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# IS 151 Lecture 9

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IS 151 Lecture 9 - UDSM 2013

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### IS 151 Lecture 9

1. 1. Functions of Combinational Logic • Basic Adders – Important in computers and many types of digital systems to process numerical data – Basic adder operations are fundamental to the study of digital systems IS 151 Digital Circuitry 1
2. 2. The Half Adder • Basic rules of binary addition –0+0=0 –0+1=1 –1+0=1 – 1 + 1 = 10 • A half-adder accepts 2 binary digits on its inputs and produces two binary digits on its outputs – a sum bit and a carry bit IS 151 Digital Circuitry 2
3. 3. The Half Adder • Logic symbol for the half-adder • Truth table for the half adder A B Cout Sum 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 IS 151 Digital Circuitry 3
4. 4. The Half Adder • Derive expressions for the Sum and Cout as functions of inputs A and B • From the truth table: – Cout is a 1 only when both A and B are 1s: Cout = AB – Sum is a 1 only if A and B are not equal: Sum = A B IS 151 Digital Circuitry 4
5. 5. The Half Adder • Half-adder logic diagram IS 151 Digital Circuitry 5
6. 6. The Full Adder • Accepts 3 inputs including an input carry and generates a sum output and an output carry • Difference between half and full-adder – the full adder has three inputs (including an input carry) while a half adder has only two inputs (without the input carry) IS 151 Digital Circuitry 6
7. 7. The Full Adder Logic symbol for a full adder IS 151 Digital Circuitry 7
8. 8. The Full Adder • Full adder truth table A B Cin Cout Sum 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 IS 151 Digital Circuitry 8
9. 9. The Full Adder • Derive expressions for the Sum and Cout as functions of inputs A, B and Cin from the truth table • Alternatively, using knowledge from the half adder: – Sum = inputs exclusively-ORed • Sum = (A B) Cin – Cout = inputs ANDed • AB + (A B)Cin IS 151 Digital Circuitry 9
10. 10. The Full Adder • From the truth table: – Sum = A’B’C + A’BC’+ AB’C’ + ABC – = A’B’C + ABC + A’BC’ + AB’C’ – = C(A’B’ + AB) + C’(A’B + AB’) – = C(AexB)’ + C’(AexB) – = CexAexB – = AexBexC IS 151 Digital Circuitry 10
11. 11. The Full Adder • Also from the truth table: – Carry out = A’BC + AB’C ABC’ + ABC – = C(A’B + AB’) + AB(C’ + C) – = C(AexB) + AB – = AB + (AexB)C IS 151 Digital Circuitry 11
12. 12. The Full Adder • Logic circuit for full-adder IS 151 Digital Circuitry 12
13. 13. The Full Adder - Exercise • Determine an alternative method for implementing the full-adder – Write SoP expressions from the truth table for Sum and Cout – Map the expressions on K-maps and write simplified expressions if any – Implement/draw the circuit diagram for the full-adder IS 151 Digital Circuitry 13
14. 14. Parallel Binary Adders • A single full adder is capable of adding two 1- bit numbers and an input carry • To add binary numbers with more than one bit, additional full-adders must be used. • Example, for a 2-bit number, 2 full adders are used, 4-bit numbers, 4 full-adders are used, etc. • The carry output of each adder is connected to the carry input of the next higher-order adder IS 151 Digital Circuitry 14
15. 15. Parallel Binary Adders • Block diagram of a basic 2-bit parallel adder General format, addition of two 2-bit numbers A2 B2 A1 B1 A B Cin A B Cin A2A1 + B2B1 Σ3Σ2Σ1 Cout (MSB) Σ3 Σ Σ2 Cout Σ Σ1 (LSB) IS 151 Digital Circuitry 15
16. 16. Parallel Binary Adders • Block diagram of a basic 4-bit parallel adder A4 B4 A3 B3 A2 B2 A1 B1 A B Cin A B Cin A B Cin A B Cin (MSB) (LSB) Cout Σ Cout Σ Cout Σ Cout Σ C4 Σ4 Σ3 IS 151 Digital Circuitry Σ2 Σ1 16
17. 17. Comparators • Basic function – to compare the magnitudes of two binary quantities to determine the relationship of those quantities • Comparators determine whether two numbers are equal or not IS 151 Digital Circuitry 17
18. 18. Comparators • Basic comparator operations The input bits are equal The input bits are not equal The input bits are equal IS 151 Digital Circuitry 18
19. 19. Comparators • To compare binary numbers containing 2 bits each, an additional Ex-OR gate is required IS 151 Digital Circuitry 19
