RU2491711C2 - Method of demodulating and filtering phase-modulated signals and apparatus for realising said method - Google Patents

Abstract

FIELD: radio engineering, communication.

SUBSTANCE: apparatus for demodulating and filtering phase-modulated signals consists of a four-terminal element, a two-terminal nonlinear element, a low-pass filter and a separating capacitor and is characterised by that a high-frequency load is connected in a transverse circuit between the output of the four-terminal element and the low-pass filter, the two-terminal nonlinear element is connected between the source of the phase-modulated signal and the input of the four-terminal element in longitudinal circuit.

EFFECT: providing simultaneous demodulation and filtration of phase-modulated signals without using a reference oscillator with conversion of the phase-modulated signal to an amplitude-phase-modulated signal using the high-frequency part of the demodulator with a given quasi-linear slope and shape of the amplitude-frequency characteristics, which increases noise-immunity of the receiver, based on the frequency dependency of the resistance of the two-terminal nonlinear element, which increases the accuracy of calculating values of parameters of circuit components.

2 cl, 4 dwg

Description

The invention relates to radio communications and can be simultaneously used for demodulation and filtering phase-modulated, as well as phase-shifted signals.

A known method of demodulating phase-modulated signals (PMS), consisting in the fact that two nonlinear elements simultaneously are 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 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. In this case, the high-frequency part of the demodulator has an arbitrary (uncontrolled) frequency response. Therefore, in the general case, there is a low noise immunity, since there is no possibility of forming the necessary shape of the frequency response.

The closest in technical essence and the achieved result (prototype) is a device for demodulating phase-modulated signals, made in the form of a frequency detector, consisting of a cascade-connected amplitude limiter, a converter of a frequency-modulated signal (HMS) into an amplitude-frequency-modulated signal (AMF) in in the form of a parallel oscillatory circuit and a conventional amplitude demodulator. Further, the process of isolating the low-frequency component is also carried out 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 signal and interference selection. 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 this device is that after converting the FMS to AFMS, the depth (coefficient) of the amplitude modulation of the AFMS is not controlled and, as a rule, is insignificant in value, 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 communication, 1986, pp. 247-252]. Another drawback is that the classical theory of radio circuits assumes that the nonlinear element is purely resistive and inertial-free, and therefore does not react to a change in the frequency of the input signal, but only responds to a change in amplitude. Meanwhile, everyday experience shows that non-linear 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 conductivity matrix, or resistances) on frequency. This is especially significant with increasing frequency, which is currently mainly sought by designers of new systems and means of radio communications. In this case, the high-frequency part of the demodulator has an arbitrary (uncontrolled) frequency response. Therefore, in the general case, there is a low noise immunity, since there is no possibility of forming the necessary shape of the frequency response.

The technical result of the invention is the simultaneous provision of demodulation and filtering of the FMS without using the reference oscillator with the conversion of the FMS to AFMS using the high-frequency part of the demodulator with a given quasilinear slope and frequency response, which increases the noise immunity of the receiver, taking into account the dependence of the resistance of a bipolar nonlinear element on frequency, which increases accuracy of calculation of values of parameters of circuit elements.

1. The specified result is achieved by the fact that in the method of demodulating and filtering the phase-modulated signals, consisting in the fact that the demodulator is switched on between the source of the phase-modulated signals and the low-frequency load and made of a four-terminal, two-pole non-linear element, a low-pass filter and a separation capacitance, the phase-modulated signal is converted in the amplitude-phase-modulated signal by applying it to the right or left slope of the amplitude-frequency characteristic, using a nonlinear element they destroy the spectrum of the amplitude-phase-modulated signal into high-frequency and low-frequency components, use the low-pass filter to extract the low-frequency information signal, use a dividing capacitance to eliminate the constant component of the low-frequency information signal, whose amplitude changes according to the law of phase change of the phase-modulated input signal, and the low-frequency information signal is fed to the low-frequency selective load in the form of a differentiating or integrating circuit It is clear that, in addition, a high-frequency load is introduced between the output of the four-terminal device and the low-pass filter into the transverse circuit, the two-pole nonlinear element is connected between the phase-modulated signal source and the four-terminal input into the longitudinal circuit, the dependences of the elements of the four-terminal resistance matrix on the frequency are selected from the conditions for the formation of the given shape and the left quasilinear amplitude slope -frequency characteristics of the high-frequency part of the demodulator using the following mathematical expressions:

x ₁₁ , x ₂₁ = -x ₁₂ are the optimal frequency dependencies of the corresponding elements of the quadripole resistance matrix taking into account the reciprocity condition; x ₂₂ is the predetermined dependence of the corresponding element of the quadripole resistance matrix on frequency; m is the given shape of the amplitude-frequency characteristic taking into account the quasilinear dependence of the transfer function module on frequency on the left slope of the amplitude-frequency characteristic of the high-frequency part of the phase demodulator; φ is the predetermined dependence of the phase of the transfer function on frequency from the condition of physical realizability of the given shape of the amplitude-frequency characteristic of the high-frequency part of the phase demodulator; r _n , x _n - given frequency dependencies of the real and imaginary components of the load resistance; r ₀ , x ₀ - given frequency dependences of the real and imaginary components of the resistance of the source of the phase-modulated signal; r, x are the given frequency dependences of the real and imaginary components of the resistance of a bipolar nonlinear element; the remaining quantities have the meaning of intermediate notation to simplify mathematical expressions.

