The Analysis of the Methods of Data Diagnostic in a Residue Number System

Abstract

The article presents the results of the analysis of the methods of data diagnostic presented in residue number system (RNS). Two practical methods of data diagnostic in RNS are investigated. Their advantages and disadvantages are shown. The main disadvantage of these methods is the lack of the efficiency in data diagnostic in RNS. The third method of the efficient diagnostic in RNS, which eliminates the above-mentioned disadvantage, has been reviewed in the article. The usage of this method can significantly increase the efficiency of data diagnostic in RNS. The main drawback of this method is a significant amount of equipment required to implement the process of data diagnostic in RNS. The method of the efficient diagnostic has been improved in terms of reducing the amount of equipment required for implementing the process of data diagnostic in RNS. The application of the improved method of the efficient diagnostics allows reducing the amount of equipment for the implementation of a diagnostic data procedure in RNS without increasing the diagnostic time. Examples of practical use of the improved method of data diagnostic in RNS are presented.

Keywords

• Alternative Set of Numbers
• Data Diagnostic
• Diagnostic Efficiency
• Error Control and Correction
• Residue Number System
• Zeroisation Procedure

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Data diagnostic in residue number system (RNS) is the process of determining the distorted residues in redundant non-positional code structure (NCS) presented in the following form ARNS  a1 || a2 || ... || ai1 || ai || ... || an || ... || ank  where n and k are the number of, respectively, informational and control bases mi i  1, n  k  of ordered mi  mi1  RNS. The diagnostic is carried out after data control, if it is necessary for the subsequent error correction. Some methods, algorithms and devices for data diagnostic in RNS have already been presented [ 1-3 ]. To monitor, diagnose and

Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). correct errors, the certain information redundancy must be introduced. Power of the information redundancy, which as in positional number system (PNS), determines the corrective abilities of the code, is estimated by the valued d mRinNS  of a minimum code distance (MCD). In RNS the value of MCD is determined by the ratio d mRinNS   k  1 [ 4-7 ]. For one control base, the value of MCD is equal to d mRinNS   2 . In accordance with the general coding theory, in RNS with a minimum code distance d mRinNS   2 the distortion of only one of the residues can be reliably established (one-time error) in NCS. For example, to correct a one-time error (in one residue) and determine double errors (in two residues) it is necessary to ensure that d mRinNS   3 [ 1, 8-12 ]. Due to the influence of RNS properties on the data processing it is possible, in some cases, to correct one-time data errors (in one NCS residue) when introducing the minimal (k = 1) information code redundancy. So, the property of the independence of the residues of NCS allows us to correct not intermediate calculation results, but final one. A typical example for this case is the possibility of implementing the data error correction procedure with one control base without stopping the intermediate computing process (during the computational process). To implement such procedure, it becomes necessary to diagnose intermediate results of calculations based on the use of the concept of an alternative set of numbers (AS) in RNS [13-19].

The purpose of the article is to study the methods of data diagnostic, presented in non-positional residue number system with one control base.

Main part. Let us consider the method of data diagnostic in RNS based on the concept of AS numbers in RNS.

The first method of diagnosis. The alternative set W A   ml , ml ,..., ml of in~ 1 2 p ~ correct number ARNS  a1 || a2 || ... || ai1 || a~1 || ai1 || ... || an || an1  can be determined by a sequential testing of each base mi i  1, n RNS. We determine the set of ~ numbers, that have the same residues for all bases of RNS, as number A , except one certain residue (base), and differ only in values of possible residues on this base. In this set there may be no correct numbers or there may be only one correct number. In ~ the last case, the number is a part of AS of number A .

The proposed method involves carrying out similar verifications for each of the information base of RNS (a control base always is a part of a set of bases of AS). The result of such sequential verifications completely and reliably determines the AS ~ ~ W A   ml , ml ,..., ml  of the incorrect number A . The disadvantage of the me1 2 p thod is the low efficiency in determining AS. This is due to the considerable time of consecutive executions of data diagnostic stages in RNS.

