2 Building Blocks. There is often the need to compare two binary values.

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Theodore Houston
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1 2 Building Blocks 2.1 Comparators There is often the need to compare two binary values. This is done using a comparator. A comparator determines whether binary values A and B are: 1. A = B 2. A < B 3. A > B Equality The XOR can be used to compare equality. It will give a 0 when the two bits are equal and a 1 when they are unequal. To get a HIGH for equality and a LOW for inequality an XNOR can be used. X Y XOR XNOR Comment Equal Not Equal Not Equal Equal One XNOR compares one bit. To compare multiple bits, we need a XNOR for each bit. The circuit to compare two nibbles (4 bits) is shown below: A0 B0 A1 B1 A2 B2 A3 B3 = MEE1004 Digital Networks and Systems 26

2 2.1.2 Integrated Circuit Comparators Many integrated circuit comparators contain outputs for: A = B A < B A > B They are designed for cascading together and contain inputs for A = B from the preceding stage A < B from the preceding stage A > B from the preceding stage The 74LS85 is a TTL 4 bit magnitude comparator. Q. How can we compare an 8 bit number? A. By cascading two 7485 comparators. One to compare the Least Significant Nibble. One to compare the Most Significant Nibble. A0 A1 A2 A3 B0 B1 B2 B3 A0 A1 A2 A3 B0 B1 B2 B3 A<Bi A=Bi A>Bi A<Bo A=Bo A>Bo A4 A5 A6 A7 B4 B5 B6 B7 A0 A1 A2 A3 B0 B1 B2 B3 A<Bi A=Bi A>Bi A<Bo A=Bo A>Bo A < B = A > B LSN MSN Stage 1 Stage 2 MEE1004 Digital Networks and Systems 27

3 2.2 Decoders Decoding is taking a code (binary, BCD, hex etc) and activating a single output representing its numeric value Method to create a decoder 1. Generate a truth table 2. Find an expression for each output. (Karnaugh) to 4 decoder A 2 bit value can activate 1 of 4 possible output lines. 0 1 LSB MSB DECODER Boolean Expressions Z0 = /A. /B Z1 = /A. B Z2 = A. /B Z3 = A. B A B Z3 Z2 Z1 Z0 Expression Z0 = /A./B Z1 = /A. B Z2 = A./B Z3 = A. B MEE1004 Digital Networks and Systems 28

4 We can then construct the circuit for each output. A B Z0 Z1 Z2 Z3 MEE1004 Digital Networks and Systems 29

5 2.3 Encoders Encoding is taking a single input representing a numeric value and converting it to its equivalent code (binary, BCD, hex etc). This is the reverse of decoding Method to create a encoder 1. Generate a truth table 2. Find an expression for each output. (Karnaugh) to 2 line binary encoder One of 4 lines is encoded into binary. 0 1 D C Encoder DECODER 2 B MSB 1 3 A LSB 0 A B C D Z1 Z Boolean Expressions Z1 = A + B Z0 = A + C Note: D is not used. MEE1004 Digital Networks and Systems 30

6 We can then construct the circuit for each output. A B Z1 C Z0 D N / C MEE1004 Digital Networks and Systems 31

7 2.4 Code Convertors Gray Code Gray code is a very useful code in electronics and is used for indicating the angular position of the shaft on a motor. Gray code allows only one bit to change when moving from one code to the next. You are familiar with this concept when writing the inputs for a Karnaugh Map. eg A B, AB, AB, AB eg 2 bit Gray code ( 00, 01, 11, 10 ) 2 bit Gray code Decimal Binary Gray Note: The Gray code can easily roll back to itself 00, 01, 11, 10 00, 01 etc. Sensors The angular position of a motor s shaft can be determined by connecting a wheel to the motor s shaft. The wheel has a code stored at different positions around it. The more codes stored on the wheel, the greater the accuracy. Each code is stored on a separate segment of the wheel. Each segment is divided into rings which allows each sector to represent a binary digit. When the motor turns, the wheel connected to the shaft also turns. Sensors in a fixed position above the wheel pick up the codes and use them to state the position of the wheel and hence the shaft. If a binary code is used then as the wheel changes from position 3 to position 0, the binary read goes from 11 to 00. If the sensors are not perfectly in line and were read on the transition from position 3 to 0, then values of 01 or 10 could occur depending upon the misalignment. Therefore there is a potential for a large error using a binary scheme. MEE1004 Digital Networks and Systems 32

