Optimal soft-output decoder for tail-biting trellis codes

• US 5,721,746 A
• Filed: 04/19/1996
• Issued: 02/24/1998
• Est. Priority Date: 04/19/1996
• Status: Expired due to Term

• Current Assignee: SES AMERICOM, Inc. (SES SA)
• Original Assignee: General Electric Company
• Inventors: Hladik, Stephen Michael, Anderson, John Bailey
• Primary Examiner(s): Baker, Stephen M.
• Application Number:
  US08/636,742
• Time in Patent Office: 676 Days
• Field of Search: 371/43
• US Class Current: 714/792
• CPC Class Codes: H03M 13/3723 using means or methods for the initialisation of the decoder
• CPC Class Codes: H03M 13/3933 Decoding in probability domain
• CPC Class Codes: H03M 13/3938 Tail-biting H03M13/2996 takes precedence
• CPC Class Codes: H03M 13/616 Matrix operations, especially for generator matrices or check matrices, e.g. column or row permutations
• CPC Class Codes: H03M 13/6583 Normalization other than scaling, e.g. by subtraction

Abstract

A circular MAP decoder for error-correcting trellis codes that employ tail biting produces soft-decision outputs provides an estimate of the probabilities of the states in the first stage of the trellis, which probabilities replace the a priori knowledge of the starting state in a conventional MAP decoder. The circular MAP decoder provides the initial state probability distribution in either of two ways. The first involves a solution to an eigenvalue problem for which the resulting eigenvector is the desired initial state probability distribution; with knowledge of the starting state, the circular MAP decoder performs the rest of the decoding according to the MAP decoding algorithm. The second is based on a recursion for which the iterations converge to a starting state distribution. After sufficient iterations, a state on the circular sequence of states is known with high probability, and the circular MAP decoder performs the rest of the decoding according to the MAP decoding algorithm.

Claims

1. A decoder for a tall-biting trellis code generated by an encoder, which decoder decodes by determining the joint probability that the state of the encoder at time t, S_t, is m and that a sequence of L channel outputs having values Y₁^L ={y₁, . . . , y_L } is received, as represented by λ

_t (m)=P{S_t =m;

Y₁^L }, said trellis code having M encoder states, said decoder determining L probability matrices γ

_t, one at each of L trellis levels, the elements of said probability matrices being defined by;

space="preserve" listing-type="equation">γ

.sub.t (ij)=P{state j at time t-1;

y.sub.t /state i at time t.sub.1 };

by determining row vectors α

_t having M joint probability elements defined by;

space="preserve" listing-type="equation">α

.sub.t (j)=P{state j at time t;

y.sub.1, . . . , y.sub.t };

and by determining column vectors β

_t having M conditional probability elements defined by;

space="preserve" listing-type="equation">β

.sub.t (j)=P{y.sub.t+1, . . . y.sub.L /state j at time t}

for j=0, 1, . . . , (M-1), said decoder comprising;

a γ

_t calculator for receiving said channel outputs, the channel transition probability R(Y_t, X), the probability p_t (m/m'"'"') that the encoder makes a transition from state m'"'"' to m at time t, and the probability q_t (X/m'"'"', m) that the encoder'"'"'s output symbol is X given that the previous encoder state is m'"'"' and the present encoder state is m, and for determining the scalar elements of said probability matrices γ

_t therefrom;

a γ

_t matrix product calculator for receiving said scalar elements from said γ

_t calculator and computing a matrix product γ

₁ γ

₂ . . . γ

_L therefrom;

a normalized eigenvector computer for receiving said matrix product γ

₁ γ

₂ . . . γ

_L and computing a normalized eigenvector α

₀ corresponding to the largest eigenvalue P{Y₁^L } of said matrix product;

a α

_t matrix product calculator for receiving said normalized eigenvector α

₀ and forming the succeeding α

_t by a forward recursion as follows;

