values of the frequency change of the input phase-modulated signal. This provides a given depth of amplitude modulation AFMS, which increases the noise immunity of the receiver. At the same time, the AFMS spectrum is destroyed by a nonlinear element 3 connected between the signal source and the input of the four-terminal network into the transverse circuit. Each two-terminal network is formed in such a way that their resistances at two frequencies are equal. In the present invention, they are formed from parallel-connected parallel L ₁ , C ₁ , and serial L ₂ , C ₂ oscillatory circuits (Fig. 4). As a result, the low-frequency oscillation, the amplitude of which varies according to the law of phase change of the input FMS, is allocated to the low-frequency load 6.

Let us prove the feasibility of implementing these properties.

где U_H, ω_H - амплитуда и частота несущего высокочастотного колебания; m_φ - индекс фазовой модуляции; φ₀ - начальная фаза; Ω - частота первичного информационного низкочастотного сигнала. Поскольку частота определяется производной от фазы, то одновременно с изменением фазы в фазомодулированном колебании будет происходить изменение частоты по закону

Let the phase-modulated oscillation act on the input of the demodulator

where U _H , ω _H is the amplitude and frequency of the carrier high-frequency oscillations; m _φ - phase modulation index; φ ₀ is the initial phase; Ω is the frequency of the primary information low-frequency signal. Since the frequency is determined by the derivative of the phase, then simultaneously with the phase change in the phase-modulated oscillation, the frequency will change according to the law

If the limits of the frequency change of the phase-modulated signal (PMS) do not go beyond the boundaries of the left slope of the amplitude-frequency characteristic (AFC) of the demodulator, then the FMS will be converted to an amplitude-modulated signal (AFMS). In this case, the amplitude of the AFMS will change according to the law (minus sin (Ωt)). Therefore, in order for the law of the amplitude change of the AFMS to repeat the law of the phase change of the FMS, it is necessary to integrate the converted signal, i.e. apply to the integrating circuit. If the boundaries of the FMS frequency change do not go beyond the right slope of the frequency response of the demodulator, then the FMS will also be converted to an amplitude-phase modulated signal (AFMS). In this case, the amplitude of the AFMS will change according to the law (sin (Ωt)). Therefore, in order for the law of the amplitude change of the AFMS to repeat the law of the phase change of the FMS, it is necessary to differentiate the converted signal, i.e. apply to the differentiating circuit.

Thus, the main task when creating a phase demodulator is to provide the conditions under which the left or right slopes of the frequency response of the demodulator are formed in a given frequency band, the boundaries of which coincide or are comparable with the extreme values of the frequencies of the frequency range of the FMS.

The input modulated high-frequency signal S _in and converted using the high-frequency part of the demodulator (before the low-pass filter) the high-frequency signal S _out are interconnected as follows:

S _o = S ₂₁ S _I, where the input and output signal refers to the input and output voltages; S ₂₁ - gear ratio.

Consider phase-modulated oscillations in two states characterized by extreme values of the amplitude range of the AFMS on a nonlinear element.

;

(модуль АФМС в двух состояниях различен);

. Таким образом, на выходе высокочастотной части демодулятора модули коэффициента передачи и входного сигнала перемножаются, а их фазы складываются. Выходные напряжения в двух состояниях связаны между собой следующим образом:We write the indicated physical quantities in two states in complex form

;

(AFMS module in two states is different);

. Thus, at the output of the high-frequency part of the demodulator, the transmission coefficient and input signal modules are multiplied, and their phases are added up. The output voltages in two states are interconnected as follows:

. Для уменьшения фазовых искажений целесообразно положить

Since the input FMS module does not depend on changes in its phase and frequency, then in (1)

. To reduce phase distortion, it is advisable to put

Отношение модулей коэффициентов передачи высокочастотной части демодулятора m₂₁ связано с глубиной амплитудной модуляции АФМС следующим образом:

при

или

We introduce the following notation:

The ratio of the transmission coefficient modules of the high-frequency part of the demodulator m ₂₁ is related to the depth of amplitude modulation of the AFMS as follows:

at

or

moreover, for m ₂₁ > 1 and f ₁ > f ₂ the left slope of the frequency response is formed, for f ₁ <f _{2 the} right one is formed, for m ₂₁ <1 and f ₁ > f ₂ the right slope of the frequency response is formed, for f ₁ > f ₂ left,

Let the complex resistances of the high-frequency load z _{n1, n2} = r _{n1, n2} + jx _{n1, n2} and the signal source z _01.02 = r _01.02 + jx _01.02 as well as the conductivity of the nonlinear element y _1,2 = g _1,2 + jb _1,2 at extreme values of the AFMS frequency are known.

