How to EncodE and dEcodE InformatIon

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IT Zone sequential, whereas the 2421 and 5211 codes are not sequential. Some features of weighted codes Except 8421 and 74-2-1 codes, for all other weighted codes some of the decimal number may be coded in more than one forms. For example, decimal 4 in 2421 code may be represented as 1010 or 0100. That’s why 2421 and 4221 are known as self-complementing codes. A property known as self-complementing is applied to select a code out of different options. The property is stated as follows: If ‘D’ is the given decimal number, you may code it in any option, but then decimal number 9-D must be coded taking the option such that the code is bit-wise complement of the code of D. For example, if decimal 4 in 2421 is coded as 0100, then 9-4=5 must be coded as 1011. See that 1011 is bit-wise complement of 0100. We have shown the code in Table II applying the self-complementing property. A necessary condition of self-complementing property is that the sum of weights in the code is 9. The codes 8421 and 74-2-1 are not selfcomplementing. Non-weighted code In many cases, there may be some requirement for non-weighted codes (see Table III), particularly from design point of view. Non-weighted codes are codes that are not positionally weighted. That is, each position within the binary number is not assigned a fixed value. The examples of a few nonweighted codes are excess-3 code, 1-to- 2 code and BCDP (BCD with parity). Excess-3 is a non-weighted code used to express decimal numbers. The code derives its name from the fact that each binary code is the corresponding 8421 code plus 0011(3). Some features of nonweighted code Excess-3 code is just BCD with plus 3 weight. If ‘d’ is the decimal number, d+3 is coded in 8421 to get the excess-3 code. This code is the most important non-weighted code in digital logic design. 1-to-2 code has a unique feature. No code has less than one and greater than two 1’s. BCDP (with odd parity) is designed so as to make any code having odd numbers of 1’s. This is done with selection of parity bit, which is the right-most bit of the code. Gray code/binary reflected code Table II Weighted Codes Decimal digit 2421 code 74-2-1 code 4221 code Weight 2 4 2 1 Weight 7 4 -2 -1 Weight 4 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 1 2 0 0 1 0 0 1 1 0 0 0 1 0 3 0 0 1 1 0 1 0 1 0 0 1 1 4 0 1 0 0 0 1 0 0 0 1 1 0 5 1 0 1 1 1 0 1 0 1 0 0 1 6 1 1 0 0 1 0 0 1 1 1 0 0 7 1 1 0 1 1 0 0 0 1 1 0 1 8 1 1 1 0 1 1 1 1 1 1 1 0 9 1 1 1 1 1 1 1 0 1 1 1 1 Decimal digit Table III Non-weighted Codes Excess-3 code (XS-3 code) 1-to-2 code Gray code is a very important nonweighted code. An N-bit gray code is a sequence of all the N-bit binary numbers, ordered in such a way that each binary number differs from its predecessor and successor by exactly 1 bit (and the first and last differ by one bit also). For example, one-bit gray code 0, 1 and three-bit gray code 000, 001, 011, 010, 110, 111, 101, 100 (differs from 000 by 1 bit). The sequence 00, 11, 01, 10 is not a two-bit gray code as the first and second elements differ by two bits. Table IV shows a 4-bit code in comparison with a binary number. In the column of decimal number, it is seen that from 00 to 09 if you move from number N to number N+1, there is a change in only one digit. But when you move from 09 to 10, there are changes in two digit positions. Look at the column of reflected decimal number: as you move from N to N+1, always there is change in one digit position. In binary representation as we move from N to N+1, there may be many changes in bits. For example, as you go from 3 to 4, there are as many as three changes in bit position. But, look at the reflected binary or gray code: as you move from any N to N+1, always there is only one change in bit position. This is the property of gray code or reflected binary code. Gray code is also known as a variable weighted code and is cyclic. The gray code is called ‘reflected binary,’ because the first eight values compare with those of the last eight values, but in reverse order. The gray code belongs to a class of codes called ‘minimum change codes,’ in which only one bit in the code changes when moving from one code to the next. The gray code is a reflective digital code which has a special property that any two subsequent numbers’ codes differ by only one bit. It is also called ‘unitdistance code.’ Gray code has an important application in digital control systems. Let BCDP (with odd parity) 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 1 0 0 0 0 1 0 2 0 1 0 1 0 0 1 1 0 0 1 0 0 3 0 1 1 0 0 1 0 0 0 0 1 1 1 4 0 1 1 1 0 1 0 1 0 1 0 0 0 5 1 0 0 0 0 1 1 0 0 1 0 1 1 6 1 0 0 1 1 0 0 0 0 1 1 0 1 7 1 0 1 0 1 0 0 1 0 1 1 1 0 8 1 0 1 1 1 0 1 0 1 0 0 0 0 9 1 1 0 1 1 1 0 0 1 0 0 1 1 72 • september 2010 • electronics for you www.efymag.com
