Computer Organization By Dr. 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 Bhatele Revised By: Dr. M. Khamis FAll 2008
Volts 1.8 0 1 0 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 analog voltages 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 as the logical values “true” and “false.” Number 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 Number 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. Number Systems and Boolean Algebra (1 x 102) + (6 x 101) + (2 x 100) + (3 x 10-1) + (7 x 10-2) + (5 x 10-3) = 162.375
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: Number Systems and Boolean Algebra (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
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 = 0.750 0.750 x 2 = 1.500 0.500 x 2 = 1.000 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 Number 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 Number Systems and Boolean Algebra 0.375 x 10 = 3.750 0.750 x 10 = 7.500 0.500 x 10 = 5.000
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. Number Systems and Boolean Algebra
Hex Binary Hex Binary Hex Binary Hex Binary 0 0000 4 0100 8 1000 C 1100 1 0001 5 0101 9 1001 D 1101 2 0010 6 0110 A 1010 E 1110 3 0011 7 0111 B 1011 F 1111 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 = 261 . 3516 = 001001100001 . 001101012 Number Systems and Boolean Algebra 10110100.0010112 = 10110100 . 001011002 = B4 . 2C16
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: Convert each bit in the value into its complement (1 to 0 and vice versa) 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 corresponding bits. This operation continues until adding all of the corresponding bits in the numbers.
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 corresponding bits. 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. Number 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 Number Systems and Boolean Algebra
Volts 1.8 0 True False Boolean values • Earlier, we used electrical voltages to represent two discrete values 1 and 0, from which binary numbers can be formed. • It’s also possible to think of voltages as representing two logical values, true and false. • For simplicity, we often still write digits instead: • 1 is true • 0is false • We will use this interpretation along with special operations to design functions and hardware for doing arbitrary computations. Number Systems and Boolean Algebra
An expression is finite but not unique A function table is unique but infinite f(x,y) = 2x + y = x + x + y = 2(x + y/2) = ... 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. Number Systems and Boolean Algebra
Basic Boolean operations • There are three basic operations for logical values. NOT (complement) on one input AND (product) of two inputs OR (sum) of two inputs Operation: Expression: xy, or xy x + y x’ Number Systems and Boolean Algebra Truth table:
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’) Number Systems and Boolean Algebra
f(x,y,z) = (x + y’)z + x’ f(0,0,0) = (0 + 1)0 + 1 = 1 f(0,0,1) = (0 + 1)1 + 1 = 1 f(0,1,0) = (0 + 0)0 + 1 = 1 f(0,1,1) = (0 + 0)1 + 1 = 1 f(1,0,0) = (1 + 1)0 + 0 = 0 f(1,0,1) = (1 + 1)1 + 0 = 1 f(1,1,0) = (1 + 0)0 + 0 = 0 f(1,1,1) = (1 + 0)1 + 0 = 1 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. Number 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. NOT (complement) on one input AND (product) of two inputs OR (sum) of two inputs Operation: Number Systems and Boolean Algebra Expression: xy, or xy x + y x’ Logic gate:
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’ Number 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’s inputs and outputs Number Systems and Boolean Algebra Find a Boolean expression for the circuit Find a truth table for the circuit
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. Number 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’ Number Systems and Boolean Algebra • Can we make this circuit “better”? • Cheaper: fewer gates • Faster: fewer delays from inputs to outputs
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 Number 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 magentaaxioms deal with the complement operation • Blue axioms (especially 15) are different from regular algebra Number 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 Number 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’) Number 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 Number 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 ] Number 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 Number 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: Number Systems and Boolean Algebra x + x’y = (x + x’)(x + y) [ Distributive ] = 1 (x + y) [ x + x’ = 1 ] = x + y [ Axiom 3 ]
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) Number 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] Number 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 Number Systems and Boolean Algebra
Parity Checking Parity checking is used to check the correctness or erroneous of data. Circuit used to raise 1 on its output when its input data contain even number of 1’s is called even parity circuit, otherwise it called odd parity circuit. Parity checking is used with memory to ensure the read data is exactly as written one or otherwise, it interrupts the main processor. Also, it is used within the modem to ensure the correct arrival of the received data.
Parity Checking with the Main Memory Data RAM Parity RAM To interrut processor Parity cct Parity cct
How is parity circuits detect the error? The table depicts the expected output of the parity checking during reading data (yellow circuit). Note that the output will be always1 when memory is defected, otherwise, it will be 0.
Design parity checking for 3-inputs For sake of simplicity 3-inputs are considered. The truth table will be as shown. The output (F) of the even parity circuit is 1 when the number of the 1’s in the inputs are even. F= xyz’ + xy’z + x’yz + x’y’z’ (Draw the corresponding circuit for the above logic expression to get the odd parity circuit for 3-inputs)
How to add binary numbers Consider adding two 1-bit binary numbers x and y 0+0 = 0 0+1 = 1 1+0 = 1 1+1 = 10 Carry is x AND y Sum is x XOR y The circuit to compute this is called a half-adder
The half-adder Sum = x XOR y Carry = x AND y
Using half adders We can then use a half-adder to compute the sum of two Boolean numbers 1 0 0 1 1 0 0 + 1 1 1 0 ? 0 1 0
How to fix this We need to create an adder that can take a carry bit as an additional input Inputs: x, y, carry in Outputs: sum, carry out This is called a full adder Will add x and y with a half-adder Will add the sum of that to the carry in What about the carry out? It’s 1 if either (or both): x+y = 10 x+y = 01 and carry in = 1