RU2470477C1 - Method of all-fiber interferometer signal demodulation - Google Patents

Abstract

FIELD: physics, communication.

SUBSTANCE: invention relates to optical measurements of physical magnitudes by means of all-fiber interferometers. Two proposed method comprise modulation of phase difference in interferometer by harmonic signal, converting obtained signal in sequence of digital counts, and computing phase difference. In compliance with first version, signal is converted at frequency three times higher than that of phase difference modulation in interferometer to form triple of counts from all three counts of all periods. Thereafter, unknown phase difference for every triple is computed. In compliance with second version, triples are formed from three adjacent counts with shifting every next triple by one count.

EFFECT: simplified computation and implementation.

3 cl

Description

The invention relates to the field of optical methods for measuring physical quantities using fiber interferometers, including for measuring mechanical and acoustic vibrations, as well as data collection lines based on them.

Known "Interrogation device for fiber optic lines using two slopes" [US Patent No. 6778720]. The method of signal demodulation used in this device includes auxiliary modulation of the phase difference of light waves in interferometers by a harmonic signal, converting the output signal of a photodetector installed at the output of the line into a sequence of digital samples, selecting five samples on each of the two slopes of the signal's periodic dependence on time , the calculation of the selected samples of the desired phase difference of the light waves in the interferometer, the calculation of the amplitude of the auxiliary modulation for the rest of accounts and its subsequent adjustment. The disadvantages of this method are the complexity of its implementation, the use in each calculation cycle of at least ten samples of the signal, the need to accurately maintain the amplitude of the auxiliary modulation and the uniformity of the shoulder difference of all interferometers included in the line.

The well-known "Four-stage discrete method of demodulating phase shifts for lines of fiber-optic sensors" [US Patent No. 6122057]. The method includes auxiliary phase modulation of the interferometer by a harmonic signal, the formation of four signals that are integrals of the output signal of the photodetector for different periods of time, converting the integrated signals into a sequence of digital samples, calculating the desired phase difference of the light waves in the interferometer. The disadvantages of this method are the complexity of its implementation, the need to form four integrated signals, additional calculation of the correction factor to adjust the calculated values of the phase difference.

The well-known "Method and device for demodulating the output signals of the interferometer high accuracy", selected for the prototype [US Patent No. 6556509]. The demodulation method includes modulating the phase difference of the light waves in the interferometer according to a harmonic law with amplitude, converting the periodic signal received from the photodetector installed at the output of the interferometer (the signal period is equal to the period of the auxiliary modulation) into a stream of digital samples to obtain twelve samples during each period. Samples are recorded at regular intervals, the sampling frequency is twelve times higher than the modulation frequency. For each signal period, select six samples out of twelve and calculate from these six samples the values of the desired phase difference once per period using formulas obtained for the modulation amplitude. After that, they calculate the remaining six samples of the modulation amplitude, check the equality of the amplitude to the value of the radians, and adjust the amplitude.

The disadvantages of the prototype are the complexity of the calculations, the complexity of the method, the need to accurately maintain the amplitude of the auxiliary harmonic modulation equal to π radians, the requirement of equality of the difference of the shoulders of all interferometers when using the method for demodulating signals in a line of several interferometers.

The objectives of the demodulation method are to simplify calculations and implementation, to eliminate the requirement for a certain value of the amplitude of the auxiliary modulation and the equality of the difference in the lengths of the arms of the interferometers combined in a line.

To solve this problem, two variants of a method for demodulating a signal of a fiber interferometer are proposed.

The first version of the demodulation method includes modulating the phase difference in the interferometer with a harmonic signal, converting the signal with a frequency three times the frequency of modulating the phase difference in the interferometer, form triples of samples from all three samples of each period, after which the required phase difference is calculated for each triple of formulas:

