Outline - University of California, Berkeleybwrcs.eecs.berkeley.edu/Classes/EE290C_S04/lectures/...What is the best path to take to: Hayward? Mountain View Palo Alto (5) Milpitas Union

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EE290C - Spring 2004Advanced Topics in Circuit DesignHigh-Speed Electrical Interfaces

Lecture 12Components

Viterbi DecoderBorivoje NikolicMarch 4, 2004.

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Outline

Viterbi decoderHow it worksWhat is it used forDigital implementations

Analog implementations

2

3

Viterbi AlgorithmExample of dynamic programming [Bellman’57]Invented by A. Viterbi in 1967Explained by Forney in 1972,1973Used for:

Decoding convolutional codesDecoding trellis codesMaximum likelihood detectionSpeech recognition, etc.

Types:Hard-input, hard-outputSoft-input, hard-outputSoft-input, soft-output

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Jared’s Trip to Berkeley

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5

Jared’s Trip to Berkeley

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Trellis

States + edgesNo loopsWeights in minutes

Mountain View

PaloAlto

MilpitasUnionCity

San Mateo

Hayward Oakland

San Francisco

Berkeley

5

15

10

15

15 1515

20 20

2040

20

40

4

7

Shortest Time to Get to Berkeley?

What is the best path to take to:Union City?Hayward?

Mountain View

PaloAlto

MilpitasUnionCity

San Mateo

Hayward Oakland

San Francisco

Berkeley

5

15

10

15

15 1515

20 20

2040

20

40

Choose the minimum cost at each point (state)

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What is the best path to take to:Union City?

Mountain View

PaloAlto(5)

Milpitas(15)

UnionCity(20)

San Mateo

Hayward Oakland

San Francisco

Berkeley

5

15

10

15

20

20

5

9


What is the best path to take to:Hayward?

Mountain View

PaloAlto(5)

Milpitas

UnionCity(20)

San Mateo

Hayward(35) Oakland

San Francisco

Berkeley

5

15

15

15

20 20

20

40

10


Mountain View

PaloAlto(5)

Milpitas

UnionCity(20)

San Mateo

Hayward(35)

Oakland(50)

San Francisco

Berkeley(65)

5

15

15

15 1515

20 20

6

11

Digital Baseband Transceiver

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Convolutional Codes

Adding redundancy

D D

+

+

di di-1di-2

Channel

0110

1+D2

1+D+D2

0111

0100

(00, 11, 10, 10)

Generators:G1 = 101G2 = 111

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13

State Transition Diagrams

00

1101

10

0/00

1/11

0/01

1/10

0/11

1/00

0/10

1/01

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Error Sequences

a = 00

b = 01

d = 11

c = 10

00

11

00

01

10

01

10

11

a b c

d

a

w=2 w=2w=0

w=1

w=1

w=1

w=1

Minimum distance error event is 4 bits

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15

Decisions at Each Step

00

01

10

11

i i+1 i+2

State (path)metrics

Branchmetrics

In Gaussian channel:bm = (yk – sk)2

In BSC:bm = dH(yk, sk) = |yk – sk|

dH is Hamming distanceIf received yk = 10dH(10, 00) = 1dH(10, 01) = 2

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Trellis Diagram

00

01

10

11

i i+1 i+2

Time-indexed state diagram

i+3 i+4

0/00

1/11

0/00

1/11

0/11

1/00

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The Viterbi Algorithm

Illustrated by 2-state trellis

sm n1 1−

sm n2 1−

sm n1

sm n2

timetn−1 tn

bm1

bm2 bm3

bm4

( )( )

sm sm bm sm bm

sm sm bm sm bm

n n n

n n n

1 1 1 2 3

2 1 2 2 4

1 1

1 1

= + +

= + +

− −

− −

min ,

min ,

Add AddCompareSelect

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Viterbi Decoder

Implements Viterbi algorithm

Three main componentsBranch metric calculationPath metric accumulation (add-compare-select recursion)Survivor path decode

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Viterbi Decoder

D

Branchmetrics

unit

Add-Compare-

Select

Survivorpath

decode

Branch metrics unitCalculates the distances between the received signal and the ideal signals