The second method of diagnosis. This method is also based on the determination of AS W A   ml , ml ,..., ml . In this case, the whole procedure of diagnosing ~

1 2 p NCS is carried out by simultaneous and parallel calculation of all possible projections A~i RNS  a1 || a2 ||... || ai1 || ai1 ||... || an || an1  of the incorrect number ~ ARNS  a1 || a2 || ... || ai1 || a~l || ai1 ||... || an || an1  , and their subsequent comparison with the value of M   in1 mi without the redundant numeric information interval ( information volume of code words ) 0  M  1 given in RNS. It is proved in [ 1, 7, 8 ], that the necessary and sufficient condition of the entry of the bases of RNS in AS W A   ml , ml ,..., ml  of number ~ 1 2 p ~ ARNS  a1 || a2 || ... || ai1 || a~l || ai1 ||... || an || an1  is correctness of  A~i RNS  M   ~ for its projection Ai RNS  a1 || a2 ||... || ai1 || ai1 ||... || an || an1  . Parallelization of the procedure of calculating all possible projections A~i RNS  a1 || a2 ||... || ai1 || ai1 ||... || an || an1  of the incorrect number ~ ARNS  a1 || a2 || ... || ai1 || a~i || ai1 || ... || an || an1  reduces the time of AS determination and increases the efficiency of diagnosing data in RNS.

Let us consider the following example of data diagnostic based on the usage of the second method. ~

Example 1. Let us determine the AS of the number ARNS  0 || 0 || 0 || 0 || 5 , which is defined in RNS by the information m1  3 , m2  4 , m3  5 , m4  7 and control bases mk  m5  11. Wherein M   in1 mi   i41 mi  420 and the full range 0  M 0  1 of coded words equals to M 0  M  mn1  420 11  4620 (Table 1).

At first, the procedure of controlling number ARNS  0 || 0 || 0 || 0 || 5 is carried out by the known method [ 1, 18, 19 ]. According to the standard control procedure we determine the value of the original number in PNS. In the end of the control it is determined that APNS  3360  M  420 . In this case, assuming the occurrence of only one-time (in one residue number) errors, it can be concluded that the considered ~ number A3360  0 || 0 || 0 || 0 || 5 is incorrect, i.e., one of the number residues is dis~ torted. Then the procedure of determining AS A3360  0 || 0 || 0 || 0 || 5 is realized (Table 1). For the number ARNS  0 || 0 || 0 || 0 || 5 not distorted residues have been determined. They are a2  0 and a3  0 . The values of residues on the bases m , 1 m4 and m5 , i.e., residues a1  0 , a4  0 and a5  5 may be incorrect. In this case, for the number ARNS  0 || 0 || 0 || 0 || 5 AS will be equal to the set of RNS basesW A  m1, m4 , m5.

~ A in PNS

The use of the second method of data diagnostic in RNS allows us to speed up the proc~ess of determining AS W A~   ml1 , ml2 ,..., mlp  of the number ARNS  a1 || a2 || ... || ai1 || a~l || ai1 ||... || an || an1  , due to the possibility of p~arallel determination of projections A~ j of incorrect number ARNS  a1 || a2 || ... || ai1 || a~l || ai1 ||... || an || an1  . It should be noted, that for the second method the procedure of determining the number of AS includes such b~asic operations as transferring ARNS  a1 || a2 || ... || ai1 || a~l || ai1 ||... || an || an1  from RNS to PNS; converting projections A~i RNS  a1 || a2 ||... || ai1 || ai1 ||... || an || an1  of the incorrect num~ ber ARNS from RNS to PNS and the operation of comparing them with the value M . In RNS the listed operations refer to non-positional operations, the implementation of which is very consuming both in time and hardware.

The known methods of diagnosing in RNS have the common drawback, that is the low efficiency of data diagnostic. This reduces the effectiveness of RNS usage for rapid implementation of integer-valued operations.

