Ceng 241 digital design 1 lecture 7
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CENG 241 Digital Design 1 Lecture 7. Amirali Baniasadi [email protected] 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.

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CENG 241 Digital Design 1 Lecture 7

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Ceng 241 digital design 1 lecture 7

CENG 241Digital Design 1Lecture 7

Amirali Baniasadi

[email protected]


Ceng 241 digital design 1 lecture 7

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


Decimal adder

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


Decimal adder1

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.


Ceng 241 digital design 1 lecture 7

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


Ceng 241 digital design 1 lecture 7

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


Magnitude comparators

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?)


Magnitude comparators1

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


Ceng 241 digital design 1 lecture 7

Magnitude Comparators

A3=B3 ?

X3A2’B2


Decoders

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.


Decoders truth table

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


Ceng 241 digital design 1 lecture 7

Decoders: AND implementation


Ceng 241 digital design 1 lecture 7

2-to-4 Decoder: NAND implementation

Decoder is enabled when E=0


Ceng 241 digital design 1 lecture 7

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


Combinational logic implementation

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)


Ceng 241 digital design 1 lecture 7

Combinational Logic implementation


Encoders

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.


Encoders truth table

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


Priority encoders

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


Ceng 241 digital design 1 lecture 7

Priority Encoders: Map


Ceng 241 digital design 1 lecture 7

Priority Encoders: Circuit


Multiplexers

Multiplexers

  • Multiplexer: selects one binary input from many selections

  • example: 2-to-1 MUX


4 to 1 mux

4-to-1 MUX

Directs 1 of the 4 inputs to the output


Multi bit selection logic

Multi-bit selection logic

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


Implementing boolean functions w mux

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.


Implementing boolean functions w mux1

Implementing Boolean functions w/ MUX


Ceng 241 digital design 1 lecture 7

Implementing Boolean functions w/ MUX


Summary

Summary

  • Reading up to page 156


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