Upload
leonard-craig
View
217
Download
0
Embed Size (px)
Citation preview
Computer Organization
ByDr. M. Khamis
Mrs. Dua’a Al Sinari
Computer Organization
The course is aimed at designing the different computer components (circuits) and connecting these components in a way to achieve the goals of a specific architectures. Computer (hardware) consists of processor, memory and I/O
units. Processor itself consists of Arithmetic Logic Unit (ALU) and
Control unit.All the above units are designed using primitive logic circuits.
Course Objectives Understanding the basic Laws of Boolean algebra. Designing and using the basic logic devices. Understanding the operation of the main computer
units and their design. Interconnecting the various computer units to achieve
the specific architecture. Presenting the attributes of the different architectures. Programming specific architecture using its instruction
set (machine instruction). Explaining the Interaction between Computer hardware
and the operating system.
Course outline
The course will consist of two parts: The first part is Logic design: in which the primitive
components, by which the different devices are designed, are presented.
The second part is intended for interconnecting the components presented in first part in a way to build a logical system (computer organization).
Part 1: Logic Design
Introduction to number systems and arithmetic operations in binary system.
Combinational circuits: Logic Gates (AND, OR, NOT, NOR, NAND and XOR), in this
regards we will give the truth tables and symbols for each . Laws of Boolean algebra, deriving logical expression and
simplification. Karnaugh maps and its use for simplification . half and full adders and binary coded decimal adders
Part 1: Logic Design devices include:
Decoder. Encoder. Multiplexers/ De-multiplexer. Comparator.
Sequential circuits include : Flip/Flops and counter Design Mealy and MOORE machines.
Part 2: Computer Organization
chapter 3: computer system chapter 7: Input/output chapter 8: Operating System Support. chapter 9: Computer Arithmetic chapter 10: Instruction Sets. chapter 11: :Instruction Sets: Addressing Modes and
Format. chapter 12: CPU structure chapter 16 : Control Unit chapter 17: Micro Programmed Control Unit.
NUMBERS AND BOOLEAN ALGEBRA
Author: Abhinav BhateleRevised By: Dr. M. KhamisFAll 2008
NUMBER SYSTEMS To get started, we’ll discuss one of the fundamental concepts
underlying digital computer design:
Deep down inside, computers work with just 1s and 0s.
Computers use voltages to represent information. In modern CPUs the voltage is usually limited to 1.6-1.8V to minimize power consumption.
It’s convenient for us to translate these analogvoltages into the discrete, or digital, values 1 and 0.
But how can binary system be useful for anything? First, we’ll see how to represent numbers with
just 1s and 0s. Then we’ll introduce special operations
for computing with 1s and 0s, by treating them asthe logical values “true” and “false.”
Volts1.8
0
1
0
June 10th, 2008
9
Num
ber Systems and Boolean Algebra
TODAY’S LECTURE
Number systems Review of binary number representation How to convert between binary and decimal representations Octal and Hex representations
Basic boolean operations AND, OR and NOT The idea of “Truth Table” Boolean functions and expressions Truth table for Boolean expressions
June 10th, 2008
10
Num
ber Systems and Boolean Algebra
DECIMAL REVIEW
Numbers consist of a bunch of digits, each with a weight
These weights are all powers of the base, which is 10. We can rewrite this:
To find the decimal value of a number, multiply each digit by its weight and sum the products.
1 6 2 . 3 7 5 Digits 100 10 1 1/10 1/100 1/1000 Weights
1 6 2 . 3 7 5 Digits 102 101 100 10-1 10-2 10-3 Weights
(1 x 102) + (6 x 101) + (2 x 100) + (3 x 10-1) + (7 x 10-2) + (5 x 10-3) = 162.375
June 10th, 2008
11
Num
ber Systems and Boolean Algebra
CONVERTING BINARY TO DECIMAL We can use the same trick to convert binary, or base 2, numbers to
decimal. This time, the weights are powers of 2. Example: 1101.01 in binary
The decimal value is:
1 1 0 1 . 0 1 Binary digits, or bits 23 22 21 20 2-1 2-2 Weights (in base 2)
(1 x 23) + (1 x 22) + (0 x 21) + (1 x 20) + (0 x 2-1) + (1 x 2-2) = 8 + 4 + 0 + 1 + 0 + 0.25 = 13.25
Powers of 2: Useful abbreviations:
20 = 1 24 = 16 28 = 256 K = 210 = 1,02421 = 2 25 = 32 29 = 512 M = 220 = 1,048,57622 = 4 26 = 64 210 = 1024 G = 230 = 1,073,741,82423 = 8 27 = 128
June 10th, 2008
12
Num
ber Systems and Boolean Algebra
CONVERTING DECIMAL TO BINARY To convert a decimal integer into binary, keep dividing by 2 until the
quotient is 0. Collect the remainders in reverse order. To convert a fraction, keep multiplying the fractional part by 2 until
it becomes 0. Collect the integer parts in forward order. Example: 162.375:
So, 162.37510 = 10100010.0112
162 / 2 = 81 rem 0 81 / 2 = 40 rem 1 40 / 2 = 20 rem 0 20 / 2 = 10 rem 0 10 / 2 = 5 rem 0 5 / 2 = 2 rem 1 2 / 2 = 1 rem 0 1 / 2 = 0 rem 1
0.375 x 2 = 0.7500.750 x 2 = 1.5000.500 x 2 = 1.000
June 10th, 2008
13
Num
ber Systems and Boolean Algebra
WHY DOES THIS WORK? This works for converting from decimal to any base Why? Think about converting 162.375 from decimal
to decimal.
