DIGITAL LOGIC : Sum Of Products

DIGITAL  LOGIC

Subtopic  : Sum of Products (SOP)

What is Sum of Products?

Sum of products is any mathematical expressions in which addition operator is applied to two or  more subexpressions each of which is an application of a multiplication operator. The most common case would be scalar addition and multiplication.

e.g . : ab+cd

But the terms is also used for other kinds of operators with similar properties  such as AND or OR in Boolean Algebra.

e.g.  : (a AND b) or ( c AND d)

All Boolean equation can be represented in two forms :  Sum of Products and Product of Sums. But we will be focusing only on Sum of Products because it is easier.

Sum of Products(SOP)

→ Combination of input values that produced 1s is converted into equivalent variables, ANDed together then ORed with other combination variables with the same output.

 Now I will show you guys an example on how to find SOP  from truth table.

The  equation for the truth table is  F = A’B’+ABC’D

The truth table :

A	B	C	D	F
0	0	0	0	1
0	0	0	1	1
0	0	1	0	1
0	0	1	1	1
0	1	0	0	0
0	1	0	1	0
0	1	1	0	0
0	1	1	1	0
1	0	0	0	0
1	0	0	1	0
1	0	1	0	0
1	0	1	1	0
1	1	0	0	0
1	1	0	1	1
1	1	1	0	0
1	1	1	1	0

Now, based on the equation,what you should do is to choose those equations which has the  value 1 as the product and convert it into equation.

note : A = 1   A’=0

first row :

A=0 B=0 C=0 D=0 when converted into equation would be ,

A’B’C’D’

Second row :

A=0  B=0  C=0  D=1  when converted into equations would be,

A’B’C’D

Third row :

A=0  B=0  C=1  D=0 when converted into equations would be,

A’B’CD’

Fourth row :

A=0  B=0  C=1  D=1 when converted into equations would be,

A’B’CD

Fourteenth row :

A=1  B=1  C=0  D=1 when converted into equations would be,

ABC’D

Next, what you do is you combine all of them into one equation.

SOP Equation :

F = (A’B’C’D’)+( A’B’C’D)+( A’B’CD’)+( A’B’CD)+( ABC’D)

*The equation can also be used to build truth table*

Incase you guys dont understand, I've inserted a good tutorial video from Youtube. I do not own the video. It is for academic purpose only. I hope you guys find this post helpful.

By NURUL HIDAYAH BINTI ISMAIL  (B031310124) J

INPUT/OUTPUT

Assalamualaikum and hello guys!
This post will discuss about new topic.This topic is about input and output computer organization.In this topic there all have six subtopic.But today i will not discuss all the subtopics.I will just discuss about accessing I/O devices.

OUTLINE

>Accessing I/O devices

>Interrupts                

>Direct memory access

>Buses                      

>Interface circuits       

>Standard I/O            

interfaces          

  

CONTENT COVERAGE

ACCESSING I/O DEVICES

>Single-bus structure

• The bus enable all the devices connected to it to exchange information.
• consists of three sets of lines used to :- 

         - carry address 

         - data

         - control signals

• Each I/O devices is assigned a unique set of addresses

I/O MAPPING

Memory mapped I/O

>Devices and memory share an address space 

>I/O looks just like memory read/write 

>No special commands for I/O

• large selection of memory access commands available

Isolated I/O

>Separate address spaces

>Need I/O or memory select lines 

>Specials commands for I/O

• limited set

MEMORY-MAPPED I/O

>I/O devices and the memory share the address space,so the arrangement is called      memory-mapped I/O

>Any machine instruction that can access memory can be used to transfer data or from an  I/O device using memory-mapped

>Most computer system use memory-mapped I/O

>Some processor have special IN and OUT instructions to perform I/O transfers

I/O INTERFACE FOR AN INPUT DEVICE

NURFARAH ALIA BINTI KAMARUDDIN 

B031310432

MEMORY ORGANIZATION (PART 1)

Assalamualaikum and Hello guys!

This post we will discuss about Cache Memory.

CACHE MEMORY 

• A typical computer program flows in a straight-line fashion with program loops and subroutine calls encountered frequently.
• If the active portions of the program and data are placed in a fast small memory,the average memory access time can be reduced.The memory known as Cache memory.
• The performance of cache memory is frequently measured in terms of a quantity called hit ratio.
• When the CPU refers to memory and finds the word in cache,its said to produce a hit.
• If the word is not found in cache,its in main memory and its counts as a miss.

CHARACTERISTIC of CACHE MEMORY

• It is fast access time.
• Mapping process is a transformation of data from main memory to cache memory.
• 3 types of mapping process are: 

  

    ASSOCIATIVE MAPPING 

The fastest and most flexible cache organization uses in associative memory.There have stores both the address and content (data) of the memory word.If the address is found corresponding data is read otherwise main memory is accessed for the word.This is expensive compared to random-access memories.