Where

a = u ⁽⁰⁾ (C ₁ -C ₂ ) + u ⁽¹⁾ (C ₂ -C ₀ ) + u ⁽²⁾ (C ₀ -C ₁ ), b = u ⁽⁰⁾ (S ₁ -S ₂ ) + u ⁽¹⁾ (S ₂ -S ₀ ) + u ⁽²⁾ (S ₀ -S ₁ ),

where u ⁽⁰⁾ , u ⁽¹⁾ , u ⁽²⁾ are the values of the samples in the triple, С ₀ = cos [φ ⁽⁰⁾ ], C ₁ = cos [φ ⁽¹⁾ ], C ₂ = cos [φ ^{( 2)} ] and S ₀ = sin [φ ⁽⁰⁾ ], S ₁ = sin [φ ⁽¹⁾ ], S ₂ = sin [φ ⁽²⁾ ],

where φ ⁽⁰⁾ = δφ _m sin [θ ₀ ], φ ⁽¹⁾ = δφ _m sin [θ ₁ ], φ ⁽²⁾ = δφ _m sin [θ ₂ ],

where δφ _m is the amplitude of the modulating signal, θ ₀ is the phase delay of the first reference relative to the beginning of the modulation period, θ ₁ = θ ₀ + 2π / 3, θ ₂ = θ ₀ + 4π / 3. The triples of samples can be formed from all three samples of the same period with the omission of M-1 periods, where M is an integer> 1.

The second variant of the method for demodulating the signal of a fiber interferometer includes modulating the phase difference in the interferometer with a harmonic signal, converting the signal with a frequency three times the frequency of modulating the phase difference in the interferometer, forming triples of samples that combine every three neighboring samples with a shift by one sample, then calculating the desired phase differences are for each triple according to the formulas:

Where

a = u ⁽⁰⁾ (C ₁ -C ₂ ) + u ⁽¹⁾ (C ₂ -C ₀ ) + u ⁽²⁾ (C ₀ -C ₁ ), b = u ⁽⁰⁾ (S ₁ -S ₂ ) + u ⁽¹⁾ (S ₂ -S ₀ ) + u ⁽²⁾ (S ₀ -S ₁ ),

where u ⁽⁰⁾ , u ⁽¹⁾ , u ⁽²⁾ are the values of the samples in the triple, С ₀ = cos [φ ⁽⁰⁾ ], C ₁ = cos [φ ⁽¹⁾ ], C ₂ = cos [φ ^{( 2)} ] and S ₀ = sin [φ ⁽⁰⁾ ], S ₁ = sin [φ ⁽¹⁾ ], S ₂ = sin [φ ⁽²⁾ ],

where for the first triple - φ ⁽⁰⁾ = δφ _m sin [θ ₀ ], φ ⁽¹⁾ = δφ _m sin [θ ₁ ], φ ⁽²⁾ = δφ _m sin [θ ₂ ], for the second triple φ ^{(0 )} = δφ _m sin [θ ₁ ], φ ⁽¹⁾ = δφ _m sin [θ ₂ ], φ ⁽²⁾ = δφ _m sin [θ ₀ ], for the third triple φ ⁽⁰⁾ = δφ _m sin [θ ₂ ], φ ⁽¹⁾ = δφ _m sin [θ ₀ ], φ ⁽²⁾ = δφ _m sin [θ ₁ ],

where δφ _m is the amplitude of the modulating signal, θ ₀ is the phase delay of the first reference relative to the beginning of the modulation period, θ ₁ = θ ₀ + 2π / 3, θ ₂ = θ ₀ + 4π / 3, after which the phase calculation process is repeated.

The choice of conversion frequency is determined by the required minimum number of samples during each modulation period.

In the case when the frequency of change in the desired phase difference is small, it is advisable to form triples of samples from all three samples of the same period with the omission of M-1 periods, which will reduce the average amount of data processed per unit time, M times, where M is an integer > 1.

In the case when the frequency of the desired phase difference is large, it is advisable to form triples of samples that combine every three neighboring samples with a shift by one sample and use different sets of coefficients to avoid signal distortion.

In the proposed method, the frequency of converting the received periodic signal into a sequence of digital samples and calculating each value of the desired phase difference across the three samples allows you to reduce the amount of data and simplify the calculations.

The mathematical formulas used to calculate the desired phase difference are applicable for arbitrary modulation amplitude, which simplifies the implementation and removes the requirement of the same shoulder difference of all interferometers in the line.

To modulate the phase difference in the interferometer, we use a harmonic signal

where δφ _m and f _m are the amplitude and frequency of the modulating signal.

It is known that the voltage obtained from a photodetector installed at the output of a two-arm fiber interferometer, in the presence of modulation φ (t), has the form

where φ _C is the desired phase difference, which must be determined as a result of demodulation, U ₀ and U _m are the constant component and the amplitude of the interference signal.

