# Estimating the efficiency of methods for error detection in information storage devices for measurement techniques

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The equipment and time costs and the functional reliability of information storage devices with advancedmethods for detection of single and double errors in measurement technology are compared with those forexisting methods of data monitoring.Keywords: single and double errors, information storage and transfer devices.

The efficiency of computerized data acquisition systems depends to a great extent on the choice of techniques formonitoring the information to be processed (error detection methods) [1]. The most important characteristics of an data mon-itoring technique are the error detection coefficient, along with the equipment cost (probability of failure-free operation) ofthe control scheme and the time expenditure that characterize the degree of influence of the control system on the responsetime of the device being controlled [2]. Thus, in constructing computerized data acquisition systems it is necessary to chosemethods for controlling information that satisfy these characteristics to the greatest extent. The criterion for this choice is thefunctional reliability [3]:

Df(t) = [Psd(t)Pcs(t) + P1Pdet1(1 Psd(t)Pcs(t)) + P2Pdet2(1 Psd(t)Pcs(t)]Pdo, (1)

where Psd(t), Pcs(t), and Pdo(t) are the probabilities of failure-free operation of the storage device, control scheme, and deci-sion organ, respectively; P1 and P2 are the occurrence probabilities, and Pdet1 and Pdet2 are the detection probabilities, respec-tively, for single and double errors.

Thus, in organizing the control of the information to be processed in computerized data acquisition systems, it isnecessary to find the most effective means of controlling the greatest operational reliability for the multiplicity of specifieddetected errors, number of information bits, operating time, and the least equipment and time costs for error detection:

D(t) = Dmax(t) /d = dsp, k = ksp, Cmin, tmin, top = tsp,

where D(t) and Dmax(t) are the functional reliability of the information storage device of the computerized data acquisition sys-tem and its maximum value; d and dsp are the multiplicity of the detected error, and k and ksp are the number of data bits and

Measurement Techniques, Vol. 55, No. 3, June, 2012

ESTIMATING THE EFFICIENCY OF METHODSFOR ERROR DETECTION IN INFORMATIONSTORAGE DEVICES FOR MEASUREMENT TECHNIQUES

GENERAL PROBLEMS OF METROLOGY AND MEASUREMENT TECHNIQUE

K. Yu. Borisov, Al. A. Pavlov,A. A. Pavlov, P. A. Pavlov,A. N. Tsarkov, and O. V. Khoruzhenko

UDC 519.725(047)

Serpukhov Military Institute of the Rocket Forces, Serpukhov, Russia; e-mail: Pavlov_iif@mail.ru. Translated from Izmeritelnaya Tekhnika,No. 3, pp. 1318, March, 2012. Original article submitted April 25, 2011.

0543-1972/12/5503-0236 2012 Springer Science+Business Media, Inc.236

their maximum values; Cmin are the minimal equipment cost for the means of control; tmin are the minimum time expendituresfor error detection; top and tsp are the operating time for the computerized data acquisition system and its specified value.

Here we solve this problem for choosing a method of data monitoring that detects single and double errors in a 12-bitset of binary data. For this purpose, we compare the efficiency and reliability indices for information storage devices employ-ing existing and advanced data monitoring techniques.

Estimating the Equipment and Time Costs of Means for Monitoring Information Storage Devices inMeasurement Technology. The advanced technique for error detection in information storage devices for measurement sys-tems encodes the information in accordance with the following rules [4]:

1) the binary set is broken up into blocks of information (let the number of data bits be a multiple of three) with threebits in each block:

Y = x1x2x3, y1y2y3, ..., z1z2z3; (2)

2) values of two control bits are formulated in accordance with the rule

r1 = x1 x2 y1 y2 ... z1 z2,

r2 = x2 x3 y2 y3 ... z2 z3.

The coding produces the code set

Yc = x1x2x3, y1y2y3, ..., z1z2z3 x1x2x2x3

y1y2y2y3...

z1z2z2z3

or

Yc = x1x2x3, y1y2y3, ..., z1z2z3, r1r2.

237

p

Fig. 1. Functional diagram of a coding device for parity monitoring.

It is best to make a comparative estimate of the equipment and time costs required for using an existing error detec-tion method (the most widely used control techniques for parity mod 3, the Hamming code) and the advanced method con-sidered here for a specific example.

Suppose it is required to detect errors in a 12-bit binary set. In this case, we have functional schemes for encoding(decoding) devices with the existing and advanced error detection methods, respectively.

