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Understanding Decimal Adders, Comparators, Decoders, Encoders, and Multiplexers in Digital Design

This lecture covers essential components in digital design, focusing on decimal adders, magnitude comparators, decoders, encoders, and multiplexers. It delves into the functionality of a BCD adder where decimal numbers are processed in binary code. The lecture explains how to correct results in BCD addition, compares magnitude of binary numbers using comparators, and describes the operation of decoders and encoders in converting binary values. Additionally, it explores implementing Boolean functions using multiplexers, providing insights into combinational logic circuits.

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Understanding Decimal Adders, Comparators, Decoders, Encoders, and Multiplexers in Digital Design

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  1. CENG 241Digital Design 1Lecture 6 Amirali Baniasadi amirali@ece.uvic.ca

  2. 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

  3. 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.

  4. 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

  5. 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

  6. 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?)

  7. 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

  8. Magnitude Comparators A3=B3 ? X3A2’B2

  9. 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.

  10. 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

  11. Decoders: AND implementation

  12. 2-to-4 Decoder: NAND implementation Decoder is enabled when E=0

  13. 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

  14. 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)

  15. Combinational Logic implementation

  16. 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.

  17. 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

  18. 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

  19. Priority Encoders: Map

  20. Priority Encoders: Circuit

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

  22. 4-to-1 MUX Directs 1 of the 4 inputs to the output

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

  24. 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.

  25. Implementing Boolean functions w/ MUX

  26. Implementing Boolean functions w/ MUX

  27. Summary • Reading up to page 154

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