Chapter 3 Computer Organization Boolean Algebra and Digital Logic

Boolean Operators

1. AND 2. OR 3. NOT

SOP Kmap

1. Group the biggest possible group of 1s 2. Drop the term that differs between the groups 3. Any terms remaining that have an input of 0 turn to NOTs 4. Repeat for all groups until expression is complete EX: A`CD + ABCD + AB`C` + AB`C goes to AB`+CD

unsigned integer

A data type that stores positive integer values as ordinary binary numbers; its value is always assumed to be positive.

D Flip-Flop

A flip-flop with one data input that stores the value of that input signal in the internal memory when the clock edge occurs *stores 1 bit of data Output value of 1 means circuit is currently storing a value of 1

mealy machine

A state machine whose output is determined by both the sequential logic and the combinational logic of the machine. associates an output for each transition. outputs are a function of it's current state and it's input

moore machine

A state machine whose output is determined only by the sequential logic of the machine. Output is a function of only its current state. outputs depend only on the present state

redundancy law

AB + A`C+BC = AB + A`C

DeMorgan's Law

AND form: (xy)` = x` + y` OR form: (x+y)`= x`y`

associative law

AND form: (xy)z = x(yz) OR form: (x+y)+z = x + (y+z)

null (dominance) law

AND form: 0x = 0 OR form: 1 + x = 1

identity law

AND form: 1x = x OR form- 0+x=x

distributive law

AND form: x + (yz) = (x+y)(x+z) OR form: x(y+z) = xy + xz

integrated circuit (IC)

An electronic circuit having many components, such as transistors, diodes, resistors, and capacitors, in a single package.

Edge-triggered

Certain triggering that require a change in the clock signal as a condition for switching states EX: flip-flop

operator priority

Prioritized as 1. () 2. NOT 3. AND 4. OR

absorption law

AND form: x(x+y)=x OR form: x+xy=x

double complement law

AND form: x`` = x

idempotent law

AND form: xx = x OR form: x + x = x

Inverse Law

AND form: xx` = 0 OR form: x + x` = 1

Commutative law

AND form: xy = yz OR form: x + y = y+x

data word

The number of bits that a CPU retrieves from a memory for processing in a one machine cycle. When all other conditions are equal, a machine with a larger data word is faster

parity generator

a circuit that creates the necessary parity bit to add to a word

deterministic finite automata

a finite automaton that has at most one transition from a state for each input symbol and no empty transitions. Abbreviated DFA.

majority function

a function that, when given 3 inputs, outputs a 0 if less than half of its inputs are 1 and a 1 if at least half of its inputs are 1

karnaugh maps

a graphical way to represent boolean functions Used as a table to enumerate the values of a given boolean expression for different input value Rows and columns correspond to the possible values of the function's inputs (from truth table) Each cell represents the outputs of the function for those possible inputs

essential prime implicant

a group of 1s that contains at least one 1 that is not part of any prime implicant

prime implicant

a group of 1s that isn't contained in any other groups of 1s

finite state machine

a machine that consists of a fixed set of possible states with a set of allowable inputs that change the state and a set of possible outcomes EX: traffic stop: Goes from green to yellow to red, but can only go from red to green

convolutional coding

a method that operates on an incoming serial bit stream, generating an encoded serial output stream that enables it to correct errors continuously

implicant

a rectangular grouping of 1s that contain any power of 2 (1,2,4,8)

half-adder

a simple combinational circuit, can only add two bits together

embedded systems

a single programmed microcomputer chip

chip

a small electronic device consisting of the necessary electronic components (transistors, resistors, and capacitors) to implement various gates *components are etched directly on the chip, allowing it to require less power

flip-flop

a storage element used for sequential circuits *State of the flip flop is a function of the previous inputs to the circuit *Output depends on the current inputs and the current state of the circuit

hamming code

a type of forward error correction that uses blocks of data (block coding) to compute the necessary redundant bits

boolean variables

a variable which stores a Boolean value (true or false). *Usually represented with a choice of ABCD or XYZ *can be reordered and regrouped

memory addresses

addresses in a computer is specified as binary numbers 1. when a memory address is referenced, computer has to determine the actual address (done using a decoder) 2. When figuring out how many bits are needed, use the base 2 then use how many addresses you have and solve for x

boolean algebra

algebra for the manipulation of objects that can take on only 2 values (true and false) but can be any pair of values *natural way to represent digital information *0 = false, 1 = true