EXAMPLE FOR
ASSOCIATIVE MAPPING



DIRECT MAPPING

The direct mapping cache organization uses the n-bit address to access the main memory and the k-bit to access the cache.Each word in cache consists of the data word and associated tag.

EXAMPLE FOR
DIRECT MAPPING





SET- ASSOCIATIVE
MAPPING
In which each word of cache can store two or more words of memory under the same index address.Each data word is stored together with its tag and the number of tag-data items in one word of cache is said to form a set.

EXAMPLE FOR
SET-ASSOCIATIVE MAPPING

AHMAD AMNI FARHAN BIN MISHADI
B031310306

       

ARITHMETIC FOR COMPUTERS (Part 2)

HELLO AND ASSALAMUALAIKUM DEAR READERS :)

This post will discuss about the IEEE Standard 725 floating-point formats. Sorry for the bad audio quality, so do turn up the volume & wear your headphones/earphones. Below this video link, there is a link where you can actually download the PowerPoint file. If you have any questions or doubts that you would like to ask us, do not hesitate to post your comments down below. We will try our best to answer them for you. Thank you for reading this post, and have a good day. :)

https://www.dropbox.com/s/i7od0z0x5lt0a2m/Presentation2.pptx

WAN NUR IZZIE BINTI WAN RUSLIM

B031310315

MEMORY ORGANIZATION

Assalamualaikum and Hello!

We will discuss new chapter : Memory Organization.

THE SUBTOPICS
IN MEMORY ORGANIZATION

In the introduction we will learn about memory,memory heirarchy and memory management system.In this topic we will know how the memory communicates directly with CPU.We can also know how many operating system are designed.

INTRODUCTION

In main memory topic we will learn the Principal Technology,Static RAM and Dynamic RAM.

The Static RAM and Dynamic RAM are operating mode which is available to The Integrated circuit chip.

MAIN MEMORY

NURFARAH ALIA BINTI KAMARUDDIN

B031310432

NURUL HIDAYAH BINTI ISMAILL

B031310124

ARITHMETIC FOR COMPUTERS (Part 1)

Assalamualaikum and Hello everyone.

Let's start our discussion with Arithmetic for Computers.

This chapter will include with :

• Operations on Integers
• Floating point real numbers

INTEGER ADDITION

 Computers represent integers in binary (base2)

3 = 11, 4 = 100, 5 = 101, 30 = 11110

      + Natural since only two values are represented

• Addition, etc. take place as usual (carry the 1, etc.)

        

        17 = 10001

        +5 = 101

____________________

        22 = 10110

• Some old machines use decimal (base10) with only 0/1

30 = 011 000

– Unnatural for digial logic, implementation complicated & slow.

1st Grade: Decimal Addition

  1 111111

 43 = 00101011

+29 = 00011101

___________________

 72 = 01001000

• Overflow if result out of range

- Adding +ve and -ve operands,no overflow

- Adding two +ve operands

• Overflow if result sign is 1

- Adding two -ve operands

• Overflow if result sign is 0

INTEGER SUTRACTION

Add negation of second operand

           0 = 100000000

         -10 = 00001010

      _____________________

        -10 = 011110110

• Now, add the conventional way…

         -10 = 11110110

        +10 = 00001010

      ____________________

          0 = 100000000

• Overflow if result out of range

              - Subtracting two +ve or two -ve operands, no overflow

• Subtracting +ve from -ve operand

              - Overflow if result sign is 0

• Subtracting -ve from =ve operand

              - Overflow if result sign is 1

DEALING WITH OVERFLOW

•  Some languages ignore overflow
• Other languages require raising an exception

                - Use MIPS add, addi, sub instructuctions

                - On overflow, invoke exception handler

                - Save PC in exception program counter (EPC) register

                - Jump to predefined handler acccess

MULTIPLICATION'

If you use the last row in multiplication, you should get exactly the same result as using the first row (

after all they represent the same number.

DIVISION

Divisiion in MIPS. MIPS supports multiplication and division using existing hardware, primarily the ALU and shifter. MIPS needs one extra hardware component - a 64-bit register able to supportsll and sra instructions. The upper (high) 32 bits of the register contains the remainder resulting from division. This is moved into a register in the MIPS register stack (e.g., $t0) by the mfhi command. The lower 32 bits of the 64-bit register contains the quotient resulting from division. This is moved into a register in the MIPS register stack by the mflo command.

In MIPS assembly language code, signed division is supported by the div instruction and unsigned division, by the divu instruction. MIPS hardware does not check for division by zero. Thus, divide-by-zero exception must be detected and handled in system software. A similar comment holds for overflow or underflow resulting from division.

FP ADDER HARDWARE

• Much more complex than integer adder
• Doing it in one clock cycle would take too long
• Much longer than integer operations
• Slower clock would penalize all instructions

Name : MARMI'ATI AL - AZIMAH BINTI AZHARI

Matrix no : B031310334

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