Error Detection/Correction

Section 1.7Section 3.9

Bonus Material: Hamming Code

ASCII Code

Format effector:control layout

CommunicationControlCharacters: frame a text message.

ASCII Examples

• ASCII A=1000001• ASCII T=1010100

ASCII Code

1011001 (Y)1001110 (N)

If the probability of a bit flipping event is 1%, what is the likely hood that 4 bits are flipped simultaneously?

Parity Bit

• ASCII characters are stored one per byte (8 bits)

• The leftmost bit is called the parity bit

• A parity bit is an extra bit included with a message to make the total number of 1’s either even or odd.

Examples of Parity Bit

• Even Parity– ASCII A=01000001– ASCII T=11010100

• Odd Parity– ASCII A=11000001– ASCII T=01010100

Signal Transmission Algorithm

• (Even Parity System)• A parity bit is generated and attached

to the raw data• An eight-bit sequence including the

parity bit are sent.• The parity of each character is

checked at the receiving end. • If the parity of the received character

is not even, then at least one bit has changed value during transmission. The sender must retransmit the signal.

Parity Generator

• The circuit that generates the parity bit in the transmitter is called a parity generator.

(Truth Table)

Parity Checker

• The Circuit that checks the parity in the receiver is called a parity checker.

Limitation of Parity Checking

(1) (1)

Hardware implementation

• Review of two-terminal XOR/XNOR• Three terminal XOR/XNOR• Hardware Implementation

Two-terminal XOR

– Equal to 1 if x and y differ in value– Alternative description: equal to 1 if an

odd number of variables equal to 1– Characteristics:

Gate Level Implementation of XOR

x⊕𝑦=𝑥 𝑦 ′+𝑥 ′ 𝑦

Alternative Implementation of XOR

• (A’B’)’=A+B

Parity Generator

• The circuit that generates the parity bit in the transmitter is called a parity generator.

(Truth Table)

Three-Terminal XOR

• equal to 1 if there is an odd number of variables equal to 1

Four-Input Odd Function

Parity Error Check

z P

Since , if P=0, the same circuit can be usedas a 3-bit even parity generator.

Error Correction

• Hamming Code• Use check bits to correct error

Raw Data

Notation: (bit 1, bit 2, bit 3)000001010011100101110111

Add Check Bits

Notation: (bit 1, bit 2, bit 3, bit 4, bit 5, bit 6)CC0C00CC0C01CC0C10CC0C11CC1C00CC1C01CC1C10CC1C11

Generate the First Check Bit

0C0C000C0C011C0C101C0C111C1C001C1C010C1C100C1C11

Check bit 1 looks at bit 3 and bit 5.Check bit 1=

Generate the Second Check Bit

000C00010C01100C10110C11111C00101C01011C10001C11

Check bit 2 looks at bit 3 and bit 6.Check bit 2=

Generate the third Check Bit

000000010101100110110011111000101101011110001011

Check bit 4 looks at bit 6 and bit 5.Check bit 4=

Hamming Code

000000010101100110110011111000101101011110001011

Blue: Check bitsBlack: Data bits

Error in a Data Bit

• Data Bit: 100• 111000• Error occurs in the 6th bit, we get

111001 instead.• Expected check bit 1:. OK• Expected check bit 2:. Problem!• Expected check bit 4:. Problem!• The bad bit is 2+4=6!

Error in the Check Bit

• Data Bit: 100, Check Bit: 110• 111000 (No error)• Error occurs in the 1th bit, we get

011000 instead.• Expected check bit 1:. Problem!• Expected check bit 2:. OK• Expected check bit 4:. OK• The bad bit is bit number 1.