A slow change in φ _С , U _0, and U _m is assumed as compared with the oscillations of the modulating signal, and during one modulation period these values can be considered constant. This condition can be formally expressed by the following inequalities:

We transform the received signal (2) into a sequence of samples u _n with a sampling frequency f _d three times the modulation frequency f _d = 3 · f _M ,

where n is the reference number from the beginning of the first modulation period (n = 0, 1, 2, ...), θ ₀ = 2πf _M Δt is the phase delay of the first reference relative to the beginning of the modulation period, t _n is the time instant of the n-th reference interference signal, φ _n - values of the modulating signal φ (t) at time t _n .

To describe the generated signals, it is convenient to introduce into consideration the values of the argument modulating the phase of harmonic oscillation (1) at the moments of registration of samples t _n

To calculate the value of the desired phase difference φ _{C, we} select three samples of the interference signal. This condition is determined by the fact that function (2), which describes the modulated interference signal, has three unknown parameters, which, starting from condition (3), are assumed to be constant during the modulation period: the desired phase difference φ _С , U ₀ and U _m .

Denoting the three samples of the signal during one modulation period {u ⁽⁰⁾ , u ⁽¹⁾ , u ⁽²⁾ } and writing down their values according to formula (4), we obtain a system of equations, the solution of which leads to the expressions for the desired phase difference φ _C in the range [-π, π]:

Where

a = u ⁽⁰⁾ (C ₁ -C ₂ ) + u ⁽¹⁾ (C ₂ -C ₀ ) + u ⁽²⁾ (C ₀ -C ₁ ), b = u ⁽⁰⁾ (S ₁ -S ₂ ) + u ⁽¹⁾ (S ₂ -S ₀ ) + u ⁽²⁾ (S ₀ -S ₁ ),

where C ₀ = cos [φ ⁽⁰⁾ ], C ₁ = cos [φ ⁽¹⁾ ], C ₂ = cos [φ ⁽²⁾ ] and S ₀ = sin [φ ⁽⁰⁾ ], S ₁ = sin [ φ ⁽¹⁾ ], S ₂ = sin [φ ⁽²⁾ ],

where φ ⁽⁰⁾ = δφ _m sin [θ ₀ ], φ ⁽¹⁾ = δφ _m sin [θ ₁ ], φ ⁽²⁾ = δφ _m sin [θ ₂ ].

The triples of samples can be formed from all three samples of the same period, with the omission of M-1 periods, where M is an integer> 1.

The second version of the demodulation method is similar to the first, but the triples {u ⁽⁰⁾ , u ⁽¹⁾ , u ⁽²⁾ } are formed from three neighboring samples with a shift in each subsequent triple by one sample.

Formula (6), which calculates the values of the desired phase difference φ _C , does not change, but in the calculations use three options for the coefficients corresponding to three different sets {φ ⁽⁰⁾ , φ ⁽¹⁾ , φ ⁽²⁾ }:

for the first three {φ ⁽⁰⁾ , φ ⁽¹⁾ , φ ⁽²⁾ } = {δφ _m sin [θ ₀ ], δφ _m sin [θ ₁ ], δφ _m sin [θ ₂ ]},

for the second triple {φ ⁽⁰⁾ , φ ⁽¹⁾ , φ ⁽²⁾ } = {δφ _m sin [θ ₁ ], δφ _m sin [θ ₂ ], δφ _m sin [θ ₀ ]},

for the third triple {φ ⁽⁰⁾ , φ ⁽¹⁾ , φ ⁽²⁾ } = {δφ _m sin [θ ₂ ], δφ _m sin [θ ₀ ], δφ _m sin [θ ₁ ]}.

Each of these three sets {φ ⁽⁰⁾ , φ ⁽¹⁾ , φ ⁽²⁾ } corresponds to its own set of coefficients {С ₀ , С ₁ , С ₂ , S ₀ , S ₁ , S ₂ }, calculated by the same formulas : C ₀ = cos [φ ⁽⁰⁾ ], C ₁ = cos [φ ⁽¹⁾ ], C ₂ = cos [φ ⁽²⁾ ] and S ₀ = sin [φ ⁽⁰⁾ ], S ₁ = sin [ φ ⁽¹⁾ ], S ₂ = sin [φ ⁽²⁾ ].

After the third triple, the sets {φ ⁽⁰⁾ , φ ⁽¹⁾ , φ ⁽²⁾ } and the process of calculating the phase difference are repeated for the 4th, 5th and 6th triples, etc.

