Chapter 1

Digital Systems and Binary Numbers

Key Concepts :● Why digital Over Analog Systems ?● Number - Base Conversions● Complements of Numbers● Signed Binary Number Systems● OverFlow Concept● Binary and Decimal Codes● BCD Addition● Gray Code● Parity Check

Why Digital Over Analog ?

1.Reduced Cost2.More Reliable3.Programmable (so has major

application)4.Selective ( Akin to a Research Scientist

vs a Payroll Schedule)5. Implemented via Electronic Components

Number Conversions Convert From X ---- > Y Method

Any System(r) To Decimal Multiply Coefficients before Decimal by r ^ index. And Coefficients after Decimal by r^(-index) where index starts from -1 after decimal and 0 before.

Decimal to Any System (r) Divide Decimal by r Repeatedly and Collect remainders. The Final Answer is From Last Remainder Collected to First.

Binary To Octal Group in 3s

Binary To Hexadecimal Group in 4s

Convertion of 75.375 (decimal) to Binary

Therefore the answer is : (001011.011)

Complements

Complements of Numbers

Radix Complement (r^n- N )

Diminished Radix Complement ((r^n-1)-N )

2’s , 10’s 1’s , 9’s

*

Signed Binary NumbersWe need to represent these symbols using bits

– Convention:• 0 positive

1 negative• The leftmost bit position is used as a sign bit

– In signed representation, bits to the right of sign bit is the number

– In unsigned representation, the leftmost bit is a part of the number (the most significant bit (MSB))

*

Example

– 01011 → (unsigned binary)– → (signed binary)– 11011 → (unsigned binary)– → (signed binary)– This method is called “signed-magnitude” and is rarely used in

digital systems (if at all)

• In computers, a negative number is represented by the complement of its absolute value.

• Signed-complement system– positive numbers have always “0” in the MSB position– negative numbers have always “1” in the MSB position

*

Signed Number Representation

Signed magnitude One’s complement Two’s complement

000 +0 000 +0 000 0

001 +1 001 +1 001 +1

010 +2 010 +2 010 +2

011 +3 011 +3 011 +3

100 -0 111 -0 111 -1

101 -1 110 -1 110 -2

110 -2 101 -2 101 -3

111 -3 100 -3 100 -4

8,4,2,1 and Excess Three Coding

● BCD : Involves each digit being assigned appropriate binary code.

Eg : 123 = 0001-0010-0011● In BCD Addition if sum is greater

than 10, then 0110 (6) added .● 8,4,2,1 Coding is a Weighted

Code● Excess 3 Code: Is Binary Code

+3 . Therefore it is NOT A weighted Code.

● Excess 3 is a SELF COMPLIMENTING CODE.

GRAY CODE ● Used when digital data to analog data

is converted

● Only one bit in group changes from one number to another

● Used in cases where normal binary sequence may produce error

● Non Weighted Code

Parity Check

Used Often in Transmission Of Messages.

If message is received : ACK returned.

Else ,NAK returned.

By :

Debarati Das 1PI13CS052

PES University