Chapter 3 Convolutional Codes and Trellis Coded Modulation
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1 Chapter 3 Convolutional Codes and Trellis Coded Modulation 3. Encoder Structure and Trellis Representation 3. Systematic Convolutional Codes 3.3 Viterbi Decoding Algorithm 3.4 BCJR Decoding Algorithm 3.5 Trellis Coded Modulation
2 3. Encoder Structure and Trellis Representation Introduction Encoder: contains memory (order m: m memory units); Output: encoder output at time unit t depends on the input and the memory units status at time unit t; By increasing the memory order m, one can increase the convolutional code s minimum distance (d min ) and achieve low bit error rate performance (P b ); Decoding Methods: Viterbi algorithm []: Maximum Likelihood (ML) decoding algorithm; Bahl, Cocke, Jelinek, and Raviv (BCJR) [] algorithm: Maximum A Posteriori Probability (MAP) decoding algorithm, used for iterative decoding process, e.g. turbo decoding. [] A. J. Viterbi, Error bounds for convolutional codes and an asymptotically optimum decoding algorithm, IEEE Trans. Inform. Theory, IT-3, 6-69, April, 967. [] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, Optimal decoding of linear codes for minimizing symbol error rate, IEEE Trans, Inform. Theory, IT-; 84-87, March, 974.
3 3. Encoder Structure and Trellis Representation - The (7, 5) 8 conv. code Encoder structure: c Input a S S Encoding Process: c (Initialised state s s = ) At time t At time t At time t 3 Code rate: ½; Memory: m = ; Constraint length: m + = 3 Output calculation: c = a S S ; c = a S ; Registers update: S = S. S = a.
4 3. Encoder Structure and Trellis Representation At time t 4 At time t 6 At time t 5 Input sequence [m m m 3 m 4 m 5 m 6 ] = [ ]
5 3. Encoder Structure and Trellis Representation A state transition diagram of the (7, 5) 8 conv. code / / d / / State definition (s s ) a = b = c = c b d = / / a / / Interpretation of the state diagram c Input bit () / output bits () The current state of the encoder is c. If the input bit is, it will output and the next state of the encoder is a. a
6 3. Encoder Structure and Trellis Representation Tree Representation of the (7, 5) 8 conv. code Time unit: 3 4 Initialised state: a a b a b c d a b c d a b c d Tree diagram interpretation: The current state of the encoder is b. If the input bit is, the output will be, and the next state of the encoder is c. Input bit as Input bit as State after transition Output from transition Example 3. Determine the codeword that corresponds to message [ ]
7 3. Encoder Structure and Trellis Representation Trellis of the (7, 5) 8 conv. code State Table IN Current State Next State Out ID Trellis Remark: A trellis tells the state transition and IN/OUT relationship. It can be used to yield a convolutional codeword of a sequential input. Example 3. Use the above trellis to determine the codeword that corresponds to message [ ]. OUT IN: IN: 4 5 6
8 3. Encoder Structure and Trellis Representation A number of conv. codes (7, 5) 8 conv. code (5, 3) 8 conv. code 4 states 8 states (3, 35) 8 conv. code (7, 33) 8 conv. code 6 states 64 states Remark: A convolutional code s error-correction capability improves by increasing the number of the encoder states.
9 3. Encoder Structure and Trellis Representation Remark: The encoder structure can also be represented by generator sequences or transfer functions. Example 3.3:
10 3. Systematic Convolutional Codes - The (7, 5) 8 conv. code s systematic counterpart is: (7, 5) 8 conv. code c a (, 5/7) 8 conv. code c c a S S S S Nonsystematic code c Systematic code Encoding and Registers updating rules: [S S ] are initialization as [ ]; c = a ; (systematic feature) feedback = S S ; c = a feedback S ; S = S ; S = a feedback; Remark: Systematic encoding structure is important for iterative decoding, e.g., the decoding of turbo codes.
11 3. Systematic Convolutional Codes For the (, 5/7) 8 conv. code State Table IN Current State Next State Out ID OUT Trellis IN: IN:
12 3.3 Viterbi Decoding Algorithm Let us extend the trellis of the (7, 5) 8 conv. code as if there is a sequential input. - Such an extension results in a Viterbi trellis - A path in the Viterbi trellis represents a convolutional codeword that corresponds to a sequential input (message).
13 3.3 Viterbi Decoding Algorithm - Branch metrics: Hamming distance between a transition branch s output and the corresponding received symbol (or bits). - Path metrics: Accumulated Hamming distance of the previous branch metrics.
14 3.3 Viterbi Decoding Algorithm Example 3.4. Given the (7, 5) 8 conv. code as in Examples The transmitted codeword is Step : Calculate all the branch metrics.
15 3.3 Viterbi Decoding Algorithm Step : Calculate the path metrics When two paths join in a node, keep the smaller accumulated Hamming distance. When the two joining paths give the same accumulated Hamming distance, pick up one randomly.
