maximum likelihood decoding techniques notes

7/29/2019 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



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Convolutional Encoder (rate K=3)



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Encoder Trellis Diagram (rate K=3)



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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



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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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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


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