RU2524843C2 - Method of measuring time of arrival of signal and apparatus for realising said method - Google Patents

Abstract

FIELD: physics, computer engineering.

SUBSTANCE: invention relates to means of measuring the time of arrival of two-position angular manipulated signals at a receiving position. The technical result is achieved by eliminating receiving frequency uncertainty, which eliminates the need for two-dimensional search of arguments of the maximum of a two-dimensional discrete cross-correlation function (DCCF) for the two-position angular manipulated signal, thereby avoiding the search for the argument of the maximum of a one-dimensional DCCF. A two-level modulating signal is generated and differentially decoded. Each symbol of the differentially decoded signal is mapped onto a corresponding number of readings of the received signal converted to digital relative to the time scale of the receiving position by repeating values of the differentially decoded signal with increase in sampling frequency thereof to the sampling frequency of the received signal converted to digital. A second digital data stream is generated, which is converted using fast Fourier transform to values of a second spectrum.

EFFECT: high computational efficiency and high measurement accuracy.

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Description

The invention relates to radio engineering and can be used to measure the time of arrival of signals with on-off angle manipulation at the receiving position.

The measurement of the arrival time (VP) of signals at a receiving position with known coordinates is of great importance in systems of long-distance space communications with spacecraft (SC) of the Mars Polarlander, Mars Pathfinder type [1] or with the Voyager and Galileo type spacecraft [2], including to determine the motion parameters of such spacecraft.

A number of methods are known for measuring VP signals, including from spaced receiving positions [3-6], based on finding the arguments for the maximum of a two-dimensional cross-correlation function, which potentially give statistically optimal, most plausible estimates. However, the potential accuracy of the methods [3-6] is not feasible in practice due to the lack of continuously adjustable standards of time and frequency [7, 8].

A number of digital methods are known for measuring VP signals from spaced receiving positions [8], based on finding the argument of the maximum of the cross-correlation function or the argument of the minimum of the differential discrete cross-correlation functions (DKKF), allowing to realize the potential accuracy of the methods [3-6] by eliminating discreteness errors by parabolic interpolation of the neighborhoods of the maximum or minimum of the corresponding functions. However, digital methods for measuring VP signals [8–9] are not applicable in the presence of uncertainty in the frequency of reception, which occurs in systems of long-distance space communications with spacecraft. Another disadvantage of the digital measurement methods presented in [8] is the low computational efficiency on large data sample sizes, since they are based on the direct calculation of cross-correlation functions (with this method of finding cross-correlation functions, the number of multiplication operations is proportional to the square of the length (n ) data sampling, such proportionality is usually denoted as O (n ² )).

When measuring the airborne signal for determining the distance to the spacecraft, the uncertainty in the reception frequency inevitably arises, which can be caused both by the difference in the reference frequencies on the spacecraft and on the ground, as well as due to the Doppler effect caused by the mutual movement of the ground receiving position and the spacecraft. Therefore, in such cases, when measuring the IP signal, it is necessary to find the maximum of the two-dimensional cross-correlation function with the simultaneous resolution of the frequency uncertainty and, thus, measuring the reception frequency or the difference in the reception frequency (RFI).

In patents [10-12], a number of digital methods are presented for jointly measuring the difference in arrival time (RWP) and RFI of signals from spaced receiving positions, based on finding the arguments of the maximum of two-dimensional DKKF (DKKF), which allow measurements in the presence of movement of the signal source or receiving position . The main goal set by the author of patents [10-12] is devoted to reducing the flow of information transmitted between spaced receiving positions, and to compensate for the systematic error arising from the correlation of the compressed and reference signals. However, in the methods of joint measurement of RWP and RFI [10-12], the issues of eliminating discreteness errors are not resolved (the correlation between the RVP and RFI [13] does not allow the approaches described in [8] to be applied to joint measurements in an obvious way) and the issues of increasing computational efficiency , since the cross-correlation functions in methods [10-12] are generated by the direct method through the convolution of two data sequences, as in [8], but due to n-fold repetition (to ensure the search for the maximum argument in the frequency domain) if the number of multiplication operations increases and is proportional to the cube of the length (n) of the data sample, i.e. O (n ³ ).

Closest to the proposed method for measuring the VP of signals by the totality of the actions used on the signal is the method [14], based on finding the arguments of the maximum DKKF (computationally significantly more efficient than analogues), adopted as a prototype.

According to this method:

1. Receive a signal at two spaced receiving positions.

2. Convert the signal received from one receiving position to the first digital data stream, which digitally represents the signal as a series of values of the time function.

3. Convert the signal received from another receiving position into a second digital data stream, which digitally represents the signal as a series of values of the time function.

4. Using the fast Fourier transform (FFT), the aforementioned first digital data stream is converted into the values of the first spectrum, which represents the signal as a series of values of the frequency function S ₁ (k), where k is an integer index variable that varies within the data length.

5. Using the FFT, the mentioned second digital data stream is converted into the values of the second spectrum, which represents the signal as a series of values of the frequency function S ₂ (k).

6. Mutually multiply the spectrum values of one of the signals with the complex conjugate values of the spectrum of another signal to generate a cross spectrum.