ASM

algorithmic state machine expresses algorithms that administer an FSM from one state to another Consists of blocks that contain a state box, a label, and optionally condition and output boxes Each block has 1 entry point and at least 1 exist point Moore type outputs (circuit signals) indicated inside the state block Mealy type outputs- indicated in the oval output box

level triggered

allowed to change state whenever the clock signal is either high or low

black box

allows us to ignore details of the actual gates and shows us the inputs and outputs of the circuits

convolutional code

an error correcting code that uses parity bits in such a way that only the parity bits (not the parity bits + the message) needs to be transmitted *An encoding process whereby the output is a function of the input and some number of bits previously received. Input is overlapped over itself to form a stream of output symbols

SR flip flop

basic memory unit S stands for set, R stands for reset Combination of 1 and 1 is undefined in this type of flip flop

parity checker

checks to make sure proper parity (odd or even) is present in the word, detecting an error if the parity bit is incorrect

encoder

combinational circuit with 2^n input lines with n output lines *opposite of decoder

boolean expressions

combining the variables and operators yield B_______ E___________

product of sums form

consist of ORed variables that are ANDed together EX: (x+y)(x+z`)(y+z`)(y+z)

sequential circuits

defines its output as a function of both its current inputs and its previous inputs Output depends on past inputs

machines

devices that respond to a set of stimuli (events) by generating predictable responses (actions) based on a history of prior events (current state)

logic families

different types of logic devices that belong to different production technologies Include TTL, NMOS/PMOS, CMOS and ECL

boolean duality principle

each function/property has an AND form and an OR form

synchronous

edge triggered use clocks to order events

XOR

exclusive or Returns false if both inputs are equal, and returns true otherwise

canonical form

for any given boolean function thee is a unique standardized form *Different standards that designers use *2 most common are the sum of products form and the product of sums form

boolean function

has one or more input values and yields a result based on the input values in the set

minterm

if a product terms includes all of the variables exactly once (complemented or not complemented) EX: for 2 inputs (x and y), m________ would be x'y`, x`y, xy`, xy Represents all the possible input combinations for the function

gates

implement each of the basic logic functions 1. building blocks for digital design 2. small electronic device that computes various functions of two-valued signals 3. basically just implements a simple boolean function

characteristic table

indicates what the next state should be based on the inputs and the current state Q Q(t) = current state Q(t+1) = next state or the state the flip flop should enter after the clock has been pulses

bit masking

individual bits in a byte are stripped (set to 0) according to a specified pattern 1. good for processing individual bits in a byte 2. Place a 1 in the position of each bit you want to keep, set the others to 0

Asynchronous

level triggered. become active any input value changes

bit shifting

moves the bits of a word or byte one position to the left or right *Shifting a bit to the left takes it to the position of the next higher power of two *When the bits of an unsigned integer are shifted to the left by one position, has the same effect as multiplying that integer by 2

NAND

produces complementary output to AND Always true unless both inputs are 1,then it returns false *Universal gate because any electronic circuit can be made using only this type of gate Cheap to build and easier to build complex circuits

NOR

produces complementary output to OR Always false unless both inputs are 0, then it returns true

viterbi decoder

reads the bit that have been output by a conventional encoder and compares the symbol stream read with a set of probable symbol stream

AND

represented as a dot or no symbol at all *Basically a boolean product EX: xy = x*y= x and y