The third recent designed method of data diagnosis is presented in [ 2, 7, 8 ]. Its usage allows increasing the efficiency of diagnosing in RNS. The essence of the developed method of improving the efficiency of diagnosing data in RNS is that AS ~ ~ W A   ml , ml ,..., ml  of the number ARNS is determined not in the whole inter1 2 p ~ val  jM ,  j  1M , which contains the incorrect number ARNS , but only in a small AH   ARNS  A~RHNS   M , where ~ H   0 || 0 || ... || 0 || n1  is a number reduced to zero in RNS. The essence of numerical interval ~ ARNS reducing to zero in RNS is to replace ~ ARNS  a1 || a2 || ... || ai1 || a~l || ai1 ||... || an || an1  with the original the number number A~RHNS  0 || 0 || ... || 0 || n1  , by using a sequence of transformations, by which any intermediate number does not go beyond the working range 0  M  1 . zeroisation procedure can be implemented by various methods. The essence of all these methods is that some minimum ZC(i) numbers, so called zeroisatio constants (ZC), are sequentially subtracted from the initial number ~ ~ ARNS  a1 || a2 || ... || ai1 || a~1 || ai1 || ... || an ||... || ank until the number ARNS is converted into the number A~RHNS  0 || 0 || ... || 0 || n1  and the value of the number ~ ARNS does not go beyond the range 0, M  . Geometrically, zeroisation procedure corresponds to the offset of the original number ~ ARNS  a1 || a2 || ... || ai1 || a~l || ai1 ||... || an || an1  to the left edge jM of its numeric range  jM , j 1M . Thus, to eliminate the redundancy of AS ~ ~ W A   ml , ml ,..., ml , by reducing the interval range of the number ARNS , the 1 2 p values ARHNS  0 || 0 || ... || 0 || n1  and AH   ARNS  A~ H   mod M have to be ~

RNS pre-defined. It can be conveniently demonstrated for particular RNS.

As an example, for RNS defined by the bases m1  2, m2  3, m3  mn1  5 M  2  3  6; M 0  2  3  5  30 (Table 2), in accordance with the distribution of errors in the intervals of the working range 0, M  [ 1 ], for each interval A to PNS  jM , j  1M  two-entry tables are preliminarily compiled. Tables 3 of the correspondence of W A~   n1; AH   .

A in RNS As it was noted above AS W A~  ml1 , ml2 ,..., mlp  numbers are determ~ined not on the whole range  jM ,  j  1M , which contains the incorrect number A , but only on the numerical range AH  . The method of on-line data diagnostic in RNS is presented in Fig. 1.

The considered method allows reducing the time of data diagnostic in RNS. The time to diagnose data is reduced, firstly, by eliminating non-positional operations such as converting numbers from RNS to PNS and comparing numbers, and, secondly, by using a single-entry tabular sampling of AS value. The proposed method of the rapid diagnostic of data errors improves the overall efficiency of using non-positional code structures in RNS. The drawback of the considered method of rapid data diagnostics in RNS is the considerable amount of equipment required for its implementation due to the large vol~ umes ( A   n1  1 is a memory unit) of the memory (MMU) realizing function  n1 ; AH  . We propose the following improvements in order to reduce the amount of the necessary equipment to implement the method of rapid diagnostic.

The essence of the improvements is to decrease in half the amount of the required equipment for the implementation of MMU content. This allows reducing the total amount of the required equipment for the implementation of the procedure for error diagnosing in NCS presented in RNS [20-22].

This is done by using the symmetry properties of the numerical data of the complete MMU table (Table 7) relative to the point with coordinate M 0  M 1 , that cor2 responds to the value m2, m3 and is analytically expressed (1) in the following way: W A  1  n1; AH     2  mn1   n1 ;M  1  AH    ~ a two-entry (two-coordinate)

There

1   n1  mn1  1 . Each pair of values  n1 and AH  corresponds to a specific set of AS bases.

2. By means of a set of reduction to zero constants ZC(i) initial incorrect ~ ARNS  a1 || a2 || ... || ai1 || a~l || ai1 || ... || an || an1 number converted (reduced to zero) to AH   0 || 0 || ... || 0 || n1  number. We obtain value  n1 that corresponds to ~ the first coordinate in the lookup table W A   n1; AH  .

3. AH   ARNS  A~RHNS  is determined. Therefore we obtain value AH  of the ~

~ second coordinate it the lookup table W A   n1; AH  .

4. According to obtained values of two coordinates AH  and  n1 we refer to the two~ entry lookup table W A   n1; AH   from which the specific value of AS W A~   ml1 , ml2 ,..., mlp  of incorrect number ~ ARNS  a1 || a2 || ... || ai1 || a~l || ai1 || ... || an || an1  in RNS is determined. The correctness of (1) can be easily shown by using the results of the lemma on the distribution of the terms of number sequence Ais  a1 , a2 ,..., ai1 , s, ai1 ,..., an , an1  in the numerical range 0, M 0  , where s  0,1,..., mi1 i  1, n  1 [ 1, 7, 8 ]. Basing on (1), the content of MMU for the proposed method of data diagnostic in RNS is presented in Table 4. Table 5 presents the characteristics Zi of quadrant numbers from the completed Table 3 of MMU data and Table 6 presents the attributes of quadrant numbers of the shortened Table 4 of MMU data. In Table 7 there are the values of numerical ranges for finding the MMU input numbers and the correspondent data attributes formed by the group of decoders.

