1 A Simple ALU Binary Logic. 2 Outline l Binary Logic l Representation of Logic Gates l Constructing...

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A Simple ALU

Binary Logic

2

Outline

Binary Logic Representation of Logic Gates Constructing a 1-bit adder Constructing an n-bit adder

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Binary Logic

A computer works using digital electronics. There are only High and Low voltages All computer logic is based on the

manipulation of these Abstractly,

» High is interpreted as 1 (true)» Low is interpreted as 0 (false)

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Logic Circuit

A logic circuit takes in a number of input lines (A, B, C, ...)» and has a number of output lines (X,Y,Z,...)

AB

C

X

Y

Z

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Logic Basics Every expression evaluates to 1 or 0 NOT

» ~A (“A bar”)– is the complement of A, i.e. 1-A

OR» A + B

– is 1 if and only if at least one of A or B is 1, else 0

AND» A.B

– is 1 if and only if both A and B are 1, else 0

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Truth Tables

A B A+B A.B0 0 0 00 1 1 01 0 1 01 1 1 1

A ~A0 11 0

A B C A.B A.C B+C A.(B+C) A.B+A.C0 0 0 0 0 0 0 00 0 1 0 0 1 0 00 1 0 0 0 1 0 00 1 1 0 0 1 0 01 0 0 0 0 0 0 01 0 1 0 1 1 1 11 1 0 1 0 1 1 11 1 1 1 1 1 1 1

A.(B+C)=A.B+A.C

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Gates

A

BA.B AND Gate

A

BA+B OR Gate

A ~A Inverter (NOT)

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Representations of Integers

Positive integers are represented by strings of binary digits - of fixed length given by the word length (eg, 32 bits).

Numbers in the range 0 to 232-1 can be represented.

0131

LSBMSB

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Negative Numbers We have to have a way to represent negative

numbers and 0.» If the word length is n, then there are 2n bit-patterns

possible - there cannot be an equal number of positive and negative numbers represented.

Suppose n=3, and consider the following:» 000two = 0ten 100two = -4ten

» 001two = 1ten 101two = -3ten

» 010two = 2ten 110two = -2ten

» 011two = 3ten 111two = -1ten

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Two’s Complement

For an n-bit word, the MSB is called the ‘sign bit’, which if 1 determines a negative number.

An n-bit word: (bn-1bn-2...b1b0)two represents the decimal number:

Eg, if n=4, 1101 = -8 + 4 + 1 = -3. Eg, if n=5, 10110 = -16 + 2 + 4 = -10

2 21

10

2n

nii

nb b

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Negating a Number

A simple technique for negating a number:» Invert each bit, and then add 1.

Convert:-» 10110

01001 + 1 = 01010 = 2ten+8ten = 10ten

» 1101 0010 + 1 = 0011 = 1ten + 2ten = 3ten

» 0011 1100 + 1 = 1101 = -8ten+4ten+1ten = -3ten

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Arithmetic

Binary arithmetic, addition and subtraction are similar to decimal:» work from least significant bit to most significant bit

– using carries where necessary.

A fundamental issue on a computer is the fixed word size» The operands and result is dependent on the word

length, and sometimes the result will be too small (negative) or too large (positive) to be represented.

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Addition and Subtraction Examples, n=4 (OVERFLOW)

0110+ 6ten

0111 7ten

------- ------

1101 = -8+4+1 = -3ten

Subtraction0111-0110 = 0111+1010 (two’s complement) =

0001 = 1ten

Overflow (Patterson)

° Examples: 7 + 3 = 10 but ... - 4 - 5 = - 9 but ...

2’s ComplementBinaryDecimal

0 0000

1 0001

2 0010

3 0011

0000

1111

1110

1101

Decimal

0

-1

-2

-3

4 0100

5 0101

6 0110

7 0111

1100

1011

1010

1001

-4

-5

-6

-7

1000-8

0 1 1 1

0 0 1 1+

1 0 1 0

1

1 1 0 0

1 0 1 1+

0 1 1 1

110

7

3

1

- 6

- 4

- 5

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Overflow Detection (Patterson)

° Overflow: the result is too large (or too small) to represent properly

• Example: - 8 < = 4-bit binary number <= 7

° When adding operands with different signs, overflow cannot occur!

° Overflow occurs when adding:

• 2 positive numbers and the sum is negative

• 2 negative numbers and the sum is positive

0 1 1 1

0 0 1 1+

1 0 1 0

1

1 1 0 0

1 0 1 1+

0 1 1 1

110

7

3

1

-6

- 4

- 5

7

0

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Designing a 1-bit Adder

A 1-bit adder must follow the following rules:-» a b R C» 0 0 0 0» 0 1 1 0» 1 0 1 0» 1 1 0 1

a, b inputs, R result, C carry.

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Logic Circuit for 1-bit Adder

Notice that » R = (~a).b + a.(~b)» C = a.b a

b

r

c

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n-bit Adder

Problem with previous circuit is that there is no way to combine a sequence of adders together » the carry of one addition must be passed to

the next. Need a structure as:

carryin

a

b r

carryout

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Truth Table

a b carryin result carryout0 0 0 0 00 0 1 1 00 1 0 1 01 0 0 1 01 0 1 0 11 1 0 0 10 1 1 0 11 1 1 1 1

Cout = a.(~b).c + a.b.(~c) + (~a).b.c + a.b.c = a.c+b.c+a.bResult = (~a).(~b).c + (~a).b.(~c) + a.(~b).(~c) + a.b.c

Logic diagramscan be constructedfor each of carryout and Result, using the results below.

Logic Diagrams for CarryOut and Sum (Patterson)

° CarryOut = B & CarryIn | A & CarryIn | A & B

° Result = A XOR B XOR CarryIn

CarryIn

CarryOut

A

B

A

B

CarryIn

Result

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Link 1-bit adders for n-bitcarryin

a

b r0

carryout/carry in

a

b r1

carryout

A chain of 1-bitadders is formedwhere the carryout of eachbecomes the carryin of thenext.The result is the sequenceof results from LSB to MSB.

A0

B0

1-bitALU

Result0

CarryIn0

CarryOut0

A1

B1

1-bitALU

Result1

CarryIn1

CarryOut1

A2

B2

1-bitALU

Result2

CarryIn2

CarryOut2

A3

B3

1-bitALU

Result3

CarryIn3

CarryOut3

(Patterson)

Notice howthe 1-bit addersare cascadedtogether.

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Summary

Binary logic Number representations

» two’s complement Binary arithmetic Constructing a 1-bit adder Constructing an n-bit adder ...next time --- floating point.