space="preserve" listing-type="equation">α

.sub.t =α

.sub.t-1 γ

.sub.t, t=1, . . . , L;

memory for storing said probability matrices γ

_t and said row vectors α

_t ;

a β

_t matrix product calculator for providing said column vectors by initializing β

_L =(1, 1, 1, . . . ,

     1)^T and forming the preceding β

_t by a backward recursion as follows;

space="preserve" listing-type="equation">β

.sub.t =γ

.sub.t+1 β

.sub.t+1, t=L-1, . . . , 1;

an element-by-element product calculator for forming the joint probability vectors λ

t, the elements of which are said joint probabilities λ

_t (ij), by multiplying the elements of said row vectors by the elements of said column vectors as follows;

space="preserve" listing-type="equation">λ

.sub.t (i)=α

.sub.t (i) β

.sub.t (i), all i, t=1, . . . , L; and

a decoded bit value probability calculator for determining from λ

_t the probability that a given data bit inputted to the encoder at time=t is equal to zero, the data bit being the m^th of k data bits, and providing a soft output as a function of said probability.

2. The decoder of claim 1, further comprising a threshold decision device for receiving the probability that the data bit inputted to the encoder at time=t is equal to zero and for implementing a decision rule to provide decoded output bits therefrom.

3. The decoder of claim 1 wherein said tail-biting trellis code comprises a convolutional code.

4. The decoder of claim 1 comprising a bounded distance decoder.

5. A decoder for a tail-biting trellis code generated by an encoder, which decoder decodes by determining the joint probability that the state of the encoder at time t, S_t, is m and that a sequence of L channel outputs having values Y₁^L ={y₁, . . . , y₁ } is received, as represented by λ

_t (m)=P{S_t =m, Y₁^L }, said trellis code having M encoder states, said decoder determining L conditional probability matrices γ

_t, one at each of L trellis levels, the elements of said probability matrices being defined by;

space="preserve" listing-type="equation">γ

.sub.t (ij)=P{state j at time t;

y.sub.t /state i at time t-1};

by determining row vectors α

_t having M joint probability elements defined by;

space="preserve" listing-type="equation">α

.sub.t (j)=P{state j at time t;

y.sub.1, . . . , y.sub.t };

and by determining column vectors β

_t having M conditional probability elements defined by;

space="preserve" listing-type="equation">β

.sub.t (j)=P{y.sub.t+1, . . . y.sub.L /state j at time t}

for j=0, 1, . . . , (M-1), said decoder comprising;

a γ

_t calculator for receiving said channel outputs, the channel transition probability R(Y_t, X), the probability p_t (m/m'"'"') that the encoder makes a transition from state m'"'"' to m at time t, and the probability q_t (X/m'"'"', m) that the encoder'"'"'s output symbol is X given that the previous encoder state is m'"'"' and the present encoder state is m, and for determining the scalar elements of said probability matrices γ

_t therefrom;

a α

_t matrix product calculator for receiving said scalar elements of γ

_t from said γ

_t calculator and providing said row vectors α

_t ;

a β

_t matrix product calculator for providing said column vectors β

_t ;

an element-by-element product calculator for forming the joint probability vectors λ

_t, the elements of which are said joint probabilities λ

_t (i,j);

said α

_t matrix product calculator, said β

_t matrix product calculator, and said element-by-element product calculator forming said vectors α

_t, β

_t, and λ

_t, respectively, by;

(i.a) starting with an initial α

₀ equal to (1/M, . . . , 1/M), calculating L times the forward recursion;

space="preserve" listing-type="equation">α

.sub.t =α

.sub.t-1 γ

.sub.t, t=1, . . . , L;

and normalizing the results so that the elements of each α

_t sum to unity, retaining all L α

_t vectors;

