RU2510572C1 - Pseudorandom code scale - Google Patents

Abstract

FIELD: electricity.

SUBSTANCE: pseudorandom code scale includes an information track made in form of gradations of a pseudorandom binary sequence with maximum period length M=2ⁿ-1, n information reading elements, arranged along the information track with angular spacing which is a multiple of the value of the quantum of the scale δ=360°/M, with the possibility of obtaining therefrom M different n-bit code combinations, k correcting reading elements, arranged along the information track with possibility of obtaining therefrom, along with n information reading elements, M different (n+k)-bit code combinations which represent a Hamming code with detection and correction of a single error, a control reading element arranged along the information track with possibility of obtaining therefrom, along with (n+k) reading elements, M different (n+k+1)-bit code combinations which represent a Hamming code with correction of a single error and detection of a double error, outputs of n information reading elements.

EFFECT: high information reliability of the pseudorandom code scale owing to generation therefrom of correcting codes with correction of single errors and detection of double errors.

1 dwg, 7 tbl

Description

The invention relates to measuring equipment, in particular to analog-to-digital conversion, and in particular to the code scales of angular displacement converters to code.

Known pseudo-random code scale [1], containing an information track made in the form of gradations of a pseudo-random binary sequence of a maximum period length M = 2 ⁿ -1, n reading elements, an additional control reading element.

A disadvantage of the known pseudo-random code scale is low information reliability, since it does not provide the ability to correct single and detect double errors.

Closest to the technical solution and chosen by the authors for the prototype is a pseudo-random code scale [2] containing an information track made in the form of gradations of a pseudo-random binary sequence of maximum period length M = 2 ⁿ -1 constructed by means of a primitive polynomial h (x) of degree n, where n is the bit depth of the scale, n of information reading elements located along the information track with angular steps that are multiples of the quantum of the scale δ = 360 ° / M, with the possibility of obtaining from them M different n bit single combinations [3], k corrective reading elements located along the information track with the possibility of receiving together with n information reading elements M different (n + k) -bit code combinations, which are a Hamming code with detection and correction of a single error.

The disadvantage of the prototype is the low information reliability, since it does not provide the ability to detect double errors.

The present invention solves the problem of increasing the information reliability of a pseudo-random code scale by generating corrective codes from it with the detection of double errors while maintaining the ability to correct single errors.

To achieve a technical result, a pseudo-random code scale containing an information track made in the form of gradations of a pseudo-random binary sequence of maximum period length M = 2 ⁿ -1, constructed by means of a primitive polynomial h (x) of degree n, where n is the bit depth of the scale, n information reading elements placed along the information track with angular steps that are multiples of the quantum of the scale δ = 360 ° / M, with the possibility of obtaining from them M different n bit code combinations, k correcting reading x elements located along the information track with the possibility of receiving together with n information reading elements M different (n + k) -bit code combinations, which are a Hamming code with the detection and correction of a single error, is equipped with a control reading element located along the information track with the possibility of receiving from it together with (n + k) readout elements M different (n + k + 1) -bit code combinations representing a Hamming code with correction of single and zheniem double error, n outputs information sensing elements outputs k correcting sensing elements and an output controlling the read element are the outputs of the pseudorandom code scale.

New in the present invention is the provision of a pseudo-random code scale with a control reading element, the corresponding placement of which along the information track of the scale allows to receive together with n information reading elements and k correct reading elements M of various (n + k + 1) -bit code combinations representing Hamming code with the correction of a single and the detection of a double error.

The set of essential features in the present invention has improved the information reliability of the pseudo-random code scale.

As a result of this, we can conclude that the invention has an inventive step and allows to obtain a technical result.

The invention is new, since the prior art on available sources of information did not reveal analogues with a similar set of features.

The invention is industrially applicable, as it can be used in all areas where high-precision positional determination of the angular position of an object is required using angular displacement transducers of increased information reliability based on the claimed pseudo-random code scales.

The invention is illustrated in the drawing, which shows a linear scan of a circular four-row pseudo-random code scale.

