Multiplexer

A multiplexer performs the function of selecting the input on any one of 'n' input lines and feeding this input to one output line.
Multiplexers are used as one method of reducing the number of integrated circuit packages required by a particular circuit design. This in turn reduces the cost of the system.
Assume that we have four lines, C0, C1, C2 and C3, which are to be multiplexed on a single line, Output (f). The four input lines are also known as the Data Inputs. Since there are four inputs, we will need two additional inputs to the multiplexer, known as the Select Inputs, to select which of the C inputs is to appear at the output. Call these select lines A and B.

The circuit symbol for the above multiplexer is:

Written by:Su’aidah Binti Mokhtar

Flip Flop

What is Flip Flop???

  Ø Flip flop (latch) is a circuit that has two stable states and can be used to store state      
      information.

  Ø Flip-flops can be either simple (transparent or asynchronous) or clocked          
      (synchronous).

Type of Flip Flop

Ø SR Flip-flop - (Set / Reset)

Ø T flip-flop (Triggered / Toggle)

Ø D type flip-flop (Delay)

Ø JK flip-flop

SR Flip Flop ( Set / Reset)

Input: Set and Reset

Output: Q and Q’(Q’ = reverse of Q)

Operating Characteristic Table:

T flip-flop (Triggered / Toggle)
Input: T (trigger)Outputs: Q and Q'

Operating Characteristic Table:

D type flip-flop (Delay)
Input: DOutputs: Q and Q'

Operating Characteristic Table:

JK flip-flop
Input: DOutputs: Q and Q'

Operating Characteristic Table:

Written by
Name: Woon Pei Wen,Loo Shi Min
No. Matrik: B031210273,B031210106

      

Number System

Arithmetic for computer

NUMBER SYSTEM   

A number system is a basic symbol to represent a set of quantities.

1.1~ Basic types of number system

1.2 ~ Number system base:       
The maximum number that can be represented on the single digit or number is called a base.

Ø DECIMAL NUMBER

ü Most widely used by modern civilizations.

ü The prefix “deci-” stands for 10

ü Base of 10 because:

·       There are 10 symbols that represent quantities (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

·       Each place value in a decimal number is a

power of 10

ü The value of assigned weight is composed by 10digits starting from 0 until 9.

ü The positive and negative values are determined by their position weight structure(power of a number):

  Ø BINARY NUMBER

ü most basic number system that most machines (and electricaldevices) use to communicate.

ü Base of 2 because each position in the number represent an incremental number with a base of 2.

ü Since the number system is represented in “twos”, there are only 2 numbers that can be a value in each position of the base-2 number. 

ü Each position can only contain a 0 or a 1.

ü The least significant bit (LSB) and most significant bit (MSB) is depend on the size of binary number.

Ø   HEXADECIMAL

ü  The hexadecimal number system is used as an intermediary system in computers, such as arepresentation of memory addresses or a representation of colors

ü Base of 16 because each position in the number represents an incremental number with a base of 16.

ü Since the number system is represented in “sixteen”, there are only 10 numbers and 6 letters (0-9, A-F) that can be a value in each position of the base-16 number.

1.3 ~ Number system conversion : 

Converting Binary Numbers to Decimal

•Step1                                                               – Starting with the 1s place, write the binary place value over each digit in the binary number being converted :  

•Step2                                                                   – Add up all of the place values that have a “1” in the binary number

*EXAMPLE

Convert the following binary number to decimal number :  

                                    

Converting decimal Numbers to binary

• There are two methods that can be used to convert decimal numbers to binary:

  v Repeated subtraction method

  v Repeated division method

• Both methods produce the same result and you should use whichever one you are most comfortable with .

Repeated subtraction method

As an explanation of the repeated subtraction method, let’s convert the decimal number 204 to binary.

Step 1:

• Starting with the 1s place, write down all of the binary place values in order until you get to the first binary place value that is GREATER THAN the decimal number you are trying to convert.

Step 2:

• Mark out the largest place value (it just tells us how many place values we need).

Step 3:

• Subtract the largest place value from the decimal number.  Place a “1” under that   place value.

Step 4:

• For the rest of the place values, try to subtract each one from the previous result.

     – If you can, place a “1” under that place value.

     – If you can’t, place a “0” under that place value.

Step 5:

• Repeat Step 4 until all of the place values have been processed.

    – The resulting set of 1s and 0s is the binary equivalent of the      decimal number you started with.