2. This result is achieved by the fact that in the device for demodulating and filtering phase-modulated signals connected between the source of phase-modulated signals and the low-frequency load in the form of an integrating circuit and consisting of a four-terminal, two-pole non-linear element, a low-pass filter and a separation capacitance, in addition between the four-terminal output and the filter high-frequency load is introduced into the transverse circuit; a bipolar nonlinear element is connected between the source phase-modulated of the signal and the input of the four-terminal into the longitudinal circuit, the four-terminal is made of a L-shaped connection of two two-terminal with resistances x _1n , x _2n , each of the two-terminal is formed of two series-parallel parallel oscillatory circuits of elements with parameters L _1k , C _1k , L _2k , C _2k , the values of which are determined in accordance with the following mathematical expressions:

m _n - given values of the transfer function modules at given four frequencies ω _n = 2πf _n from the conditions for the formation of a given shape and a quasilinear left slope of the amplitude-frequency characteristic of the high-frequency part of the phase demodulator; φ _n - the given values of the phases of the transfer function at the given four frequencies ω _n = 22πf _n from the condition of physical realizability of the given form of the amplitude-frequency characteristic of the high-frequency part of the phase demodulator; r _0n , r _0n - set values of the real and imaginary components of the resistance of the source of the phase-modulated signal at the specified four frequencies; r _nn - set values of the real and imaginary components of the resistance of the high-frequency load at the specified four frequencies; r _n , x _n - given values of the real and imaginary components of the resistance of a bipolar nonlinear element at given four frequencies; k = 1,2 is the number of the two-terminal L-shaped connection; n = 1, 2, 3, 4 - frequency number; the remaining quantities have the meaning of intermediate notation to simplify mathematical expressions.

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, which implements the proposed method according to claim 1.

Figure 3 shows the structural diagram of the four-terminal network (matching filtering device (SFU)) included in the proposed device according to claim 2, shown in figure 2.

Figure 4 shows a schematic diagram of the two-terminal network included in the structural diagram of the four-terminal network shown in figure 3.

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 _n , C _n

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. 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 (decomposed) into high-frequency and low-frequency components. The latter are allocated using a low-pass filter (integrating circuit) 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 high-frequency amplitude-modulated oscillation generated by the AFMS.

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 amplitude modulation coefficient of the latter is negligible. This is due to the large width of the FMS spectrum, i.e. with low quality factor of a contour. On the other hand, the narrower the bandwidth of the circuit, the greater the distortion of the received signal. The high-frequency part of the demodulator has an arbitrary (uncontrolled) frequency response. Therefore, in the general case, there is low noise immunity, since there is no possibility of forming the necessary shape of the frequency response, which would simultaneously convert the FMS to AFMS and filter the FMS.

The high-frequency part of the structural diagram of the generalized proposed device according to claim 2 (figure 2) consists of a cascade-connected FMS source with resistance z ₀ 1, a bipolar nonlinear element with resistance z3, reactive four-terminal 2 and a high-frequency load with resistance z _n 7. Low-frequency part block diagram contains a low-pass filter 4, a separation capacitance 5 and a low-frequency load 6.

The principle of operation of this device is that when applying the FMS from source 1 as a result of a special choice of the parameters of the resistance matrix of the two-terminal network 2 from the conditions for providing a given quasilinear slope of the frequency response (with a given slope slope and a given shape of the frequency response), the FMS is converted to AFMS with a given amplitude coefficient AFMS modulations. Due to this, a minimum of distortion of the input signal is achieved. At the same time, the AFMS spectrum is destroyed by a nonlinear element 3 included in the longitudinal circuit between the FMS source and the input of a four-terminal network, between the output of which and the low-pass filter a high-frequency load 7 is turned on, the low-pass filter 4 selects a low-frequency component, the constant component is eliminated using a separation capacitor 5. The ability to choose the location of the inclusion of a nonlinear element additionally varies the size of the physical realizability region and the passband. Since the left slope of the frequency response is selected, it is necessary to use an integrating circuit as a low-frequency load, which is implemented using a low-pass filter. As a result, a low-frequency oscillation, the amplitude of which varies according to the law of phase change of the input FMS, is allocated at a low-frequency load 6. The four-terminal is made in the form of an L-shaped connection of two two-terminal with resistances x ₁ (8), x ₂ (9) (Fig. 3). Each of the two-terminal devices is made in the form of two parallel loops connected in series (Fig. 4). The values of inductances and capacitances are selected from the condition that the real values of the transfer function modules of the high-frequency part of the phase demodulator coincide with the given values of the transfer function modules at four predetermined frequencies. A reasonable choice of the relative position of these frequencies ensures the implementation of the required shape of the frequency response and the specified slope of the quasilinear left slope of the frequency response. As a result, the conversion of FMS to AFMS and filtering of FMS using the entire high-frequency part of the phase demodulator is simultaneously ensured.

Let us prove the feasibility of implementing these properties.