Each division strips off the rightmost digit (the remainder). The quotient represents the remaining digits in the number.
Similarly, to convert fractions, each multiplication strips off the leftmost digit (the integer part). The fraction represents the remaining digits.
162 / 10 = 16 rem 2 16 / 10 = 1 rem 6 1 / 10 = 0 rem 1
0.375 x 10 = 3.7500.750 x 10 = 7.5000.500 x 10 = 5.000
June 10th, 2008
14
Num
ber Systems and Boolean Algebra
BASE 16 IS USEFUL TOO The hexadecimal system uses 16 digits:
0 1 2 3 4 5 6 7 8 9 A B C D E F You can convert between base 10 and base
16 using techniques like the ones we just showed for converting between decimal and binary.
For our purposes, base 16 is most useful as a “shorthand” notation for binary numbers. Since 16 = 24, one hexadecimal digit is
equivalent to 4 binary digits. It’s often easier to work with a number like B4
instead of 10110100. Hex is frequently used to specify things like
32-bit IP addresses and 24-bit colors.
Decimal Binary Hex 0 0000 0 1 0001 1 2 0010 2 3 0011 3 4 0100 4 5 0101 5 6 0110 6 7 0111 7 8 1000 8 9 1001 9
10 1010 A 11 1011 B 12 1100 C 13 1101 D 14 1110 E 15 1111 F
Decimal Binary Hex 0 0000 0 1 0001 1 2 0010 2 3 0011 3 4 0100 4 5 0101 5 6 0110 6 7 0111 7 8 1000 8 9 1001 9
10 1010 A 11 1011 B 12 1100 C 13 1101 D 14 1110 E 15 1111 F
June 10th, 2008
15
Num
ber Systems and Boolean Algebra
BINARY AND HEXADECIMAL CONVERSIONS Converting from hexadecimal to binary is easy: just replace each hex digit
with its equivalent 4-bit binary sequence.
To convert from binary to hex, make groups of 4 bits, starting from the binary point. Add 0s to the ends of the number if needed. Then, just convert each bit group to its corresponding hex digit.
261.3516 = 2 6 1 . 3 516
= 0010 0110 0001 . 0011 01012
10110100.0010112 = 1011 0100 . 0010 11002
= B 4 . 2 C16
1111F1011B0111700113
1110E1010A0110600102
1101D100190101500011
1100C100080100400000
BinaryHexBinaryHexBinaryHexBinaryHex
June 10th, 2008
16
Num
ber Systems and Boolean Algebra
2’s complement Binary number can be represented using sign and
magnitude. If N bits are used to represent the number, then the last bit is used to hold the sign of the number while the other (N-1) bits are used to represent the value.
0 is used for +ve sign and 1 is used for -ve sign. To get the 2’s complement for any number follow the
following two steps:1. Convert each bit in the value into its complement (1 to 0
and vice versa)2. Add 1 to the result of step 1.
Binary addition & subtraction
If the number is +ve keep it in sign and magnitude form, otherwise represent the number (magnitude only) using its 2’s complement.
Add the binary numbers in the ordinary way as the decimal numbers.
The addition in decimal makes carry 1 for the next digit for each 10 collected in the sum, and the reset which will be less than 10 is left as result of the addition of the corresponding bits.
This operation continues until adding all bits with its corresponding bits in the other number.
Binary addition & subtraction (Continued)
The addition in binary is exactly the same as decimal with only one difference, which is, carry 1 is taken for the next digit for each 2 collected in the sum, and the reset which is less than 2 is left as result of the addition of the corresponding bit.
The addition is continued for all bits including the sign bit, and in order to get correct answer the number must be represented in enough number of bits.