7. The spectrum of one of the signals S _i (k), where i = 1, 2 is the number of one of the two spectra, is converted in frequency to the values of m, selected according to the requirements for resolution of the RFI, from an integer series of values from unity to (N-1 ), where N is the length of the processed digital data stream, with the creation of a sequence of frequency-converted spectra S _{i, m} (k) obtained from the original spectral components of the signal according to the following rule:

$$S_{i,m}\left(k\right)=\{\begin{array}{c}{S}_i\left(k-m\right),m\le k<\frac{N}{2}\\ S_i\left(k-m\right)+S_i\left(k+m\right),\frac{N}{2}-k\le m\le k<\frac{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="00000037.png"\; state="\{\{state\}\}">N</figure-callout>}}{2}\\ 0k<m<\frac{N}{2}-k\\ S_i\left(k+m\right),\frac{N}{2}-m<k<m<\frac{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="00000037.png"\; state="\{\{state\}\}">N</figure-callout>}}{2},\end{array}$$

where i is the number of one of the two received signals.

8. The values of the spectra S _{i, m} (k) of the frequency-converted signals are mutually multiplied with the complex conjugate values of the spectrum of the second of the two source signals to generate multiple cross-spectra.

9. Calculate the DKKF signal using the inverse FFT multiple cross-spectra.

10. Determine the difference in the time of arrival and frequency of reception of signals as the arguments of the maximum DCCF signal.

In fact, the method described above [14] implements a computationally efficient, in comparison with analogs [10-12], finding the arguments of the maximum DKKF using fast Fourier transforms when calculating the DKKF based on the Wiener-Khinchin theorem [15], which determines the relationship between the spectrum and signal correlation function.

In the description of the prototype [14], an estimate of the amount of computational costs for finding a two-dimensional cross-correlation function in the prototype method and in analogues is presented, showing that ordering data of length n = 1024 with k = n in the prototype method for finding the DQFF requires about O (k · n · log ₂ n + k · n) or 10 million operations (page 10 of the description of the prototype method), and in analogs it is required from O (3 · k · n · log ₂ n + k · n) to O ( n ³ ) or from 30 million to a billion operations (page 5 of the description of the prototype method).

Despite a significant reduction in computing costs compared with analogues, one of the disadvantages of the prototype method is the lack of computing efficiency. Another disadvantage of the prototype method is, like analogs [10-12], a long time to find the maximum argument, that is, a long measurement time.

The prototype device [14] contains the first means of receiving signals connected to the device for determining the arguments of the maximum DKKF through sequentially connected the first analog-to-digital Converter (ADC), the first FFT processor, a cross-spectrum computer and a second FFT processor. The output of the DCDCF maximum argument determination device is the output of the measurement device. Between the second signal receiving means and the second input of the cross-spectrum calculator, a second ADC and a third FFT processor are connected in series. Between the output of the first FFT processor and the third input of the cross-spectrum calculator, an arithmetic device is turned on.

The disadvantage of the prototype device is the lack of speed in determining RFI and RVP signals. Another disadvantage of the prototype device is the long time it takes to find the maximum argument, that is, a long measurement time.

The technical result of the invention is to increase computational efficiency by eliminating the uncertainty of the reception frequency, which eliminates the need for a two-dimensional search for the arguments of the maximum DKKF for a signal with two-position angular manipulation and, therefore, by searching for the argument of the maximum of the one-dimensional DKKF.

The technical result is achieved by the fact that in the method of measuring the time of arrival of a signal with on-off angle manipulation, including receiving a signal, analog-to-digital conversion of the received signal into a first digital data stream, representing the signal as a series of values of the time function, converted to digital form, using the fast Fourier transform (FFT) for the two signals representing the two signals as a series of discrete spectral values S _i (k), where k - integer index variable which varies within the length data, and i = 1, 2 is the index variable, which means the serial number of the digital data stream converted using the FFT to the spectrum values, the mutual multiplication of the spectrum values of one of the signals with the complex conjugate values of the spectrum of another signal, which generates a cross spectrum, calculates a discrete cross the correlation function (DKKF) of the signal using the inverse FFT, determining the time of arrival (VP) of the signal as an argument of the maximum of the DKKF signal, according to the invention, quadrature decomposition of the first digit a straight data stream relative to the nominal center frequency of the modulated signal into a plurality of in-phase and quadrature samples corresponding to the specified stream, the obtained in-phase and quadrature samples are independently low-pass filtered with a cutoff frequency corresponding to the modulation signal manipulation speed, and the set of current signal phases as complex number arguments is obtained as real parts of which use the corresponding filtered in-phase readings, and as imaginary - respectively The filtered filtered quadrature samples of the signal, delay the set of received current signal phases by the duration of the modulating signal symbol, subtract modulo 2π from each received current phase the corresponding value of the delayed current signal phase, and the resulting differential digital data stream is converted using the FFT into the values of the first frequency function S ₁ (k), a two-level modulating signal is generated and differentially decoded, each symbol of the differentially decoded signal is mapped to the corresponding number of samples of the digitized received signal relative to the time scale of the receiving position and thus form the second digital data stream, which is converted using the FFT into the values of the second frequency function S ₂ (k).

Another technical result of the invention is the elimination of measurement error caused by the multiple duration of the two-level modulating symbol and the sampling frequency of the analog-to-digital conversion of the received signal.