The inventive pseudo-random code scale contains an information code track 1, information reading elements 2, 3, 4, 5 (n = 4), correcting reading elements 6, 7, 8 (k = 3) and a control reading element 9.

To clarify the essence of the invention, we give some theoretical background.

In [1], code scales used in the invention, called pseudorandom random codes (PSCS), and constructed on the basis of the theory of M-sequences, are considered. PCSS have only one information code track, made in accordance with the symbols of the M-sequence a = a ₀ a ₁ ... a _M-1 and n information reading elements (SE) located along the track. Reading elements make it possible to obtain, with a full turn of the scale M = 2 ⁿ -1, various n-bit code combinations, which provides a resolution of 8–360 ° / M for the PCSS. In general terms, the problem of placing informational SCs on the PCSS was solved in [3].

To generate an M-sequence with a period of M = 2 ⁿ -1, a primitive irreducible polynomial h (x) of degree n with Galois field coefficients GF (2) is used [4], ie

$$h\left(x\right)={\displaystyle \sum _{i=0}^n{h}_i{x}^i,\left(\mathrm{one}\right)}$$

where h ₀ = h _n = 1, and h _i = 0.1 for 0 <i <n.

The symbols of the M-sequence a _{n + j} satisfy the recurrence expression

означает суммирование по модулю два, а индексы при символах М-последовательности берутся по модулю M. Начальные значения символов M-последовательности а ₀ а ₁…a _n-1 могут выбираться произвольно, за исключением нулевой комбинации. Для определенности при построении круговой ПСКШ символы М-последовательности а ₀ а ₁…a_M-1 отображаются на информационной дорожке по ходу часовой стрелки.where is the sign $$\Xi$$

means summation modulo two, and indices for symbols of the M-sequence are taken modulo M. The initial values of the symbols of the M-sequence a ₀ a ₁ ... a _n-1 can be chosen arbitrarily, with the exception of the zero combination. For definiteness, when constructing a circular PCSS, the symbols of the M-sequence а ₀ а ₁ ... a _{M-1 are} displayed on the information track clockwise.

M-sequences belong to the class of cyclic codes and can be specified using the generating polynomial g (x) = (x ^M +1) / h (x), where h (x) is determined in accordance with the expression (1), M = 2 ⁿ -one.

For each M-sequence of length M, there are exactly M different cyclic shifts that can be obtained by multiplying the generating polynomial g (x) by x ^j , where j = 0, 1, ..., M-1.

Since PCSS are constructed in accordance with the symbols of the M-sequence, it is possible by cyclic shifts to determine the order in which n information reading elements are placed on the scale, i.e. m-th SE, m = 1, 2, ..., n, corresponds to the j _mth cyclic shift x ^jm g (x) of the M-sequence.

Then the polynomial that determines the order of placement of n informational FEs on a scale is

$$r\left(x\right)={\displaystyle \sum _{m=\mathrm{one}}^n{x}^{j_m}}$$

where j _m ∈ {0, 1, ..., M-1}.

Setting j ₁ = 0, according to polynomial (3), we obtain the positions of the 2nd, 3rd, ..., n-th information SCs shifted relative to the first SC by j ₂ , j ₃ , ..., j _n quanta of the information track of the scale, respectively.

Table 1 shows the polynomials h (x) up to n = 12 inclusive, which can be used to generate the corresponding M-sequences [4].

Table 1 n h (x) M = 2 ⁿ -1 one x + 1 one 2 x ² + x + 1 3 3 x ³ + x + 1 7 four x ⁴ + x + 1 fifteen 5 x ⁵ + x ² +1 31 6 x ⁶ + x + 1 63 7 x ⁷ + x + 1 127 8 x ⁸ + x ⁶ + x ⁵ + x + 1 255 9 x ⁹ + x ⁴ +1 511 10 x ¹⁰ + x ³ +1 1023 eleven x ¹¹ + x ² +1 2047 12 X ¹² + x ⁷ + x ⁴ + x ³ +1 4095

Let us explain the option of building PCSS, shown in the drawing.