16 3.3 Viterbi Decoding Algorithm Step 3: Pick up the minimal path metric and trace back to determine the message. Tracing rules: () Trellis connection; () The previous path metric should NOT be greater than the current path metric; (3) The tracing route should match the trellis transition ID.,,, 3,5, 3,6,6 3,3,3,5,7,,4, 3,5,4,6 3,8 Decoding output:
17 3.3 Viterbi Decoding Algorithm Branch Metrics Table Path Metrics Table Trellis Transition ID Table
18 3.3 Viterbi Decoding Algorithm Soft-decision Viterbi decoding While we are performing the hard-decision Viterbi decoding, we have the scenario that two joining paths yield the same accumulated Hamming distance. This would cause decoding ambiguity and performance penalty; Such a performance loss can be compensated by utilizing soft-decision decoding, e.g., soft-decision Viterbi decoding Modulation and Demodulation (e.g., BPSK) Modulation: mapping binary information into a transmitted symbol; Demodulation: determining the binary information with a received symbol; information (-, ) (, ) Channel (-, ) (.5,.9) (, ) Received symbol Transmitted symbol Modulation Demodulation
19 3.3 Viterbi Decoding Algorithm Modulation and Demodulation (e.g., BPSK) (.5,.9) (-, ) (, ) Demodulation Received symbol Hard-decision: the information bit is. The Hamming distance becomes the Viterbi decoding metrics; Soft-decision: the information bit has Pr. of.7 being and Pr. of.3 bing. The Euclidean distance (or probability) becomes the Viterbi decoding metrics; Euclidean Distance Definition: The Euclidean distance between points p and q is the length of the line segment connecting them. p(x, y ) q(x, y ) d Eud x) ( y ) ( x y
20 3.3 Viterbi Decoding Algorithm Example 3.5. Given the (7, 5) 8 conv. code as in Examples The transmitted codeword is After BPSK modulation, the transmitted symbols are: (, ), (, ), (-, ), (-, ), (, ), (-, ), (, ), (-, ), (, ), (, ). After the channel, the received symbols are: (.8,.), (., -.4), (-.3,.3), (-.9, -.), (-.5,.4), (-.,.), (.,.4), (-.7, -.), (.,.), (.9,.3).
21 3.3 Viterbi Decoding Algorithm Step : Calculate all the branch metrics. (.8,.) (., -.4) (-.3,.3) (-.5,.4) (.,.4) (.,.) (-.9, -.) (-.,.) (-.7, -.) (.9,.3) (, ) (-, ).93.94
22 3.3 Viterbi Decoding Algorithm Step : Calculate the path metrics. (.8,.) (., -.4) (-.3,.3) (-.5,.4) (.,.4) (.,.) (-.9, -.) (-.,.) (-.7, -.) (.9,.3) When two paths join in a node, keep the smaller accumulated Euclidean distance.
23 3.3 Viterbi Decoding Algorithm Step 3: Pick up the minimal path metric and trace back to determine the message. Tracing rules: The same as hard-decision Viterbi decoding algorithm. (.8,.) (., -.4) (-.3,.3) (-.5,.4) (.,.4) (.,.) (-.9, -.) (-.,.) (-.7, -.) (.9,.3) , , , ,5 3.8,5 3.64, ,.98, 4.8, 4.84, ,6..54, , , , , , ,8 Decoding output:
24 3.3 Viterbi Decoding Algorithm Branch Metrics Table Path Metrics Table Trellis Transition ID Table
25 3.3 Viterbi Decoding Algorithm Free distance of convolutional code - A convolutional code s performance is determined by its free distance. - Free distance - With knowing is also a convolutional codeword. Hence, it is the minimum weight of all finite length paths in the Viterbi trellis that diverge from and emerge with the all zero state.
26 3.3 Viterbi Decoding Algorithm Hence, it is the minimum weight of all finite length paths in the Viterbi trellis that diverge from and emerge with the all zero state.
27 3.3 Viterbi Decoding Algorithm Remark: Convolutional code is more competent in correcting spread errors, but not bursty errors. error
28 3.4 BCJR Decoding Algorithm - Hard-decision Viterbi algorithm: a Hard-In-Hard-Out (HIHO) decoding. Soft-decision Viterbi algorithm: a Soft-In-Hard-Out (SIHO) decoding. BCJR Algorithm: a Soft-In-Soft-Out (SISO) decoding. - A Soft-In-Soft-Out (SISO) decoding algorithm that takes probabilities as the input and delivers probabilities as the output.
