Maximum likelihood decoding techniques notes
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Transcript of Maximum likelihood decoding techniques notes
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Maximum Likelihood
Decoding
Unit 4
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Channel Models: Hard vs Soft decisions
Binary Symmetric Channel (BSC)- Discrete memoryless channel, binary i/p & o/p and symmetrictransition probabilities
- hard decision channel
- U(m) is chosen closest in Hamming distance to Z
- From U(m) , U(m) is chosen for which distance to Z is minimum
(0 |1) (1| 0)
(1 |1) (0 | 0) 1
P P p
P P p
MDCT Unit 4: ML Decoding
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Gaussian Channel
-Each demodulator o/p symbol is a value from continuous
alphabet
-Symbol cannot be labeled as a correct or incorrect detection
decision-Maximizing P(Z|U(m)) is equiv. to maximizing inner product
between codeword sequence U(m) and Z
-Decoder chooses U(m) if it maximizes
-Equivalent to choosing U(m) closest in Euclidean distance to Z
-Soft decision channel
Channel Models: Hard vs Soft decisions
( )
1 1
nm
ji ji
i jz u
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Viterbi Decoding Algorithm
Performs ML decoding with reduced computational load
special structure in code trellis Calculates measure of similarity or distance received
signal (ti) and all trellis paths entering each state (ti)
Trellis paths which could not be the candidate for ML
choices are not considered Surviving path - The path with best metric is chosen when
two paths enter the same state, and performed for all states- least likely paths are eliminated
Optimum path: expressed choosing codeword withmaximum likelihood metric orminimum distance metric
Advantage: Complexity is not a function of no. of symbolsin codeword sequence
MDCT Unit 4: ML Decoding
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Convolutional Encoder (rate K=3)
MDCT Unit 4: ML Decoding
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Encoder Trellis Diagram (rate K=3)
MDCT Unit 4: ML Decoding
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Decoder Trellis Diagram (rate K=3)
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The basis
Any two paths merge to a single state, one path is
eliminated in search of an optimum path
Path metrics for two merging paths
MDCT Unit 4: ML Decoding
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Decoding Principle
At each time ti, 2K-1 states are present in trellis and
can be entered by means of two paths
Decoding computes the metric for two paths entering
each state and eliminating one of them Computation is done for each of the 2K-1 states at time
ti and decoder moves to time ti+1 and the process is
repeated
At any time the winning path metric for each state is
termed as state metric for that state at that time
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Selection of Survivor paths
Survivors at t2 Survivors at t3
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Selection of Survivor paths (Contd.)
Metric comparisons at t4 Survivors at t4
Only one surviving path between time t1 and t2and it is termed as common stem
Transition occurred between 0010 and since it
is due to input bit 1, decoder output is 1
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Selection of Survivor paths (Contd.)
Metric comparisons at t5 Survivors at t5
MDCT Unit 4: ML Decoding
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Selection of Survivor paths (Contd.)
Metric comparisons at t6 Survivors at t6
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Sequential Decoding
Proposed by Wozencraft and modified by Fano Generates hypothesis about transmitted codeword
sequence and computes metric between thesehypothesis and received signal
Goes forward still metric indicates its choices arelikely; else goes backward changing hypothesis stillfinding a likely one
Can work with both hard and soft decisions, howeversoft decisions are normally avoided since largestorage elements are used and also complexity
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Convolutional Encoder (rate K=3)
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Tree Diagram (rate K=3)
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Sequential Decoding Example
MDCT Unit 4: ML Decoding