The technical result is achieved in that in the method for measuring the time of arrival of a signal according to the invention for a signal for which the signal symbol duration is not a multiple of the sampling frequency of the analog-to-digital conversion of the received signal, the filtered in-phase and quadrature samples are additionally decimated and interpolated to provide a frequency multiplicity of the samples of the first digital data stream symbol duration of a two-level modulating signal.

Another technical result of the invention is the elimination of the discreteness error due to the fact that the neighborhood of the main maximum of the discrete cross-correlation signal function has the shape of a parabola [13, p. 8-9], and due to the development of an optimal, in the mean-square sense, analytical method of estimation parameters of the parabola and determining the argument of its maximum.

определяют по следующему правилу:The technical result is achieved by the fact that in the method of measuring the time of arrival of the signal according to the invention, to the left and to the right of the maximum argument DKKF signal is selected by L points of the specified function, the DKKF values for the selected points and maximum points are combined in the order of time in a column vector c, from units form a column vector v _{0 of} the same dimension as vector c, corresponding to the selected points of the DKKF, the arguments are expressed through the index variable l = 0, ± 1, ..., ± L, where the indices l with a negative sign correspond to the times d about the DKKF maximum point, the zero index l corresponds to the DKKF maximum point, and the l indices with a positive sign correspond to the times after the DKKF maximum point, the arguments of the selected DKKF points are ordered as well as the components of the vector c and are combined into the column vector v ₁ , the squares of the components of the vector v ₁ similarly combine into a column vector v ₂ , and the specified value of the VP $${\widehat{\tau }}_M$$

determined by the following rule:

$${\widehat{\tau }}_M=\frac{\left(S^2-NQ\right)v_{\mathrm{one}}^{T}c}{2{\left(NS{v}_2-S^2{v}_0\right)}^{T}c}{T}_s,$$

- скалярное произведение векторов v₁ и с, коэффициент S определяется выражениемwhere N = 2L + 1 is the dimension of the generated vectors, the superscript ^T denotes the transpose of the vector, T _s is the signal sampling period, $$v_{\mathrm{one}}^{T}c$$

- a scalar product of the vectors v ₁ and, S ratio is determined by the expression

the coefficient Q is determined by another expression

a (NSv ₂ -S ² v ₀ ) ^T c is the scalar product of the difference vector (NSv ₂ -S ² v ₀ ) with the vector с.

The method is implemented by a signal arrival measuring device with on-off angle manipulation according to claims 1 and 2, comprising a signal receiving means, the input of which is an input to a measuring device connected to an analog-to-digital converter (ADC), a first FFT processor, a cross-spectrum calculator, connected in series, a second FFT processor and a device for determining the arguments of the maximum of the discrete cross-correlation function (DKKF) of the signal, the output of which is the output of the measurement device, while calculating to the second input a third FFT processor is connected according to the invention, between the ADC output and the input of the first FFT processor, a quadrature signal decomposition device, a first low-pass filter (LPF), a read-only memory (ROM) and a subtractor modulo 2π between the second output are connected in series the quadrature decomposition of the signal and the second input of the ROM includes a second low-pass filter, the subtracting input of the subtraction device is connected to the output of the ROM through the delay element for the duration of the signal symbol, and the input is third its FFT processor is connected to the output of the shaper of a two-level modulating signal through a series-connected differential decoder and expander of the sampling frequency.

Another technical result of the invention is the elimination of discreteness errors in the device due to the fact that the neighborhood of the main maximum of the DCQF signal has the shape of a parabola, and due to the analytical method for estimating the parameters of the parabola and determining the argument of its maximum.

The technical result is achieved in that an arithmetic device is additionally connected to the output of the device for measuring the signal arrival with on-off angle manipulation according to the invention to the output of the device for determining the maximum argument of the discrete cross-correlation function of the signal.

Figure 1 presents an example of the full phase of the signal with on-off phase shift keying in the presence of frequency detuning.

Figure 2 presents an example of a phase difference of signals with a shift by the duration of the symbol T.

Figure 3 shows the structural diagram of a device in which the proposed method is implemented.

According to the proposed method:

1. Receive a signal.

2. The received signal is converted into a first digital data stream representing the signal as a series of values of a time function converted to digital form.

3. Quadrature decomposition of the first digital data stream is carried out relative to the nominal center frequency of the modulated signal into a plurality of in-phase and quadrature samples corresponding to said stream.

4. The obtained in-phase and quadrature samples are independently low-pass filtered with a cutoff frequency corresponding to the manipulation speed of the modulating signal.

5. A plurality of current signal phases are obtained as arguments of complex numbers, the corresponding filtered common-mode samples are used as the real part, and the filtered filtered quadrature samples of the signal are used as the imaginary part.

6. Delay the set of received current signal phases for the duration of the modulating signal symbol.

7. Subtract modulo 2π from each received current phase the corresponding value of the delayed current phase of the signal.

8. Using the fast Fourier transform (FFT), the resulting differential digital data stream is converted as the first digital data stream into discrete values of the first spectrum S ₁ (k), where k is an integer index variable that varies within the data length.

9. Form a two-level modulating signal.

10. Differentially decode the generated signal.

11. Each symbol of the differentially decoded signal is mapped to the corresponding number of samples of the digitized received signal relative to the time scale of the receiving position by repeating the values of the differentially decoded signal with increasing its sampling frequency to the sampling frequency of the digitalized received signal and, thus, form a second digital data stream.