To implement the demodulation method, it is necessary to register fluctuations φ _C (t) caused by the action on the interferometer in the frequency range not exceeding the 5 kHz band and amplitude not more than 1 radian (for example, acoustic or vibration effects).

According to the first variant of the method, we modulate the phase difference in the interferometer with a harmonic signal with a frequency f _M = 50 kHz and an amplitude δφ _m = 2 radians. Moreover, for the maximum frequency and amplitude of the measured signal F _C = 5 kHz, the inequalities of condition (3) take the form

f _M / F _С = 10 >> 1 and (dφ _С / dt) = 2π · 1 · 5 = 10π << 2πf _М = 100π,

those. conditions are met.

The conversion of the received periodic signal into a sequence of digital samples is performed with a frequency f _d = 3f _M = 150 kHz, i.e. three times the modulation frequency. In this case, we assume that the shift of the initial reference relative to the beginning of the period corresponds to the phase shift θ ₀ = 0.2 radians (i.e., Δt = 0.637 μs).

Let the parameters of the interference signal have the values U ₀ = 1, U _m = 1 for a certain period of modulation, and the desired phase is φ _C = 2 radians. Then, according to (4), a triple of samples will be formed for this period

{u ⁽⁰⁾ , u ⁽¹⁾ , u ⁽²⁾ } = {0.264, 0.063, 1.995}

To calculate the desired phase difference, we use fixed coefficients:

φ ⁽⁰⁾ = δφ _m sin [θ ₀ ] = 0.397, φ ⁽¹⁾ = δφ _m sin [θ ₁ ] = 1.499, φ ⁽²⁾ = δφ _m sin [θ ₂ ] = - 1.896 rad;

C ₀ = cos [φ ⁽⁰⁾ ] = 0.922, C ₁ = cos [φ ⁽¹⁾ ] = 0.072, C ₂ = cos [φ ⁽²⁾ ] = - 0.32;

S ₀ = sin [φ ⁽⁰⁾ ] = 0.387, S ₁ = sin [φ ⁽¹⁾ ] = 0.997, S ₂ = sin [φ ⁽²⁾ ] = - 0.948.

As a result of the calculation according to formula (6) for the indicated triple, we obtain:

a = u ⁽⁰⁾ (C ₁ -C ₂ ) + u ⁽¹⁾ (C ₂ -C ₀ ) + u ⁽²⁾ (C ₀ -C ₁ ) = 0.264 (0.072 + 0.32) +0.063 (-0.32- 0.922) +

1.995 (0.922-0.072) = 1.721;

b = u ⁽⁰⁾ (S ₁ -S ₂ ) + u ⁽¹⁾ (S ₂ -S ₀ ) + u ⁽²⁾ (S ₀ -S ₁ ) = 0.264 (0.997 + 0.948) +0.063 (-0.948- 0.387) +

1.995 (0.387-0.997) = - 0.788;

a / b = -2.185; arctg (a / b) = - 1.42;

φ _С = arctan (a / b) + π = 2 radians.

Thus, the correct value of the desired phase difference φ _{C was} obtained.

This example illustrates the calculation of the desired phase difference φ _C in the case of the formation of triples from the samples of each period, and in the case of the formation of triples from all samples of the same period with the omission of M-1 periods, where M is an integer> 1, for example, when M = 2, we omit every second period.

In the second version of the method, we perform all the actions similarly to the first, but we form triples of samples that combine every three neighboring samples with a shift of one sample. For example, let for some two periods of modulation the parameters of the interference signal have the values U ₀ = 1, U _m = 1, and the desired phase is φ _C = 2 radians. Then, according to (4) for these two periods, the following values of samples will be fixed

0.264; 0.063; 1.995; 0.264; 0.063; 1.995.