In the example, for simplicity, n = 4 was taken and, accordingly, a primitive irreducible polynomial h (x) = x ⁴ + x + 1 was chosen from Table 1, where h ₀ = h ₁ = h ₄ = 1, h ₂ = h ₃ = 0. Here, the period of the M-sequence is M = 2 ⁴ -1 = 15, and the M-sequence itself is a = a ₀ a ₁ a ₂ a ₃ a ₄ a ₅ a ₆ a ₇ a ₈ a ₉ a ₁₀ a ₁₁ a ₁₂ a ₁₃ a ₁₄ = 000100110101111. For the initial values of the M-sequence a ₀ = a ₁ = a ₁ = 0, a ₃ = 1, the remaining characters of the sequence are obtained in accordance with the recurrence relation (2), which in this example has the form a _{4 + j} = a ₁ + _j ⊕ a _j , j = 0, 1, ..., 10. Placement of four information reading elements SE ₁ (position 2 in the drawing), SE ₂ (position 3 in the drawing), SE ₃ (position 4 in the drawing) and SE ₄ (in the drawing position 5) along the information track of the scale is defined according to (3) by the polynomial r (x) = 1 + x + x ² + x ³ .

When constructing the information track 1, the M-sequence with a period of M = 15 should be plotted on the scale in the form of passive (zeros of the M-sequence) and active (units of the M-sequence) sections of the information track, for example, clockwise, and on the information track scale is applied only one period of the M-sequence. M-sequence with a period of M = 2 ⁿ -1 determines the number of quanta of the information track of the scale, which in this example is equal to M = 15. Hence, the quantum value is δ = 360 ° / M = 360 ° / 15 = 24 °. Information reading elements, number 4, should be placed along the information track according to r (x) with a step equal to the value of one quantum of the scale δ, for example, clockwise. Note that other options for placing informational solar cells along the information track of the scale are possible [3].

Consistently fixing four-digit code combination by informational SEs when moving the scale one quantum counterclockwise, we get 15 different four-digit code combinations. These code combinations corresponding to 15 different angular positions of the PCSS are shown in Table 2.

table 2 PSKSH position number SE SE ₂ SE ₃ SE ₄ Decimal equivalent of code 0 0 0 0 one one one 0 0 one 0 2 2 0 one 0 0 four 3 one 0 0 one 9 four 0 0 one one 3 5 0 one one 0 6 6 one one 0 one 13 7 one 0 one 0 10 8 0 one 0 one 5 9 one 0 one one eleven 10 0 one one one 7 eleven one one one one fifteen 12 one one one 0 fourteen 13 one one 0 0 12 fourteen one 0 0 0 8

Since the technical result of the invention is achieved through the use of Hamming correcting codes with the correction of single and double errors, we show how such a code should be implemented in the present invention. In the present invention, by errors we mean the malfunctioning of reading elements (for example, their failure).

In the development and creation of a number of error-correcting codes, a significant role is given to various methods of checking for the parity of received code combinations. In the early 1950s, Hamming [5] proposed a code in which control characters were placed in a code combination not arbitrarily, but in strictly defined places, which naturally facilitated decoding.

A system was developed to conduct verification of the transmitted encoded message, including an algorithm for determining the syndrome of the error, indicating not only the presence of an error, but also the number of the distorted code position.

Two Hamming code models were most widely used: a code with detection and correction of a single error (minimum code distance d = 3) and a code with correction of a single error and detection of a double (d = 4).

To synthesize the Hamming code, it is necessary to solve the following problems.

1. Determine the number of control characters that provide the specified requirements for noise immunity.

2. To establish at what positions of the code combination the control symbols should be placed and which positions the information symbols will occupy.

3. Having collected the layout of the code combination, determine the value of each control character.

4. Create code combinations that include both control and information symbols.

5. Give a check algorithm to determine the presence and place of the error. First, in relation to the invention, we consider the synthesis of a Hamming code with d = 3.

The number of corrective FEs k is equal to the number of control characters in the Hamming code and is determined from the relation

$$k=]{\log }_2\{(n+\mathrm{one})+]{\log }_2(n+\mathrm{one})[\}[,\left(\mathrm{four}\right)$$

where n is the number of informational FEs equal to the number of informational symbols in the Hamming code, and the sign] ... [means rounding to the nearest larger integer.