The equipment cost of the coding device for parity monitoring (Fig. 1) consists of eleven nonequivalence elements.We express the equipment costs of a single nonequivalence element in terms of the simplest (two-input) logical elements (LE).Assuming that construction of a single nonequivalence element requires four of the simplest logical elements, then the equip-ment cost of the coding device for parity monitoring is 44 LE, while the equipment cost for the decision organ if 48 LE, where44 of them are used in the construction of the coding device and 4 in a nonequivalence element for comparison of controlbits. Here the time costs, determined by the largest number of nonequivalence elements in the path followed by the signal inthe decision organ, are 10 (2 per nonequivalence element), and is the time for the signal to pass through an LE.

A functional diagram of a pyramid reduction scheme (coding device for mod 3 control) for a 12-bit binary set is shownin Fig. 2. In this case, the equipment cost consists of 10 one-bit adders. Since the construction of a single one-bit binary adderwith a minimum number of the simplest logical elements requires 9 LE, the equipment cost of the reduction scheme is 90 LE.The equipment cost of the deciding organ is 98 LE.

The time cost for a single one-bit adder is 6 (6 of the simplest LE lie along the signal propagation path), and willbe 12 given the cyclical transfer. Since the pyramid reduction scheme contains three arrays, the time costs for the codingdevice increase to 36. The time cost in the decision organ (given time spent for the comparison scheme) is 38.

To construct a functional coding diagram based on a Hamming code, we use a table to generate the control bits for12 data bits (Table 1). When a Hamming code is used, the number of control bits is given by r0 = log2(n + 1). Here r0 = 5.

A functional diagram for generating the first control bit is shown in Fig. 3.

238

Fig. 2. A pyramid mod 3 reduction scheme for a 12-bit number.

Thirty five nonequivalence elements are needed to generate five control bits. In constructing a decision organ(synthesized only for error detection), 40 elements will be required for the output unit, i.e., the 35 elements are supple-mented by 5 for comparing the values of the signals from the input and output coding units.

In this case, the maximum number of nonequivalence elements along the path followed by the signal in the codingdevice is four. If these equipment costs are expressed in terms of simplest logical elements, and it is assumed that the timefor the signal to pass in each nonequivalence element is 2, then the equipment cost for detection of single errors by theHamming code will be 160 LE, while the time cost will be 8.

This Hamming code makes it possible to detect 100% of single errors and 18% of double errors. In order to enhancethe detection capability of the code, a modified Hamming code is used with an additional control bit to check simultaneous-ly for parity of the data and control bits. In this case, the additional equipment cost of the decoding device for a modifiedHamming code to detect single and double errors (without correction of the single errors) is 16 nonequivalence elements andthe total cost for constructing the decision organ (including an additional comparison element) increases to 57 nonequiva-lence elements or 228 LE. The largest time expenditures are associated with the generation of the additional control bit andconsist of 6 nonequivalence elements or 12.

Figure 4 is a functional diagram for the coding device for this error detection method with two control checks inaccordance with Eq. (2).

239

TABLE 1. Generating Control Bits for a Hamming Code

In this case, the equipment cost for the coding device is 14 nonequivalence elements (56 LE), the equipment costfor the decision organ is 64 LE, the time cost is 10, and the detection coefficient for double errors is Pdet12 = 0.74.

When the proposed error detection method with two control bits, PM2, is used for coding a set Yc = x1x2x3, y1y2y3, ...,z1z2z3, s1s2s3, r1r2 of 12 data bits, 26 double errors constructed relative to an error-free code set Yc0 = 000 000 000 000 00 werenot detected (Table 2).

An analysis of this matrix shows that to detect 100% of the double errors would require the use of three additionalcontrol bits to carry out the checks:

r3 = x1 x2 x3 y1 y2 y3,;

r4 = y1 y2 y3 z1 z2 z3;

240

Fig. 3. Functional diagram for generating a control bit for a Hamming code.

Fig. 4. A functional diagram of the coding device for the proposed error detectionmethod with two control bits.

r5 = s1 s2 s3.

In this case, check r3 makes it possible to detect the 16 double errors lying in rows 2, 3, 4, 5, 8, 9, 10, 11, 14, 15,16, 17, 19, 20, 23, 24, i.e., the largest number of double errors in the higher bits; check r4, the 7 double errors in rows 1, 6,7, 12, 13, 21, 25; and check r5, the three double errors in rows 12, 22, 26. The equipment cost of the coding device for themodified code PM2M1 with a single additional control bit, r3, is 19 nonequivalence elements, i.e., 76 LE, the equipment costfor constructing the decision organ is 88 LE, the time cost is 10, and the detection coefficient Pdet12M1 = 0.9. The equipmentcost of the coding device for the modified code PM2M2 with two additional control bits r3, r4 is 24 nonequivalence elements,i.e., 96 LE, the equipment cost for constructing the decision organ is 112 LE, the time cost is 10, and the detection coeffi-cient Pdet12M2 = 0.975.