12. Using the FFT, said second digital data stream is converted to the values of the second spectrum S ₂ (k).

13. Mutually multiply the spectrum values of one of the signals with the complex conjugate values of the spectrum of another signal to generate a cross spectrum.

14. The discrete cross-correlation function (DKKF) of the signal is calculated using the inverse FFT cross-spectrum.

15. Determine the time of arrival of the signal as an argument to the maximum DKKF signal.

We show that in the proposed method for measuring the time of arrival of a signal with two-position angular manipulation, it is possible to exclude the need for a two-dimensional search when finding the argument of the maximum DKKF.

A signal with on-off phase shift keying is described as follows [24]:

$$s\left(t\right)=Ab\left(t\right)\cos {\omega }_{c}t,-\infty \le t\le \infty ,\left(\mathrm{one}\right)$$

$$b\left(t\right)={\displaystyle \sum _{k=-\infty }^{\infty }{b}_{k}p\left(t-kT\right)},\left(2\right)$$

where b _k ∈ {-1, + 1}, p (t) is a rectangular pulse with a unit amplitude of duration T, defined on the interval [-T / 2, T / 2], A is the signal amplitude, and ω _c is the carrier frequency . The full phase of such a signal

$$F\left(t\right)={\phi }_0+{\omega }_{c}t+{\displaystyle \sum _{k=-\infty }^{\infty }{\theta }_{k}p\left(t-kT\right)},\left(3\right)$$

where θ _k ∈ {0, π} and is related to b _{k by the} relation

$${\theta }_k=\frac{\pi }{2}\left(\mathrm{one}-b_k\right).\left(\mathrm{four}\right)$$

In the quadrature decomposition of signal (1), the carrier frequency of which is a priori unknown, in the full phase of the signal there will almost always be a frequency mismatch between the frequency of the signal carrier ω _c and the nominal frequency ω ₀ , relative to which the quadrature decomposition is performed, i.e., the component will be present in the full phase of the signal linearly dependent on time, which corresponds to the second term of expression (3), which does not allow direct use of the expression for the full phase for the formation of DKKF.

Figure 1 presents an example of the full phase of a signal with on-off phase shift keying in the presence of frequency detuning, showing the influence of a component that linearly depends on time.

From the example shown in Fig. 1, it can be seen that in the presence of a frequency detuning, the complete phase of the signal is not identical to the modulating function of the signal, and it cannot be directly used for one-dimensional correlation processing when determining the signal arrival time.

The phase difference with a shift by the duration of the symbol T eliminates the frequency detuning as a component that linearly depends on time $$\begin{array}{l}\psi \left(t\right)=F\left(t\right)-F\left(t-T\right)=\\ ={\omega }_{0}t+{\displaystyle \sum _{k=-\infty }^{\infty }{\theta }_{k}p\left(t-kT\right)-\left({\omega }_0\left(t-T\right)+{\displaystyle \sum _{k=-\infty }^{\infty }{\theta }_{k}p\left(t-\left(k+\mathrm{one}\right)T\right)}\right)}=\\ ={\omega }_{0}T+{\displaystyle \sum _{k=-\infty }^{\infty }{\theta }_{k}p\left(t-kT\right)-{\displaystyle \sum _{k=-\infty }^{\infty }{\theta }_{k-\mathrm{one}}p\left(t-kT\right)}={\omega }_{0}T+{\displaystyle \sum _{k=-\infty }^{\infty }\left({\theta }_k-{\theta }_{k-\mathrm{one}}\right)p\left(t-kT\right)}}.\left(5\right)\end{array}$$

As can be seen from expression (5), the phase difference Ψ (t), with the exception of the constant bias ω ₀ T, determined by the magnitude of the frequency detuning, uniquely corresponds to the differentially decoded modulating function of the signal (see figure 2).

Since the constant bias does not affect the argument of the cross-correlation function [19], one-dimensional correlation processing of the differential-decoded signal modulating function with the signal phase difference function Ψ (t) will allow us to determine the signal arrival time as an argument of the maximum cross-correlation function of these signals. A similar result can be shown for the signal with manipulation of the minimum shift [24].

According to the second variant of the proposed method, eliminating the measurement error caused by the non-multiplicity of the duration of the two-level modulating symbol and the sampling frequency of the analog-to-digital conversion of the received signal:

1. Receive a signal.

2. The received signal is converted into a first digital data stream representing the signal as a series of values of a time function converted to digital form.

3. Quadrature decomposition of the first digital data stream is carried out relative to the nominal center frequency of the modulated signal into a plurality of in-phase and quadrature samples corresponding to said stream.

4. The obtained in-phase and quadrature samples are independently low-pass filtered with a cutoff frequency corresponding to the manipulation speed of the modulating signal.

5. The filtered in-phase and quadrature samples are additionally decimated and interpolated to provide a frequency multiplicity of the samples of the first digital data stream of the signal symbol duration.

6. A plurality of current signal phases are obtained as arguments of complex numbers, the corresponding in-phase samples are used as the real part, and the corresponding quadrature samples of the signal are used as the imaginary part.

7. The set of received current signal phases is delayed by the duration of the modulating signal symbol.

8. Subtract modulo 2π from each received current phase the corresponding value of the delayed current phase of the signal.

