Codes

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Codes

• ELEC 311
• Digital Logic and Circuits
• Dr. Ron Hayne
• Images Courtesy of John Wakerly and Prentice-Hall

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Binary-Coded Decimal (BCD)

• Encodes digits 0 thru 9
• 4-bit unsigned binary
• 0000 thru 1001
• 6 unused code words
• 1010 thru 1111
• Packed BCD
• 8-bit byte
• 2 BCD digits

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BCD Arithmetic

• Correction made if result exceeds 1001
• Add 6
• Produces carry

36 0011 0110 27 0010 0111 63 0101
1101 0000 0110 0110 0011
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Decimal Codes
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Device Select Encoding
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Code Adjacency

• Mechanical boundary problem

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3-bit Reflected Gray Code

• Angle Codeword
• 0-44 0 0 0
• 45-89 0 0 1
• 90-134 0 1 1
• 135-179 0 1 0
• 180-224 1 1 0
• 225-269 1 1 1
• 270-314 1 0 1
• 315-359 1 0 0
• Only 1 bit changes between code words

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Gray Code
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Character Codes

• ASCII (7-bits)
• American Standard Code for Information
  Interchange

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Code Distance

• Hamming Distance (HD)
• number of bits by which two binary words differ
• HD( 101, 100 ) 1
• HD( 111, 000 ) 3
• Code Distance (Dmin)
• minimum Hamming Distance between any two code
  words

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Distance-2 Code
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Parity Codes

• Distance-2

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Even Parity Generator-Checker

• Compute parity bit at source
• Ps I1 ? I2 ? I3 ? I4
• Transmit information bits plus parity bit
• Recompute parity bit at receiver
• PR I1 ? I2 ? I3 ? I4
• Compare transmitted and received parity
• Error PS ? PR

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Even Parity Generator-Checker
Source
Receiver
Channel
I1
I1
I2
I2
I3
I3
I4
I4
PR
Error
PS
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Hamming Codes

• Overlapping Parity
• Distance-3
• Error detection and correction

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Hamming Parity Generation
P2 I4 ? I3 ? I2 P1 I4 ? I3 ? I1 P0 I4 ?
I2 ? I1
S2 I4 ? I3 ? I2 ? P2 S1 I4 ? I3 ? I1 ?
P1 S0 I4 ? I2 ? I1 ? P0
Syndrome S2S1S0 indicates bit position in error
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Hamming Code Sizes
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CRC Codes

• Cyclic-Redundancy-Check (CRC) Codes
• Include Hamming Codes
• Applications
• Disk Drives
• Data Networks
• Burst-error detecting properties

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RAID

• Redundant Array of Inexpensive Disks

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Checksum Codes

• Expansion of parity from bits to bytes
• 8-bit information bytes
• 0 to 255
• Checksum
• modulo-256 addition of information bytes
• single check byte
• detect single byte error

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Serial Data Transmission

• Data transmitted (or stored) one bit at a time
• Bit Rate / Clock Frequency
• bits per second (bps)
• cycles per second (Hz)

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Serial Line Codes

• Non-Return-to-Zero (NRZ)
• Non-Return-to-Zero Invert-on-1s (NRZI)
• Return-to-Zero (RZ)
• Bipolar Return-to-Zero (BPRZ)
• Manchester (diphase) Code
• Digital Phase-Locked Loop (DPLL)
• Recovers clock signal from serial data stream

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Serial Line Codes
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Summary

• Decimal Codes
• Gray Code
• Parity Codes
• Serial Codes