The third additional check, r5, makes 100% double error detection possible. The equipment cost of the coding devicefor the modified code PM2M3 is 26 nonequivalence elements, i.e., 104 LE, the equipment cost for constructing the decisionorgan is 124 LE, and the time cost is 10. In this case, for an equal number of control bits (five), compared to the modifiedcode PM1M2 (which also detects 100% of the double errors), the modified code PM2M3 for 100% double error detection hasa lower equipment cost and the time cost of 10 is lower than for the Hamming code with a smaller number of control bits.

For clarity, Table 3 lists the efficiency indices both for the proposed and the most widely used error detection meth-ods for 12-bit data.

Table 3 shows that the advanced methods for monitoring data have lower equipment and time costs than the exist-ing methods for the same detection capability.

Comparative Estimate of the Functional Reliability of Data Storage Devices Using These Methods of ErrorDetection. Since functional reliability is determined under conditions such that single and double errors can occur, it is nec-essary to include the probabilities that they will occur.

241

TABLE 2. A Code Set Yc

Based on experience with semiconductor memory devices, the probability of a single error is P1 = 0.8 and of a dou-ble error, P2 = 0.2. (The probabilities of higher error multiplicities are neglected here because they are small.) In this case,according to Eq. (1), the functional reliability of a data storage device with single and double error detection is given by

Df(t) = [Psd(t)Pcs(t) + 0.8Pdet1(1 Psd(t)Pcs(t)) + 0.2Pdet2(1 Psd(t)Pcs(t)]Pdo. (3)

We now calculate the functional reliability for the example of a storage device containing 1000 bit words (bytes) ofmemory. This requires calculating the probabilities of failure-free operation in Eq. (3).

Since a single memory element is made up of four of the simplest LE (a trigger, including four two-input LE), theequipment cost of the memory device is 48000 LE. Then, since the failure rate for a single logical element is i = 1107 h1,the probability of failure-free operation of the entire device will be

The probability of failure-free operation of the control scheme Pcsi(t) for a given (ith) method of error detection isdetermined by the equipment cost of the control scheme Ccsi, including the equipment cost for constructing a coding deviceCcdi and the memory elements for the storage device Csdi to store the values of the control bits:

The probability of failure-free operation of the decision organ Pdoi(t) for a given (ith) method of data monitoring isdetermined by the equipment cost Cdoi, including the equipment costs for the output coding device and the comparisonschemes (in turn, the number of nonequivalence elements in the comparison scheme equals the number of control bits):

P tiC ti

do edo( ) .=

10 7

P tiC ti

cs ecs( ) .=

10 7

P t tsd e( ) .= 48000 10 7

Name of methodEfficiency indices for the control method

Pdet1 Pdet2 r0 Ccd, LE Csd, LE Ccs, LE Cdo, LE C, LE

Parity control (mod 2) 1 1 10 44 4000 4044 48 4092mod 3 control 1 0.5 2 38 90 8000 8090 98 8278

mod 5 control 1 0.75 3 12000

Hamming code 1 0.18 5 8 160 20000 20140 180 20480

Modified Hamming code 1 1 6 12 204 24000 24240 228 24672

PM2 1 0.74 2 10 56 8000 8056 64 8176

PM2M1 1 0.9 3 10 76 12000 12076 88 12240

PM2M2 1 0.975 4 10 96 16000 16096 112 16304

PM2M3 1 1 5 10 104 20000 20104 124 20332

Note: Ccd, Csd, Ccs, Cdo, and C are the equipment costs, respectively, for construction of the coding device, storage of the values of the control bits in amemory device, for the control scheme, for the decision organ, and the total for organizing control of the data; PM2 is the proposed method for monitor-ing data with two control bits; PM2M1, PM2M2, and PM2M3 are the modified methods for data control with two control bits and with one, two, and threeadditional control bits.

242

TABLE 3. Comparison of the Efficiency of the Proposed Methods with Existing Methods

Using the values for the equipment costs given in Table 2, we now find the corresponding probabilities of failure-freeoperation for these methods of data monitoring. Substituting the calculated probabilities in Eq. (3), we obtain the functionalreliabilities of the data storage devices when these monitoring methods are used.

The comparative functional reliabilities of this device when it is monitored by the proposed and existing methodsare plotted in Fig. 5. This figure shows that the proposed method for data monitoring with three additional control bits canprovide greater functional reliability of data storage devices in computerized data acquisition systems with lower equipmentand time costs; that is, it is more efficient.

REFERENCES

1. K. L. Kulikovskii and V. Ya. Kuper, Methods and Means of Measurement [in Russian], Energo...

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