9. Using the fast Fourier transform (FFT), the resulting differential digital data stream is converted as the first digital data stream into discrete values of the first spectrum S ₁ (k), where k is an integer index variable that varies within the data length.

10. Form a two-level modulating signal.

11. Differentially decode the generated signal.

12. Each symbol of the differentially decoded signal is mapped to the corresponding number of samples of the digitized received signal relative to the time scale of the receiving position by repeating the values of the differentially decoded signal with increasing its sampling frequency to the sampling frequency of the digitalized received signal and, thus, form a second digital data stream.

13. Using the FFT, said second digital data stream is converted to the values of the second spectrum S ₂ (k).

14. Mutually multiply the spectrum values of one of the signals with the complex conjugate values of the spectrum of another signal to generate a cross spectrum.

15. Calculate the discrete cross-correlation function (DKKF) signal using the inverse FFT cross-spectrum.

16. Determine the time of arrival of the signal as an argument to the maximum DKKF signal.

Slowness of the duration of a two-level modulating symbol and the sampling frequency of the analog-to-digital conversion of the received signal can be eliminated, for example, by polyphase filtering [18], which allows, due to interpolation and decimation, to provide an integer number of signal samples for the symbol duration over a finite time interval.

According to another variant of the proposed method, eliminating the discreteness error:

1. Receive a signal.

2. The received signal is converted into a first digital data stream representing the signal as a series of values of a time function converted to digital form.

3. Quadrature decomposition of the first digital data stream is carried out relative to the nominal center frequency of the modulated signal into a plurality of in-phase and quadrature samples corresponding to said stream.

4. The obtained in-phase and quadrature samples are independently low-pass filtered with a cutoff frequency corresponding to the manipulation speed of the modulating signal.

5. A plurality of current signal phases are obtained as arguments of complex numbers, the corresponding filtered common-mode samples are used as the real part, and the filtered filtered quadrature samples of the signal are used as the imaginary part.

6. Delay the set of received current signal phases for the duration of the modulating signal symbol.

7. Subtract modulo 2π from each received current phase the corresponding value of the delayed current phase of the signal.

8. Using the fast Fourier transform (FFT), the resulting differential digital data stream is converted as the first digital data stream into discrete values of the first frequency function S ₁ (k), where k is an integer index variable that varies within the data length.

9. Form a two-level modulating signal.

10. Differentially decode the generated signal.

11. Each symbol of the differentially decoded signal is mapped to the corresponding number of samples of the digitized received signal relative to the time scale of the receiving position by repeating the values of the differentially decoded signal with increasing its sampling frequency to the sampling frequency of the digitalized received signal and, thus, form a second digital data stream.

12. Using the FFT, said second digital data stream is converted to the values of the second spectrum S ₂ (k).

13. Mutually multiply the spectrum values of one of the signals with the complex conjugate values of the spectrum of another signal to generate a cross spectrum.

14. The discrete cross-correlation function (DKKF) of the signal is calculated using the inverse FFT cross-spectrum.

15. Determine the time of arrival of the signal as an argument to the maximum DKKF signal.

16. To the left and right of the argument of the maximum DKKF signal select L points of the specified function.

17. The values of DKKF for the selected points and maximum points are combined in the order of temporal sequence in the column vector c.

18. Of the units form the column vector v _{0 of} the same dimension as the vector c.

19. The arguments corresponding to the selected DKKF points are expressed through the index variable l = 0, ± 1, ..., ± L, where the indices l with a negative sign correspond to the times to the maximum point of the DKKF, the zero index l corresponds to the maximum of the DKKF, and the indices l with a positive sign correspond to times after the maximum point of DKKF.

20. The arguments of the selected DKKF points are ordered as well as the components of the vector c, and are combined into a column vector v ₁ .

21. The squares of the components of the vector v _{1 are} likewise combined into a column vector v ₂ .

:22. Determine, by the following rule, the specified value of VP $${\widehat{\tau }}_M$$

:

$${\widehat{\tau }}_M=\frac{\left(S^2-NQ\right)v_{\mathrm{one}}^{T}c}{2{\left(NS{v}_2-S^2{v}_0\right)}^{T}c}{T}_s,$$

- скалярное произведение векторов v₁ и с, коэффициент S определяется выражениемwhere N = 2L + 1 is the dimension of the generated vectors, the superscript ^T denotes the transpose of the vector, T _s is the signal sampling period, $$v_{\mathrm{one}}^{T}c$$

- a scalar product of the vectors v ₁ and, S ratio is determined by the expression

the coefficient Q is determined by another expression

and (NSv ₂ -S ² v ₀ ) ^T c is the scalar product of the difference vector (NSv ₂ -S ² v ₀ ) with the vector с.

When digitally transforming a continuous signal, discreteness errors inevitably appear [17-18], their influence can be significantly reduced by interpolating the DCQF in the vicinity of the extremum, which does not entail significant computational costs arising from the interpolation of the signal as a whole.

A variant of the proposed method, which eliminates the discreteness error when estimating the DQF parameters, is based on the fact that in the vicinity of the maximum this function has the shape of a parabola [13, p. 8-9], estimating the parameters of which as a result of solving a system of linear equations, we can clarify the value of the arrival time as an argument of the maximum DKKF, since DKKF is selective from the cross-correlation function in continuous time.