IT Zone Decimal Table IV Gray/Reflected Code Reflected decimal Binary b 3 b 2 b 1 b 0 us take the example of a fan regulator which is digitally controlled. You want to switch from position 3 to 4. If you follow the normal binary system, you need to change three bit positions to switch from 3 to 4. But how? Change three positions one after another as shown in Table V. In gray code, there is change in only one position, so there will be no oscillation as such. The code is called reflected because it can be generated in the following manner: Take the gray code 0, 1. Write it forwards, then backwards: 0, 1, 1, 0. Then prepend 0’s Reflected binary/gray code g 3 g 2 g 1 g 0 00 00 0000 0000 01 01 0001 0001 02 02 0010 0011 03 03 0011 0010 04 04 0100 0110 05 05 0101 0111 06 06 0110 0101 07 07 0111 0100 08 08 1000 1100 09 09 1001 1101 10 19 1010 1111 11 18 1011 1110 12 17 1100 1010 13 16 1101 1011 14 15 1110 1001 15 14 1111 1000 16 13 17 12 18 11 19 10 Table V Illustration of Property of Gray Code to the first half and 1’s to the second half: 00, 01, 11, 10. Continuing, write 00, 01, 11, 10, 10, 11, 01, 00 to obtain 000, 001, 011, 010, 110, 111, 101, 100. Each iteration therefore doubles the number of codes. Algorithm for binary-to-gray conversion This is very simple: 1. The most significant bit of the binary number is the most significant bit of the gray code 2. Add (using modulo 2, i.e., ignoring carry) the next significant bit of the binary number to the next significant bit of the binary number to obtain the next gray code bit. 3. Repeat step 2 until all bits of the binary bits have been added modulo 2. The resultant number is Present number Action Next number 0011 (=3) Starting position Change in the first position from right 0010 (=2) 0010(=2) Change in the second position from right 0000(=0) 0000(=0) Change in the third position from right 0100(=4) Final position Comment: So for a change from regulating position 3 to 4, the fan will go a round of 3 to 2 to 0 to 4. This will cause huge oscillation in the circuit, which is not a good design. the gray code equivalent of the binary number. Assume binary number b 3 b 2 b 1 b 0 = 0011. Conversion into gray is done as follows: g 3 = b 3 =0; g 2 = b 3 +b 2 =0+0=0; g 1 =b 2 +b 1 =1+0=1; and g 0 = b 1 +b 0 =1+1=0 (ignore ‘carry’) Gray-to-binary conversion Assume gray code g 3 g 2 g 1 g 0 =0010. It can be converted into binary as follows: b 3 =g 3 /2=remainder 0; b 2 = [g 3 +g 2 ]/2=remainder 0; b 1 = [g 3 +g 2 +g 1 ]/2=remainder 1; b 0 = [g 3 +g 2 +g 1 +g 0 ]/2=remainder 1. Thus equivalent binary is 0011. Excess-3 gray code In many applications, it is desirable to use a BCD as well as unit distance. Excess-3 gray code is such a code. The values for 0 and 9 differ in only one bit, and so do all values for successive numbers. Outputs from linear devices or angular encoders may be coded in excess-3 gray code to obtain multi-digit BCD numbers. The code obtained from excess-3 code by applying conversion rule is shown below: Decimal Excess-3 gray code 0 0010 1 0110 2 0111 3 0101 4 0100 5 1100 6 1101 7 1111 8 1110 9 1010 Error correction and detection codes When a binary message made of strings of 0’s and 1’s is transmitted from the source to the destination, the message is corrupted by the noise during transmission. The corrupted message becomes erroneous by conversion from transmitted 0 to received 1 or from transmitted 1 to received 0. The error-correction code (ECC) and the error-detection code (EDC) are used to rectify the errors. In EDCs and ECCs, redundant check bits are pended with original message bits to design the codes. EDCs detect presence of errors. ECCs detect and correct the errors. Typically, transmission errors are of two types: random and burst. An error is called random if bits in the error are randomly distributed over the code. Burst error occurs when bits in the error are clustered together over the code. For transmitted byte 01010101, the examples of random error and burst error may be as below www.efymag.com electronics for you • september 2010 • 73
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