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The Viterbi Algorithm. Application of Dynamic Programming-the Principle of Optimality -Search of Citation Index -213 references since 1998 Applications Telecommunications Convolutional codes-Trellis codes Inter-symbol interference in Digital Transmission Continuous phase transmission

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the viterbi algorithm
The Viterbi Algorithm
  • Application of Dynamic Programming-the Principle of Optimality
  • -Search of Citation Index -213 references since 1998
  • Applications
    • Telecommunications
      • Convolutional codes-Trellis codes
      • Inter-symbol interference in Digital Transmission
      • Continuous phase transmission
      • Magnetic Recording-Partial Response Signaling-
      • Divers others
        • Image restoration
        • Rainfall prediction
        • Gene sequencing
        • Character recognition
milestones
Milestones
  • Viterbi (1967) decoding convolutional codes
  • Omura (1968) VA optimal
  • Kobayashi (1971) Magnetic recording
  • Forney (1973) Classic survey recognizing the generality of the VA
  • Rabiner (1989) Influential survey paper of hidden Markov chains
example principle of optimality

Professor X chooses an optimum

path on his trip to lunch

Find optimal path to each bridge

Example-Principle of Optimality

Perish

Publish

N

1.2

.2

.7

1.2

.5

N

Faculty

Club

1.2

.5

.3

EE Bld

Optimal: 6 adds

Brute force:8 adds

N bridges

Optimal: 4(N+1) adds

Brute force: (N-1)2N adds

.5

S

1.0

.8

.8

S

.8

digital transmission with convolutional codes
Digital Transmission with Convolutional Codes

Information

Source

Convolutional

Encoder

BSC

p

p

Information

Sink

Viterbi

Algorithm

maximum a posteriori map estimate
Maximum a Posteriori (MAP) Estimate

Brute force = Exponential Growth with N

convolutional codes encoding a sequence
Convolutional codes-Encoding a sequence

(output,input)

efficiency=input/output

Example(3,1) code

Output

111 100 010 110 011 001 000

Input

110100

T

T

trellis representation
Trellis Representation

State output Next state

0 input

s1s2

1 input

000

111

00

01

10

11

00

01

10

11

001

110

011

100

010

101

key step
Key step!

Redundant

Linear growth in N

deciding previous state
Deciding Previous State

State i-1

State i

00

00

1

4

4

Search previous states

10

2

2

viterbi algorithm shortest path to detect sequence
Viterbi Algorithm-shortest pathto detect sequence

First step

s0

s1

s2

s3

Trace though successive states

shortest path-Hamming distance to s0

Trellis codes-Euclidean distance

Optimum Sequence Detection

inter symbol interference
Inter-symbol Interference

Transmitter

Channel

Equalizer

VA

Decisions

viterbi algorithm for isi
Viterbi Algorithm for ISI

Magic Iteration

State = number of symbols in memory

magnetic recording
Magnetic Recording

Magnetization pattern

Magnetic flux passes over heads

Differentiation of pulses

Controlled ISI

Same model applies to

Partial Response signaling

Output

Sample

continuous phase fsk
Continuous Phase FSK

odd number ½ cycles

Whole cycles

merges and state reduction
Merges and State Reduction

Optimal paths through trellis

All paths merge

Force merges to reduce complexity

Computations order of (No states)2

Carry only high probability states

slide19

Image Restoration

Input Pixel

Effect of Blurring

Blurring Analogous to ISI

row scan
Row Scan

VA for optimal row sequence

Known state transitions

And Decision Feed back

Utilized for state reduction

hidden markov chain
Hidden Markov Chain
  • Data suggests Markovian structure
  • Estimate initial state probabilities
  • Estimate transition probabilities
  • VA used for estimation of Probabilities
  • Iteration
rainfall prediction
Rainfall Prediction

Rainy

wet

Rainy

dry

Transition probabilities estimated

from Rainfall data using VA.

No rain

Showery

wet

Showery

dry

Rainfall observations

dna sequencing
DNA Sequencing
  • DNA-double helix
    • Sequences of four nucleotides, A,T,C and G
    • Pairing between strands
    • Bonding
  • Genes
    • Made up of Cordons, i.e. triplets of adjacent nucleotides
    • Overlapping of genes

Nucleotide sequence

CGGATTC

Gene 1

Cordon A in three genes

Gene 2

Gene 3

hidden markov chain tracking genes
Hidden Markov ChainTracking genes

S

M1

P1

S-start first cordon of gene

P1-4- +1,…,+4 from start

G- Gene

E-stop

H-gap

M1-4-1,…,-4 from start

M2

P2

M3

P3

Initial and

Transition

Probabilities known

M4

P4

.

G

H

E

recognizing handwritten chinese characters

Text-line images

Recognizing Handwritten Chinese Characters

Estimate stroke width

Set up m X n grid

Estimate initial and transition probabilities

Detect possible segmentation paths by VA

Next Slide

Results

example segmenting handwritten characters
Example Segmenting Handwritten Characters

Eliminating

Redundant

Paths

All possible

segmentation

paths

Removal of

Overlapping

paths

Discarding

near paths