The equation describing the vertex of the cross-correlation function C _XY (τ) as a parabola has the form

$$C_{XY}\left(\tau \right)\approx a{\mathrm{<figure-callout\; id="2"\; label="\tau "\; filenames="00000037.png"\; state="\{\{state\}\}">\tau </figure-callout>}}^2+b\tau +c\left(6\right)$$

For given samples c _{n of the} cross-correlation function C _XY (τ)

$$c_n=C_{XY}\left({\tau }_n\right)+{\xi }_n,{\tau }_n=n\cdot T_s,\left(7\right)$$

where ξ _n is the error in specifying the CF samples, T _s is the sampling period.

For parabola (6), the estimate of the extremum point (in this case, the maximum, based on the meaning of the problem and the assumption of uniqueness of the extremum in the considered neighborhood) is determined by the relation

$${\widehat{\tau }}_M=\frac{-\widehat{b}}{2\widehat{a}}.\left(8\right)$$

The estimates for the coefficients a, b in (8) can be obtained from a linear system of the form

$$\underset{V}{\underbrace{\left[\begin{array}{ccc}\mathrm{one}& -\mathrm{<figure-callout\; id="2"\; label="K\; K"\; filenames="00000037.png"\; state="\{\{state\}\}">K\; </figure-callout>}& K^{}{}^2\\ ...& ...& ...\\ \mathrm{one}& -\mathrm{one}& \mathrm{one}\\ \mathrm{one}& 0& 0\\ \mathrm{one}& \mathrm{one}& \mathrm{one}\\ ...& ...& ...\\ \mathrm{one}& K& K^2\end{array}\right]}}\underset{\theta }{\underbrace{\left[\begin{array}{c}c\\ b{T}_s\\ a{T}_s^2\end{array}\right]}}+\xi =\underset{c}{\underbrace{\left[\begin{array}{c}{c}_{-k}\\ ...\\ c_{-\mathrm{one}}\\ c_0\\ c_{\mathrm{one}}\\ ...\\ c_K\end{array}\right]}}.\left(9\right)$$

where ξ is the vector of errors in specifying the samples of the cross-correlation function (7). In addition, an odd number of source data (7) is assumed in (9) and symmetric indexing n = -K, - (K-1), ..., -1, 0, 1, 2, ..., K is accepted, and although these assumptions are not essential for a general least squares (LSM) estimate, they do not detract from the generality, and their meaning will be explained later.

According to [22] OLS, the estimate of the coefficient vector for (9) and the delay estimate corresponding to (8) are of the form

$$\widehat{\theta }=V^{\text{†}}c,\Rightarrow {\widehat{\tau }}_M=\frac{-{\widehat{\theta }}_2}{2{\widehat{\theta }}_3}{T}_s,\left(10\right)$$

where V ^† is a pseudo-inverse matrix, which, if there is W = (V ^T V) ^-1, has the form V ^† = WV ^T [22-23].

, определяемую в (8) за требуемое решение, заметим, что для ее получения в указанном виде должна быть использована операция обращения матрицы, что не всегда вычислительно эффективно, особенно при реализации слежения за задержкой в реальном времени. Матрица V^† не зависит от исходных измерений (7) и может быть рассчитана априорно, но она зависит от объема данных N. При таком подходе необходимо хранить набор матриц для различных N либо требовать работы с всегда фиксированным объемом, что тоже может оказаться не оптимальным. Кроме того, при вычислении оценки $${\widehat{\tau }}_M$$

согласно первому выражению в (10) осуществляются затраты на оценку $${\widehat{\theta }}_1$$

не используемую далее. С целью повышения вычислительной эффективности получим для величины $${\widehat{\tau }}_M$$

тождественные соотношения видаAccepting grade $${\widehat{\tau }}_M$$

defined in (8) for the required solution, we note that to obtain it in the indicated form, the matrix inversion operation should be used, which is not always computationally effective, especially when real-time delay tracking is implemented. The matrix V ^† does not depend on the initial measurements (7) and can be calculated a priori, but it depends on the amount of data N. With this approach, it is necessary to store a set of matrices for different N or to require work with an always fixed volume, which may also not be optimal. In addition, when calculating the score $${\widehat{\tau }}_M$$

according to the first expression in (10), the costs of assessment $${\widehat{\theta }}_{\mathrm{one}}$$

not used further. In order to increase computational efficiency, we obtain for $${\widehat{\tau }}_M$$

identical relations of the form

$${\widehat{\tau }}_M=\frac{-{\beta }_{\mathrm{<figure-callout\; id="2"\; label="N\; T\; c"\; filenames="00000037.png"\; state="\{\{state\}\}">N\; </figure-callout>}}^{T}c}{2{\alpha }_N^{T}c}{T}_s,\left(\mathrm{eleven}\right)$$

where α _N , β _N are weighted N-vectors defined by explicit arithmetic dependences on N.

