Lecture 3Combinational Circuits

Lecture 3Combinational Circuits

TopicsTopics Basic Gates (3.1) Adders: half-adder, full-adder Gray Code (2.11) N-cube (2.14) Error Correcting codes (2.15) Boolean Algebra (4.1) Combinational-Circuit Analysis (4.2)

August 28, 2003

CSCE 211 Digital Design

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OverviewOverview

Last TimeLast Time Conversion of Fractions decimal base-r Representations of Integers

Unsigned, signed magnitude, one’s complement, two’s complement Arithmetic BCD, representations of characters BCD, unicode

On last Time’s Slides(what we didn’t get to)On last Time’s Slides(what we didn’t get to) Basic gates (well almost on transparencies) Adder circuits

NewNew Boolean algebra Combinational circuits

VHDLVHDL ButNot Table 4-36 page 277

History: George BooleHistory: George Boole

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Basic GatesBasic Gates

ANDAND

OROR

NOTNOT

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Basic GatesBasic Gates

NANDNAND

NORNOR

XORXOR

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Half Adder CircuitHalf Adder Circuit

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Full AdderFull Adder

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Ripple Carry AdderRipple Carry Adder

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Gray CodeGray Code

In some mechanical devices you want to encode In some mechanical devices you want to encode positions as binary strings in such a way that positions as binary strings in such a way that positions close to each other are represented by positions close to each other are represented by strings that are close together. strings that are close together.

Gray code – adjacent position differ in only one bitGray code – adjacent position differ in only one bit 000 001 011 010 110 111 101 100

Figure 2-6 encoding diskFigure 2-6 encoding disk

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Construction of Gray CodesConstruction of Gray Codes

Gray Codes are reflectiveGray Codes are reflective

Algorithm for construction of Gray Codes on n-bitsAlgorithm for construction of Gray Codes on n-bits

1.1. A 1-bit Gray code has two words 0 and 1.A 1-bit Gray code has two words 0 and 1.

2.2. The first 2n words of an (n+1)-bit gray code are the The first 2n words of an (n+1)-bit gray code are the words of the n-bit gray code with a leading 0 added.words of the n-bit gray code with a leading 0 added.

3.3. The next 2n words of an (n+1)-bit gray code are the The next 2n words of an (n+1)-bit gray code are the words of the n-bit gray code but written in reverse words of the n-bit gray code but written in reverse order with a leading 1 added.order with a leading 1 added.

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Construction of Gray CodesConstruction of Gray Codes

1-bit gray code1-bit gray code 0 1

2-bit gray code2-bit gray code 00 01 11 10

3-bit gray code3-bit gray code 0 00 0 01 0 11 0 10 1 10 1 11 1 01 1 00

4-bit gray code4-bit gray code 0 000

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Gray Code Generation Algorithm 2Gray Code Generation Algorithm 2

AlgorithmAlgorithm

1.1. The bits of an n-bit binary and n-bit Gray-code code The bits of an n-bit binary and n-bit Gray-code code word are numbered from right to left from 0 to n-1.word are numbered from right to left from 0 to n-1.

2.2. Bit i of a Gray-code word is 0 if bits i and i+1 of the Bit i of a Gray-code word is 0 if bits i and i+1 of the corresponding binary code word are the same, else corresponding binary code word are the same, else bit i is 1. (When i = n-1, bit n is considered to be 0.)bit i is 1. (When i = n-1, bit n is considered to be 0.)

ExampleExample

Given 0100 1010 Given 0100 1010

Add fake 0 on front bit n is 0Add fake 0 on front bit n is 0

0 0100 1010 0 0100 1010 0110 1111 0110 1111

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Codes representing charactersCodes representing characters

ASCII (American Standard Code for Information ASCII (American Standard Code for Information Interchange) – 8 bits = 7 + parityInterchange) – 8 bits = 7 + parity

Unicode 16 bitsUnicode 16 bits

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N-cubesN-cubes

1-cube1-cube

2-cube2-cube

3-cube3-cube

Traversing in gray-code orderTraversing in gray-code order

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Parity BitsParity Bits

Even parity bit – parity bit is set so that the number of Even parity bit – parity bit is set so that the number of ones is evenones is even

Odd parity bitOdd parity bit

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Error Correcting codesError Correcting codes

For an n-bit code, consider the hypercube of dimension nFor an n-bit code, consider the hypercube of dimension n

Choose some subset of the nodes as code words.Choose some subset of the nodes as code words.

