SpaceTime Trellis Code STTC

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Space-Time Trellis Code (STTC)

• Bahador Amiri
• Winter 2006

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Contents

• Coding
• Block codes versus Convolutional codes
• Convolutional coding and trellis diagram
• Trellis (MLSD) decoding
• Viterbi decoding
• Space-Time Trellis Codes (STTC)
• Introduction
• Encoding structure
• Design of STTC
• Rank Determinant criteria
• Trace Criterion
• Performance evaluation
• Rank Determinant criteria
• Trace Criterion

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Part I Coding
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Block codes versus convolutional codes

• Block codes
• k-digit of information, n-digit ( ) of
  codeword, redundant check digits.
• No memory between blocks, the encoding of each
  data block is independent of past and future
  blocks.
• Convolutional codes
• Each n-bit codeword block depends on the current
  information digits and on past m information
  blocks.
• The parameter m is called the memory order.
• is the number of bits that the
  decoder must consider.

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Road Map

• Coding
• Block codes versus Convolutional codes
• Convolution coding and trellis diagram
• Trellis (MLSD) decoding
• Viterbi decoding
• Space-Time Trellis Codes (STTC)
• Introduction
• Encoding structure
• Design of STTC
• Rank Determinant criteria
• Trace Criterion
• Performance evaluation
• Rank Determinant criteria
• Trace Criterion

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Convolutional Coding and Trellis diagram (I)

• current ---------------gt NEXT
• State State
• BA C CBA(XY) next BA
• -----------------------------------
• 00 0 000 (00) gt 00
• 00 1 100 (11) gt 10
• 01 0 001 (11) gt 00
• 01 1 101 (00) gt 10
• 10 0 010 (10) gt 01
• 10 1 110 (01) gt 11
• 11 0 011 (01) gt 01
• 11 1 111 (10) gt 11

• 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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Convolutional Coding and Trellis diagram (II)

• Message to be transmitted
• 1 0 0 1 1 1 0 1 1
• Encoded message
• 11 10 11 11 01 10 01 00 01 11

• BA C CBA (XY) next BA 00 1 100
  (11) 10 10 0 010 (10) 01
  01 0 001 (11) 00 00 1 100
  (11) 10 10 1 110 (01) 11
  11 1 111 (10) 11 11 0 011
  (01) 01 01 1 101 (00) 10
  10 1 110 (01) 11 ----- end flush
  with zeros ----- 11 0 011 (01) 01
  01 0 001 (11) 00

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Convolutional Coding and Trellis diagram (III)

• Trellis Diagram
• For each incoming bit there are only two
  possible branches to a new state

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Convolutional Coding and Trellis diagram (IV)


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Road Map

• Coding
• Block codes versus Convolutional codes
• Convolution coding and trellis diagram
• Trellis (MLSD) decoding
• Viterbi decoding
• Space-Time Trellis Codes (STTC)
• Introduction
• Encoding structure
• Design of STTC
• Rank Determinant criteria
• Trace Criterion
• Performance evaluation
• Rank Determinant criteria
• Trace Criterion

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Trellis (MLSD) decoding (I)

• By continuing same example
• First, I will suppose that with do not have any
  noise, then the exact transmitted message will
  receive at the receiver (no error). I will show
  that recovery of our information bits from the
  codeword by Trellis diagram is easy.
• Next, I will show how can we correct one or more
  errors with Trellis decoding which is based on
  Maximum Likelihood Sequence Detection (MLSD).

• Received encoded message 11 10 11 11 01 10 01 00
  01 01 11
• BA C CBA(XY) next BA
• 00 0 000(00) gt 00
• 00 1 100(11) gt 10
• 01 0 001(11) gt 00
• 01 1 101(00) gt 10
• 10 0 010(10) gt 01
• 10 1 110(01) gt 11
• 11 0 011(01) gt 01
• 11 1 111(10) gt 11

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Trellis (MLSD) decoding (II)

• Finding the best path

• 1 0 0 1 1 1 0 1
  1 0 0
• Decoded Data Bits (continuous path)

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Trellis (MLSD) decoding (III)

• Error Correction Example
• If the received signal had some error, then the
  Trellis diagram will not be continuous and we
  need to use the maximum likelihood theory the
  find the encoded transmitted signal and
  consequently the information bits.
• Received message 11 10 10 11 01 10 01 00
  01 01 11
• Transmitted message 11 10 11 11 01 10 01 00
  01 01 11

• 1 0 ? 1
  1 1 0 1 1 0 0
• Error Data Sequence (not a continuous path)

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Trellis (MLSD) decoding (IV)

• 1 0 0 1 1 1 0 1
  1 0 0
• Corrected Data Bits (Continuous path)

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Trellis (MLSD) decoding (V)
Received data 11 10 10 11 01 10 01 00 01 01
11 Repaired data 11 10 11 11 01 10 01 00 01
01 11
Difference 00 00 01 00 00 00 00 00 00
00 00