NOT

represented as an overscore (-) or prime (`) EX: x` and read as n__ x

OR

represented by plus sign, referred to as boolean sum *Expression is x + y

sum of products

requires the expression to be a collection of ANDed variables that are ORed together EX: F(x,y,z) = xy + yz` + xyz

JK flip flop

same thing as an SR flip flop except that when both inputs are 1, the circuit negates the current state

multiplexer

selects binary info from one of many input lines and directs it to a single output line 1. Selection of a particular input line is controlled by a set of selection variables or control lines 2. Only one input is routed through the circuit to the output line, other inputs are cut off 3. basically just a data selector

truth table

shows the relationship between the input values and the result of a specific boolean operator or function on the input variables

transistors

small electrical devices that could receive and amplify radio signals *basically just digital switches

SSI chips

the first integrated circuits only contained up to 100 electronic components per chip

digital circuits

the physical components that perform arithmetic operations or make choices in a computer, constructed from a number of primitive elements called gates

ripple carry adder

the sequential generation of carries ripple through the adder stages

check board configuration

this is when there are 4 individual 1's that cannot be grouped together S = A XOR B XOR C

feedback

to remember a past state, simply means the output of a circuit is fed back as input to the same circuit

ttl

transistor logic replaces all diodes originally found in IC's with bipolar transistors 0-.8V = logic 0 2_5 V = logic 1

ULSI

ultra large scale integration Has more than 1 million electronic components per chip

combinational logic

used to build circuits that contain basic boolean operators, inputs and outputs 1. an output with c________ logic is always based on the given inputs 2. Output of c_____________- circuit = function of inputs 3. Can have several outputs and each output represents a different boolean function

clocks

used to order events A circuit that emits a series of pulses with a precise pulse width and a precise interval between consecutive pulses Interval is called the clock cycle time, measured in MHz or GHz Used to decide when to update the state of the circuit

decoder

uses the inputs and their respective values to select one specific output line has n input lines and 2^n output lines 2. one unique output line is set to 1, while other output lines are set to 0

POS Kmaps

1. Group the biggest possible groups of 0's 2. Drop the term that differs 3. Any terms that have an input of 1 turn to NOTs 4. Make terms with the sums 5. Repeat for all terms until expression is complete EX: (A+C)(A`+B`+C)(A+C`+D)(A`+B`+D)

boolean identities

1. Identity law 2. Null (dominance) law 3. Idempotent Law 4. Inverse Law 5. Commutative law 6. Associative law 7. Distributive law 8. Absorption law 9. DeMorgan's Law 10. Double complement law 11. Redundancy Law

KMAP rules

1. Only 1s can be grouped 2. can only group 1s in the kmap if they are in the same row or in the same column (can't be diagonal) 3. Can groups 1's if the total number in the group is a power of 2 4. Make the groups as large as possible and all 1's must be in a group 6. Overlapping and wraparound groups are allowed 7. Write down the function

using a memory circuit

1. an address is asserted on S0 and S1 2. Write enable (WE) is set to high 3. The decoder using S0 and S1 enables only one AND gate which selects a given word in memory 4. The line from step 3 combines with the clock and write enable select only one word 5. write gate enabled in step 4 drives the clock for the selected word 6. when the clock pulses, the word on the input lines is loaded into the D flip flops

analyzing a circuit

1. determine the input equations for the flip flops 2. determine the output equations for the outputs 3. find the net state equations for each flip flop (using the input equations) 4. use kMAPS to simplify the next state equations 5. Construct the truth table

designing a circuit

1. read the problem carefully to determine the input and output values 2. establish a truth table that shows the output for all possible inputs 3. convert the truth table into a boolean expression 4. simplify the boolean expression

adding binary numbers

1. uses 3 inputs (x,y, carry in) and 2 outputs (sum and carryout) 2. use a full adder and feed the carry out of one circuit into the carry in of the circuit immediately to its left

4by3 memory

4 D flipflops 2 control line decoder results in 4 output lines to represent the words