Suppose the distance between any two two code words is at least 3.Suppose the distance between any two two code words is at least 3.

Now consider transmission errors.Now consider transmission errors.

Then if there is an error in transmitting just one bit then the distance Then if there is an error in transmitting just one bit then the distance from the received word to one code word is one, distances to from the received word to one code word is one, distances to other code words are at least two.other code words are at least two.

Single error correcting, double error detecting.Single error correcting, double error detecting.

Such codes are called Hamming codes after their inventor Richard Such codes are called Hamming codes after their inventor Richard Hamming. Hamming.

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Boolean AlgebraBoolean Algebra

George Boole (1854) invented a two valued algebraGeorge Boole (1854) invented a two valued algebra

To “give expression … to the fundamental laws of To “give expression … to the fundamental laws of reasoning in the symbolic language of a Calculus.”reasoning in the symbolic language of a Calculus.”

1938 Claude Shannon at Bell Labs noted that this 1938 Claude Shannon at Bell Labs noted that this Boolean logic could be used to describe switching Boolean logic could be used to describe switching circuits. (Switching Algebra)circuits. (Switching Algebra)

In Shannon’s view a relay had two positions open and In Shannon’s view a relay had two positions open and closed and collections of relays satisfied the closed and collections of relays satisfied the properties of Boolean algebra.properties of Boolean algebra.

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Boolean Algebra AxiomsBoolean Algebra Axioms

The axioms of a mathematical system are a minimal set The axioms of a mathematical system are a minimal set of properties that are assumed to be true.of properties that are assumed to be true.

Axioms of Boolean AlgebraAxioms of Boolean Algebra

1.1. X = 0 if X != 1 X = 0 if X != 1 X=1 if X!=0X=1 if X!=0

2.2. If X = 0, then X’ = 1If X = 0, then X’ = 1 if X=1 then X’=0if X=1 then X’=0

3.3. 0 . 0 = 00 . 0 = 0 1 + 1 = 11 + 1 = 1

4.4. 1 . 1 = 11 . 1 = 1 0 + 0 = 00 + 0 = 0

5.5. 0 . 1 = 1 . 0 = 00 . 1 = 1 . 0 = 0 1 + 0 = 0 + 1 = 11 + 0 = 0 + 1 = 1

Axioms 1-5 and 1-5’ completely define Boolean algebra.Axioms 1-5 and 1-5’ completely define Boolean algebra.

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Boolean Algebra TheoremsBoolean Algebra Theorems

Proofs by perfect inductionProofs by perfect induction

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More TheoremsMore Theorems

N.B. T8N.B. T8, T10, T11, T10, T11

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DualityDuality

Swap 0 & 1, AND & ORSwap 0 & 1, AND & OR Result: Theorems still true

Why?Why? Each axiom (A1-A5) has a dual (A1-A5

Counterexample:Counterexample:X + X X + X Y = X (T9) Y = X (T9)X X X + Y = X (dual) X + Y = X (dual)X + Y = X (T3X + Y = X (T3))????????????????????????

X + (X Y) = X (T9)X (X + Y) = X (dual)(X X) + (X Y) = X (T8)X + (X Y) = X (T3)parentheses,operator precedence!

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N-variable TheoremsN-variable Theorems

Prove using finite inductionProve using finite induction

Most important: DeMorgan theoremsMost important: DeMorgan theorems

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Combinational Circuit AnalysisCombinational Circuit Analysis

A combinational circuit is one whose outputs are a A combinational circuit is one whose outputs are a function of its inputs and only its inputs.function of its inputs and only its inputs.

These circuits can be analyzed using:These circuits can be analyzed using:

1.1. Truth tablesTruth tables

2.2. Algebraic equationsAlgebraic equations

3.3. Logic diagramsLogic diagrams

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Truth TablesTruth Tables