где U_н, ω_н - амплитуда и частота несущего высокочастотного колебания; m_φ - индекс фазовой модуляци; φ_o - 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 следующему закону: $$\omega \left(t\right)=\frac{d\varphi }{dt}=\frac{d}{dt}\left[{\omega }_{н}t+{\varphi }_o+m_{\varphi }\text{cos}\left(\text{Ωt}\right)\right]={\omega }_{н}-{\text{Ωm}}_{\varphi }\text{sin}\left(\text{Ωt}\right)$$

. Поэтому, если ФМС подать на правый склон АЧХ высокочастотной части демодулятора, то произойдет преобразование ФМС в АФМС. При этом амплитуда АФМС будет изменяться по закону (sin(Ωt). Следовательно, для того чтобы закон изменения амплитуды АФМС повторял закон изменения фазы ФМС, необходимо продифференцировать огибающую преобразованного сигнала, т.е. подать на дифференцирующую цепь. Если ФМС подать на левый склон АЧХ, то также произойдет преобразование ФМС в АФМС. При этом амплитуда АФМС будет изменяться по закону (минус sin(Ωt)). Следовательно, для того чтобы закон изменения амплитуды АФМС повторял закон изменения фазы ФМС, необходимо проинтегрировать огибающую преобразованного сигнала, т.е. подать на интегрирующую цепь. Этот случай более предпочтителен, потому что интегрирующая цепь (фильтр нижних частот) не шунтирует низкочастотную цепь.Let the phase-modulated oscillation act on the input of the demodulator $$U_{FM}(t)=U_n\text{cos}\left[{\omega }_{\text{n}}\text{t}+{\phi }_{\text{o}}+m_{\phi }\text{cos}\left(\text{Ωt}\right)\right],$$

where U _n , ω _n - the amplitude and frequency of the carrier high-frequency oscillations; m _φ — phase modulation index; φ _o 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 at the same time as the phase changes in the phase-modulated oscillation, the frequency changes according to the following law: $$\omega \left(t\right)=\frac{d\varphi }{dt}=\frac{d}{dt}\left[{\omega }_{n}t+{\varphi }_o+m_{\varphi }\text{cos}\left(\text{Ωt}\right)\right]={\omega }_n-{\text{Ωm}}_{\varphi }\text{sin}\left(\text{Ωt}\right)$$

. Therefore, if the FMS is applied to the right slope of the frequency response of the high-frequency part of the demodulator, then the FMS will be converted to AFMS. In this case, the amplitude of the AFMS will change according to the law (sin (Ωt). Therefore, in order for the law of amplitude variation of the AFMS to repeat the law of change in the phase of the FMS, it is necessary to differentiate the envelope of the converted signal, that is, apply to the differentiating circuit. If the FMS is applied to the left slope The frequency response will also transform the FMS to AFMS, and the amplitude of the AFMS will change according to the law (minus sin (Ωt)) .Therefore, in order for the law of amplitude change of the AFMS to repeat the law of change of the phase of the FMS, it is necessary to integrate the envelope formed by the signal, i.e. to apply an integrating circuit. This case is more preferable because the integrating circuit (low pass filter) is not shunts the low-frequency circuit.

The input modulated high-frequency signal S _in and converted using the high-frequency part of the demodulator (before the low-pass filter) S _out are interconnected as follows: S _out = S ₂₁ S _in , where the input and output signal means the input and output voltages; S ₂₁ - gear ratio:

$$S_{21}=m\left[\text{cos}\left(\varphi \right)+j\text{sin}\left(\varphi \right)\right],\left(\mathrm{one}\right)$$

where m, φ are the given dependences of the modulus and phase of the transmission coefficient on frequency (the argument ω = 2πf is omitted for simplicity).

, где m_1,2 - заданные значения модулей коэффициентов передачи высокочастотной части демодулятора на двух заданных частотах левого склона АЧХ.The amplitude modulation coefficient AFMS can be determined as follows: $$M=\frac{\left|m_{\mathrm{one}}-{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="00000070.png"\; state="\{\{state\}\}">m</figure-callout>}}_2\right|}{m_{\mathrm{one}}+{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="00000070.png"\; state="\{\{state\}\}">m</figure-callout>}}_2}$$

where m _1,2 are the specified values of the moduli of the transmission coefficients of the high-frequency part of the demodulator at two given frequencies of the left slope of the frequency response.

Let the dependences of the resistance of the signal source Z ₀ = r ₀ + jx ₀ , the load Z _n = r _n + jx _n and the bipolar nonlinear element Z = r + jx on frequency be known (the argument ω = 2πf is omitted for simplicity).

It is required to determine the frequency characteristics of the four-terminal and two-terminal, from which the four-terminal is formed, as well as the minimum number of elements and parameter values of the two-terminal, for which the given frequency dependences of the modules m and phases φ of the transmission coefficient would be provided (1).

Let the quadrupole contain only reactive elements. Thus, taking into account the reciprocity condition (x ₁₂ = -x ₂₁ ), the SFU can be characterized by a resistance matrix

$$X=\left[\begin{array}{cc}j{x}_{\mathrm{eleven}}& -j{x}_{21}\\ j{x}_{21}& j{x}_{22}\end{array}\right]\left(2\right)$$

and the corresponding classical transfer matrix:

$${\mathrm{BUT}}_{\mathrm{from}f\mathrm{at}}=\left[\begin{array}{cc}\frac{x_{\mathrm{eleven}}}{x_{21}}& j\frac{\left|x\right|}{x_{21}}\\ -j\frac{\mathrm{one}}{x_{21}}& -\frac{x_{22}}{x_{21}}\end{array}\right],\left(3\right)$$

- определитель матрицы (2).Where $$\left|x\right|=-x_{\mathrm{eleven}}{x}_{22}-x_{21}^2$$

is the determinant of the matrix (2).