• Maximum Likelihood

Received data 11 10 10 11 01 10 01 00 01 01
11 Repaired data 11 01 01 00 01 10 01 00 01
01 11
Difference 00 11 11 11 00 00 00 00 00
00 00
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Road Map

• Coding
• Block codes versus Convolutional codes
• Convolution coding and trellis diagram
• Trellis (MLSD) decoding
• Viterbi decoding
• Space-Time Trellis Codes (STTC)
• Introduction
• Encoding structure
• Design of STTC
• Rank Determinant criteria
• Trace Criterion
• Performance evaluation
• Rank Determinant criteria
• Trace Criterion

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Viterbi Decoding (I)

• Long message or more error numbers
  more complexity

Received data 11 01 11 11 00 01 01 11
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Viterbi Decoding (II)
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Part II Space-Time Trellis Codes (STTC)
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Road Map

• Coding
• Block codes versus Convolutional codes
• Convolution coding and trellis diagram
• Trellis (MLSD) decoding
• Viterbi decoding
• Space-Time Trellis Codes (STTC)
• Introduction
• Encoding structure
• Design of STTC
• Rank Determinant criteria
• Trace Criterion
• Performance evaluation
• Rank Determinant criteria
• Trace Criterion

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Introduction

• Space-Time Block Codes achieve
  maximum possible diversity
  advantage but no coding gain and
  bandwidth expansion
• advantage simplicity
• Space-Time Trellis Codes (STTC) joint
  design of error control coding,
  Modulation, transmit
• 
  and receive diversity
• complexity
  coding gain, spectral efficiency,
• 
  and diversity improvement

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Road Map

• Coding
• Block codes versus Convolutional codes
• Convolution coding and trellis diagram
• Trellis (MLSD) decoding
• Viterbi decoding
• Space-Time Trellis Codes (STTC)
• Introduction
• Encoding structure
• Design of STTC
• Rank Determinant criteria
• Trace Criterion
• Performance evaluation
• Rank Determinant criteria
• Trace Criterion

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Encoding structure (I)

• M-PSK modulation with transmit antennas
  input
• 
  
• 
  where
• 
  adder is
  modulo M
• 
  m
  feed-forward shift register
• 
  

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Encoding structure (II)

• is the output of encoder at time t for i-th
  antenna
• is total memory order of encoder
• is memory order of k-th branch of encoder
• Is the total number of states for the trellis
  encoder
• are generator sequences, and
  they can describe the encoder.

mod M, i 1,2,,nT
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Road Map

• Coding
• Block codes versus Convolutional codes
• Convolution coding and trellis diagram
• Trellis (MLSD) decoding
• Viterbi decoding
• Space-Time Trellis Codes (STTC)
• Introduction
• Encoding structure
• Design of STTC
• Rank Determinant criteria
• Trace Criterion
• Performance evaluation
• Rank Determinant criteria
• Trace Criterion

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Design of STTC (I)

• Goal design optimum space-time trellis code for
  given number of transmit antenna and memory order

• best set of encoder coefficients to minimize the
  error probability

Codeword difference matrix

• rank of B

• Code design depends on r and number of received
  antennas

• rank determinant criteria
• trace criterion

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Design of STTC (II)

• Table Find r choose
  criteria

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Rank Determinant criteria
(minimum diversity)
Rank Determinant criteria

• Maximize the minimum rank r of matrix
  over all pairs of distinct codewords
• Maximize the minimum product, , of
  matrix along the pairs of distinct
  codewords with the minimum rank

By V. Tarokh, N. Seshadri and R. Calderbank (TSC)
1998
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Trace Criterion
(minimum diversity)
Trace Criterion

• Make sure that the minimum rank r of matrix
  over all pairs of distinct codewords in such
  that
• Maximize the minimum trace of matrix
  among all pairs of distinct codewords

By B. Vucetic, J. Yuan and Z. Chen 2001
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Comparison of TSC and Trace (TR) criteria
Comparison of TSC and Trace criteria for 4-PSK
STTC modulation 2 transmit antennas
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Road Map

• Coding
• Block codes versus Convolutional codes
• Convolution coding and trellis diagram
• Trellis (MLSD) decoding
• Viterbi decoding
• Space-Time Trellis Codes (STTC)
• Introduction
• Encoding structure
• Design of STTC
• Rank Determinant criteria
• Trace Criterion
• Performance evaluation
• Rank Determinant criteria
• Trace Criterion

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Performance evaluation (I)

• 130 symbols frame
• Increasing the receiver antennas results the
  coding advantage by increasing the number of
  states
• is the number of states

• 4-PSK with rate 2 b/s/Hz, one receive and two
  transmit antennas

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Performance evaluation (II)

• 4-PSK with rate 2 b/s/Hz, two receive and two
  transmit antennas

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Performance evaluation (III)

• Comparing TSC Trace criteria
• Slow fading channels is supposed

• Performance comparison of the 8-state 8-PSK STTCM

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Questions?