When implementing this method of data diagnostic in RNS [21, 22], in the diagnostic scheme the module of determining characteristics is intended for to form and ~ use the characteristics Z1  Z 4 of quadrant numbers A   n1  1 of the completed data table MMU W A   ml1 , ml2 ,..., mlp  (Table 3). The characteristics are formed ~ by means of a group of decoders (Table 4) and a combination of OR elements. Using the values Z1  Z 4 , according to input data  n1 and A~ , the AS W A~    ml1 , ml2 ,..., mlp  is determined by shortened table A~    n12 1  of

MMU data (Table 4).

m2 , m3 The characteristics Z1  Z 4 are applied as follows (Table 7): Z1 and Z 2 are the characteristics of finding a distorted A~RNS number in the numerical ranges 1  mn1  1 and 2 2 mn1  1  mn1  1 respectively; Z 3 and Z 4 - characteristics of finding a distorted number A~RNS in the numerical ranges 0  M  1  1 and M  M  1 respectively. 2 2 For the second (II) and the third (III) quadrants, shortened Table 6, AS W A values ~ are determined by formula W A  F  n1 ; AH  .

~

1

W A~    ml1 , ml2 ,..., mlp  For the first (I) and the fourth (IV) quadrants of the completed Table 3, according to the values of the shortened Table 4, AS W A values are determined by formula (2): ~ The method of rapid data diagnostic in RNS is presented in Fig. 2.

1. A two-entry (two-coordinate) table of AS W A   n1 ; AH   values of the ~ AH  corresponds to a specific set of AS bases.

MMU content is compiled where 1   n1  mn2  1 Each pair of values  n1 and 2 2. By means of a set of reduction to zero constants ZC(i) initial incorrect A~RNS  a1 || a2 || ... || ai1 || a~l || ai1 || ... || an || an1  number is converted (reduced to zero) into the following ARHNS  0 || 0 || ... || 0 || n1  number.

3. The analysis of the magnitude of obtained  n1 value. If the condition 1   n1  mn1  1/ 2 is not met, i.e.  n1  mn1  1 then the subtraction 2 mn1   n1 mod mn1 is performed. The value of mn1   n1 mod mn1 is the first coordinate of W A   n1 ; AH   table.

~ 4. AH   ARNS  A~RHNS  is determined. Therefore we obtain value AH  of the ~ second coordinate it the lookup table W A   n1 ; AH  .

~ 5. According to obtained values of two coordinates AH  and  n1 we refer to the two-entry lookup table W A   n1 ; AH   from which the specific value of AS ~ W A~    ml1 , ml2 ,..., ml p  of incorrect number ~ ARNS  a1 || a2 || ... || ai1 || a~l || ai1 || ... || an || an1  in RNS is determined.

To check the obtained diagnostic result W A~  FRES  n1;A~, which is determined ~ by the shortened Table 4 of W A MMU of the dimension A~   n1  1 , the values  2  W A~  FTEST  n1; A~ are used, which are determined by the completed Table 3 of ~ MMU data of the dimension A   n1  1.

Examples of using the method of rapid data diagnostic in RNS In accordance with Fig. 2, let us present the examples 2-4 [21, 22] of using the method of on-line data diagnostic in RNS determined by bases m1  2, m2  3, m3  mn1  5 ; M  2  3  6 ; M 0  2  3 5  30 (Table 2). Tables 8 and 9 present some zeroisation constants for the corresponding RNS basis. code 001) is fed to the input of the decoder, from the output of which the value  n1  1 is fed to the input of the corresponding element OR in the unitary code. The value A~ H   1 (in binary code 001) is fed to the input of the fourth decoder, from the output of which the value A~H   1is fed to the input of the corresponding OR element in the unitary code (Table 7). The value  n1  1 (in the binary code 001) is fed to the decoder, from the output of which the value 1 in a unitary code, through a corresponding OR element, is fed to the first input of the first groups of MMU inputs. At the same time, the value A~H   1 (in binary code 001) is fed to the input of the second decoder, from the output of which value 1 in the unitary code is fed to the first ~ input of the second group of MMU inputs (Table 4). In accordance with the W A data of MMU (Table 4), we obtain W A~  m3 as the result of the procedure. Therefore W A~  FRES  n1; A~  FRES 1;1  m3.