Using expression (4), we calculate the number of corrective solar cells k for n = 4, ..., 12. The corresponding calculations are presented in Table 3.

Table 3 Number of informational SEs n four 5 6 7 8 9 10 eleven 12 The number of corrective solar cells k 3 four four four four four four four 5

Next, the places where the control bits should be located in the common code combination are determined. The control characters must be a binary number (error syndrome), which would indicate the number of the error position. As a result of the first private parity check, the symbol of the first (lower) category of the syndrome is obtained, as a result of the second check, the symbol of the second, etc. So, if the error syndrome is represented in the form of a binary number, and the corresponding decimal equivalents are written nearby, then we get Table 4.

Table 4 Error syndrome Decimal equivalent Error syndrome Decimal equivalent 00001 one 01010 10 00010 2 01011 eleven 00011 3 01100 12 00100 four 01101 13 00101 5 01110 fourteen 00110 6 01111 fifteen 00111 7 10,000 16 01000 8 10001 17 01001 9

Next, the numbers of the positions involved in each parity check are written out sequentially.

The first check should include those positions that contain a unit in the lower order. Based on table 3, it will be 1, 3, 5, 9, 11, 13, 15, 17.

The second check should include those positions that contain a unit in the second category. According to Table 3, these will be 2, 3, 6, 7, 10, 11, 14, 15.

The third check should include 4, 5, 6, 7, 12, 13, 14, 15 positions.

The fourth check should be attended by 8, 9, 10, 11, 12, 13, 14, 15 positions.

The fifth check should be attended by 16, 17 positions.

As a result, we obtain Table 5.

Table 5 Verification Number Item numbers covered by this check First one 3 5 7 9 eleven 13 fifteen 17 Second 2 3 6 7 10 eleven fourteen fifteen Third four 5 6 7 12 13 fourteen fifteen Fourth 8 9 10 eleven 12 13 fourteen fifteen Fifth 16 17

Analyzing table 4, we can conclude that the control characters K _m should be placed at the following positions:

K ₁ at position 1, i.e. 2 ⁰ ;

K ₂ at position 2, i.e. 2 ¹ ;

K ₃ at position 4, i.e. 2 ² ;

K ₄ at position 8, i.e. 2 ³ ;

K ₅ at position 16, i.e. 2 ⁴ .

Positions 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17 should occupy information symbols And ₁₁ , And ₁₀ , And ₉ , And ₈ , And ₇ , And ₆ , And ₅ , respectively. And ₄ , And ₃ , And ₂ , And ₁ , And ₀ .

In relation to the invention, this result must be applied as follows.

Symbols And ₀ should be formed from the first informational SE, from the second - And ₁ , ..., from the twelfth - And ₁₁ , from the first corrective SC should be formed K ₁ symbols, from the second - K ₂ , ... symbols, from the fifth - K ₅ symbols.

In accordance with the foregoing, the layout of the Hamming code for d = 3 should look as follows.

Item No. one 2 3 four 5 6 7 8 9 10 eleven 12 13 fourteen fifteen 16 17 Symbol K ₁ K ₂ And ₁₁ K ₃ And ₁₀ And ₉ And ₈ K ₄ And ₇ And ₆ And ₅ And ₄ And ₃ And ₂ And ₁ K ₅ And ₀

The definition of the values of each control character is determined by the following expressions.

K ₁ = AND ₁₁ ⊕ AND ₁₀ ⊕ AND ₈ ⊕ AND ₇ ⊕ AND ₅ ⊕ AND ₃ ⊕ AND ₁ ⊕ AND ₀ ;

K ₂ = AND ₁₁ ⊕ AND ₉ ⊕ AND ₈ ⊕ AND ₆ ⊕ AND ₅ ⊕ AND ₂ ⊕ AND ₁ ;

K ₃ = AND ₁₀ ⊕ AND ₉ ⊕ AND ₈ ⊕ AND ₄ ⊕ AND ₃ ⊕ AND ₂ ⊕ AND ₁ ;

K ₄ = AND ₇ ⊕ AND ₆ ⊕ AND ₅ ⊕ AND ₄ ⊕ AND ₃ ⊕ AND ₂ ⊕ AND ₁ ;

K ₅ = And ₀ .