* EXAMPLE                                              

 Convert the following decimal number to binary using repeated   subtraction method :

 a)     189₁₀

b)     38.3125₁₀=38₁₀+0.3125₁₀

Repeated division method

Step1:                                                                 

• Divide the decimal number you’re trying to convert by 2 in regular long division until you have a final remainder.

Step2:                                                                 

• Use the first remainder as the LEAST SIGNIFICANT DIGIT of the binary number.

Step3:                                                                         

•Divide the quotient you got from the first division operation until you have a final remainder.

Step4:                                                                       

•Use the remainder as the next digit of the binary number.

Step5:                                                                  

• Repeat Steps 3 & 4 as many times as necessary until you get a quotient that can’t be divided by 2.

Step6:                                                                             

•Use the last remainder (the one that can’t be divided by 2) as the MOST SIGNIFICANT digit.

* EXAMPLE                                                          

Convert the following decimal number to binary using repeated division method

a)     339₁₀

b)     59.625₁₀

Converting Binary Numbers to Hexadecimal

•Step1:                                                      

– Starting with the LEAST SIGNIFICANT digit, mark off the digits in groups of 4.                                                                                               – For example, to convert 110101001011 to hexadecimal, mark off the digits in groups of four.                                                                        

 • 1101 | 0100 | 1011

•Step2:                                                      

– Convert each group of four digits to its hexadecimal character.

•HelpfulHint                                                 

– The last group on the left can have anywhere from 1 to 4 binary digits group.                                                                       

 – If it will help you see the pattern, you can fill in enough leading zeroes to make the last group on the left have four digits.                       

 – For example, 1 1 0 | 0 1 1 1 | 1 0 0 1 could be written as
 0 1 1 0 | 0 1 1 1 | 1 0 0 1

* EXAMPLE                                                         Convert the following binary number to hexadecimal:                        a)         1111101₂=  01111101₂

b)     10011111001.101₂=010011111001.0100₂

Converting Hexadecimal to Binary Numbers                           

• Converting hexadecimal numbers to binary is just the reverse operation of converting binary to hexadecimal.                               

 • Just convert each hexadecimal digit to its four-bit binary pattern. The resulting set of 1s and 0s is the binary equivalent of the hexadecimal number.:

* EXAMPLE                                                          Convert the following hexadecimal to binary number:        

a)     AF025E₁₆  

b)     3B5D2A7₁₆

Converting Hexadecimal to Decimal

• Step 1                                                               

– Starting with the 1s place, write the hexadecimal place value over each digit in the hexadecimal being converted

• Step 2                                                                   

– Add up all of the place values that have a value in the hexadecimal

* EXAMPLE                                                          

Convert the following hexadecimal to decimal:

a)      3DA4₁₆

b)     123D.2₁₆ 

Converting Decimal to Hexadecimal

• There are two methods to choose from:                                               – Do a decimal-to-binary conversion and then a binary-to-hexadecimal conversion.                                                                          – Do a direct conversion using the repeated division method.                          • Since this is a conversion to hexadecimal, 16 is the divisor each time.

decimal-to-binary conversion and then a binary-to-hexadecimal conversion

* EXAMPLE                                                         

Convert 136₁₀ to hexadecimal

Step 1: convert it to binary number

Step 2: convert the binary number to hexadecimal

direct conversion using the repeated division method

* EXAMPLE   

Convert 836₁₀ to hexadecimal

1.4~ 2’S complement number:  

       Two's complement is a mathematical operation on binary numbers AND allows the use of binary arithmetic operations on signed integers, yielding the correct 2's complement results.

1^st and 2nd Complement of Binary Numbers  

ü They are important because they permit the representation of negative numbers

ü The method of 2^nd complement arithmetic is commonly used in computers handle negative numbers.

ü Assume that a microprocessor have 8 register bits, figure below show the sign bit or the most significant bit ( MSB ) if the MSB bit is 0 , then the number is positive (+) . Conversely, if the MSB is 1 , then the number is negative (-). The others remaining 7 bits are represent as the magnitude numbers.

1st Complement of Binary Numbers

v The 1^st complement of a binary number is found by changing all 1s to 0s and all 0s to 1s by using parallel inverters (NOT circuits).

v Example :

2nd Complement of Binary Numbers

 v The 2^nd complement of a binary number is found by adding 1 to LSB of the 1s complement.

v Example :

v Another method for finding 2^nd complement of a binary number by:

§  Start at the right with the LSB and write the bits as they are up to and including the first 1

§  Take the 1^st complement of the remaining bits

wWritten by:Woon Pei Wen,Loo Shi Min
MMatrix no.:B031210273,B031210106