(i.b) letting α

₀ equal α

_L from step (i.a) and, starting at t =1, calculating

space="preserve" listing-type="equation">α

.sub.t =α

.sub.t-1 γ

.sub.t, t=1, . . . , L.sub.w.sbsb.min ;

where L_w.sbsb.min is a predetermined minimum number of trellis stages;

normalizing the results so that the elements of each α

_t sum to unity, retaining only the most recent set of α

_t '"'"'s found by the recursion in steps (i.a) and (i.b) and the ##EQU15## found in step (i.a);

(i.c) comparing ##EQU16## from step (i.b) to the ##EQU17## found in step (i.a) and proceeding to step (ii) if within a tolerance range;

otherwise, continuing to step (i.d);

(i.d) letting t=t+1 and calculating α

_t =α

_t-1 β

_t, normalizing the results of the recursion so that the elements of each α

_t sum to unity, retaining only the most recent set of L α

_t '"'"'s calculated and this α

_t found previously in step (i.a).

(i.e) comparing the α

_t to the most recent, previously-calculated α

_t from steps (i.a), (i.b) and (i.d) and proceeding to step (ii) if within a tolerance range, continuing with step (i.d) if the two most recent vectors do not agree to within said tolerance range and if the number of recursions does not exceed a predetermined maximum;

proceeding to step (ii) otherwise;

(ii) initializing β

_L =(1, 1, 1, . . . ,

     1)^T and forming the preceding β

_t by a backward recursion as follows;

space="preserve" listing-type="equation">β

.sub.t =γ

.sub.t+1 β

.sub.t+1, t=L-1, . . . , 1;

and normalizing the results of the recursion so that the elements of each β

_t sum to unity, retaining all L β

_t vectors;

(iii) forming the joint probability vectors λ

_t, the elements of which are said joint probabilities λ

_t, by multiplying the elements of said row vectors by the elements of said column vectors as follows;

space="preserve" listing-type="equation">λ

.sub.t (i)=α

.sub.t (i)β

.sub.t (i), all i, t=1, . . . , L;

memory for storing said probability matrices and said row vectors; and

a decoded bit value probability calculator for determining from λ

_t the probability that a given data bit inputted to the encoder at time=t is equal to zero, the data bit being the m^th of k data bits, and providing a soft output as a function of said probability.

6. The decoder of claim 5, further comprising a threshold decision device for receiving the probability that the data bit inputted to the encoder at time=t is equal to zero and for implementing a decision rule to provide decoded output bits therefrom.

7. The decoder of claim 5 wherein said tail-biting trellis code comprises a convolutional code.

8. The decoder of claim 5 comprising a bounded distance decoder.

9. The decoder of claim 5 wherein the encoder encodes a block of data bits, said data bits being grouped into k-bit symbols for encoding, the predetermined maximum number of recursions comprising two times the number of k-bit input symbols in a block of data bits.

10. A decoder for a tail-biting trellis code generated by an encoder, which decoder decodes by determining the joint probability that the state of the encoder at time t, S_t, is m and that a sequence of L channel outputs having values Y₁^L ={y₁, . . . , y_L } is received, as represented by λ

_t (m)=P{S_t =m;

Y₁^L }, said trellis code having M encoder states, said decoder determining L conditional probability matrices γ

_t, one at each of L trellis levels, the elements of said probability matrices being defined by;

space="preserve" listing-type="equation">γ

.sub.t (ij)=P{state j at time t;

y.sub.t /state i at time t-1};

by determining row vectors α

_t having M joint probability elements defined by;

space="preserve" listing-type="equation">α

.sub.t (j)=P{state j at time t;

y.sub.1, . . . , y.sub.t };

and by determining column vectors β

_t having M conditional probability elements defined by;

space="preserve" listing-type="equation">β

.sub.t (j)=P{y.sub.t+1, . . . y.sub.L /state j at time t}

for j=0, 1, . . . , (M-1), said decoder comprising;

a γ

_t calculator for receiving said channel outputs, the channel transition probability R(Y_t, X), the probability p_t (m/m'"'"') that the encoder makes a transition from state m'"'"' to m at time t, and the probability q_t (X/m'"'"', m) that the encoder'"'"'s output symbol is X given that the previous encoder state is m'"'"' and the present encode state is m, and for determining the scalar elements of said probability matrices γ