A bipolar non-linear element is characterized by the following transfer matrix:

$${\mathrm{BUT}}_{\mathrm{at}\mathrm{uh}}=\left[\begin{array}{cc}\mathrm{one}& Z\\ 0& \mathrm{one}\end{array}\right].\left(\mathrm{four}\right)$$

We multiply matrices (4) and (3) and taking into account normalization conditions [Feldstein A.L., Yavich L.R. Microwave four-terminal and eight-terminal synthesis. M .: Communication, 1971, p. 34-36], we obtain the expression for the normalized transfer matrix of the high-frequency part of the entire device:

$${\mathrm{BUT}}_{\mathrm{at}}=\frac{x_{22}}{x_{21}}\left[\begin{array}{cc}\sqrt{\frac{Z_n}{Z_0}}\left(\frac{x_{\mathrm{eleven}}}{x_{22}}+j\frac{Z}{X_{22}}\right)& \frac{\mathrm{one}}{\sqrt{Z_0{Z}_n}}\left(Z-j\frac{\left|x\right|}{x_{22}}\right)\\ j\sqrt{Z_0{Z}_n}\frac{\mathrm{one}}{X_{22}}& \sqrt{\frac{Z_0}{Z_n}}\end{array}\right].\left(5\right)$$

Using the well-known relations between the elements of the classical transfer matrix and the elements of the scattering matrix [Feldstein A.L., Yavich L.R. Microwave four-terminal and eight-terminal synthesis. M .: Communication, 1971, p. 39], taking into account (5), we obtain the expression for the transmission coefficient:

$$S_{21}=\frac{2x_{21}\sqrt{Z_0{Z}_n}}{\left(x_{\mathrm{eleven}}-jZ\right)Z_n-Z_0\left(x_{22}+j{Z}_n\right)+j\left|x\right|-Z{x}_{22}}.\left(6\right)$$

; $$b_1=\pm \sqrt{\frac{\sqrt{x^2+y^2-x}}{2}}$$

; $$x=r_{о}{r}_{н}-x_{о}{x}_{н}$$

; $$y=r_{о}{x}_{н}+x_{о}{r}_{н}$$

.The radical expression in (6) can be represented as a complex number a ₁ + jb ₁ , where $$a_{\mathrm{one}}=\pm \sqrt{\frac{\sqrt{x^2+{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="00000070.png"\; state="\{\{state\}\}">y</figure-callout>}}^2+x}}{2}}$$

; $$b_{\mathrm{one}}=\pm \sqrt{\frac{\sqrt{x^2+y^2-\mathrm{<figure-callout\; id="2"\; label="x"\; filenames="00000070.png"\; state="\{\{state\}\}">x</figure-callout>}}}{2}}$$

; $$x=r_{\mathrm{about}}{r}_n-x_{\mathrm{about}}{x}_n$$

; $$y=r_{\mathrm{about}}{x}_n+x_{\mathrm{about}}{r}_n$$

.

последнее выражение изменяется a₁=r_н; b₁=x_н.After denormalizing the transmission coefficient (6) by multiplying by $$\sqrt{\frac{z_n}{z_o}}$$

the last expression changes a ₁ = r _n; b ₁ = x _n

.The denormalized transmission coefficient is associated with a physically feasible transfer function as follows $$H^{I,II}(j\omega )=\frac{\mathrm{one}}{2}{S}_{21}^{I,II}$$

.

To obtain the relationships between the elements of the resistance matrix, which are optimal by the criterion for ensuring the given frequency dependences of the modules and phases of the transfer function of the high-frequency part of the demodulator, we substitute (6) in (1) and after separating the real and imaginary parts from each other, we obtain a system of two equations:

$$\begin{array}{c}-2x_{21}{a}_{\mathrm{one}}+m\cos \varphi \left[r_n{x}_{\mathrm{eleven}}-x_{22}(r_0+r)+A\right]-m\sin \varphi \left[x_n{x}_{\mathrm{eleven}}-x_{22}(x_0+x+\left|x\right|+B\right]=0;\\ -2x_{21}{b}_{\mathrm{one}}+m\cos \varphi \left[x_n{x}_{\mathrm{eleven}}-x_{22}(x_0+x)+\left|x\right|+B\right]+m\sin \varphi \left[r_n{x}_{\mathrm{eleven}}-x_{22}(r_0+r)+A\right]=0\end{array}(7)$$

; $$B=-r_{н}r+x_{н}x-r_0{r}_{н}+x_0{x}_{н}$$

.Where $$A=r_{n}x+x_{n}r+r_0{x}_n+x_0{r}_n$$

; $$B=-r_{n}r+x_{n}x-r_0{r}_n+x_0{x}_n$$

.