Add-Compare-Select unitAccumulates path metrics

Survivor path decodeKeeps track of the path through the trellis

Inputs Decoded bits

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Calculating Branch Metrics

Assume G1 and G2 both output a 0Soft decoder inputs are 0.1 and 0.25 insteadBranch metrics:

bm00 = |0 – 0.1| + |0 – 0.25| = 0.35bm01 = |0 – 0.1| + |1 – 0.25| = 0.85bm10 = |1 – 0.1| + |0 – 0.25| = 1.15bm11 = |1 – 0.1| + |1 – 0.25| = 1.65

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Calculating Branch Metrics

Euclidean distances (AWGN channel)bmi = (yk – sk)2

sk are usually integersbmi = yk

2 -2Ciyk + Ci2

Since yk terms are common to all branch metrics they drop out in ACS comparison, and can be eliminated.

Ci2 are precomputed and 2Ciyk are shifts and adds.

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Conventional add-compare-select unit

Add-Compare-Select Recursion

Eight state trellis

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Adder

Adder

Subtractor

2:1 Multiplexer

Register

sm1 sm2

bm1

bm2

MSB

DecisionNew State Metric

5

5

8 8

8 8

8

24


sm20

bm20

sm21

bm21

sm22

bm22

sm23

bm23

sm24

bm24

sm25

sm26

sm27

+

+

+

+

+

+

+

+

+ −

−

−

−

−

−

−

−

+

+

+

+

+

+

+

sm10

sm0

sm1

sm2

sm3

sm14 sm

4

bm10

bm11

bm12

bm13

bm14

sm15 sm

5

sm16 sm

6

sm17 sm

7

diff

13

25


Bit

1 2 3 4 5 6 70

Select

Register

Register

Time

Add

CC

CC

CC

CC

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Radix-4 add-compare-select unit


One step lookaheadapplied to an eight-state trellis

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Add-Compare-Select RecursionAdder

Adder

Subtractor

4:1 Multiplexer

Register

sm1 sm2

bm1

bm2

MSB

DecisionNew State Metric

Adder

Subtractor

Subtractor

sm3

Sele

ct

MSB

MSB

bm3

Adder

Subtractor

Subtractor

Subtractor

sm4

bm4

MSB

MSB

MSB

Inhi

bit

sm3

Inhi

bit s

m4

5

5

5

5

8 8 8 8

8 8 8 8

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4-way ACS:2 additions + 6 comparisons

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Concurrent add and compare

Sridharan, Carley, JSSC’00

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29


Retimed ACS

Fettweiss, Globecom’95

30Parallel compare-select-add unit


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31Area comparison of various ACS structures


32Power comparison of various ACS structures


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Extreme ACSs

LSB-first computation MSB-first computation

Fettweiss, Meyr, TCom’90Yeung, Rabaey, ISSCC’95Drawings by Haratsch

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RNS-Based ACS

si ∈ {0, 1, 2}, and are two-bit encoded as si ∈ {00, 01, 11}

∑−

==

1

02

w

i

iisS

smx0,y0

bm0

s0

co0 sx1

sy1smx1,y1

bm1

s1

co1 sx2

sy2smx2,y2

bm2

s2

co2

sx7

sy7smx0,y0

bm7 = 0

s7

co7

sx0 = 00

sy0 = s0

normalization

c6

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Bit-Level Pipelined ACS

ADD3 ADD3

ADD2 ADD2ADD1 ADD1

Clk

ADD0 ADD0

ADD3

ADD2

SEL3 SEL3SEL2 SEL2

SEL1

SEL0

Yeung, Rabaey, ISSCC’95

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Analog ACS

Fukahori, ISSCC’98

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Interleaved Analog ACS

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State metrics sm i and sm j represented on a number circle

Metric Normalization

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0

1

2

3

nn−1n−2n−3n−4n−5

4-state trellis example

Survivor Path Decode

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Register Exchange

Documents

Outline - University of California, Berkeleybwrcs.eecs.berkeley.edu/Classes/EE290C_S04/lectures/...What is the best path to take to: Hayward? Mountain View Palo Alto (5) Milpitas Union