Let us explain the above theoretical reasoning by the example of a four-digit pseudo-random code scale.

From table 3 for n = 4 we find k = 3.

From Table 5, we have the numbers of positions that participated in each parity check.

Verification No. No. of items covered by this check one 1, 3, 5, 7 2 2, 3, 6, 7 3 4, 5, 6, 7

The control symbols should be placed at the following positions: K ₁ - at position 1, K ₂ - at position 2, K ₃ - at position 4. Positions 3, 5, 6, 7 will occupy respectively the information symbols I ₃ , I ₂ , I ₁ , And ₀ .

The layout of the Hamming code in this case will look as follows.

Item No. one 2 3 four 5 6 7 Symbol K ₁ K ₂ And ₃ K ₃ And ₂ And ₁ And ₀

Symbols And ₀ should be formed from the first informational SE, from the second - And ₁ , from the third - And ₂ , from the fourth - And ₃ , from the first corrective SE should be formed the symbols K ₁ , from the second - symbols K ₂ , from the third - symbols K ₃ .

To obtain a seven-digit Hamming code from the scale, we determine the placement positions along the information track of the three corrective SCs.

The initial data for the calculation are the expressions for determining the control symbols K ₁ = AND ₃ ⊕ AND ₂ ⊕ AND ₀ , K ₂ = AND ₃ ⊕ AND ₁ ⊕ AND ₀ , K ₃ = AND ₂ ⊕ AND ₁ ⊕ AND ₀ , as well as the polynomial, determining the placement along the information track of the code disk of four informational SEs r (x) = 1 + x + x ² + x ³ .

To determine the position of the first corrective FE by the expression for K ₁ , taking into account r (x), we form a polynomial that will have the form r ₁ (x) = 1 + x ² + x ³ .

Next, we divide this polynomial modulo two from the side of the lower degrees into the primitive polynomial h (x) = x ⁴ + x + 1 to obtain the remainder as a monomial, and the degree of the monomial is taken modulo M-15, i.e.

$$\frac{\begin{array}{l}\mathrm{one}+x^2+x^3\\ \mathrm{one}+x+x^{\mathrm{four}}\end{array}}{\frac{\begin{array}{l}x+x^2+x^3+x^{\mathrm{four}}\\ x+x^2+x^5\end{array}}{\frac{\begin{array}{l}{x}^3+x^{\mathrm{four}}+x^5\\ x^3+x^{\mathrm{four}}+x_7\end{array}}{\frac{\begin{array}{l}{x}^5+x^7\\ x^5+x^6+x^9\end{array}}{\frac{\begin{array}{l}{x}^6+x^7+x^9\\ x^6+x^7+x^{10}\end{array}}{\frac{\begin{array}{l}{x}^9+x^{10}\\ x^9+x^{10}+x^{13}\end{array}}{x^{13}}}}}}}|\; |\; |\frac{\mathrm{one}+x+x^{\mathrm{four}}}{\mathrm{one}+x+x^3+x^5+x^6+x^9}$$

The degree of the monomial determines the placement position on the scale of the first corrective SC, and it should be shifted relative to the first information SC by 13 quanta of the scale δ clockwise.

To determine the position of the second corrective FE by the expression for K ₂ , taking into account r (x), we form a polynomial that will have the form r ₂ (x) = 1 + x + x ³ . Next, we divide this polynomial modulo two from the side of the lower degrees into the primitive polynomial h (x) = x ⁴ + x + 1 to obtain the remainder as a monomial, and the degree of the monomial is taken modulo M = 15, i.e.

$$\frac{\begin{array}{l}\mathrm{one}+x+x^3\\ \mathrm{one}+x+x^{\mathrm{four}}\end{array}}{\frac{\begin{array}{l}{x}^3+x^{\mathrm{four}}\\ x^3+x^{\mathrm{four}}+x^7\end{array}}{x^7}}|\; |\; |\frac{\mathrm{one}+x+x^{\mathrm{four}}}{\mathrm{one}+x^3}$$

The degree of the monomial determines the position of the second corrective SC on the scale, and it should be shifted relative to the first information SC by 7 quanta of the scale δ clockwise.