Solution (7) takes the form of two relationships between the elements of the resistance matrix of the four-terminal network or the approximating functions of the frequency dependences of these elements:

$$x_{22}=D{x}_{\mathrm{eleven}}+E+x_n+\frac{x_{21}G}{m};\left(8\right)$$

, $$x_{21}=\frac{H-x_{\mathrm{eleven}}G}{2m}\pm \sqrt{-\left(D-\frac{{\mathrm{<figure-callout\; id="2"\; label="G"\; filenames="00000070.png"\; state="\{\{state\}\}">G</figure-callout>}}^2}{\mathrm{four}{m}^2}\right)x_{\mathrm{eleven}}^2-2x_{\mathrm{eleven}}\left(E+\frac{HG}{\mathrm{four}{m}^2}\right)+F+\frac{{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="00000070.png"\; state="\{\{state\}\}">H</figure-callout>}}^2}{\mathrm{four}{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="00000070.png"\; state="\{\{state\}\}">m</figure-callout>}}^2}}$$

,

; $$E=\frac{r_{н}\left(x_0+x\right)}{r_0+r}$$

; $$F=\frac{-r_{н}\left[{\left(x_0+x\right)}^2+{\left(r_0+r\right)}^2\right]}{r_0+r};$$

Where $$D=\frac{r_n}{r_0+r}$$

; $$E=\frac{r_n\left(x_0+x\right)}{r_0+r}$$

; $$F=\frac{-r_n\left[{\left(x_0+x\right)}^2+{\left(r_0+r\right)}^2\right]}{r_0+r};$$

; $$H=\frac{\left[r_{н}\left(x_0+x\right)-x_{н}\left(r_0+r\right)\right]\text{cos}\varphi +\left[{\text{r}}_{\text{н}}\left(r_0+r\right)+x_{н}\left(x_0+x\right)\right]\text{sin}\varphi }{{\text{r}}_{\text{0}}+\text{r}}$$

$$G=-\frac{r_n\text{cos}\varphi +{\text{x}}_{\text{n}}\text{sin}\varphi }{r_0+r}$$

; $$H=\frac{\left[r_n\left(x_0+x\right)-x_n\left(r_0+r\right)\right]\text{cos}\varphi +\left[{\text{r}}_{\text{n}}\left(r_0+r\right)+x_n\left(x_0+x\right)\right]\text{sin}\varphi }{{\text{r}}_{\text{0}}+\text{r}}$$

Since the information is enclosed in the envelope of the AFMS, the frequency dependence of the transfer function module m on the left slope of the frequency response must be linear. The frequency dependence of the phase φ of the transfer function can be arbitrarily chosen, since the information is enclosed in the envelope of the AFMS, or based on any other physical considerations. In this invention, it is selected from the condition of physical feasibility, determined by the positivity of the radical expression in (8).

In order to determine the optimal dependences of the resistance of the two-terminal circuits that form the SFU according to criterion (1), it is necessary to select a typical SFU circuit, find the elements of its resistance matrix, substitute them in (8), and solve the resulting system of two equations for the resistance of two two-terminal devices. If the number of two-terminal circuits of the selected SFU circuit is more than two, then the frequency characteristics of the remaining two-terminal circuits can be chosen arbitrarily or based on any other physical considerations.

In accordance with the above algorithm, approximating functions of the frequency dependences of the resistance of the two-terminal L-shaped connection of two two-terminal are obtained (Fig.3):

; $$x_{2n}=\frac{-\left(F{m}_n+x_{1n}{m}_n\left(x_{нn}-E\right)+H{x}_{1n}\right)}{x_{1n}{m}_n};\left(9\right)$$

$$x_{\mathrm{one}n}=\frac{H-2E{m}_n+Q}{2\left(m_n+D{m}_n+G\right)}$$

; $$x_{2n}=\frac{-\left(F{m}_n+x_{\mathrm{one}n}{m}_n\left(x_{nn}-E\right)+H{x}_{\mathrm{one}n}\right)}{x_{\mathrm{one}n}{m}_n};\left(9\right)$$

, $$Q=\pm \sqrt{\mathrm{four}{m}_n^2\left(F-r_{nn}^2\right)+\mathrm{four}{m}_n\left(FG-EH\right)+{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="00000070.png"\; state="\{\{state\}\}">H</figure-callout>}}^2}$$

,

where the index n is introduced to indicate the frequency number in the interests of the implementation of approximating functions (9) by the interpolation method. In the coefficients D, E, F, G, H (8), it is also necessary to introduce the index n in the quantities r ₀ , x ₀ , r _n , x _n , r, x. To implement optimal approximations (9) by interpolation, it is necessary to form two-terminal networks with resistances x _1n , x _2n from at least N (the number of interpolation frequencies) of the reactive elements, find expressions for their resistances, equate them with the optimal values of two-terminal resistances at given frequencies, determined by formulas (9) and solve the thus formed system of N equations with respect to N selected parameters of reactive elements. The values of the parameters of the remaining elements can be chosen arbitrarily or on the basis of any other physical considerations, for example, from the condition of physical realizability. Let each of the two-terminal networks with resistances x _1n , x _{2n be} formed from two parallel-connected parallel circuits L _1k , C _1k , L _2k , C _2k (figure 4) (k = 1, 2 is the number of the two-terminal network (figure 3)). For N = 4, we compose two systems of four equations:

Realization of optimal approximations of the frequency characteristics of a four-terminal network in the form of a L-shaped link (9) using (11) with a reasonable choice of the positions of the given frequencies relative to each other ω ₁ -ω ₂ , ω ₁ -ω ₃ , ω ₁ -ω ₄ , ω ₂ - ω ₃ , ω ₂ -ω ₄ , ω ₃ -ω ₄ provides an increase in the quasilinear portion of the left slope of the frequency response of the high-frequency part of the phase demodulator and the required shape of the frequency response for arbitrarily given frequency dependences r ₀ , x ₀ , r _n, x _n r, x, which determine the values of the resistances of the FMS source r _0n , x _0n , load r _нn , x _нn and nonlinear element r _n , x _n per even predefined frequencies.