8 If a gray code was used then as the wheel changes from position 3 to position 0, the binary read goes from 10 to 00. If the sensors were misaligned and read on a transition from position 3 to position 0 then it could only read 10 or 00. This eliminates the problem. Converting Binary to Gray Code. 1. The Most Significant Bit (MSB) in the Gray code is the same as the binary number. 2. Going from the MSB to LSB (left to right), add the current bit to the next bit (right) ignoring the carry. The result of the addition is the gray code bit. For example to find the gray code for Binary Action Gray Bit MSB Therefore the Gray code for is B0 G0 B1 G1 B2 G2 B3 G3 (MSB) MEE1004 Digital Networks and Systems 33

9 Converting Gray Code to Binary. 1. The Most Significant Bit (MSB) in the Gray code is the same as the binary number. 2. Going from the MSB to LSB (left to right), add the current gray code bit to the last binary bit found ignoring the carry. The result of the addition is the binary code bit. For example to convert a gray code of to binary Gray Code Previous Action Binary Binary N/A MSB Therefore the Binary for a Gray code of is G0 B0 G1 B1 G2 G3 B2 B3 (MSB) MEE1004 Digital Networks and Systems 34

10 2.4.2 BCD to Binary BCD is a binary code that is used to represent each decimal digit. Therefore each decimal digit is represented with 0000 (0) to 1001 (9). Eg Weight: Decimal: 2 9 BCD: Binary: Therefore we must assign a weight to each bit. Wherever there is a 1 in the BCD digit we add the weight. BCD Bit Weight Binary BCD 29 is Weight Binary = Integrated Circuit Converter The is a 6 bit BCD to binary converter. MEE1004 Digital Networks and Systems 35

11 2.4.3 Binary to BCD BCD is a binary code that is used to represent each decimal digit. Therefore each decimal digit is represented with 0000 (0) to 1001 (9) A 4 bit binary code has 16 values 0000 (0) to 1111 (15). The binary numbers from 0000 (0) to 1001 (9) can represent their values. The binary numbers from 1010 (10) to 1111 (15) can be converted to BCD by adding 6. Note that this will generate a carry. Example 10: = (1) 0000 BCD: 1_0 15: = (1) 0101 BCD: 1_5 The circuit to perform the binary to BCD conversion is given below. It consists of a magnitude comparator and an adder. A B C D LO LO A1 A2 A3 A4 B1 B2 B3 B4 S1 S2 S3 S4 7483A LSB MSB LO C0 C4 9 HI LO LO HI LO HI LO A0 A1 A2 A3 B0 B1 B2 B3 A<Bi A=Bi A>Bi 7485 A<Bo A=Bo A>Bo Integrated Circuit Converter The is a 6 bit binary to BCD converter. MEE1004 Digital Networks and Systems 36

12 2.4.4 BCD to 7 segment The 7 segment display is widely used as a numeric display in many everyday devices All the digits can be constructed out of 7 lines arranged in an 8 pattern: In a 7 segment display each line is made up out of a LED or LCD bar. The bar is switched on according to pattern required by the BCD digit. A code converter is used to convert from BCD to the 7 segments. Integrated Circuit Converter The 7447 is the BCD to 7 segment converter. Exercise Draw the circuits to the BCD to 7 segment converter. There should be 7 equations. Hint: Create the truth table for each segment, minimise using Karnuagh maps, and draw the resulting circuit. MEE1004 Digital Networks and Systems 37

13 2.5 Multiplexers A multiplexer is a circuit that allows digital information from several sources to be routed onto a single line. This is the equivalent of a digital switch. Data is input into the lines D0 D1. Data select inputs determine which input is connected to the output. A multiplexor is also known as Data Selector. A 4 to 1 line multiplexor is shown below. D0 D1 D2 Z D3 S1 S0 The control lines S0 & S1 can have 4 binary settings. Therefore the output (Z) can be connected to any one of the inputs (D0 to D3). Truth Table for 4 to 1 line multiplexor S1 S0 Z 0 0 D0 0 1 D1 1 0 D2 1 1 D3 MEE1004 Digital Networks and Systems 38

