Digital System IED 12303

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Chapter ONE: Digital Electronic IED 12303


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Chapter ONE

1. In digital electronics numeral system is anessential process used by computer in

order to communicate from one part toanother for entire system.

2. The system can use, with the most

common ones are: binary (base-2), octal(base-8), decimal (base 10) andhexadecimal (base 16).


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1. Decimal : 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

2. Binary : 0, 1, 0,0,1,1,0.

3. Octal : 0, 1, 2, 3, 4, 5, 6 and 7.

4. Hexadecimal : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B,C, D, E, F.


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1. All of the number system can beencountered in every digital system which

is embedded in their systems for instancesoftware.

2. Since all electronic gadget today are usingmicroprocessor, hence the understanding

of number system is a must.


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1. All of the number in the system can beconverted among them. For instance,

Decimal to Binary, Binary to Decimal,Binary to Hexadecimal ect.

2. The conversion of these numeral systems

are subjected to the mathematical rules.


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Binary to Decimal


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1. Least Significant Bit (LSB) and MostSignificant Bit (MSB) are need to be

determined in certain numeral system.

Binary to Decimal


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1. Conversion example. Convert thefollowing Binary number to Decimal

number.

Binary to Decimal

(256) + (64) + (32) + (4) + (1) = 35710


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1. ExerciseConvert the followings Binary system to

Decimal :

i) 10012ii) 101102

iii) 1101001012iv) 11001011002v) 11001001112

Binary to Decimal


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1. So far, the Binary number is converted toDecimal numeral system. However, the

process can be reversed from Decimal toBinary as well.

2. Example: Convert Decimal 35710 to Binary

Decimal to Binary

h


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1. Solution

2 357 result 178 reminder 12 178 result 89 reminder 0

2 89 result 44 reminder 1


2 22 result 11 reminder 02 11 result 5 reminder 1




Decimal to Binary

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2. ExerciseConvert the following Decimal numbers to

Binary number.

i) 76ii) 278iii) 403iv) 657v) 860

Decimal to Binary

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3. ExerciseConvert the following Octal numbers to

Decimal number.

i) 76ii) 278iii) 403iv) 657v) 860

Decimal to Octal

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The Octal numeral is based on 8 system. Theconversion from Octal to Decimal can be

done with following technique.

Example: Convert 3458 to base 10 numeralsystem.

345 octal = (3 * 82) + (4 * 81) + (5 * 80) = (3 *64) + (4 * 8) + (5 * 1) = 229

Octal to Decimal

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Binary operations when dealing with binaryare often occur including Additional,

Subtraction, Multiplication and Division.

These operations are embedded in computersystems, however the process are notobviously visible because their process onlyoccur in operation system.

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Example : Suppose add the binary numbers11112 and 1102 similar to the way we add the

decimal numbers 1510and 610.

Solution :

Additional

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Additional

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AdditionalExercise

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SubtractionHow to subtract the binary numbers101012and 11102

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SubtractionExercise

1. 101 112. 11001 11013. 1011001 -1010014. 1101001 - 11000015. 11010101 10100101

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Multiplication

How to multiply the binary numbers11112and 10112

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Multiplication


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Multiplication


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C apte O

Multiplication


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p

Multiplication



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Multiplication

Exercise

1. 10100 * 112. 110001 * 1113. 1100100 * 1014. 110111 * 100011

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Division

How to divide 1000012by 1102

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p

Division


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p

Division


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Division


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Division


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Division


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Division


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Division

Exercise

1. 111100 / 1012. 1010010 / 1003. 1111101 / 104. 11001000 / 1005. 100101100 / 1000


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Floating point

Convert 2.62510to Binary.0.625 x 2 = 1.25 1 (MSB)0.25 x 2 = 0.5 00.5 x 2 = 1.0 1 (LSB)

Hence the answer is 10.101

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Floating point

Convert 0.4062510to Binary.0.40625 x 2 = 0.8125 0 (MSB)0.8125 x 2 = 1.625 10.625 x 2 = 1.25 10.25 x 2 = 0.5 00.5 x 2 = 1 1 (LSB)

Hence the answer is 0.01101


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1s Complement

Representing a signed number with 1'scomplement is done by changing all thebits that are 1 to 0 and all the bits that

are 0 to 1. Reversing the digits in thisway is also called complementing anumber.

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1s Complement

How can we represent the number -510in 1's complement?

1.First, we write the positive value of thenumber in binary.

0101 (+5)

2.Next, we reverse each bit of the number so 1's

become 0's and 0's become 1's

1010 (-5)

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1s Complement

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1s Complement

Here is a quick summary of how

to find the 1's complement

representation of any decimal

number x.

1.If x is positive, simply

convert xto binary.

2.If x is negative, write the

positive value of xin binary

3.Reverse each bit.

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2s Complement

Representing a signed numberwith 2's complement is done byadding 1 to the 1's complement

representation of the number.

1. First, we write the positiveof the number in binary.

2. Next, we reverse each bit toget the 1's complement.

3. Last, we add 1 to thenumber.

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2s Complement

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Addition

If 2 Two's Complementnumbers are added, and theyboth have the same sign (both

positive or both negative), thenoverflow occurs if and only ifthe result has the opposite sign.Overflow never occurs when

adding operands with differentsigns.

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Unsigned bit

1. The binary numbers can berepresented in positive andnegative binary number.

2. The unsigned binary bit is akind of binary number thatuse for positive binary only.

3. Example, +25 express in 8-bit binary system.

ans : 0001 1001

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Signed bit

1. The signed binary system isuse to indicate whether theis said to be Negative or

Positive.2. It can be seen that, in natureall numbers are in positiveand negative number.