The proposed technical solutions are new, because from publicly available information, there is no known method and device for simultaneously performing FMS demodulation and filtering, which will convert the FMS to AFMS on the enlarged quasilinear section of the left slope of the frequency response and the given shape of the frequency response, while the device consists of a nonlinear bipolar element included in the longitudinal circuit (sequentially) between the FMS source and the input of the reactive four-terminal, high-frequency load introduced between the output of the four-terminal and a low-pass filter in the transverse circuit, and the four-terminal network is made in the form of a L-shaped connection of two reactive two-terminal networks, each of which is formed of two series-connected parallel oscillatory circuits, the parameter values 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 known technical solutions that the claimed sequence of operations (switching on a bipolar nonlinear element between the FMS source and the input of the four-terminal device, switching on the high-frequency load between the output of the four-terminal device and the low-pass filter in the transverse circuit , the execution of a four-terminal reactive in the form of a L-shaped connection of two two-terminal networks, their formation from two after of parallel connected oscillatory circuits with the choice of their parameter values from the condition of increasing the quasilinear portion of the left slope of the frequency response and ensuring the given shape of the frequency response, simultaneously filters and converts the PMF to AFMS without a reference signal source.

The proposed technical solutions are practically applicable, since for their implementation semiconductor diodes, inductances and capacitors, commercially available from the industry, formed in the claimed reactive four-terminal circuit can be used. The values of inductances and capacitances can be uniquely determined using mathematical expressions given in the claims.

The technical and economic efficiency of the proposed device is to simultaneously provide filtering and demodulation of the input FMS without a reference signal source, which helps to increase noise immunity and reduce the cost of the phase demodulator.

Claims (2)

1. A method of demodulating and filtering phase-modulated signals, which consists in the fact that the demodulator is switched on between a source of phase-modulated signals and a low-frequency load and is made of a four-terminal device, a two-pole nonlinear element, a low-pass filter and a separation capacitance, a phase-modulated signal is converted into an amplitude-phase-modulated signal by its supply to the right or left slope of the amplitude-frequency characteristic, using a nonlinear element destroy the spectrum of amplitude-phase modulation signal to the high-frequency and low-frequency components, using the low-pass filter, the information low-frequency signal is extracted, the dc component of the information low-frequency signal is eliminated with the help of a separation capacitance, the amplitude of which changes according to the law of phase change of the phase-modulated input signal, the information low-frequency signal is applied to the low-frequency selective load in the form differentiating or integrating circuit, respectively, characterized in that between the output even a high-frequency load and a low-pass filter into the transverse circuit, a high-frequency load is introduced, a two-pole non-linear element is connected between the phase-modulated signal source and the four-terminal input into the longitudinal circuit, the frequency dependence of the four-terminal resistance matrix elements is selected from the conditions for the formation of the given shape and the left quasilinear slope of the amplitude-frequency characteristic of the high-frequency part demodulator using the following mathematical expressions:
$$x_{22}=D{x}_{\mathrm{eleven}}+E+x_n+\frac{x_{21}G}{m}$$

; $$x_{21}=\frac{H-x_{\mathrm{eleven}}G}{2m}\pm \sqrt{-\left(D-\frac{G^2}{\mathrm{four}{m}^2}\right)x_{\mathrm{eleven}}^2-2x_{\mathrm{eleven}}\left(E+\frac{HG}{\mathrm{four}{m}^{{}_2}}\right)+F+\frac{H^2}{\mathrm{four}{m}^2}}$$

,
Where $$D=\frac{r_n}{r_0+r}$$

; $$E=\frac{r_n\left(x_0+x\right)}{r_0+r}$$

; $$F=\frac{-r_n\left[{\left(x_0+x\right)}^2+{\left(r_0+r\right)}^2\right]}{r_0+r};$$

$$G=-\frac{r_n\cos \phi +x_n\sin \phi }{r_0+r};$$

$$H=\frac{\left[r_n\left(x_0+x\right)-x_n\left(r_0+r\right)\right]\cos \phi +\left[r_n\left(r_0+r\right)\right]+x_n\left(x_0+x\right)]\sin \phi }{r_0+r};$$

x ₁₁ , x ₂₁ = -x ₁₂ are the optimal frequency dependencies of the corresponding elements of the quadripole resistance matrix taking into account the reciprocity condition; x ₂₂ is the predetermined dependence of the corresponding element of the quadripole resistance matrix on frequency; m is the given shape of the amplitude-frequency characteristic taking into account the quasilinear dependence of the transfer function module on frequency on the left slope of the amplitude-frequency characteristic of the high-frequency part of the phase demodulator; φ is the predetermined dependence of the phase of the transfer function on frequency from the condition of physical realizability of the given shape of the amplitude-frequency characteristic of the high-frequency part of the phase demodulator; r _n , x _n - given frequency dependencies of the real and imaginary components of the load resistance; r ₀ , x ₀ - given frequency dependences of the real and imaginary components of the resistance of the source of the phase-modulated signal; r, x are the given frequency dependences of the real and imaginary components of the resistance of a bipolar nonlinear element; the remaining quantities have the meaning of intermediate notation to simplify mathematical expressions. 2. A device for demodulating and filtering phase-modulated signals, connected between the source of phase-modulated signals and a low-frequency load in the form of an integrating circuit and consisting of a four-terminal, two-pole nonlinear element, a low-pass filter and a separation capacitance, characterized in that between the output of the four-terminal and a low-pass filter in the transverse a high-frequency load is introduced, a bipolar non-linear element is connected between the phase-modulated signal source and the four-terminal input and in the longitudinal circuit, the four-terminal network is made of a L-shaped connection of two two-terminal networks with resistances x _1n , x _2n , each of the two-terminal networks is formed of two parallel-connected parallel oscillatory circuits of elements with parameters L _1k , C _1k , L _2k , C _2k , values which are defined in accordance with the following mathematical expressions:
$$L_{\mathrm{one}k}=\frac{e_{\mathrm{one}}{x}_{k2}+h_{\mathrm{one}}{x}_{k\mathrm{one}}}{{\omega }_{\mathrm{one}}{\omega }_2\left({\omega }_{\mathrm{one}}^2-{\omega }_2^2\right)\left(B-A\right)}$$

