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# CENG 241 Digital Design 1 Lecture 7

CENG 241 Digital Design 1 Lecture 7. Amirali Baniasadi amirali@ece.uvic.ca. 4-bit by 3-bit Binary Multiplier. B3 B2 B1 B0 A2 A1 A0 A0B3 A0B2 A0B1 A0B0 A1B3 A1B2 A1B1 A1B0 A2B3 A2B2 A2B1 A2B0.

## CENG 241 Digital Design 1 Lecture 7

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1. CENG 241Digital Design 1Lecture 7 Amirali Baniasadi amirali@ece.uvic.ca

2. 4-bit by 3-bit Binary Multiplier B3 B2 B1 B0 A2 A1 A0 A0B3 A0B2 A0B1 A0B0 A1B3 A1B2 A1B1 A1B0 A2B3 A2B2 A2B1 A2B0

3. Decimal adder • When dealing with decimal numbers BCD code is used. • A decimal adders requires at least 9 inputs and 5 outputs. • BCD adder: each input does not exceed 9, the output can not exceed 19 • How are decimal numbers presented in BCD? • Decimal Binary BCD • 9 1001 1001 • 19 10011 (0001)(1001) • 1 9

4. Decimal Adder • Decimal numbers should be represented in binary code number. • Example: BCD adder • Suppose we apply two BCD numbers to a binary adder then: • The result will be in binary and ranges from 0 through 19. • Binary sum: K(carry) Z8 Z4 Z2 Z1 • BCD sum : C(carry) S8 S4 S2 S1 • For numbers equal or less than 1001 binary and BCD are identical. • For numbers more than 1001, we should add 6(0110) to binary to get BCD. • example: 10011(binary) = 11001(BCD) =19 • ADD 6 to correct.

5. BCD adder Numbers that need correction (add 6) are: 01010 (10) 01011 (11) 01100 (12) 01101 (13) 01110 (14) 01111 (15) 10000 (16) 10001 (17) 10010 (18) 10011 (19) Decides to add 6? Adds 6

6. BCD adder Numbers that need correction (add 6) are: K Z8 Z4 Z2 Z1 0 1 0 1 0 (10) 0 1 0 1 1 (11) 0 1 1 0 0 (12) 0 1 1 0 1 (13) 0 1 1 1 0 (14) 0 1 1 1 1 (15) 1 0 0 0 0 (16) 1 0 0 0 1 (17) 1 0 0 1 0 (18) 1 0 0 1 1 (19) C = K + Z8Z4 +Z8Z2

7. Magnitude Comparators • Compares two numbers, determines their relative magnitude. • We look at a 4-bit magnitude comparator; • A=A3A2A1A0, B=B3B2B1B0 • Two numbers are equal if all bits are equal. • A=B if A3=B3 AND A2=B2 AND A1=B1 AND A0=B0 • Xi= AiBi + Ai’Bi’ ; Ai=Bi Xi=1 (remember exclusive NOR?)

8. Magnitude Comparators • How do we know if A>B? • 1.Compare bits starting from the most significant pair of digits • 2.If the two are equal, compare the next lower significant bits • 3.Continue until a pair of unequal digits are reached • 4.Once the unequal digits are reached, A>B if Ai=1 and Bi=0, A<B if Ai=0 and Bi = 1 • A>B = A3B3’+X3A2B2’+X3X2A1B1’+X3X2X1A0B0’ • A<B = A3’B3+X3A2’B2+X3X2A1’B1+X3X2X1A0’B0 • Xi=1 if Ai=Bi

9. Magnitude Comparators A3=B3 ? X3A2’B2

10. Decoders • A decoder converts binary information from n input lines to a maximum of 2n output lines • Also known as n-to-m line decoders where m< 2n • Example 3-to-8 decoders.

11. Decoders: Truth Table • X Y Z D0 D1 D2 D3 D4 D5 D6 D7 • 0 0 0 1 0 0 0 0 0 0 0 • 0 0 1 0 1 0 0 0 0 0 0 • 0 1 0 0 0 1 0 0 0 0 0 • 0 1 1 0 0 0 1 0 0 0 0 • 1 0 0 0 0 0 0 1 0 0 0 • 1 0 1 0 0 0 0 0 1 0 0 • 1 1 0 0 0 0 0 0 0 1 0 • 1 1 1 0 0 0 0 0 0 0 1

12. Decoders: AND implementation

13. 2-to-4 Decoder: NAND implementation Decoder is enabled when E=0

14. How to build bigger decoders? We can combine two 3-to-8 decoders to build a 4-to-16 decoder. Generates from 0000 to 0111 Generates from 1000 to 1111

15. Combinational Logic implementation • A decoder provides the 2n minterms of n input variables. • Any function is can be expressed in sum of minterms. • Use a decoder to make the minterms and an external OR gate to make the sum. • Example: consider a full adder. • S(x,y,z) = Σ(1,2,4,7) • C(x,y,z) = Σ (3,5,6,7)

16. Combinational Logic implementation

17. Encoders • Encoders perform the inverse operation of a decoder: • Encoders have 2n input lines and n output line. • Output lines generate the binary code corresponding to the input value.

18. Encoders: Truth Table • OutputsInputs • X Y Z D0 D1 D2 D3 D4 D5 D6 D7 • 0 0 0 1 0 0 0 0 0 0 0 • 0 0 1 0 1 0 0 0 0 0 0 • 0 1 0 0 0 1 0 0 0 0 0 • 0 1 1 0 0 0 1 0 0 0 0 • 1 0 0 0 0 0 0 1 0 0 0 • 1 0 1 0 0 0 0 0 1 0 0 • 1 1 0 0 0 0 0 0 0 1 0 • 1 1 1 0 0 0 0 0 0 0 1 • z=D1+D3+D5+D7 y=D2+D3+D6+D7 x=D4+D5+D6+D7

19. Priority Encoders • Encoder limitations: • If two inputs are active, the output is undefined. • Solution: we need to take into account priority. • What if all inputs are 0? • Solution: we need a valid bit • Input Output • D0 D1 D2 D3 x y v • 0 0 0 0 X X 0 • 1 0 0 0 0 0 1 • X 1 0 0 0 1 1 • X X 1 0 1 0 1 • X X X 1 1 1 1

20. Priority Encoders: Map

21. Priority Encoders: Circuit

22. Multiplexers • Multiplexer: selects one binary input from many selections • example: 2-to-1 MUX

23. 4-to-1 MUX Directs 1 of the 4 inputs to the output

24. Multi-bit selection logic • Multiplexers can be combined with common selection inputs to support multi-bit selection logic

25. Implementing Boolean functions w/ MUX • General rules for implementing any Boolean function with n variables: • Use a multiplexer with n-1 selection inputs and 2 n-1 data inputs • List the truth tabel • Apply the first n-1 variables to the selection inputs of multiplexer • For each combination evaluate the output as a function of the last variable. • The function can be 0, 1 the variable or the complement of the variable.

26. Implementing Boolean functions w/ MUX

27. Implementing Boolean functions w/ MUX

28. Summary • Reading up to page 156

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