Considering in (9) the matrix V = [v ₀ v ₁ v ₂ ] as a collection of its columns, the definitions of which are obvious from (9), we note that, due to symmetry, the following properties hold

Then the matrix W used in estimate (10) and requiring inversion takes the form

$$W={\left(V^{T}V\right)}^{-\mathrm{one}}={\left[\begin{array}{ccc}N& 0& S\\ 0& S& 0\\ S& 0& Q\end{array}\right]}^{-\mathrm{one}}=\frac{\mathrm{one}}{D}\left[\begin{array}{ccc}SQ& 0& -{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="00000037.png"\; state="\{\{state\}\}">S</figure-callout>}}^2\\ 0& NQ-{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="00000037.png"\; state="\{\{state\}\}">S</figure-callout>}}^2& 0\\ -{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="00000037.png"\; state="\{\{state\}\}">S</figure-callout>}}^2& 0& NS\end{array}\right],\left(13\right)$$

where D = det (V ^T V) = S (NQ-S ² ) is a determinant, the value of which for later can not be calculated.

, $${\alpha }_N^T$$

являются второй и третьей строками матрицы V^† соответственно, получимIt follows from definitions (10) and (11) that the desired weight functions $${\beta }_N^T$$

, $${\alpha }_N^T$$

are the second and third rows of the matrix V ^†, respectively, we obtain

$${\beta }_N^T=\frac{NQ-S^2}{D}{v}_{\mathrm{one}}^T,{\alpha }_N^T=\frac{\mathrm{one}}{D}\left(NS{v}_2^T-S^2{v}_0^T\right).\left(\mathrm{fourteen}\right)$$

Then, taking into account (14), from definition (11) we can find the updated desired estimate of the arrival time in the form

$${\widehat{\tau }}_M=\frac{\left(S^2-NQ\right)v_{\mathrm{one}}^{T}c}{2{\left(NS{v}_2-S^2{v}_0\right)}^{T}c}{T}_s.\left(\mathrm{fifteen}\right)$$

Thus, with the appropriate choice of points in the vicinity of the DQF maximum, which is, in discrete time, a sample function for the cross-correlation function C _XY (τ), with the corresponding formation of a vector c from the selected DQF samples, with the formation of the selected samples and their squares from the arguments vectors v ₁ and v ₂ , as well as with the formation of the vector v ₀ from units, you can find an updated estimate of the time of arrival using rule (15).

The sequence of actions obtained on the basis of the transformations derived here above the signal implements the patented method.

A device for implementing a method of measuring the time of arrival of a signal with on-off angle manipulation (see Figure 3) comprises a signal receiving means 1 connected to an analog-to-digital converter 2 (ADC), a first FFT processor 3, a cross-spectrum calculator 4, a second FFT processor, connected in series 5 and a device for determining the maximum argument of a discrete cross-correlation function (DKKF) of the signal 6. The input of the signal receiving means 1 is the input of the measuring device. The output of device 6 is the output of the measurement device. A third FFT processor 7 is connected to the second input of the cross-spectrum calculator 4. Between the ADC 2 output and the input of the first FFT processor 3, the quadrature decomposition of signal 8, the first low-pass filter (LPF) 9, read-only memory (ROM) 10, and the device a subtraction modulo 2π 11. Between the second output of the quadrature decomposition device of signal 8 and the second input of the ROM 10, a second low-pass filter is connected 12. The subtractive input of the subtractor 11 is connected to the output of the ROM 10 through a delay element for the symbol duration 13. Sign drove third FFT processor 7 is connected to the output of the two-level modulation signal generator 14 through a series connected differential decoder 15 and expander 16, the sampling frequency.

In another embodiment of the device for measuring the time of arrival of the signal to the output of the device for determining the argument of the maximum DKKF signal 6, an arithmetic device 17 is additionally connected.

The proposed device for implementing the method of measuring the time of arrival of a signal with on-off angle manipulation operates as follows.

Means for receiving signals 1 receives a signal. From the output of the first signal receiving means 1, the received signal is fed to the input of the ADC 2 and the signal is converted into a first digital data stream, which digitally represents the signal as a series of values of the time function. The quadrature decomposition device of signal 8 performs quadrature decomposition of the first digital data stream relative to the nominal center frequency of the modulated signal into a plurality of in-phase and quadrature samples corresponding to the specified stream. The obtained in-phase and quadrature samples independently low-pass filter the low-pass filters 9 and 12 with a cutoff frequency corresponding to the speed of manipulation of the modulating signal. In ROM 10, a plurality of current signal phases are obtained as arguments of complex numbers, the corresponding filtered common-mode samples are used as the real part, and the corresponding filtered quadrature signal samples are used as the imaginary part. The delay element 13 delays the set of received current signal phases by the duration of the modulating signal symbol. In the subtractor 11, the corresponding value of the delayed current phase of the signal is subtracted modulo 2π from each received current phase. In the first FFT processor 3, the resulting differential digital data stream is converted as the first digital data stream into discrete values of the first frequency function S ₁ (k), where k is an integer index variable that varies within the data length. Shaper 14 generates a two-level modulating signal. The generated signal is differentially decoded by the decoder 15. Each symbol of the differentially decoded signal is mapped to the corresponding number of samples of the digitized received signal relative to the time scale of the receiving position by increasing the sampling frequency using the sampling frequency expander 16 and, thus, a second digital data stream is generated. In the third processor, the FFT 7 converts said second digital data stream into the values of the second spectrum, which represents the signal as a series of values of the frequency function S ₂ (k). In the cross-spectral calculator 4, the spectrum values of one of the signals are mutually multiplied with the complex conjugate values of the spectrum of the other signal coming from the first 3 and third 7 FFT processors to generate the cross spectrum. The output signal from the calculator 4 is fed to the input of the second FFT processor 5, in which the DCFF signal is calculated using the inverse FFT cross-spectrum. Received DKKF signal from the output of the second processor FFT 5 is fed to the input of device 2, where they determine the time of arrival of the signal as an argument of the maximum DKKF signal.