3. The procedure of makingthe bit signed is if thenumber is Positive, let theleft most bit is 0while 1 ifthe number is negative.

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2s Complement


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Signed bit

1. The numbers can beexpressed in 3 ways

i) Sign-Magnitude system

ii) 1scomplement systemiii) 2scomplement system

2. Example: Express the

decimal -39 by using thosethree sign system.

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Signed bit

1. Sign-Magnitude system+ 39 = 0010 0111-39 = 1010 0111

Sign bit is zero if+ve

Sign bit is ONE if

-ve2. 1scomplement system+ 39 = 0010 0111-39 = 1101 1000

3. 2scomplement system+ 39 = 0010 0111-39 = 1101 1000

+ 1

= 1101 1001

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Signed bit

1. The unsigned-magnitudesystem is apply for positivenumber only.

2. The range for unsignednumber is as follows :

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unsigned bit

1. Example: If 4, 8 bits systemare used, determine theirmaximum and minimum

allowed.2. N=4, 0 to 2N-1 = 24-1= 15 or0000 to 1111

3. N=8, 0 to 2N-1 = 28-1= 255 or

0000 000 to 1111 1111

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Signed bit

1. The number system can bepresented in three ways ofthe signed binary forms,

that is Signed-MagnitudeSystem, 1s complement and2scomplement system.

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Signed bit

1. Sign-Magnitude system+ 39 = 0010 0111-39 = 1010 0111

Sign bit is zero if+ve

Sign bit is ONE if

-ve2. 1scomplement system+ 39 = 0010 0111-39 = 1101 1000

3. 2scomplement system+ 39 = 0010 0111-39 = 1101 1000

+ 1

= 1101 1001

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Signed bit

1. Determining the decimalvalue for signed numbers.

Signed binary number canbe determined as follows :Let the binary number is

0010 0111 = +39

(1 x 2^5 ) + ( 1 x 2^2) + (1 x 2^1 )+ ( 1 x 2^ 0) = 32 + 4 + 2 + 1= + 39

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Signed bit

Let the signed binary number is1010 0111 = -39

(-1 x 2^ 7) + (1 x 2^5 ) + ( 1 x 2^2)+ (1 x 2^1 ) + ( 1 x 2^ 0) = -127+32 + 4 + 2 + 1= - 39 ( since the MSB is bit 1 )

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Signed bit

1. Determining the decimal valuefor signed 1s complementnumbers.

Let the 1s complement binarynumber is1101 1000 = -39

(-1 x 2^ 7 ) + (1 x 2^6 ) + (1 x 2^4 )+ ( 1 x 2^3)= -128+64+ 4 + 2 = - 40Adding 1to the answer gives -39

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Signed bit

1. Determining the decimal valuefor signed 2s complementnumbers.

Let the 2s complement signedbinary number is

1101 1001 = -39

(-1 x 2^ 7 ) + (1 x 2^6 ) + (1 x 2^4 )+ ( 1 x 2^3) + ( 1 x 2^0)= -128+64 +16+ 8+1

= - 39

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1. Adding two positive

numbers must give apositive result unlessoverflow occurred.

2. Adding two negativenumbers must give anegative result unlessoverflow occurred.

3. The overflow occurs becausethe operation addition andsubtraction is over theminimum and maximum

allowed by its N bit system.


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1. (+A) + (+B) = -C OR2. (-A) + (-B) = +C

3. Example : 7 + 4 ( use 8 bitsigned binary system)

0000 0111

+ 0000 01000000 1011 = ( +11) correct

1. Example : 16 -24 ( use 8 bit

signed binary system)0001 0000

+ 1110 10001111 1000 = ( - 8) correct

Sign bit iscorrect

Sign bit is

correct

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1. (+A) + (+B) = -C OR2. (-A) + (-B) = +C

3. Example : 125 + 58 ( use 8 bitsigned binary system)

0111 1101

+ 0011 10101011 0111 = ( -72) incorrect

1. Example : -88 -41 ( use 8 bit

signed binary system)0010 1000

+ 0010 10100101 0010 = (- 46) incorrect

Sign bit isincorrect

Sign bit is

incorrect

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Signed bit

1. The logic gates are primaryelectronics invention indigital era. Gates in

electronics including OR,AND, INVERTER ect.2. Logic gates are comprises of

analog electronic devicessuch as transistors anddiodes.


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1. Digital circuitry may include

Multiplexer, Register,Counter ect.

2. The gates can be embeddedonto chip.

3. SSI: Small-scale Integration,Gates< 102. MSl: Medium-scale

Integration,10 10004. VLSI: Very Large-scale

Integration, Gates>100000

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1. A truth table shows how a

logic circuit's outputresponds to variouscombinations of the inputs,using logic 1 for true and

logic 0 for false. Allpermutations of the inputsare listed on the left, and theoutput of the circuit is listedon the right.

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1. The deriving of Booleanexpression is essential whendealing with logic gate.

2. Boolean expression can beobtained by using standardlogic gate principals.

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1. Any complex Booleanexpression however can besimplified into simplest one

depending on the expressionitself.

2. The simplicity of theexpression leads the

expression more compactand easy to read.

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1. On the other hand, thesimplify version of Booleanexpression can be achieved

by obeying the set of rulesstated on the previous slide.

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Digital System IED 12303

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