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Homework Due

• Book work chapter 4• How is the palindrome program coming along?

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Logic Microoperations

• Counter-part to the arithmetic microoperations we looked at last week

• Similar to Boolean functions (expressions) except that the operands are registers (bit strings) rather than individual Boolean variables– This is exactly what you implemented in code

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Register Transfer Language

• We need some new symbols to avoid (or cause) confusion

R1 R2R1011010110100111001001010

Bitwise AND

R1 R2R1011010110100111001101111

Bitwise OR

R2R1 0100111010010001

Bitwise NOT

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Context

will always be a bitwise AND microop.

will always be a bitwise OR microop.

will always be a single bit AND

will be a single bit OR in a Boolean function (never a bitwise OR microop.)

will be an arithmetic ADD in an arithmetic function

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Context

R1 R2,R1P + Q: R4 R5R3

What operation is this?

What operation is this?

What operation is this?

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Logical Microoperations

• Given two binary variables there are 16 different logical microoperations that can be performed– How can this be given that there are only 4

combinations of input values?

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Logical Microoperations

Inputs Outputs

x y F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15

0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1

1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

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Logical Microoperations

• Some of them are not intuitive (or useful)

• They can all be created using AND, OR, NOT, and XOR functions– You should be able to create the table of

functions (page 110, Table 4-6)

• How would you go about doing this?

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Hardware Implementation

• We know how to implement Boolean functions using combination logic

• How do we implement them to perform bitwise microoperations?

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Hardware Implementation

0

1

2

3

S0

S1

Fi

Ai

Bi

4x1MUX

• If you need n-bit microoperations (n-bit registers) you have n of these circuits

• …with a lot of logic gates!

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Hardware Implementation

• But doesn’t that MUX only implement 4 expressions?

• What about the other 12?• We can use a bigger MUX or realize that

many of the other 12 are useless and the ones that are useful can be derived from the 4 that we implemented, keeping hardware costs down

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But Why?

• What can be done with logic microoperations?– Most useful when we consider the bits of a

given register as separate signals

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• In Java if we need 8 boolean variables we generally do something like this…

• This may be very inefficient dependent on how the Java compiler generates the byte code

Logic Microoperations

boolean b7 = false;boolean b6 = false;…boolean b0 = false;

ORboolean b[] = new boolean[8];for (int i = 0; i < 8; ++i) {b[i] = false;

}

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• A more memory efficient way to represent the same information would be…

• This is referred to as “bit packing”

• But how do we get to the individual bits within the byte?

Logic Microoperations

byte b = 0;

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Logic Microoperations

• Through bitwise logic operations– Selective-set– Selective-complement– Selective-clear– Mask– Insert– Clear

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• We want to set certain bits to 1 while leaving the others unchanged

• Any ideas?– Bitwise OR

Selective-Set

01101010? 1111000011111010

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Selective-Complement

• We want to complement (invert) certain bits while leaving the others unchanged

• Any ideas?– Bitwise XOR

01101010? 1111000010011010

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Selective-Clear

• We want to clear (0) certain bits while leaving the others unchanged

• Any ideas?– Bitwise A AND B’

01101010? 1111000000001010

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Mask

• Similar to Selective-clear but the “selection” is done with 0’s rather than 1’s

• Any ideas?– Bitwise AND

01101010? 1111000001100000

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Insert

• Insert (replace) a group of bits into the existing bit string while leaving other bits unchanged

• Any ideas?– Mask followed by bitwise OR

01101010? 1101xxxx11011010

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Clear

• Compare the two values and if they are equal, produce a string of 0’s

• Any ideas?– Bitwise XOR with self

01101010? 0110101000000000

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Usage

• These types of operations are especially useful when designing hardware

• But, they are also heavily used in software– Device drivers (s/w that talks to h/w)– Embedded systems (small programmable h/w

devices)– High speed applications such as games or other

video applications

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Shift Microoperations

• Serial modification/transfer of bits similar to the shift registers we looked at previously

• Three basic types of shift– Logical shift– Circular shift (often called rotation)– Arithmetic shift

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Logical Shift

• Acts like a shift register with a 0 shifted in to the vacated bit position– Shifts can be to the left or the right– Bits that are shifted “off the end” are lost

1110101011010100

01110101

R1 ← shl R1

R1 ← shr R1

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Circular Shift

• Acts like a shift register with the end bit shifted in to the vacated bit position– Shifts can be to the left or the right– No information is lost

1110101011010101

01110101

R1 ← cil R1

R1 ← cir R1

R1 ← rotl R1

R1 ← rotr R1

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Arithmetic Shift

• Shifts are performed with special attention paid to the MSB (the sign bit – 2’s complement notation)– Left arithmetic shifts correspond to integer divide by

powers of 2• Division must leave the sign bit unchanged

• Division cannot causes an overflow

– Right arithmetic shifts correspond to integer multiply by powers of 2

• Multiplication must leave the sign bit unchanged

• Multiplication must be overflow aware

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Arithmetic Shift Right

• Leaves the sign bit unchanged and shifts the number (including the sign bit) to the right

• Did it work as expected?

10101010 11010101R1 ← ashr R1

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Arithmetic Shift Left

• Inserts 0 into LSB (why?)• If MSB changes after the shift an overflow has

occurred (why?)• Can predict overflow by comparing MSB (bit n-1)

to next bit (bit n-2)

• Did it work as expected?

• Did it work as expected?

11101010 11010100R1 ← ashl R1

01101010 11010100R1 ← ashl R1

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Arithmetic Shift Left

• What is the “overflow rule”?

• How can you implement it?– Check before performing the shift

• Bn-1 XOR Bn-2

• If 0 there is no overflow, if 1 there is an overflow with a sign reversal

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Implementation

• The arithmetic, logic, and shift microoperations are generally implemented in a single “unit” – the ALU (Arithmetic-Logic Unit)

• The ALU is generally made up of combinational circuits (as opposed to sequential circuits)– Why should this be the case?

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ALU Block DiagramStage i

Arithmetic Circuit Stage i

Logic Circuit Stage i

4x1MUX

S3

S2

S1

S0

shl

Cin

Cout

Bi

Ai

Ai-1

Ai+1

shr

Fi

select

0123

• Combinational circuits that we looked at previously are inserted into the boxes

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Table of MicrooperationsOperation Select Resultant operation/function

S3 S2 S1 S0 Cin Operation Function

0 0 0 0 0 F=A Transfer

0 0 0 0 1 F=A+1 Increment

0 0 0 1 0 F=A+B Add

0 0 0 1 1 F=A+B+1 Add w/carry

0 0 1 0 0 F=A+B’ Sub w/borrow

0 0 1 0 1 F=A+B’+1 Subtract

0 0 1 1 0 F=A-1 Decrement

0 0 1 1 1 F=A Transfer

0 1 0 0 x F=A B AND

0 1 0 1 x F=A B OR

0 1 1 0 x F=A B XOR

0 1 1 1 x F=A’ Complement

1 0 x x x F=shr A Shift right

1 1 x x x F= shl A Shift left

Homework

• 4-12, 4-17, 4-18, 4-19, 4-20, 4-21, 4-22• Read chapter 5

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