20. 20. Comparators • The two LSBs of the two numbers are compared by gate G1, and the MSBs are compared by gate G2 – Binary number A = A1A0 – Binary number B = B1B0 • If the two numbers are equal, their corresponding bits are the same, and the output of each Ex-OR gate is a 0 • The 0s are inverted using the inverter, producing 1s • The 1s are ANDed, producing a 1, which indicates equality IS 151 Digital Circuitry 20
21. 21. Comparators • If the corresponding sets of bits are not equal, i.e. A0≠B0, or A1≠B1, a 1 occurs on that Ex-OR gate output (G1 or G2) • In order to produce a single output indicating an equality or inequality of two numbers, two inverters and an AND gate are used • When the two inputs are not equal, a 1,0 or 0,1 appears on the inputs of the AND gate, producing a 0, indicating inequality IS 151 Digital Circuitry 21
22. 22. Comparators - Examples • Determine whether A and B are equal (or not) by following the logic levels through the circuit – a) A0 = 1, B0 = 0 and A1 = 1, B1 = 0 – b) A0 = 1, B0 = 1 and A1 = 1, B1 = 0 IS 151 Digital Circuitry 22
23. 23. 4-bit Comparators • To determine if two numbers are equal • If not, determine which one is greater or less than the other • Numbers – A = A3A2A1A0 – B = B3B2B1B0 IS 151 Digital Circuitry 23
24. 24. 4-bit Comparators • Determine the inequality by examining highest-order bit in each other – If A3 = 1 and B3 = 0; A > B – If A3 = 0 and B3 = 1; A < B – If A3 = B3, examine the next lower bit position for an inequality (i.e. A2 with B2) IS 151 Digital Circuitry 24
25. 25. 4-bit Comparators • The three observations are valid for each bit position in the numbers • Check for an inequality in a bit position, starting with the highest order bits • When such an inequality is found, the relationship of the two numbers is established, and any other inequalities in lower-order bit positions are ignored IS 151 Digital Circuitry 25
26. 26. Comparators - Example • Implement a circuit to compare the relationship between A and B A B A>B A=B A<B 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 0 1 For A > B (G): G = AB’ For A = B (E): E = A’B’ + AB For A < B (L): L = A’B 0 IS 151 Digital Circuitry 26
27. 27. Decoders • Function – to detect the presence of a specified combination of bits (code) on inputs and to indicate the presence of that code by a specified output level • n-input lines (to handle n bits) • 1 to 2n output lines to indicate the presence of 1 or more n-bit combinations IS 151 Digital Circuitry 27
28. 28. Decoders • Example: determine when a binary 1001 occurs on the inputs of a digital circuit • Use AND gate; make sure that all of the inputs to the AND gate are HIGH when the binary number 1001 occurs IS 151 Digital Circuitry 28
29. 29. Decoders • Exercise: develop the logic required to detect the binary code 10010 and produce an active-LOW output (0) IS 151 Digital Circuitry 29
30. 30. Decoders - Application • BCD-to-Decimal decoder – Converts each BCD code into one of 10 possible decimal digits Decimal Digit BCD Code A3 A2 A1 A0 BCD Decoding Function 0 0 0 0 0 A3’ A2’ A1’ A0’ 1 0 0 0 1 2 0 0 1 0 3 0 0 1 1 4 0 1 0 0 5 0 1 0 1 6 0 1 1 0 7 0 1 1 1 8 1 0 0 0 9 1 0 0 1 IS 151 Digital Circuitry 30
31. 31. Encoders • A combinational logic circuit that performs a reverse decoder function • Accepts an active level on one of its inputs (e.g. a decimal or octal digit) and converts it to a coded output, e.g. BCD or binary IS 151 Digital Circuitry 31
32. 32. Encoders • The Decimal-to-BCD Encoder (10-line-to-4-line encoder) – Ten inputs (one for each decimal digit) – Four outputs corresponding to the BCD code • Logic symbol for a decimal-to-BCD encoder Decimal Input DEC/BCD 0 1 2 3 1 4 2 5 4 6 8 7 8 9 BCD Output IS 151 Digital Circuitry 32
33. 33. Encoders • From the BCD/Decimal table, determine the relationship between each BCD bit and the decimal digits in order to analyse the logic. – Example – – – – A3 = 8 + 9 A2 = 4 + 5 + 6 + 7 A1 = 2 + 3 + 6 + 7 A0 = 1 + 3 + 5 + 7 + 9 IS 151 Digital Circuitry 33
34. 34. Encoders • Example: if input line 9 is HIGH (assume all others are LOW), it will produce a HIGH on A3 and A0, and a LOW on A1and A2 – which is 1001, meaning decimal 9. IS 151 Digital Circuitry 34
35. 35. Encoders - Exercise • Implement the logic circuit for the 10-line-to-4-line encoder using the logic expressions for the BCD codes A3 to A0 and inputs 0 to 9 IS 151 Digital Circuitry 35
36. 36. Logic Functions Exercise (Lab 3) • Design and test a circuit to implement the function of the full adder (refer to lecture) IS 151 Digital Circuitry 36
37. 37. • End of Lecture IS 151 Digital Circuitry 37