Any carry after the sign bit is discarded. The value of the negative result is represented in the 2’s
complement (i.e. the actual value is the 2’s complement of the result once again).
NUMBER SYSTEMS SUMMARY Computers are binary devices.
We’re forced to think in terms of base 2. Today we learned how to convert numbers between binary, decimal
and hexadecimal. Also, we have seen:
We use 0 and 1 as abstractions for analog voltages. We showed how to represent numbers using just these two signals.
Next we’ll introduce special operations for binary values and show how those correspond to circuits.
June 10th, 2008
20
Num
ber Systems and Boolean Algebra
BOOLEAN OPERATIONS
So far, we’ve talked about how arbitrary numbers can be represented using just the two binary values 1 and 0.
Now we’ll interpret voltages as the logical values “true” and “false” instead. We’ll show: How logical functions can be defined for expressing
computations How to build circuits that implement our functions in
hardware
June 10th, 2008
21
Num
ber Systems and Boolean Algebra
BOOLEAN VALUES
Earlier, we used electrical voltages to representtwo discrete values 1 and 0, from which binary numberscan be formed.
It’s also possible to think of voltages as representingtwo logical values, true and false.
For simplicity, we often still write digits instead: 1 is true 0 is false
We will use this interpretation along with special operations to design functions and hardware for doing arbitrary computations.
Volts1.8
0
True
False
June 10th, 2008
22
Num
ber Systems and Boolean Algebra
FUNCTIONS Computers take inputs and produce outputs, just like functions in math!
Logical functions can be expressed in two ways: A finite, but non-unique Boolean expression. A truth table, which will turn out to be unique and finite.
We can represent logical functions in two analogous ways too: A finite, but non-unique Boolean expression. A truth table, which will turn out to be unique and finite.
x y f (x,y)
0 0 0… … …2 2 6… … …23 41 87… … …
f(x,y) = 2x + y= x + x + y= 2(x + y/2)= ...
An expression isfinite but not unique
A function table isunique but infinite
June 10th, 2008
23
Num
ber Systems and Boolean Algebra
BASIC BOOLEAN OPERATIONS There are three basic operations for logical values.
x y xy
0 0 0
0 1 0
1 0 0
1 1 1
x y x+y
0 0 0
0 1 1
1 0 1
1 1 1
x x’
0 1
1 0
AND (product)of two inputs
OR (sum) of two inputs
NOT (complement)on one input
xy, or xy x + y x’
Operation:
Expression:
Truth table:
June 10th, 2008
24
Num
ber Systems and Boolean Algebra
BOOLEAN EXPRESSIONS We can use these basic operations to form more complex expressions:
f(x,y,z) = (x + y’)z + x’
Some terminology and notation: f is the name of the function. (x,y,z) are the input variables, each representing 1 or 0. Listing the
inputs is optional, but sometimes helpful. A literal is any occurrence of an input variable or its complement. The
function above has four literals: x, y’, z, and x’. Precedences are important, but not too difficult.
NOT has the highest precedence, followed by AND, and then OR. Fully parenthesized, the function above would be kind of messy:
f(x,y,z) = (((x +(y’))z) + x’)
June 10th, 2008
25
Num
ber Systems and Boolean Algebra
TRUTH TABLES A truth table shows all possible inputs and outputs of a function. Remember that each input variable represents either 1 or 0.
Because there are only a finite number of values (1 and 0), truth tables themselves are finite.
A function with n variables has 2n possible combinations of inputs. Inputs are listed in binary order—in this example, from 000 to 111.
x y z f (x,y,z)
0 0 0 10 0 1 10 1 0 10 1 1 11 0 0 01 0 1 11 1 0 01 1 1 1
f(0,0,0) = (0 + 1)0 + 1 = 1f(0,0,1) = (0 + 1)1 + 1 = 1f(0,1,0) = (0 + 0)0 + 1 = 1f(0,1,1) = (0 + 0)1 + 1 = 1f(1,0,0) = (1 + 1)0 + 0 = 0f(1,0,1) = (1 + 1)1 + 0 = 1f(1,1,0) = (1 + 0)0 + 0 = 0f(1,1,1) = (1 + 0)1 + 0 = 1
f(x,y,z) = (x + y’)z + x’
June 10th, 2008
26
Num
ber Systems and Boolean Algebra
PRIMITIVE LOGIC GATES
Each of our basic operations can be implemented in hardware using a primitive logic gate. Symbols for each of the logic gates are shown below. These gates output the product, sum or complement of their
inputs.