On this set of samples four triples of samples will be formed:

first {u ⁽⁰⁾ , u ⁽¹⁾ , u ⁽²⁾ } = {0.264, 0.063, 1.995}

second {u ⁽⁰⁾ , u ⁽¹⁾ , u ⁽²⁾ } = {0.063, 1.995, 0.264}

third {u ⁽⁰⁾ , u ⁽¹⁾ , u ⁽²⁾ } = {1.995, 0.264, 0.063}

fourth {u ⁽⁰⁾ , u ⁽¹⁾ , u ⁽²⁾ } = {0.264, 0.063, 1.995}

When calculating the desired phase difference, we use three sets of fixed coefficients. For the first (4th, 7th, etc.) triples:

φ ⁽⁰⁾ = δφ _m sin [θ ₀ ] = 0.397, φ ⁽¹⁾ = δφ _m sin [θ ₁ ] = 1.499, φ ⁽²⁾ = δφ _m sin [θ ₂ ] = - 1.896 rad;

C ₀ = cos [φ ⁽⁰⁾ ] = 0.922, C ₁ = cos [φ ⁽¹⁾ ] = 0.072, C ₂ = cos [φ ⁽²⁾ ] = - 0.32;

S ₀ = sin [φ ⁽⁰⁾ ] = 0.387, S ₁ = sin [φ ^(l) ] = 0.997, S ₂ = sin [φ ⁽²⁾ ] = - 0.948.

For the second (5th, 8th, etc.) triples:

φ ⁽⁰⁾ = δφ _m sin [θ ₁ ] = 1.499, φ ⁽¹⁾ = δφ _m sin [θ ₂ ] = - 1.896, φ ⁽²⁾ = δφ _m sin [θ ₀ ] = 0.397 rad;

C ₀ = cos [φ ⁽⁰⁾ ] = 0.072, C ₁ = cos [φ ⁽¹⁾ ] = - 0.32, C ₂ = cos [φ ⁽²⁾ ] = 0.922;

S ₀ = sin [φ ⁽⁰⁾ ] = 0.997, S ₁ = sin [φ ⁽¹⁾ ] = - 0.948, S ₂ = sin [φ ⁽²⁾ ] = 0.387.

For the third (6th, 9th, etc.) triples:

φ ⁽⁰⁾ = δφ _m sin [θ ₁ ] = 1.499, φ ⁽¹⁾ = δφ _m sin [θ ₂ ] = 0.397, φ ⁽²⁾ = δφ _m sin [θ ₀ ] = - 1.896 rad;

C ₀ = cos [φ ⁽⁰⁾ ] = - 0.32, C ₁ = cos [φ ⁽¹⁾ ] = 0.922, C ₂ = cos [φ ⁽²⁾ ] = 0.072;

S ₀ = sin [φ ⁽⁰⁾ ] = - 0.948, S ₁ = sin [φ ⁽¹⁾ ] = 0.387, S ₂ = sin [φ ⁽²⁾ ] = 0.997.

The calculation of the coefficients a and b according to the formulas

a = u ⁽⁰⁾ (C ₁ -C ₂ ) + u ⁽¹⁾ (C ₂ -C ₀ ) + u ⁽²⁾ (C ₀ -C ₁ ); b = u ⁽⁰⁾ (S ₁ -S ₂ ) + u ⁽¹⁾ (S ₂ -S ₀ ) + u ⁽²⁾ (S ₀ -S ₁ ); for each triple will give the following results.

For the first three:

a = 0.264 (0.072 + 0.32) +0.063 (-0.32-0.922) +1.995 (0.922-0.072) = 1.721;

b = 0.264 (0.997 + 0.948) +0.063 (-0.948-0.387) +1.995 (0.387-0.997) = - 0.788;

For the second three:

a = 0.063 (-0.32-0.922) +1.995 (0.922-0.072) +0.264 (0.072 + 0.32) = 1.721;

b = 0.063 (-0.948-0.387) +1.995 (0.387-0.997) +0.264 (0.997 + 0.948) = - 0.788;

For the third three:

a = 1.995 (0.922-0.072) +0.264 (0.072 + 0.32) +0.063 (-0.32-0.922) = 1.721;

b = 1.995 (0.387-0.997) +0.264 (0.997 + 0.948) +0.063 (-0.948-0.387) = - 0.788;

For the fourth three:

a = 0.264 (0.072 + 0.32) +0.063 (-0.32-0.922) +1.995 (0.922-0.072) = 1.721;

b = 0.264 (0.997 + 0.948) +0.063 (-0.948-0.387) +1.995 (0.387-0.997) = - 0.788;

Accordingly, for all triples, according to (6), we obtain the same values of the desired difference φ _С = arctan (a / b) + π = 2 radians, i.e. correct value of the desired phase.