; $$L_{2k}=\frac{e_2{x}_{k2}+h_2{x}_{k\mathrm{one}}}{{\omega }_{\mathrm{one}}{\omega }_2\left({\omega }_{\mathrm{one}}^2-{\omega }_2^2\right)\left(B-A\right)}$$

; $$C_{\mathrm{one}k}=\frac{A}{L_{\mathrm{one}k}}$$

; $$C_{2k}=\frac{B}{L_{2k}}$$

,
Where $$B=\frac{-y\pm \sqrt{y^2-\mathrm{four}xz}}{2x}$$

; $$A=\frac{a_{\mathrm{one}}B+b_{\mathrm{one}}}{c_{\mathrm{one}}B+d_{\mathrm{one}}}=\frac{a_{2}B+b_2}{c_{2}B+d_2}$$

;
$$x=a_2{c}_{\mathrm{one}}-a_{\mathrm{one}}{c}_2$$

; $$y=a_2{d}_{\mathrm{one}}+b_2{c}_{\mathrm{one}}-a_{\mathrm{one}}{d}_2-b_{\mathrm{one}}{c}_2$$

; $$z=b_2{d}_{\mathrm{one}}-b_{\mathrm{one}}{d}_2$$

;
$$e_{\mathrm{one}}={\omega }_{\mathrm{one}}\left(\mathrm{one}-{\omega }_2^{2}B\right)\left(\mathrm{one}+A^2{\omega }_{\mathrm{one}}^2{\omega }_2^2-A\left({\omega }_{\mathrm{one}}^2+{\omega }_2^2\right)\right)$$

;
$$h_{\mathrm{one}}=-{\omega }_2\left(\mathrm{one}-{\omega }_{\mathrm{one}}^{2}B\right)\left(\mathrm{one}+A^2{\omega }_{\mathrm{one}}^2{\omega }_2^2-A\left({\omega }_{\mathrm{one}}^2+{\omega }_2^2\right)\right)$$

;
$$e_2=-{\omega }_{\mathrm{one}}\left(\mathrm{one}-{\omega }_2^{2}A\right)\left(\mathrm{one}+B^{{}_2}{\omega }_{\mathrm{one}}^2{\omega }_2^2-B\left({\omega }_{\mathrm{one}}^2+{\omega }_2^2\right)\right)$$

;
$$h_2={\omega }_2\left(\mathrm{one}-{\omega }_{\mathrm{one}}^{2}A\right)\left(\mathrm{one}+B^2{\omega }_{\mathrm{one}}^2{\omega }_2^2-B\left({\omega }_{\mathrm{one}}^2+{\omega }_2^2\right)\right)$$

;
$$a_{\mathrm{one}}={\omega }_{\mathrm{one}}{\omega }_2{\omega }_{\mathrm{four}}^2{x}_{k\mathrm{four}}\left({\omega }_2^2-{\omega }_{\mathrm{one}}^2\right)+{\omega }_{\mathrm{one}}{\omega }_2^2{\omega }_{\mathrm{four}}{x}_{k2}\left({\omega }_{\mathrm{one}}^2-{\omega }_{\mathrm{four}}^2\right)+{\omega }_{\mathrm{one}}^2{\omega }_2{\omega }_{\mathrm{four}}{x}_{k\mathrm{one}}\left({\omega }_{\mathrm{four}}^2-{\omega }_2^2\right)$$

;
$$b_{\mathrm{one}}={\omega }_{\mathrm{one}}{\omega }_2{x}_{k\mathrm{four}}\left({\omega }_{\mathrm{one}}^2-{\omega }_2^2\right)+{\omega }_{\mathrm{one}}{\omega }_{\mathrm{four}}{x}_{k2}\left({\omega }_{\mathrm{four}}^2-{\omega }_{\mathrm{one}}^2\right)+{\omega }_2{\omega }_{\mathrm{four}}{x}_{k\mathrm{one}}\left({\omega }_2^2-{\omega }_{\mathrm{four}}^2\right)$$