_t therefrom;

a α

_t matrix product calculator for receiving said scalar elements of γ

_t from said γ

_t calculator and providing said row vectors α

_t ;

a α

_t matrix product calculator for providing said column vectors β

_t ;

an element-by-element product calculator for forming the joint probability λ

_t ;

said α

_t matrix product calculator, said β

_t matrix product calculator, and said element-by-element product calculator forming said vectors α

_t, β

_t, and λ

_t, respectively, by;

(i.a) starting with an initial α

₀ equal to (1/M, . . . , 1/M), calculating L times the forward recursion;

space="preserve" listing-type="equation">α

.sub.t =α

.sub.t-1 γ

.sub.t, t=1, . . . , L;

and normalizing the results so that the elements of each α

_t sum to unity, retaining all α

_t vectors;

(i.b) letting α

₀ equal α

_L from step (i.a) and, starting at t =1, calculating

space="preserve" listing-type="equation">α

.sub.t =α

.sub.t-1 γ

.sub.t, for t=1, 2, . . . L.sub.w ;

where the wrap depth L_w is a predetermined number of trellis stages;

normalizing the results so that the elements of each α

_t sum to unity, replacing the α

_t calculated in step (i.a) with the α

_t calculated in step (i.b) for t=1, 2, . . . , L_w ;

(ii.a) initializing β

_L =(1, 1, 1, . . . ,

     1)^T and forming the preceding β

_t by a backward recursion as follows;

space="preserve" listing-type="equation">β

.sub.t =γ

.sub.t+1 β

.sub.t+1, t=L-1, . . . , 1;

and normalizing the results of the recursion so that the elements of each β

_t sum to unity, retaining all L β

_t vectors;

(ii.b) letting β

_L+1 equal β

₁ from step (ii.a) and letting γ

_L+1, =γ

₁, starting at t=L, and calculating

space="preserve" listing-type="equation">β

.sub.t =γ

.sub.t+1 β

.sub.t+1 for t=L, (L-1), . . . (L-L.sub.w)

where the wrap depth L_w is a predetermined number of trellis stages;

normalizing the results of the recursions so that the elements of each β

_t sum to unity, replacing the β

_t calculated in step (ii.a) with the β

_t calculated in step (ii.b) for t=L, (L-1), . . . , L-(L_w +1);

(iii) forming the joint probability vectors λ

_t, the elements of which are said joint probabilities λ

_t (ij), by multiplying the elements of said row vectors by the elements of said column vectors as follows;

space="preserve" listing-type="equation">λ

.sub.t (i)=α

.sub.t (i) β

.sub.t (i), all i, t=1, . . . , L;

memory for storing said probability matrices and said row vectors; and

a decoded bit value probability calculator for determining from λ

_t the probability that a given data bit inputted to the encoder at time=t is equal to zero, the data bit being the m^th of k data bits, and providing a soft output as a function of said probability.

11. The decoder of claim 10, further comprising a threshold decision device for receiving the probability that the data bit inputted to the encoder at time=t is equal to zero and for implementing a decision rule to provide decoded output bits therefrom.

12. The decoder of claim 10 wherein said tail-biting trellis code comprises a convolutional code.

13. The decoder of claim 10 comprising a bounded distance decoder.

14. The decoder of claim 10 wherein the encoder encodes a block of data bits, said data bits being grouped into k-bit symbols for encoding, the predetermined maximum number of recursions comprising two times the number of k-bit input symbols in a block of data bits.