An example of the implementation of 2 variants of the method of demodulating the signal of a fiber interferometer demonstrates the simplification of the calculation of the desired phase difference. In the calculation, a certain value of the amplitude of the auxiliary modulation and the equality of the differences of the arm lengths of the interferometers combined in a line are not required.

Claims (3)

1. The method of demodulating the signal of a fiber interferometer, including modulating the phase difference in the interferometer with a harmonic signal, converting the received periodic signal into a sequence of digital samples, calculating the desired phase difference, characterized in that the signal is converted at a frequency three times the frequency of the modulation of the phase difference in the interferometer , form triples of samples from all three samples of each period, after which the calculation of the desired phase difference is carried out for each triple by the formulas:

Where
a = u ⁽⁰⁾ (C ₁ -C ₂ ) + u ⁽¹⁾ (C ₂ -C ₀ ) + u ⁽²⁾ (C ₀ -C ₁ ), b = u ⁽⁰⁾ (S ₁ -S ₂ ) + u ⁽¹⁾ (S ₂ -S ₀ ) + u ⁽²⁾ (S ₀ -S ₁ ),
where u ⁽⁰⁾ , u ⁽¹⁾ , u ⁽²⁾ are the values of the samples in the triple, С ₀ = cos [φ ⁽⁰⁾ ], C ₁ = cos [φ ⁽¹⁾ ], C ₂ = cos [φ ^{( 2)} ] and S ₀ = sin [φ ⁽⁰⁾ ], S ₁ = sin [φ ⁽¹⁾ ], S ₂ = sin [φ ⁽²⁾ ],
where φ ⁽⁰⁾ = δφ _m sin [θ ₀ ], φ ⁽¹⁾ = δφ _m sin [θ ₁ ], φ ⁽²⁾ = δφ _m sin [θ ₂ ],
where δφ _m is the amplitude of the modulating signal, θ ₀ is the phase delay of the first reference relative to the beginning of the modulation period, θ ₁ = θ ₀ + 2π / 3, θ ₂ = θ ₀ + 4π / 3. 2. The method according to claim 1, characterized in that triples of samples are formed from all three samples of the same period, with the omission of M-1 periods, where M is an integer> 1. 3. A method for demodulating a signal of a fiber interferometer, including modulating the phase difference in the interferometer with a harmonic signal, converting the received periodic signal into a sequence of digital samples, calculating the desired phase difference, characterized in that the signal is converted at a frequency three times the frequency of modulating the phase difference in the interferometer , form triples of samples, combining every three neighboring samples with a shift of one sample, then the calculation of the desired phase difference is carried out for each up to the three according to the formulas:

Where
a = u ⁽⁰⁾ (C ₁ -C ₂ ) + u ⁽¹⁾ (C ₂ -C ₀ ) + u ⁽²⁾ (C ₀ -C ₁ ), b = u ⁽⁰⁾ (S ₁ -S ₂ ) + u ⁽¹⁾ (S ₂ -S ₀ ) + u ⁽²⁾ (S ₀ -S ₁ ),
where u ⁽⁰⁾ , u ⁽¹⁾ , u ⁽²⁾ are the values of the samples in the triple, С ₀ = cos [φ ⁽⁰⁾ ], C ₁ = cos [φ ⁽¹⁾ ], C ₂ = cos [φ ^{( 2)} ] and S ₀ = sin [φ ⁽⁰⁾ ], S ₁ = sin [φ ⁽¹⁾ ], S ₂ = sin [φ ⁽²⁾ ],
where for the first triple - φ ⁽⁰⁾ = δφ _m sin [θ ₀ ], φ ⁽¹⁾ = δφ _m sin [θ ₁ ], φ ⁽²⁾ = δφ _m sin [θ ₂ ], for the second triple φ ^{(0 )} = δφ _m sin [θ ₁ ], φ ⁽¹⁾ = δφ _m sin [θ ₂ ], φ ⁽²⁾ = δφ _m sin [θ ₀ ], for the third triple φ ⁽⁰⁾ = δφ _m sin [θ ₂ ], φ ⁽¹⁾ = δφ _m sin [θ ₀ ], φ ⁽²⁾ = δφ _m sin [θ ₁ ],
where δφ _m is the amplitude of the modulating signal, θ ₀ is the phase delay of the first reference relative to the beginning of the modulation period, θ ₁ = θ ₀ + 2π / 3, θ ₂ = θ ₀ + 4π / 3, after which the phase calculation process is repeated.