Check (Table 3): W A~  FTEST  n1;A~  FTEST 1;1  m3.

~ ~

Example 4. Number A  0 || 0 || 4 is assumed to be diagnosed (AS W A of ~ A  0 || 0 || 4 number must be determined). First  n1  4  0 is determined. Then A~ H   A  AH   0 || 0 || 4  0 || 0 || 4  0 || 0 || 0 and therefore ~ we obtain

~ A  0 . Value  n1  4 is fed to the input of the decoder, from the output of which value  n1  4 is fed to the input of the OR element in the unitary code (Table 7).

~ The value A  0 is fed to the input of the fourth decoder, from the output of which ~ value A  0 is fed to the input of the OR element in the unitary code. The value  n1  4 (in the binary code 100) is fed to the inverter from the output of which the value mn1  n1  5  4  1 (in the binary code 001) is fed to the first decoder from the output of which the value 1 in a unitary code, through the corresponding OR element, is fed to a first input of the first group of MMU inputs (Table 4). Simultane~ ously, the value A  0 is fed to the inverter in binary cod, from the output of which ~ the value M  1  A  6  1  0  5 (in the binary code 101), through the OR element, is fed to the decoder input from the output of which the value 5 is fed to the fifth input of the second group of MMU inputs in a unitary code (Table 4). In accor~ dance with the W A data of MMU (Table 4), the result of the diagnosing is deter~ mined by the value  n1 that equals 1, and by the value A that equals 5. We obtain W A~  m2 , m3 as the result of the procedure. Therefore W A  FRES mn1  n1 ;M  1  A~  FRES 1;5  m2 , m3.

~ Check (Table 3): W A~  FTEST  n1; A~  FTEST 4;0  m m . 2 3

Conclusion According to the results of studying the methods of data diagnostic in RNS the improved method of rapid diagnostic is proposed for the practical implementation. Application of this method allows reducing the amount of the equipment required for implementing data diagnostic procedures in RNS without increasing the time of diagnosis. This is achieved by reducing the amount of equipment for completed table ~ A   n1  1 of MMU, by forming and using numerical characteristics Z1  Z 4 which show the belonging of the input numbers  n1 and A~ of the table of MMU to ~ ~ each of the four quadrants of the completed data table AS W A of the numbers A in RNS. This makes it possible to perform reliable diagnostic of the distorted number ~ A in RNS, i.e., precisely determine those bases of RNS where the residues of the correct number A have been distorted. The values of only a half (the second and the ~ third quadrants) of the completed data table AS W A of MMU are used. The examples of the practical usage of the method of diagnosis have been presented. The verification of the diagnosis of numbers in RNS, carried out by the developed method confirms the validity of the stated goal and the practical feasibility of diagnosing data in RNS. Based on the proposed diagnostic method, an algorithm of its implementation has been developed and the patentable device has been produced. A device for monitoring and diagnosing data presented in RNS has been patented in Ukraine. It should be noted that by increasing the length of the discharge grid of the calculator in RNS, the efficiency of the proposed method also increases. This can be used to solve various applied problems of computer science [25-30]. 7. Krasnobayev, V., Kuznetsov, A., Kononchenko, A., Kuznetsova, T.: Method of data control in the residue classes. In Proceedings of the Second International Workshop on Computer Modeling and Intelligent Systems (CMIS-2019). Zaporizhzhia, Ukraine, April 15-19, pp. 241–252 (2019) 8. Banerjee, S., Chakraborty, S., Dey, N., Kumar Pal, A., Ray, R.: High Payload Watermarking using Residue Number System. International Journal of Image, Graphics and Signal Processing. 7, 1–8 (2015). doi: 10.5815/ijigsp.2015.03.01 9. Krasnobayev, V., Kuznetsov, A., Zub, M., Kuznetsova, K.: Methods for comparing numbers in non-positional notation of residual classes. In Proceedings of the Second International Workshop on Computer Modeling and Intelligent Systems (CMIS-2019).

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