Consider the structural diagram of the demodulator shown in figure 2. The quadripole or matching filtering device (SFU) 2 contains only reactive elements.

Thus, taking into account the reciprocity condition (x ₁₂ = -x ₂₁ ), the SFU can be characterized by a resistance matrix

and the corresponding classical transfer matrix:

where | x | - x ₁₁ x ₂₂ -x ₂₁ ² - determinant of the matrix (2).

A nonlinear element in two states is characterized by the following transfer matrix [Feldstein A.L., Yavich L.R. Microwave four-terminal and eight-terminal synthesis. M.: Communication, 1971. S. 34-36].

We multiply matrices (4) and (3) and, taking into account Z _01.02 , Z _{H1, H2,} we write the normalized transfer matrix of the entire device:

Using the known relationships between the elements of the transmission and scattering matrices, we obtain expressions for the transmission and reflection coefficients [Feldstein A.L., Yavich L.R. Microwave four-terminal and eight-terminal synthesis. M .: Communication, 1971. S. 39]:

, гдеThe square root of (6) can be represented as a complex number

where

последнее выражение изменяется а_1,2=r_н1,н2; b_1,2=x_н1,н2.After denormalizing the transmission coefficient (6) by multiplying by

the last expression changes and _1,2 = r _{n1, n2} ; b _1,2 = x _{n1, n2} .

.The denormalized transmission coefficient is associated with a physically feasible transfer function as follows

.

After substituting (6) in (1) and separating the real and imaginary parts from each other, we obtain a system of two equations:

Solution (7) has the form:

x ₂₂ = Dx ₁₁ + E;

Where

The resulting system of two relationships (8) means that the SFU of the phase or frequency demodulator must contain at least two independent reactive elements, the parameter values of which should be determined by the following algorithm. For a specific SFU scheme, a resistance matrix is determined, which is presented in the form (2). Certain elements of the resistance matrix, functionally dependent on the parameters of a particular SFU circuit, are substituted in the relationship (8). The system of two algebraic equations thus formed is solved with respect to any two parameters chosen. If the number of elements is M> 2, then the parameter values of M-2 elements can be selected arbitrarily or based on any physical considerations, for example, from conditions for ensuring physical realizability and the largest working frequency band.

It should be noted that in solving this problem, it was assumed that the elements of the resistance matrix (2) at the two extreme frequencies of the input FMS are unchanged. This is possible only if the parameters of the elements of the SFU circuit are also constant at these frequencies. To obtain such a result, it is necessary to form each bipolar terminal of the Siberian Federal University from at least three reactive elements of type L, C. The values of the two parameters of each bipolar terminal are determined from the condition of ensuring the same resistance value at two frequencies of the AFMS. For example, for series-connected parallel L ₁ , C ₁ and serial L ₂ , C ₂ oscillatory circuits, this condition is satisfied with the following parameters:

In formulas (9), x _k is the reactance of the k-th two-terminal, which is part of the SFU, the value of which is determined by the above algorithm. With the optimal values of the two-terminal resistances selected in accordance with this algorithm, the operation of converting the FMS to AFMS with a given (left or right) slope of the frequency response in a given frequency band using the high-frequency part of the phase demodulator will be implemented even under the assumption that the non-linear element is inertialess.

In accordance with the above algorithm, formulas were determined for finding the optimal values of the resistance of the two-terminal devices of a typical SFU circuit in the form of a bridge connection of four reactive two-terminal devices:

x ₃ = x ₁ ; x ₄ = x ₂ .

The root expression in (10) is always positive.

The proposed technical solutions are new, since the FMS demodulation device that provides the left or right slope of the frequency response of the demodulator with a given ratio of transmission coefficient modules at two frequencies corresponding to the extreme values of the frequency range of the input FMS is unknown from publicly available information, which allows the conversion of FMS to AFMS with the specified amplitude amplitude modulation depth AFMS, and consisting of a nonlinear two-electrode element connected between the FMS source and input a reactive four-terminal network in a transverse circuit, and the four-terminal network is made in the form of a bridge connection of four reactive two-terminal networks, the parameters of which are determined by the corresponding mathematical expressions.

The proposed technical solutions have an inventive step, since it does not explicitly follow from the published scientific data and the known technical solutions that the claimed sequence of operations (performing a four-terminal reactive in the form of the above scheme with a choice of its parameters from the condition of providing a given depth of amplitude modulation AFMS performs FMS conversion in AFMS without the presence of a reference signal source and an oscillatory circuit.

