Digital Coding

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Digital Coding
Kevin BoldingElectrical EngineeringSeattle
Pacific University
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Digital Transmission of Analog Data
Analog signal
Sampling
Convert to discrete samples (time domain)
Quantizing
Convert to discrete levels (amplitude)
Digitaldata
Coding
Re-map to a different logical code (may expand)
Modulation
Map to a physical code at desired frequency band
Transmission
Amplify and transmit
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Coding

• Coding is the transformation of the digital
  signal into a different form
• Usually to a more rugged form that can tolerate
  errors, reducing the BER
• Usually involves expansion of the signal
• Higher bit rate may require more bandwidth

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Code Space

• PCM produces an efficient code
• All possible codes used
• Examples 256 codes use 8 bits 4096 codes use 12
  bits

• Efficient codes dont tolerate errors
• If an error causes one bit to change, it produces
  a new codeword
• New codeword is valid because all possible
  codewords are valid

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Code Space Even/Odd Parity

• Expanding the code space introduces redundancy
• Some codewords are invalid (inefficient)
• Errors may change valid codewords to invalid
  codewords

• Add up all the bits in a codeword is the total
  even or odd?
• Generate an extra bit that forces the total to be
  even
• Doubles the size of the code space
• All codewords with odd parity are invalid

01010011010
1
11110101011
0

• If a single bit is flipped, parity is changed
  from even to odd

111101010110
110101010110

• The receiver can re-check parity and find invalid
  codewords

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Block Coding

• Bits are grouped into fixed-size blocks
• Could be 8, 16, 24, etc. bits
• A computation is performed on the bits on the
  block to create the codeword

• Most block codes produce extra bits that are
  appended to the original block to form the
  codeword
• Parity is an example one block one parity bit
• More complex block codes can detect more than one
  error and/or correct errors

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Block Error Correction

• Further expand the code space so that each valid
  bit is further from its neighbors
• Each codeword now has a group of unique one
  bit-flip neighbors
• Codewords in this group can be mapped back to the
  original codeword

• Hamming codes are a class of codes that provide
  multi-bit error detection/correction
• Formulaic way of adding redundant bits to provide
  any level of detection/correction desired

• Upside/Downside Each block is independent
• Lots of errors in one block may exceed the
  correction ability
• Even if neighboring blocks have no errors

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Performance of Hamming codes
a. Baseline Binary PSKb. (7,4) Hamming coding,
hard decisionsc. (7,4) Hamming coding, soft
decisionsd. Mariner '69 deep space probe, (32,6)
bi-orthogonal codinge. Voyager probes, NASA deep
space standard (ref 6)f. Theoretical Shannon
limit on coding performance.
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Convolutional Coding

• Treat an entire (near-infinite) stream of bits as
  a continuous coding input
• Create an output bitstream that includes the
  original information, but transforms it into a
  form that has redundant information built into
  structure
• Recovery of the message can use a large portion
  of the received bitstream to correct errors

• Rather than appending parity bits,
  convolutional codes increase the number of bits
  that are output
• Requires an increase in bit rate to keep up
• For example, an encoder might substitute two
  encoded bits for each input bit
• Output data stream is at twice the bit rate

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A Simple Convolutional Encoder
C B A X Y
0 0 0 0 0
0 0 1 1 1
0 1 0 1 0
0 1 1 0 1
1 0 0 1 1
1 0 1 0 0
1 1 0 0 1
1 1 1 1 0

• A,B,C are the three most recent bits (A oldest, C
  newest)
• X and Y are the outputs
• For each new input bit, both X and Y are produced
• Output rate double the input rate

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States

• The two bit shift register holds two bits
• Four states (BA)
• Input to the FSM is C

• The state transition table can be drawn in a
  linear form
• For each set, left side is starting state, right
  side is ending state
• Arcs for inputs 0 or 1

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Outputs

• The outputs depend on the state (AB) and the
  input (C)
• Add them to the arcs in the diagram

C B A X Y
0 0 0 0 0
0 0 1 1 1
0 1 0 1 0
0 1 1 0 1
1 0 0 1 1
1 0 1 0 0
1 1 0 0 1
1 1 1 1 0
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Encoding
ENCODING DIAGRAM
Upper arc input C0, Lower arc Input C1

• Concatenate trellis state diagrams for the
  transitions and outputs for each input bit

For startstate BA00
For startstate BA01
For startstate BA10
For startstate BA11
Input 100111011
See http//www.ee.unb.ca/cgi-bin/tervo/viterbi.pl?
binary100111011 for more examples
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Encoding/Decoding Diagrams
ENCODING DIAGRAM
Upper arc input C0, Lower arc Input C1
For startstate BA00
For startstate BA01
For startstate BA10
For startstate BA11
DECODING DIAGRAM
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Decoding
DECODING DIAGRAM

• Generate diagram corresponding to possible
  transitions that create XY codes

• Find best path

Code (XY)
11
10
11
11
01
10
01
00
01
01
11
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Decoding with Errors
DECODING DIAGRAM

• What if an error occurs to the coded message?
  (Third symbol has a bit flip)

• Find best path

1 bit difference
6 bit difference
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Decoding with Errors
DECODING DIAGRAM

• What about multiple errors?

6 bit difference
2 bit difference
3 bit difference
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Viterbi Decoding Algorithm

• The goal of the decoder is to find a path through
  the trellis with the best match
• Best match fewest bit flips from received data
• Viterbi algorithm is an efficient method to find
  the best match

See http//www.ee.unb.ca/tervo/ee4253/convolution3
.htm for details
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Performance
http//home.netcom.com/chip.f/viterbi/simrslts.ht
ml
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Soft Decoding

• Soft decoding refers to two different techniques
  for improving coding performance
• Using neighboring bits to improve the reliability
  of the decoding decision
• The trellis-decoding method for convolutional
  coding is an example

• Using channel quality information from the
  demodulator
• The demodulator sees the unquantitized analog
  signal and may have additional helpful info

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Using Channel Quality Info

• If the channel quality is below a threshold when
  receiving a symbol, erase the symbol
• Erasing removes the erroneous symbol from
  consideration, so more weight can be given to the
  correct symbols

Poor quality input - erase
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Obtaining signal quality info

• The demodulator receives imperfect data, but
  quantitizes it based on the best possibility
• The error can be measured and reported

• In standard QPSK, the angle of the received
  waveform determines the binary value
• Ideally, each received symbol overlies a blue dot

• In reality, the received signals are more spread
  out

• Define a threshold that decides when to throw out
  far-off data points