To determine the position of the third corrective FE by the expression for K ₃ , taking into account r (x), we form a polynomial that will have the form r ₃ (x) = 1 + x + x ² . Next, we divide this polynomial modulo two from the side of the lower degrees into the primitive polynomial h (x) = x ⁴ + x + 1 to obtain the remainder in the form of a monomial, and the degree of the monomial is taken modulo M = 15, i.e.

$$\frac{\begin{array}{l}\mathrm{one}+x+x^2\\ \mathrm{one}+x+x^{\mathrm{four}}\end{array}}{\frac{\begin{array}{l}{x}^2+x^{\mathrm{four}}\\ x^2+x^3+x^6\end{array}}{\frac{\begin{array}{l}{x}^3+x^{\mathrm{four}}+x^6\\ x^3+x^{\mathrm{four}}+x^7\end{array}}{\frac{\begin{array}{l}{x}^6+x^7\\ x^6+x^7+x^{10}\end{array}}{x^{10}}}}}|\; |\; |\frac{\mathrm{one}+x+x^{\mathrm{four}}}{\mathrm{one}+x^2+x^3+x^6}$$

The degree of the monomial determines the position of placement on the scale of the third corrective SC, and it should be offset relative to the first information SC by 10 quanta of the scale δ clockwise.

Now the joint placement of four information reading elements СЭ ₁ , СЭ ₂ , СЭ ₃ , СЭ ₄ and three corrective reading elements КСЭ ₁ , КСЭ ₂ , КСЭ ₃ (in the drawing, positions 6, 7 and 8, respectively) along the information track of the scale will be determined by the polynomial r _sovm (x) = r (x) + r _correctly (x) = 1 + x + x ² + x ³ + x ⁷ + x ¹⁰ + x ¹³ .

Consistently fixing the SE seven-digit code combination when moving the scale one quantum counterclockwise, we get 15 different seven-digit combinations of the Hamming code with a minimum code distance d = 3. It is known [5] that such a code allows correcting a single error. These code combinations corresponding to 15 different angular positions of the PCSS, as well as the correspondence between informational FEs and informational symbols, corrective FEs and control symbols of the Hamming code, are given in Table 6.

We show how the invention solves the problem of increasing the information reliability of the pseudo-random code scale by generating a Hamming correcting code from it with the detection of double errors while maintaining the ability to correct single errors.

Using the obtained Hamming code with d = 3 as an example, we show how the Hamming code with d = 4 is constructed, which allows one to detect double errors. The number of control characters in such a code should be one more, i.e. for k _{d = 4} = k _{d = 3} +1.

Table 6 PSKSH position number SSC ₁ SSC ₂ SE ₄ SSC ₃ SE ₃ SE ₂ SE ₁ K ₁ K ₂ And ₃ K ₃ And ₂ And ₁ And ₀ 0 one one one 0 0 0 0 one one 0 0 one one 0 0 2 0 one 0 one 0 one 0 3 0 0 one one 0 0 one four 0 one one one one 0 0 5 one one 0 0 one one 0 6 0 one one 0 0 one one 7 0 one 0 0 one 0 one 8 one 0 one one 0 one 0 9 one 0 one 0 one 0 one 10 0 0 one 0 one one 0 eleven one one one one one one one 12 0 0 0 one one one one 13 one 0 0 0 0 one one fourteen one one 0 one 0 0 one

The principle of obtaining such a code is as follows:

- to each code combination one more additional control character is added, which allows for a general parity check;

- the value of the additional control character is determined on the basis of the presence in each combination of a Hamming code (i.e. taking into account the control characters) an even number of units.

In accordance with the foregoing, the layout of the Hamming code for d = 4 should look as follows.