In another embodiment of the device for measuring the time of arrival of the signal, the output signal of the device for determining the argument of maximum 6 is additionally supplied to the arithmetic device 17, which operates in accordance with rule (15) and determines the updated value of the time of arrival of the signal.

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Claims (5)

1. A method of measuring the time of arrival of a signal with two-position angular manipulation, including receiving a signal, analog-to-digital conversion of the received signal into a first digital data stream, representing the signal as a series of values of the time function, digitized, using the fast Fourier transform (FFT) for two signals representing both signals in the form of a series of discrete values of the spectrum S _i (k), where k is an integer index variable that varies within the data length, ai = 1, 2 is an index variable, I mean the serial number of the digital data stream converted using the FFT to the spectrum values, the mutual multiplication of the spectrum values of one of the signals with the complex conjugate spectral values of the other signal, which generates the cross spectrum, the calculation of the discrete cross-correlation function (DCF) of the signal using the inverse FFT time of arrival (VP) of the signal as an argument of the maximum DKKF signal, characterized in that the quadrature decomposition of the first digital data stream relative to the nominal the central frequency of the modulated signal into the set of in-phase and quadrature samples corresponding to the specified stream, the obtained in-phase and quadrature samples are independently low-pass filtered with a cut-off frequency corresponding to the modulation signal manipulation speed, and the set of current signal phases are obtained as arguments of complex numbers, the corresponding part of which is used filtered in-phase samples, and as imaginary - the corresponding filtered quadrature samples signal Ety, delay set received current signal phases on the duration of the modulation signal symbol, subtracting modulo 2π from each of the resultant current phase corresponding to the value of the delayed current signal phase, and the resulting differential digital data stream is converted using FFT values of the first spectrum S ₁ (k) form a two-level modulating signal and differentially decode, map each symbol of the differentially decoded signal to the corresponding number of samples of the converted o into the digital form of the received signal relative to the time scale of the receiving position by repeating the values of the differentially decoded signal with increasing its sampling frequency to the sampling frequency of the digitalized received signal and thus generating a second digital data stream, which is converted using the FFT into the values of the second spectrum S ₂ (k). 2. The method of measuring the time of arrival of a signal according to claim 1, characterized in that for a signal for which the duration of the signal symbol is not a multiple of the sampling frequency of the analog-to-digital conversion of the received signal, the filtered in-phase and quadrature samples are additionally decimated and interpolated to provide a frequency multiplicity of the first digital samples data stream signal symbol duration. 3. The method of measuring the time of arrival of a signal according to claims 1 or 2, characterized in that to the left and to the right of the maximum DKKF signal argument, L points of the specified function are selected, the DKKF values for the selected points and maximum points are combined in the order of time following in a column vector c, a column vector v _{0 of} the same dimension as vector c is formed from units, and the arguments corresponding to the selected points of the DCQF are expressed through the index variable l = 0, ± 1, ..., ± L, where the indices l with a negative sign correspond to the times to the maximum point DKKF, zero index l DKKF corresponds to the maximum point, and the indices l with positive sign correspond to times when the maximum point DKKF arguments DKKF selected points as well as ordering the vector components and are combined into the column vector v _1, the squares of the vector components v ₁ similarly combined in the vector column v ₂ , a specified value of VP

determined by the following rule:

,
where N = 2L + 1 is the dimension of the generated vectors, the superscript ^T denotes the transpose of the vector, T _s is the signal sampling period,

- a scalar product of the vectors v ₁ and, S ratio is determined by the expression

,
the coefficient Q is determined by another expression

,
a (NSv ₂ -S ² v ₀ ) ^T c is the scalar product of the difference vector (NSv ₂ -S ² v ₀ ) with the vector с. 4. A device for implementing a method for measuring the time of arrival of a signal with on-off angle manipulation, comprising a signal receiving means, the input of which is an input to a measuring device connected to an analog-to-digital converter (ADC), a first FFT processor, a cross-spectrum calculator, and a second processor FFT and the device for determining the arguments of the maximum of the discrete cross-correlation function (DCF) of the signal, the output of which is the output of the measuring device, while to the second input of the subtraction a third FFT processor is connected to the cross-spectral splitter, characterized in that between the ADC output and the input of the first FFT processor, a quadrature signal decomposition device, a first low-pass filter (LPF), read-only memory (ROM) and a subtraction device modulo 2π are connected between the second output device is a quadrature decomposition of the signal and the second input of the ROM includes a second low-pass filter, the subtracting input of the subtractor is connected to the output of the ROM through a delay element for the duration of the signal symbol, and the input is the third FFT processor is connected to the output of the shaper of the two-level modulating signal through a series-connected differential decoder and expander of the sampling frequency. 5. A device for implementing the method of measuring the time of arrival of a signal with on-off angle manipulation according to claim 4, characterized in that an arithmetic device is additionally connected to the output of the device for determining the maximum argument of the discrete cross-correlation signal function.

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