Logic gate:
AND (product)of two inputs
OR (sum) of two inputs
NOT (complement)on one input
xy, or xy x + y x’
Operation:
Expression:
June 10th, 2008
27
Num
ber Systems and Boolean Algebra
EXPRESSIONS AND CIRCUITS Any Boolean expression can be converted into a circuit by
combining basic gates in a relatively straightforward way. The diagram below shows the inputs and outputs of each gate. The precedences are explicit in a circuit. Clearly, we have to make
sure that the hardware does operations in the right order!(x + y’)z + x’
June 10th, 2008
28
Num
ber Systems and Boolean Algebra
CIRCUIT ANALYSIS SUMMARY After finding the circuit inputs and outputs, you can come up with either an
expression or a truth table to describe what the circuit does. You can easily convert between expressions and truth tables.
Find the circuit’sinputs and outputs
Find a Booleanexpression
for the circuit
Find a truth tablefor the circuit
June 10th, 2008
29
Num
ber Systems and Boolean Algebra
BOOLEAN OPERATIONS SUMMARY We can interpret high or low voltage as representing true or false. A variable whose value can be either 1 or 0 is called a Boolean variable. AND, OR, and NOT are the basic Boolean operations. We can express Boolean functions with either an expression or a truth
table. Every Boolean expression can be converted to a circuit.
Next, we’ll look at how Boolean algebra can help simplify expressions, which in turn will lead to simpler circuits.
June 10th, 2008
30
Num
ber Systems and Boolean Algebra
BOOLEAN ALGEBRA Last time we talked about Boolean functions, Boolean expressions, and
truth tables. Now we’ll learn how to how use Boolean algebra to simplify Booleans
expressions. Last time, we saw this expression and converted it to a circuit:
(x + y’)z + x’
Can we make this circuit “better”?• Cheaper: fewer gates• Faster: fewer delays from inputs to outputs
June 10th, 2008
31
Num
ber Systems and Boolean Algebra
EXPRESSION SIMPLIFICATION Normal mathematical expressions can be simplified using the laws of
algebra For binary systems, we can use Boolean algebra, which is superficially
similar to regular algebra There are many differences, due to
having only two values (0 and 1) to work with having a complement operation the OR operation is not the same as addition
June 10th, 2008
32
Num
ber Systems and Boolean Algebra
FORMAL DEFINITION OF BOOLEAN ALGEBRA A Boolean algebra requires
A set of elements B, which needs at least two elements (0 and 1) Two binary (two-argument) operations OR and AND A unary (one-argument) operation NOT The axioms below must always be true (textbook, p. 42)
The magenta axioms deal with the complement operation Blue axioms (especially 15) are different from regular algebra
1. x + 0 = x 2. x 1 = x 3. x + 1 = 1 4. x 0 = 0 5. x + x = x 6. x x = x 7. x + x’ = 1 8. x x’ = 0 9. (x’)’ = x
10. x + y = y + x 11. xy = yx Commutative 12. x + (y + z) = (x + y) + z 13. x(yz) = (xy)z Associative 14. x(y + z) = xy + xz 15. x + yz = (x + y)(x + z) Distributive 16. (x + y)’ = x’y’ 17. (xy)’ = x’ + y’ DeMorgan’s
June 10th, 2008
33
Num
ber Systems and Boolean Algebra
COMMENTS ON THE AXIOMS The associative laws show that there is no ambiguity about a term such as x
+ y + z or xyz, so we can introduce multiple-input primitive gates:
The left and right columns of axioms are duals exchange all ANDs with ORs, and 0s with 1s
The dual of any equation is always true
1. x + 0 = x 2. x 1 = x 3. x + 1 = 1 4. x 0 = 0 5. x + x = x 6. x x = x 7. x + x’ = 1 8. x x’ = 0 9. (x’)’ = x
10. x + y = y + x 11. xy = yx Commutative 12. x + (y + z) = (x + y) + z 13. x(yz) = (xy)z Associative 14. x(y + z) = xy + xz 15. x + yz = (x + y)(x + z) Distributive 16. (x + y)’ = x’y’ 17. (xy)’ = x’ + y’ DeMorgan’s
June 10th, 2008
34
Num
ber Systems and Boolean Algebra
ARE THESE AXIOMS FOR REAL? We can show that these axioms are valid, given the definitions of AND, OR
and NOT
The first 11 axioms are easy to see from these truth tables alone. For example, x + x’ = 1 because of the middle two lines below (where y = x’)
x y xy 0 0 0 0 1 0 1 0 0 1 1 1
x y x+y 0 0 0 0 1 1 1 0 1 1 1 1
x x’ 0 1 1 0
x y x+y 0 0 0 0 1 1 1 0 1 1 1 1
June 10th, 2008
35
Num
ber Systems and Boolean Algebra
PROVING THE REST OF THE AXIOMS We can make up truth tables to prove (both parts of) DeMorgan’s law For (x + y)’ = x’y’, we can make truth tables for (x + y)’ and for x’y’
In each table, the columns on the left (x and y) are the inputs. The columns on the right are outputs.