The proposed technical solutions are practically applicable, since semiconductor diodes (parametric diodes, pin diodes, power supply diodes, Gunn diodes, etc.), inductances and capacitances formed in the claimed circuit of reactive two-pole circuits included in the circuit can be used for their implementation the claimed four-terminal circuit. The resistance values of bipolar, inductances and capacitances can be determined using mathematical expressions given in the claims.

The technical and economic efficiency of the proposed device is to simultaneously provide the operation of converting the input FMS to AFMS with a given depth of amplitude modulation, which helps to increase noise immunity.

Claims (2)

1. A method for demodulating phase-modulated signals, which consists in applying a phase-modulated signal to a demodulator made of a linear reactive four-terminal device, a two-electrode nonlinear element and a selective load, phase-modulated signal is converted to an amplitude-phase modulated signal, phase-modulated signal is converted to an amplitude-phase modulated signal by applying this signal to the right or left slope of the frequency response, the low-frequency component of the amplitude-phase modulated This signal is fed to a differentiating or integrating circuit, respectively, using a nonlinear element, the spectrum of the amplitude-phase modulated signal is destroyed into high-frequency and low-frequency components, an information low-frequency signal is isolated with a low-pass filter, the amplitude of which changes according to the law of phase change of the phase-modulated input signal, characterized in that a nonlinear element is connected between the source of the phase-modulated signal and the four-terminal network in the transverse circuit, The phase-modulated signal is generated into an amplitude-phase-modulated signal by forming a slope of the frequency response of the demodulator with a given ratio of modules ₂₁ of transmission coefficients at two given frequencies f ₁ , f ₂ corresponding to the extreme values of the frequency change of the input phase-modulated signal, a four-terminal network consisting of at least two two-terminal reactors , the resistance values of which are selected from the condition for providing a given amplitude modulation depth M _{21 of the} amplitude-phase modulated signal la

, for m ₂₁ > 1 or

,
moreover, for m ₂₁ > 1 and f ₁ > f ₂ the left slope of the frequency response is formed, for f ₁ <f _{2 the} right one is formed, for m ₂₁ <1 and f ₁ > f ₂ the right slope of the frequency response is formed, for f ₁ > f ₂ left, while this condition is implemented by providing the following relationships between elements
x ₁₁ , x ₂₁ = -x ₁₂ , x ₂₂ quadrupole resistance matrices:

Where

;

;

;

;

r _H1 , _H2 , x _H1 , _H2 and r ₀₁ , ₀₂ , x ₀₁ , ₀₂ are the specified real and imaginary components of the resistances of the high-frequency load and the signal source at the two extreme frequencies of the phase-modulated or amplitude-phase-modulated signal; g _1,2 , b _1,2 - specified real and imaginary components of the conductivity of a two-electrode nonlinear element at two specified known frequencies; φ ₂₁ - the set double value of the phase deviation of the input phase-modulated signal. 2. A device for demodulating phase-modulated signals connected between a source of phase-modulated signals and a low-frequency load and consisting of a converter of phase-modulated signals into an amplitude-phase-modulated signal in the form of a linear reactive four-terminal device, a two-electrode nonlinear element, a low-pass filter, characterized in that a high-frequency load is introduced parallel to the output of the four-terminal device , a two-electrode nonlinear element is connected between the signal source and the four-terminal device in the pop river chain quadripole formed as the overlapped bridging four reactive two-terminal, the resistances of which are selected from the conditions of formation of AFC demodulator slope with a predetermined moduli ratio m ₂₁ transfer coefficients at two predetermined frequencies f _1, f _2, corresponding to the extreme values of the changes of the input phase modulated signal frequency using the following mathematical expressions:

x ₃ = x ₁ ; x ₄ = x ₂ ,
Where

r _H1 , _H2 , x _H1 , _H2 and r ₀₁ , ₀₂ , x ₀₁ , ₀₂ - specified real and imaginary components of the resistances of the high-frequency load and the signal source at the two extreme frequencies of the phase-modulated or amplitude-phase-modulated signal; g _1,2 , b _1,2 - specified real and imaginary components of the resistance of a two-electrode nonlinear element at two specified known frequencies; φ ₂₁ is the predetermined double value of the phase deviation of the input phase-modulated signal, with each of the two-terminal circuits being formed from sequentially connected parallel L _2k , C _2k and serial L _2k , C _2k oscillatory circuits, and the parameters of the parallel oscillatory circuit are determined using the following formulas:

where k = 1,2,3,4 is the number of the two-terminal network; x _k - resistance of four reactive two-terminal networks; L _2k , C _2k - inductances and capacitances of successive oscillatory circuits, the values of which are selected from the conditions for providing physically feasible values of inductances and capacitances L _2k , C _2k .

Images