14 2.5.1 Design of Multiplexors Using the property that X. 1 = X Selectors ANDed together to give Decide upon the number of input Data lines required 2 n = Number of data lines required 2. The number of selector lines required will be n from the above equation. 3. Generate a truth table including selector states and output. 4. For each input data line, AND it with its equivalent selector state. 5. OR the result of all the ANDs. D0 S1 S0 D1 D2 Z D3 MEE1004 Digital Networks and Systems 39

15 2.5.2 Using A Multiplexer for Combinational Logic A multiplexer can be used to replace combinational logic If the expression is complex, it may require many ICs. If the logic is replaced with a single multiplexer then only one IC is required. The concept is that the inputs from the truth table form the selector for the multiplexer. Each Data line is set to the appropriate output value from the truth table. When the selector value is set, the output is connected to the associated Data Line. Hence the output line of the multiplexer passes the output value from the truth table that is associated with the input values from the truth table. Method 1. Generate a truth table. 2. The inputs in the truth table are the selectors of the Multiplexer. 3. Each output term in the truth table corresponds to one Data Input line of the Multiplexer. 4. Tie each Data Input Line to the corresponding output value from the truth table. Example Refer to the Judges scoring system example used in module 1. Green = BC + AC + AB Truth Table A B C Green MUX D D D D D D D D7 HI A B C LO D0 D1 D2 D3 D4 D5 D6 D7 A B C Y Green The original solution requires 2 x 14 pin ICs. ( 1 x 7432, 1 x 7408 ) The multiplexer solution requires 1 x 16 pin IC. MEE1004 Digital Networks and Systems 40

16 2.6 Demultiplexers A demultiplexer is a circuit that takes a single input and routes it to only one of the possible output lines. This is the equivalent of a digital switch. This acts in reverse to a multiplexer. Data is output from lines D0 D1. Data select inputs determine which output is connected to the input. A demultiplexor is also known as Data Spreader. A 1 to 4 line demultiplexor is shown below. D0 Z D1 D2 D3 S1 S0 The control lines S0 & S1 can have 4 binary settings. Therefore the input (Z) can be connected to any one of the outputs (D0 to D3). Truth Table for 1 to 4 line demultiplexor Where Z is the input data. S1 S0 D0 D1 D2 D3 0 0 Z Z Z Z MEE1004 Digital Networks and Systems 41

17 2.6.1 Design of Demultiplexers Using the property that X. 1 = X Selectors ANDed together to give Decide upon the number of output Data lines required 2 n = Number of data lines required 2. The number of selector lines required will be n from the above equation. 3. Generate a truth table including selector states and data lines. 4. Each output data line will be the input data (Z) ANDed with its equivalent selector state. Z S1 S0 D0 D1 D2 D3 MEE1004 Digital Networks and Systems 42

18 2.7 Parity Errors can occur in the transmission of data from one computer to another. We need a method to determine if an error has occurred. Parity allows us to detect a single bit error in the data bits sent. Usually these are grouped in bytes. To use parity to detect errors we must append a parity bit to the data bits transmitted. Any group of bits may contain an even or odd number of 1 s. The parity bit is used to set the total number of 1 s to an even number or an odd number. Even parity uses the parity bit to make the total number of 1 s an even number. Odd parity uses the parity bit to make the total number of 1 s an odd number. Example Q What is value of the parity bit required to make 1011 (a) even parity (b) odd parity A 1011 contains three 1 s. This is an odd number of 1 s. (a) parity bit = 1 to make 4 x 1 s, an even number. (b) parity bit = 0 to make 3 x 1 s, an odd number. Error Detection We must know the type of parity used for the received data. The received data will have a parity bit. This can be checked by summing together all the bits in the data received. For even parity, the sum of all the bits including the parity bit will be 0. For odd parity the sum of all the bits including the parity bit will be 1. MEE1004 Digital Networks and Systems 43

19 The bits can be summed using XOR circuits in the following manner: A0 A1 Sum Summing 2 bits. A0 A1 A2 A3 Sum Summing 4 bits (nibble). A0 A1 A2 A3 Sum A4 A5 A6 A7 Summing 8 bits (byte). MEE1004 Digital Networks and Systems 44

Digital Electronics. Functions of Combinational Logic

Digital Electronics. Functions of Combinational Logic Digital Electronics Functions of Combinational Logic Half-dder Basic rules of binary addition are performed by a half adder, which has two binary inputs ( and B) and two binary outputs (Carry out and Sum).