04 Codes

BCD¶

Binary Coded Decimal

Each digit of decimal will be represented in 4bit binary

To convert a number > 9, we add 6 (tutorial)

Eg: 33 = 0011 0011

Gray Code¶

Reflection/Unit distance code

Mirror the halfway horizontally

just a way to send information in a private method

1 Bit¶

0

1

2 Bit¶

0 0 = 0

0 1 = 1

1 1 = 2

1 0 = 3

3 Bit¶

0 0 0 = 0

0 0 1

0 1 1

0 1 0

1 1 0

1 1 1

1 0 1

1 0 0

XOR gate Shortcut¶

odd one detector

Binary to Gray¶

1. convert into binary
2. do XOR of
3. Bring 1^st digit down as it is
4. do XOR of adjacent elements

Gray to Binary¶

1. bring 1^st digit down as it is
2. do XOR diagonally after that

Error detection and code correction¶

Parity is a technique to convert codes with even no of 1s or odd no of 1s by adding an extra bit.

Parity Bit is added in MSD position

we add a 1, if the number violates the parity type

Even parity generator¶

we need even no of 1s

value of binary with even no doesn't get affected

XOR gate

Eg

1. _ 0 0 1 0 0 1 (even no of 1s) 0 0 0 1 0 0 1
2. _ 0 0 1 0 1 1 (odd no of 1s) 1 0 0 1 0 1 1

Circuit contains 2 XOR gates, or we can just do with one

Even Parity Checker¶

Checking if the parity being sent along with the bits is correct or not

Circuit contains 3 XOR gates

If there is no error, c = 0 If there is error, then c = 1

Odd parity generator¶

Circuit contains 2 XOR gates and 1 final not gate

or XNOR gate

Odd parity checker¶

Disadvantage of odd and even parity¶

we cannot find out where the error is present

eg: 1001 is correct even parity but if the other end receives 0101, it is still correct by even parity, but it's not the same value

Hamming Code¶

\(A(t, m)\)

eg: A (7, 4) means 7 total bits, 4 message bits; this means there are 3 parity bits

given no \(= m_1 m_2 \dots m_m\) (b = the bit)

Advantage

1. we can generate parity
2. check parity
3. if there is any error, we can correct

No of Parities¶

\(2^p \ge p + m + 1\)

• p = no of parity bits
• m = no of message bits

final message will have (m+p) no of bits

Position of parity¶

Parities are added in the place of 2 powers, ie, \(2^0, 2^1, 2^2, 2^3, \dots\), ie

• 1^st position
• 2^nd position
• 4^th position
• 8^th position
• so on

Example¶

Value of Parity¶

For even parity

For 3 parities, create a table of 3bit input and wherever there is one is the positions of the parities

• \(P_1 \to 1, 3, 5, 7\)
• \(P_2 \to 2, 3, 6, 7\)
• \(P_3 \to 4, 5, 6, 7\)

For 4 parities, create a table of 4bit input and wherever there is one is the positions of the parities

• \(P_1 \to 1, 3, 5, 7, 9\)
• \(P_2 \to\)
• \(P_3 \to\)

• \(P_4 \to\)

• A (7, 4) (total, message) is received as 1 0 1 0 1 1 1. Determine the correct code when even parity exists.

• 1 0 0 1 1 0 1 0

Verilog for Hamming¶

m1 = D0, m2 = D1,\(m_n = D_{n-1}\)

module hamming_code(t, m);
  input[3:0] m;
  output[6:0] t;
  wire p1, p2, p3;
  assign p1 = (m[0] ^ m[1] ^ m[3]); // 1 3 5 7
  assign p2 = (m[0] ^ m[2] ^ m[3]); // 2 3 6 7
  assign p3 = (m[1] ^ m[2] ^ m[3]); // 4 5 6 7

  assign t = {p1, p2, m[0], p3, m[1], m[2], m[3]};

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