Item No. one 2 3 four 5 6 7 8 9 10 eleven 12 13 fourteen fifteen 16 17 eighteen Symbol K ₁ K ₂ And ₁₁ K ₃ And ₁₀ And ₉ And ₈ K ₄ And ₇ And ₆ And ₅ And ₄ And ₃ And ₂ And ₁ K ₅ And ₀ K _d

The definition of the value of the additional control character is determined by the following expression

К _d = К ₁ ⊕И ₁₁ ⊕К ₃ ⊕И ₁₀ ⊕И ₉ ⊕И ₈ ⊕К ₄ ⊕И ₇ ⊕И ₆ ⊕И ₅ ⊕И ₅ ⊕И ₄ ⊕И ₃ ⊕И ₂ ⊕И ₁ ⊕К ₅ ⊕and ₀ . ^ KzFIyuFideiva ^ FITPhibeizeysey ^ eizeeigey ^ Kzeio.

For the example in question, the Hamming code layout will look like this

Item No. one 2 3 four 5 6 7 8 Symbol K ₁ K ₂ And ₃ K ₃ And ₂ And ₁ And ₀ K _d

The determination of the value of the additional control character is determined by the following expression

K _d = K ₁ ⊕К ₂ ⊕И ₃ ⊕К ₃ ⊕И ₂ ⊕И ₁ ⊕И ₀ .

Thus, symbols K _d should be formed from the control SE.

To obtain an eight-digit Hamming code from the scale, we determine the position of placement along the information track of the control solar cell.

The initial data for the calculation are the expressions for determining the control symbol K _d = K ₁ ⊕K ₂ ⊕I ₃ ⊕K ₃ ⊕I ₂ ⊕I ₁ ⊕I ₀ , as well as the polynomial that determines the placement along the information track of the scale of four information and three corrective SC , i.e. r _total (x) = r (X) + r _correct (x) = 1 + x + x ² + x ³ + x ⁷ + x ¹⁰ + x ¹³ .

Further _Akad divide polynomial g (x) modulo two by the lowest degrees on a primitive polynomial h (x) = x ⁴ + x + 1 and a residue in the form of a monomial, the degree monomial taken modulo M = 15, i.e.,

$$\frac{\begin{array}{l}\mathrm{one}+x+x^2+x^3+x^7+x^{10}+x^{13}\\ \mathrm{one}+x+x^{\mathrm{four}}\end{array}}{\frac{\begin{array}{l}{x}^2+x^3+x^{\mathrm{four}}+x^7+x^{10}+x^{13}\\ x^2+x^3+x^6\end{array}}{\frac{\begin{array}{l}{x}^{\mathrm{four}}+x^6+x^7+x^{10}+x^{`13}\\ x^{\mathrm{four}}+x^5+x^8\end{array}}{\frac{\begin{array}{l}{x}^5+x+x^7+x^8+x^{10}+x^{13}\\ \frac{x^5+x^6+x^9}{\begin{array}{l}{x}^7+x^8+x^9{x}^{10}+x^{13}\\ \frac{x^7+x^8+x^{\mathrm{eleven}}}{\begin{array}{l}{x}^9+x^{10}+x^{\mathrm{eleven}}+x^{13}\\ x^9+x^{10}+x^{`13}\end{array}}\end{array}}\end{array}}{+x^{\mathrm{eleven}}}}}}|\; |\; |\frac{\mathrm{one}+x+x^{\mathrm{four}}}{\mathrm{one}+x^2+x^{\mathrm{four}}+x^5+x^7+x^9}$$

The degree of the monomial determines the position of placement on the scale of the control FE, and it should be shifted relative to the first information FE by 11 quanta of the scale δ clockwise.

Now the joint placement of four information reading elements СЭ ₁ , СЭ ₂ , СЭ ₃ , СЭ ₄ , three correcting reading elements КСЭ ₁ , КСЭ ₂ , КСЭ ₃ and the control СЭ (position 9) along the information track of the scale will be determined by the polynomial

. $$r_{\mathrm{from}\mathrm{about}\mathrm{at}m}^{*}\left(x\right)r\left(x\right)+r_{\mathrm{to}\mathrm{about}RRe\mathrm{to}t}\left(x\right)+r_{\mathrm{to}\mathrm{about}ntR}\left(x\right)=\mathrm{one}+x+x^2+x^3+x^7+x^{10}+x^{\mathrm{eleven}}+x^{13}$$

.