29 3.4 BCJR Decoding Algorithm - In a trellis (e.g., trellis of the (7, 5) 8 conv. code). (a) (a) IN: IN: (b) (c) (d) (b) (c) (d) The (IN, OUT, current state, next state) tuple happens as an entity. θ θ θ θ θ θ θ θ θ θ
30 3.4 BCJR Decoding Algorithm (a) (a) IN: IN: (b) (c) (b) (c) (d) (d) - For a rate half conv. code, - Trellis state transition probability:
31 3.4 BCJR Decoding Algorithm - Determine the state transition probabilities. (a) (a) IN: IN: (b) (c) (b) (c) BPSK (d) (d) P ch ( c ) P( y c ) ' t t t ' P ch ( c ) P( y c ) ' t t t '
32 3.4 BCJR Decoding Algorithm - Determine the probability of each beginning state. a a a IN: IN: b c d b c d b c d - Knowing the Viterbi trellis starts from the all-zero state, we initialize: - E.g., in the highlighted trellis transition
33 3.4 BCJR Decoding Algorithm - Determine the probability of each ending state. a Ω a a IN: IN: b c d b c d b c d - By ensuring after encoding, the shift registers (encoder) are restored to the all zero state (achieved by bit tailing), we can initialize: - E.g., in the highlighted trellis transition
34 3.4 BCJR Decoding Algorithm a a IN: IN: b c b c d d State transition indicated by State transition indicated by
35 3.4 BCJR Decoding Algorithm - E.g., - Decision based on the a posteriori probabilities.
36 3.4 BCJR Decoding Algorithm Example 3.6. With the same transmitted codeword and received symbols of Example 3.5, use the BCJR algorithm to decode it. With the received symbols, we can determine
37 3.4 BCJR Decoding Algorithm (.8,.) (., -.4) (-.3,.3) (-.5,.4) (.,.4) (.,.) (-.9, -.) (-.,.) (-.7, -.) (.9,.3) a b c d a b c d
38 3.4 BCJR Decoding Algorithm a b c d.... (.8,.) (., -.4) (-.3,.3) (-.5,.4) (.,.4) (.,.) (-.9, -.) (-.,.) (-.7, - (.9,.3).) a b c d
39 3.4 BCJR Decoding Algorithm (.8,.) (., -.4) (-.3,.3) (-.5,.4) (.,.4) (.,.) (-.9, -.) (-.,.) (-.7, -.) (.9,.3) a b c d a b c d Assume we know the trellis ends at state c.
40 3.4 BCJR Decoding Algorithm Step 4: Determine the a posteriori probabilities of each information bit.
41 3.4 BCJR Decoding Algorithm BER performance of (7, 5) 8 conv. code over AWGN channel using BPSK..E+.E-.E- uncoded Hard decision Viterbi Soft decision Viterbi BCJR SNR threshold BER.E-3.E-4.E-5.E E b /N (db)
42 3.4 BCJR Decoding Algorithm BER performance of different conv. code over AWGN channel using BPSK..E+.E-.E- 4-state (7, 5) 8 conv. code 8-state (5, 3) 8 conv. code 6-state (3, 35) 8 conv. code 64-state (7, 33) 8 conv. code BER.E-3.E-4.E-5.E E b /N (db)
43 3.5 Trellis Coded Modulation - Convolutional code enables reliable communications. But as a channel code, its error-correction function is on the expense of spectral efficiency. - E.g., an uncoded system using BPSK η = info bits/symbol A rate / conv. coded system using BPSK η =.5 info bits/symbol - Can we achieve reliable and yet spectrally efficient communication? Solution: Trellis Coded Modulation (TCM) that integrates a conv. code with a high order modulation [3]. [3] G. Ungerboeck, "Channel coding with multilevel/phase signals," IEEE Trans. Inform. Theory, vol. IT-8, pp , 98.
44 3.5 Trellis Coded Modulation - A general structure of the TCM scheme a a a k a k+ a k+v Rate k/(k+) conv. encoder c c c k+ Select a subset from the constellation Select a point from the subset Output symbol
45 3.5 Trellis Coded Modulation - A rate /3 TCM code. a a S S c 3 c c Select a subset from 8PSK Select a point from the subset Output Rate ½ 4-state Convolutional Code (4) (3) 8PSK Constellation (5) () () () (6) (7)
46 3.5 Trellis Coded Modulation - State table of the rate /3 TCM code a a S S S S c c c 3 8PSK sym Input Current State Next State Output Symbol
47 3.5 Trellis Coded Modulation - Set Partitioning 8PSK ' sin.765 ' 8 c 3 c c = c = c = ' 6 Original constellation Subset c = c = c = c = ' (, 4) (, 6) (, 5) (3, 7) Subset
48 3.5 Trellis Coded Modulation - Set Partitioning 8PSK By doing set partitioning, the minimum distance between point within a subset is increasing as: Δ < Δ < Δ. Original constellation Set Partitioning Subset Subset
49 3.5 Trellis Coded Modulation - Viterbi trellis of the rate /3 TCM code 6 4 For diverse/remerge transition: For parallel transition: Choose the smaller one as the free distance of the code: Remark: Bit c 3 = and c 3 = result in two parallel transition branches. By doing set partitioning, we are trying to maximize the Euclidean distance between the two parallel branches. So that the free distance of the TCM code can be maximized.
50 3.5 Trellis Coded Modulation - Asymptotic coding gain over an uncoded system. - Spectral efficiency (η) = info bits/sym. uncoded QPSK rate /3 coded 8PSK Remark: With the same transmission spectral efficiency of info bits/sym, the TCM coded system achieves 3 db coding gain over the uncoded system asymptotically.
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