5 Binary to Gray and Gray to Binary converters:

Transcription

1 5 Binary to Gray and Gray to Binary converters: Aim: To realize a binary to Grey and Grey Code to binary Converter. Components Required: Digital IC trainer kit, IC 7486 Quad 2 input EXOR The reflected binary code, also known as Gray code after Frank Gray, is a binary numeral system where two successive values differ in only one bit (binary digit). The reflected binary code was originally designed to prevent spurious output from electromechanical switches. Bell Labs researcher Frank Gray introduced the term reflected binary code in his 1947 patent application, remarking that the code had as yet no recognized name. He derived the name from the fact that it may be built up from the conventional binary code by a sort of reflection process. The code was later named after Gray by others who used it. Constructing an n-bit Gray code: The binary-reflected Gray code list for n bits can be generated recursively from the list for n1 bits by reflecting the list (i.e. listing the entries in reverse order), concatenating the original list with the reversed list, prefixing the entries in the original list with a binary 0, and then prefixing the entries in the reflected list with a binary 1. The one-bit Gray code is G1 = (0, 1). One bit list: 0 1 Reflected list : 1 0 Prefix old entries with 0: Prefix new entries with 1: Concatenated: Logic Circuit Design Lab Page 1

2 Fig. 5.1: Example, generating the n = 3 list from the n = 2 list: Fig. 5.2: Gray Code Logic Circuit Design Lab Page 2

3 Motivation: Many devices indicate position by closing and opening switches. If that device uses natural binary codes, positions 3(011) and 4(100) are next to each other but all three bits of the binary representation differ: The problem with natural binary codes is that, with physical, mechanical switches, it is very unlikely that switches will change states exactly in synchrony. In the transition between the two states explained, all three switches change state. In the brief period while all are changing, the switches will read some spurious position. Even without key bounce, the transition might look like When the switches appear to be in position 001, the observer cannot tell if that is the real position 001, or a transitional state between two other positions. If the output feeds into a sequential system, possibly via combinational logic, then the sequential system may store a false value. The reflected binary code solves this problem by changing only one switch at a time, so there is never any ambiguity of position. Gray code bits with A, Decimal values with D and Binary bits with B. Binary Code Decimal Gray Code B3 B2 B1 B0 D G3 G2 G1 G Table 5.1: Gray/Binary Code table Logic Circuit Design Lab Page 3

4 (a) G 0 = B 1.B 0 + B 1.B 0 = B 1 B 0 (b) G 1 = B 2.B 1 + B 2.B 1 = B 2 B 1 (c) G 2 = B 3.B 2 + B 3.B 2 = B 3 B 2 (d) G 3 = B 3 Fig. 5.3: KMap for binray to gray code: Logic Circuit Design Lab Page 4

5 Binary to Gray Code Converter: This same technique can be applied to make binary to gray converter. There will be 4 input bits, which represent binary and 4 output bits which represent equivalent gray code. Since we are creating binary to gray code converter so, we need to find expressions for each gray code output in terms of input binary bits. So, there will be four output bits G 3,G 2,G 1 and G 0. For these output bits the input will be different combinations of B 3,B 2,B 1 and B 0 based on minimized expression. Binary Code Gray Code B 3 B 2 B 1 B 0 G 3 G 2 G 1 G Table 5.2: Binary to Gray Code table Logic Circuit Design Lab Page 5

6 Fig. 5.4: Logic Circuit Design Lab Page 6

7 The most significant bit (MSB) in Gray is taken directly from the MSB in binary. The rest of the Gray bits comes from a XOR operation between the precedent binary bit(b(i-1)) and the current binary bit (b(i)). In the case shown in the figure below: Fig. 5.5: Binary to Gray Code Converter: Logic Circuit Design Lab Page 7

8 (a) B 0 = G 3 G 2 G 1 G 0 (b) B 1 = G 3 G 2 G 1 (c) B 2 = G 3 G 2 (d) B 3 = G 3 Fig. 5.6: Kmap for Gray code to binary conversion Logic Circuit Design Lab Page 8