Consistently fixing the SE eight-bit code combination when moving the scale one quantum counterclockwise, we get 15 different eight-bit combinations of the Hamming code with a minimum code distance d = 4. It is known [5] that such a code allows correcting a single error and detecting a double one. These code combinations, corresponding to 15 different angular positions of the PCSS, as well as the correspondence between informational FEs and informational symbols, corrective FEs and control symbols of the Hamming code, control FEs and additional control symbols of the Hamming code, are given in Table 7.

Table 7 PSKSH position number SSC ₁ SSC ₂ SE ₄ SSC ₃ SE ₃ SE ₂ SE ₁ Counter. SE K ₁ K ₂ And ₃ K ₃ And ₂ And ₁ And ₀ K _d 0 one one one 0 0 0 0 one one one 0 0 one one 0 0 one 2 0 one 0 one 0 one 0 one 3 0 0 one one 0 0 one one four 0 one one one one 0 0 0 5 one one 0 0 one one 0 0 6 0 one one 0 0 one one 0 7 0 one 0 0 one 0 one one 8 one 0 one one 0 one 0 0 9 one 0 one 0 one 0 one 0 10 0 0 one 0 one one 0 one eleven one one one one one one one one 12 0 0 0 one one one one 0 13 one 0 0 0 0 one one one fourteen one one 0 one 0 0 one 0

Thus, in the present invention, the problem of increasing the information reliability of a pseudo-random code scale is solved by generating corrective codes from it with the detection of double errors while maintaining the possibility of correcting single ones. As noted earlier, the error in the operation of the pseudo-random code scale in the present invention refers to the failure of the reading elements. Another application of the invention is its use where information from the scale should be transmitted to the processing device via a communication channel susceptible to interference.

The proposed pseudo-random code scale can be the basis for the construction of angular displacement transducers of increased information reliability. In turn, it is advisable to use such converters in various control systems for aircraft or special-purpose equipment, where ensuring the reliability of their operation is a paramount requirement.

Literature

1. Ozhiganov A.A. Pseudorandom code scales // Izv. universities of the USSR. Instrument Engineering, 1987.V.30. No. 2. S.40-43.

2. Ozhiganov A.A., Lukyanov V.D. Code scales based on recurrence sequences for displacement transducers of increased information reliability // Sensors and Systems. - Moscow, 2012. - No. 2. - S.13-17.

3. Ozhiganov A.A. Algorithm for placing reading elements on a pseudo-random code scale // Izv. universities of the USSR. Instrument Engineering, 1994. V.37. No. 2. S.22-27.

4. McWilliams F.D., Sloan N.D. Pseudorandom sequences and tables // TIIER. 1976.V.64. No. 12. S.80-95.

5. Hamming R.V. Coding Theory and Information Theory: Per. from English - M .: Radio and communications, 1983. - 176 p., Ill.

Claims (1)

A pseudo-random code scale containing an information track made in the form of gradations of a pseudo-random binary sequence of maximum period length M = 2 ⁿ -1, constructed by means of a primitive polynomial h (x) of degree n, where n is the bit depth of the scale, n of information reading elements located along the information tracks with angular steps that are multiples of the quantum of the scale δ = 360 ° / M, with the possibility of obtaining from them M different n bit code combinations, k correcting reading elements located along the information the path with the possibility of receiving together with n information reading elements M various (n + k) -bit code combinations, which are a Hamming code with the detection and correction of a single error, characterized in that the pseudo-random code scale is equipped with a control reading element located along the information track with the possibility of receiving from it together with (n + k) reading elements M different (n + k + 1) -bit code combinations representing a Hamming code with correction one by detection and double error, the outputs of n information reading elements, the outputs of k correcting reading elements and the output of the control reading element are outputs of the pseudo-random code scale.