ECE 7650: Advanced Computer Architecture

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ECE 7650 Advanced Computer Architecture

• Chapter 4
• Computer Arithmetic
• Error Detecting and Correcting Codes
• Data Compressing Codes
• Floating point Numbers

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Error Detecting And Correcting Codes(Introduction
)

• The Error Control Coding area requires extensive
  knowledge of fundamental algebras.
• Groups, rings, fields.
• Galois Field.
• Vector Spaces.
• Matrices.
• Graduate Level Courses are Offered.
• Electrical and Computer Engineering.
• Comptuer Science.
• This course provides a brief non-mathematical
  introduction.

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Error Detecting And Correcting Codes(Communicatio
ns Block Diagram)
Information Source
Channel Encoder
Modulator
Source Encoder
Channel (Transmission or Storage Medium)
Noise
Information Destination
Source Decoder
Channel Decoder
Demodulator
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Error Detecting And Correcting Codes(Information
Source)

• The Information Source can be
• Person
• Machine
• Digital Computer
• Data Terminal
• The Information Source output (to be
  communicated to the destination) can be
• Continuous Waveform
• Discrete Symbols

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Error Detecting And Correcting Codes(Source
Encoder)

• The Source Encoder transforms the Information
  Source output to a binary sequence, called the
  information sequence.
• For continuous Information Source
• A/D conversion is performed
• Number of bits per unit time is minimized
• Data Compression
• Lossless
• Signal Compression
• Lossy

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Error Detecting And Correcting Codes(Channel
Encoder)

• The Channel Encoder transforms each
  information word into a corresponding
  codeword.
• The Channel Encoder attempt to encode the
  information to protect it against from being
  corrupted by noise in the channel.

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Error Detecting And Correcting Codes(Modulator)

• The Modulator trransforms each Channel Encoder
  output to a waveform example M-ary phase shift
  keying.

BPSK
8PSK
QPSK
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Error Detecting And Correcting Codes(Types of
Errors)

• Noise can corrupt data in computer systems
• Random bit errors
• Burst errors
• Noise arises due to a variety of causes, such as
• Thermal motion of electrons in digital systems
• Nearby lightning strikes
• Starting motors
• Crosstalk
• Fading casued by multipath transmission
• Dropouts in magnetic recording, caused by defects
  and dust

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Error Detecting And Correcting Codes

• Noise power is generally a lot smaller than
  signal power
• FCC electromagnetic radiation standards
• Specifiy maximum allowable radiation from known
  sources, such as
• Power line inductive transients (starters)
• There is still a need to protect the data in
  computer systems from noise corruption.

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Error Detecting And Correcting Codes(Analogy)

• Consider the transmission of English words in a
  noisy communications channel
• Transmitter transmits January
• Receiver recieves Jamuary
• The receiver is able to detect and correct the
  word
• Transmitter transmits Affects
• Receiver recieves Bffects
• The receiver is able to detect an error cannot
  correct it
• The B could be A or E

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Error Detecting And Correcting Codes(Channel
Encoder Encoding a Source Word)

• Redundant bits also called check bits are used by
  the receiver to check whether the code word is
  valid or not.
• Check bits can be interleaved throughtout the
  word also.

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Parity EDCs(Example a 3-bit Error Detecting
Code)

• Consider 2-bit information words encoded with
  even parity
• Hamming distance between any two code words is 2.

Any single bit-error can be detected. Two errors
or any even number of errors cannot be
detected.
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Error Correcting Codes(Example a 3-bit Error
Correcting Code (ECC))

• An ECC locates and fixes erorrs.

Given a received word, the receiver chooses the
nearest (in the hamming distance sense) codeword
to the received word.
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Channel Encoder Space

• m-bits per information word
• N-bits per codeword
• n-m redundant check bits

N-dimensional space 2n unique codewords. 2m
unique information words. 2n - 2m erorr
states.
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How Error Detecting Codes Work
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Block Parity Error-Correcting Codes
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Hadamard Codes (Construction)
1 1 1 -1
Hn Hn Hn -Hn
H2 (Row) Hamming Distance of 1
Constructor
H4 (Row) Hamming Distance of 2

• Each row differs from any other row by n/2 bit
  positions, i.e., the hamming distance of the rows
  is n/2.

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Hadamard Codes (Example Order 8)
Source Code
Row
Hadamard 8-bit Codeword
000 0 001 1 010 2 011 3 100 4 101 5 110 6 000 7

• Three information bits and five redundant check
  bits.
• The hamming distance of the code is 4.
• One error can be corrected and two errors can be
  detected.

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Hadamard Codes (Example Order 8)

• D 2t 1
• t errors can be corrected.
• D/2 errors can be detected.