In this case, we only care about the columns in blue. The other “outputs” are just to help us find the blue columns.
Since both of the columns in blue are the same, this shows that (x + y)’ and x’y’ are equivalent
x y x + y (x + y)’ x y x’ y’ x’y’ 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 1 0 0 0
June 10th, 2008
36
Num
ber Systems and Boolean Algebra
SIMPLIFICATION WITH AXIOMS We can now start doing some simplifications
x’y’ + xyz + x’y= x’(y’ + y) + xyz [ Distributive; x’y’ + x’y = x’(y’ + y) ]= x’1 + xyz [ Axiom 7; y’ + y = 1 ]= x’ + xyz [ Axiom 2; x’1 = x’ ]= (x’ + x)(x’ + yz) [ Distributive ]= 1 (x’ + yz) [ Axiom 7; x’ + x = 1 ]= x’ + yz [ Axiom 2 ]
1. x + 0 = x 2. x 1 = x 3. x + 1 = 1 4. x 0 = 0 5. x + x = x 6. x x = x 7. x + x’ = 1 8. x x’ = 0 9. (x’)’ = x
10. x + y = y + x 11. xy = yx Commutative 12. x + (y + z) = (x + y) + z 13. x(yz) = (xy)z Associative 14. x(y + z) = xy + xz 15. x + yz = (x + y)(x + z) Distributive 16. (x + y)’ = x’y’ 17. (xy)’ = x’ + y’ DeMorgan’s
June 10th, 2008
37
Num
ber Systems and Boolean Algebra
LET’S COMPARE THE RESULTING CIRCUITS Here are two different
but equivalent circuits. In general the one
with fewer gates is “better”: It costs less to
build It requires less
power But we had to do
some work to find the second form
June 10th, 2008
38
Num
ber Systems and Boolean Algebra
SOME MORE LAWS Here are some more useful laws. Notice the duals again!
We can prove these laws by either
Making truth tables:
Using the axioms:
1. x + xy = x 4. x(x + y) = x 2. xy + xy’ = x 5. (x + y)(x + y’) = x 3. x + x’y = x + y 6. x(x’ + y) = xy xy + x’z + yz = xy + x’z (x + y)(x’ + z)(y + z) = (x + y)(x’ + z)
x y x’ x’y x + x’y x y x + y 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1
x + x’y = (x + x’)(x + y) [ Distributive ]= 1 (x + y) [ x + x’ = 1 ]= x + y [ Axiom 3 ]
June 10th, 2008
39
Num
ber Systems and Boolean Algebra
THE COMPLEMENT OF A FUNCTION The complement of a function always outputs 0 where the original function
outputted 1, and 1 where the original produced 0. In a truth table, we can just exchange 0s and 1s in the output column(s)
f(x,y,z) = x(y’z’ + yz)
x y z f(x,y,z) 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 0
x y z f’(x,y,z) 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 1 1 1
June 10th, 2008
40
Num
ber Systems and Boolean Algebra
COMPLEMENTING A FUNCTION ALGEBRAICALLY You can use DeMorgan’s law to keep “pushing” the complements inwards
You can also take the dual of the function, and then complement each literal If f(x,y,z) = x(y’z’ + yz)… … the dual of f is x + (y’ + z’)(y + z)… … then complementing each literal gives x’ + (y + z)(y’ + z’)… … so f’(x,y,z) = x’ + (y + z)(y’ + z’)
f(x,y,z) = x(y’z’ + yz)
f’(x,y,z) = ( x(y’z’ + yz) )’ [ complement both sides ]= x’ + (y’z’ + yz)’ [ because (xy)’ = x’ + y’ ]= x’ + (y’z’)’ (yz)’ [ because (x + y)’ = x’ y’ ]= x’ + (y + z)(y’ + z’) [ because (xy)’ = x’ + y’, twice]
June 10th, 2008
41
Num
ber Systems and Boolean Algebra
SUMMARY SO FAR So far:
A bunch of Boolean algebra trickery for simplifying expressions and circuits
The algebra guarantees us that the simplified circuit is equivalent to the original one
Next: Introducing some standard forms and terminology An alternative simplification method We’ll start using all this stuff to build and analyze bigger, more useful,
circuits
June 10th, 2008
42
Num
ber Systems and Boolean Algebra