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III. Turbo Codes - PowerPoint PPT Presentation

III. Turbo Codes. Turbo Decoding. Turbo decoders rely on probabilistic decoding of component RSC decoders Iteratively exchanging soft-output information between decoder 1 and decoder 2 before making a final deciding on the transmitted bits

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III. Turbo Codes

• Turbo decoders rely on probabilistic decoding of component RSC decoders

• Iteratively exchanging soft-output information between decoder 1 and decoder 2 before making a final deciding on the transmitted bits

• As the number of iteration grows, the decoding performance improves

u = [u1, …, uk, …,uN]: vector of N information bits

xs = [x1s, …, xks, …,xNs] : vector of N systematic bits after turbo-coding of u

x1p = [x11p, …, xk1p, …,xN1p] : vector of N parity bits from first encoder after turbo-coding of u

x2p = [x12p, …, xk2p, …,xN2p] : vector of N parity bits from second encoder after turbo-coding of u

x= [x1s, x11p, x12p, …, xNs, xN1p, xN2p] : vector of length 3N of turbo-coded bits for u

y= [y1s, y11p, y12p, …, yNs, yN1p, yN2p] : vector of 3N received symbols corresponding to turbo-coded bits of u

ys = [y1s, …, yks, …,yNs] : vector of N received symbols corresponding to systematic bits in xs

y1p = [y11p, …, yk1p, …,yN1p] : vector of N received symbols corresponding to first encoder parity bits in x1p

y2p = [y12p, …, yk2p, …,yN2p] : vector of N received symbols corresponding to second encoder parity bits in x2p

u*= [u1*, …, uk*, …,uN*] : vector of N turbo decoder decision bits corresponding to u

ys

xs

u=[u1, …, uk, …,uN]

Turbo

Decoding

y1p

x1p

RSC Encoder 1

y

x

u*

MUX

DEMUX

Channel

N-bit

Interleaver

π

y2p

x2p

RSC Encoder 2

Pr[uk|y] is known as the a posteriori probability

of kth information bit

The LLR of an information bit uk is given by:

Given that Pr[uk=+1]+Pr[uk=-1]=1

Maximum A Posteriori Algorithm

ys

xs

u=[u1, …, uk, …,uN]

Turbo

Decoding

y1p

x1p

RSC Encoder 1

y

x

u*

MUX

DEMUX

Channel

N-bit

Interleaver

π

y2p

x2p

RSC Encoder 2

ys

xs

u=[u1, …, uk, …,uN]

Turbo

Decoding

y1p

x1p

RSC Encoder 1

y

x

u*

MUX

DEMUX

Channel

N-bit

Interleaver

π

y2p

x2p

RSC Encoder 2

uk ykp

Sk

Sk-1

s0

s0

s1

s1

s2

s2

uk=-1

uk=+1

s3

s3

uk ykp

Sk

Sk-1

Define S(i) as the set of pairs of states (s’,s) such that the transition from Sk-1=s’ to Sk=s is caused by the input uk=i, i=0,1, i.e.,

S(0) ={(s0 ,s0), (s1 ,s3), (s2 ,s1), (s3 ,s2)}

S(1) ={(s0 ,s1), (s1 ,s2), (s2 ,s0), (s3 ,s3)}

s0

s0

s1

s1

s2

s2

uk=-1

uk=+1

s3

s3

Remember Bayes’ Rule

Define

depends only on Sk and is independent on Sk-1, yk and

Sk , yk depends only on Sk-1 and is independent on

S0

Sk-1

Sk

Sk+1

SN

s0

s1

s2

s3

y=

y1

yk-1

yk

yk+1

yN

Computation of αk(s)

Example

Computation of αk(s)

Forward Recursive Equation

Given the values of γk(s’,s) for all index k, the probability αk(s) can be forward recursively computed. The initial condition α0(s) depends on the initial state of the convolutional encoder

The encoder usually starts at state 0

Computation of βk(s)

Backward Recursive Equation

Given the values of γk(s’,s) for all index k, the probability βk(s) can be backward recursively computed. The initial condition βN(s) depends on the final state of the trellis

First encoder usually finishes at state s0

Second encoder usually has open trellis

Computation of γk(s’,s)

Note that

Computation of γk(s’,s)