9 Gray to Binary Code Converter: This same technique can be applied to make gray to binary converter. There will be 4 input bits, which represent gray and 4 output bits which represent equivalent binary code. Since we are creating gray to binary code converter so, we need to find expressions for each binary code output in terms of input gray bits. So, there will be four output bits B3,B2,B1 and B0. For these output bits the input will be different combinations of G3,G2,G1 and G0 based on minimized expression. Binary Code Gray Code G3 G2 G1 G0 B3 B2 B1 B Table 5.3: Gray to Binary Code table KMap reduction B 0 = G 3 G 2 G 1 G 0 (How?) B 1 = G 3.G 2.G 1 + G 3.G 2.G 1 + G 3.G 2.G 1 + G 3 G 2.G 1 B 1 = G 3 (G 2.G 1 + G 2.G 1 ) + G 3 (G 2.G 1 + G 2.G 1 ) = G 3 (G 2 G 1 ) + G 3 (G 2 G 1 ) = G 3 G 2 G 1 B 2 = G 3.G 2 + G 3.G 2 = G 3 G 2 B 3 = G 3 Logic Circuit Design Lab Page 9

10 Fig. 5.7: Logic Circuit Design Lab Page 10

11 The most significant bit (MSB) in Binary is taken directly from the MSB in Gray. The rest of the Binary bits comes from a XOR operations. Fig. 5.8: Gray to Binary Code Converter: Additional Design Questions: 1. Design a Binary to BCD code Converter. 2. Design a Binary to Grey and Grey to Binary converter using MOD control. Result: Designed and setup a binary to gray code converter and gray code to binary code converter. Logic Circuit Design Lab Page 11

12 A Answers to Additional Questions: Logic Circuit Design Lab Page 12

13 A.1 Code Converters: Binary code to Gray code and Gray code to binary converter with mode control(x).when X=0 we need to implement binary to gray code and when X=1 we need Gray code to Binary code converter. X A 2 A 1 A 0 B 2 B 1 B Table A.1: Grey to Binary and Binary to Grey Code Converter with Mode Control(X): (a) B 2 = A 2 (b) B 1 = A 2 A 1 (c) B 0 = X.A 2 (A 1 A 0 ) Fig. A.1: KMap for Binray/Gray code converter: Kmap solution for B1 B 1 = A 2.A 1 + A 2.A 1 = A 2 A 1 Logic Circuit Design Lab Page 13

14 Kmap solution for B0 B 0 = X.A 1.A 0 + X.A 1.A 0 + A 2.A 1.A 0 + A 2.A 1.A 0 + X.A 2.A 1.A 0 + X.A 2.A 1.A 0 = (X + A 2 )(A 1.A 0 + A 1.A 0 ) + X.A 2 (A 1.A 0 + A 1.A 0 ) B 0 = X.A 2 (A 1 A 0 ) + X.A 2 (A 1 A 0 ) = XA 2 (A 1 A 0 ) Fig. A.2: Binary/Gray Code converter: Logic Circuit Design Lab Page 14

15 D C B A D4 D3 D2 D1 D Table A.2: Binary to BCD converter truth table: (a) D 0 = A (b) D 1 = B.C.D + D.B (c) D 2 = D.C + CB = C(D + B) (d) D 3 = D.CB (e) D 4 = DC + DB = D(C + B) Fig. A.3: KMap for Binray to BCD code converter: Logic Circuit Design Lab Page 15

16 Fig. A.4: Binary to BCD converter: Logic Circuit Design Lab Page 16

17 Fig. A.5: Binary to BCD converter: A Binary to BCD code converter illustrating how binary numbers (1110) and (1000) is converted to BCD. Note that the 7-segment displays are connected for illustration purpose. Fig. A.6: Binary to BCD converter: Logic Circuit Design Lab Page 17

18 B LTspice Simulations. B.1 Random sequence generator: Fig. B.1: Random sequence generator 1,2,5,7: Fig. B.2: Timing diagram Random sequence generator 1,2,5,7: LTspice Simulation File: Logic Circuit Design Lab Page 18

19 Fig. B.3: Self starting random sequence generator 1,2,5,7: Fig. B.4: Timing diagram Self starting random sequence generator 1,2,5,7:: LTspice Simulation File: Logic Circuit Design Lab Page 19

20 C C.1 Everycircuit Simulations. Binary to Gray and Gray to Binary converters: (a) Binary to gray code: (b) Gray code to binary: Fig. C.1: Binary to Gray and Gray to Binary converters: (a) Binary/Gray code converter: (b) Binary to BCD code converter: Fig. C.2: Binary/Gray converter and Binary to BCD converter: Logic Circuit Design Lab Page 20