;
$$c_{\mathrm{one}}=\left[{\omega }_{\mathrm{four}}^3{x}_{k\mathrm{four}}\left({\omega }_2^2-{\omega }_{\mathrm{one}}^2\right)+{\omega }_2^3{x}_{k2}\left({\omega }_{\mathrm{one}}^2-{\omega }_{\mathrm{four}}^2\right)+{\omega }_{\mathrm{one}}^3{x}_{k\mathrm{one}}\left({\omega }_{\mathrm{four}}^2-{\omega }_2^2\right)\right]{\omega }_{\mathrm{one}}{\omega }_2{\omega }_{\mathrm{four}};$$

$$d_{\mathrm{one}}=\left[{\omega }_{\mathrm{four}}{x}_{k\mathrm{four}}\left({\omega }_{\mathrm{one}}^2-{\omega }_2^2\right)+{\omega }_2{x}_{k2}\left({\omega }_{\mathrm{four}}^2-{\omega }_{\mathrm{one}}^2\right)+{\omega }_{\mathrm{one}}{x}_{k\mathrm{one}}\left({\omega }_2^2-{\omega }_{\mathrm{four}}^2\right)\right]{\omega }_{\mathrm{one}}{\omega }_2{\omega }_{\mathrm{four}};$$

$$a_2={\omega }_{\mathrm{one}}{\omega }_2{\omega }_3^2{x}_{k3}\left({\omega }_2^2-{\omega }_{\mathrm{one}}^2\right)+{\omega }_{\mathrm{one}}{\omega }_2^2{\omega }_3{x}_{k2}\left({\omega }_{\mathrm{one}}^2-{\omega }_3^2\right)+{\omega }_{\mathrm{one}}^2{\omega }_2{\omega }_3{x}_{k\mathrm{one}}\left({\omega }_3^2-{\omega }_2^2\right)$$

;
$$b_2={\omega }_{\mathrm{one}}{\omega }_2{x}_{k3}\left({\omega }_{\mathrm{one}}^2-{\omega }_2^2\right)+{\omega }_{\mathrm{one}}{\omega }_3{x}_{k2}\left({\omega }_3^2-{\omega }_{\mathrm{one}}^2\right)+{\omega }_2{\omega }_3{x}_{k\mathrm{one}}\left({\omega }_2^2-{\omega }_3^2\right)$$

;
$$c_2=\left[{\omega }_3^3{x}_{k3}\left({\omega }_2^2-{\omega }_{\mathrm{one}}^2\right)+{\omega }_2^3{x}_{k2}\left({\omega }_{\mathrm{one}}^2-{\omega }_3^2\right)+{\omega }_{\mathrm{one}}^3{x}_{k\mathrm{one}}\left({\omega }_3^2-{\omega }_2^2\right)\right]{\omega }_{\mathrm{one}}{\omega }_2{\omega }_3;$$

$$d_2=\left[{\omega }_3{x}_{k3}\left({\omega }_{\mathrm{one}}^2-{\omega }_2^2\right)+{\omega }_2{x}_{k2}\left({\omega }_3^2-{\omega }_{\mathrm{one}}^2\right)+{\omega }_{\mathrm{one}}{x}_{k\mathrm{one}}\left({\omega }_2^2-{\omega }_3^2\right)\right]{\omega }_{\mathrm{one}}{\omega }_2{\omega }_3;$$

$$x_{\mathrm{one}n}=\frac{H-2E{m}_n+Q}{2\left(m_n+D{m}_n+G\right)}$$

; $$x_{2n}=\frac{-\left(F{m}_n+x_{\mathrm{one}n}{m}_n\left(x_{nn}-E\right)+H{x}_{\mathrm{one}n}\right)}{x_{\mathrm{one}n}{m}_n}$$

;
$$D=\frac{r_{nn}}{r_{0n}+r_n}$$

; $$E=\frac{r_{nn}\left(x_{0n}+x_n\right)}{r_{0n}+r_n}$$

;
$$Q=\pm \sqrt{\mathrm{four}{m}_n^2\left(F-r_{nn}^2\right)+\mathrm{four}{m}_n\left(FG-EH\right)+H^2}$$

;
$$F=\frac{-r_{nn}\left[{\left(x_{0n}+x_n\right)}^2+{\left(r_{0n}+r_n\right)}^2\right]}{r_{0n}+r_n};$$

$$G=-\frac{r_{nn}cos{\varphi }_n+x_{nn}sin{\varphi }_n}{r_{0n}+r_n}$$

;
$$H=\frac{\left[r_{nn}(x_{0n}+x_n)-x_{nn}(r_{0n}+r_n)\right]\cos {\phi }_n+\left[r_{nn}(r_{0n}+r_n)+x_{nn}(x_{0n}+x_n)\right]\sin {\phi }_n}{r_{0n}+r_n},$$

where m _n are the given values of the transfer function modules at the given four frequencies ω _n = 2πf _n from the conditions for the formation of the given shape and the quasilinear left slope of the amplitude-frequency characteristic of the high-frequency part of the phase demodulator; φ _n - the given values of the phases of the transfer function at the given four frequencies ω _n = 2πf _n from the condition of physical realizability of the given form of the amplitude-frequency characteristic of the high-frequency part of the phase demodulator; r _0n , x _0n - set values of the real and imaginary components of the resistance of the source of the phase-modulated signal at the specified four frequencies; r _nn , x _nn - set values of the real and imaginary components of the resistance of the high-frequency load at the specified four frequencies; r _n , x _n - given values of the real and imaginary components of the resistance of a bipolar nonlinear element at given four frequencies; k = 1, 2 is the number of the two-terminal L-shaped connection; n = 1, 2, 3, 4 - frequency number; the remaining quantities have the meaning of intermediate notation to simplify mathematical expressions.

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