15. A method for decoding a tail-biting trellis code generated by an encoder by determining the joint probability that the state of the encoder at time t, S_t, is m and that a sequence of L channel outputs having values Y₁^L ={y₁, . . . , y_L } is received, as represented by λ

_t (m)=P{S_t =m;

Y₁^L }, said trellis code having M encoder states, said method comprising steps for determining L probability matrices γ

_t, one at each of L trellis levels, the elements of said probability matrices being defined by;

space="preserve" listing-type="equation">γ

.sub.t (ij)=P{state j at time t;

y.sub.t /state i at time t-1};

and for determining row vectors α

_t having M joint probability elements defined by;

space="preserve" listing-type="equation">α

.sub.t (j)=P{state j at time t;

y.sub.1, . . . , y.sub.t };

and for determining column vectors β

_t having M conditional probability elements defined by;

space="preserve" listing-type="equation">β

.sub.t (j)=P{y.sub.t+1, . . . y.sub.L /state j at time t}

for j=0, 1, . . . , (M-1), the steps of said method comprising;

determining the scalar elements of said probability matrices γ

_t from said channel outputs, the channel transition probability R(Y_t, X), the probability p_t (m/m'"'"') that the encoder makes a transition from state m'"'"' to m at time t, and the probability q_t (X/m'"'"', m) that the encoder'"'"'s output symbol is X given that the previous encoder state is m'"'"' and the present encoder state is m;

computing a matrix product γ

₁ γ

₂ . . . γ

_L from said scalar elements of γ

_t ;

computing a normalized eigenvector α

₀ corresponding to the largest eigenvalue P{Y₁^L } of said matrix product γ

₁ γ

₂ . . . γ

_L ;

forming the succeeding α

_t by a forward recursion as follows;

space="preserve" listing-type="equation">α

.sub.t =α

.sub.t-1 γ

.sub.t, t=1, . . . , L;

providing said column vectors by initializing β

_L =(1, 1, 1, . . . ,

     1)^T and forming the preceding β

_t by a backward recursion as follows;

space="preserve" listing-type="equation">β

.sub.t =γ

.sub.t+1 β

.sub.t+1, t=L-1, . . . , 1;

forming the joint probability vectors λ

_t, the elements of which are said joint probabilities λ

_t (ij), by multiplying the elements of said row vectors by the elements of said column vectors as follows;

space="preserve" listing-type="equation">λ

.sub.t (i)=α

.sub.t (i) β

.sub.t (i), all i, t=1, . . . , L; and

determining from λ

t the probability that a given data bit inputted to the encoder at time=t is equal to zero, the data bit being the m^th of k data bits, and providing a soft output as a function of said probability.

16. The method of claim 15, further comprising the step of implementing a decision role to provide decoded output bits from the probability that the dam bit inputted to the encoder at time=t is equal to zero.

17. The method of claim 15 wherein said tail-biting trellis code comprises a convolutional code.

18. A method for decoding a tail-biting trellis code generated by an encoder by determining the joint probability that the state of the encoder at time t, S_t, is m and that a sequence of L channel outputs having values Y₁^L ={y₁, . . . , y_L } is received, as represented by λ

_t (m)=P{S_t =m;

Y₁^L }, said trellis code having M encoder states, said method comprising steps for determining L probability matrices γ

_t one at each of L trellis levels, the elements of said probability matrices being defined by;

space="preserve" listing-type="equation">γ

.sub.t (ij)=P{state j at time t-1;

y.sub.t /state i at time t-1};

and for determining row vectors α

_t having M joint probability elements defined by;

space="preserve" listing-type="equation">α

.sub.t (j)=P{state j at time t;

y.sub.1, . . . , y.sub.t };

and for determining column vectors β

_t having M conditional probability elements defined by;

space="preserve" listing-type="equation">β

.sub.t (j)=P{y.sub.t+1, . . . y.sub.L /state j at time t}

for j=0, 1, . . . , (M-1), the steps of said method comprising;

determining the scalar elements of said probability matrices γ

_t from said channel outputs, the channel transition probability R(Y_t, X), the probability p_t (m/m'"'"') that the encoder makes a transition from state m'"'"' to m at time t, and the probability q_t (X/m'"'"', m) that the encoder'"'"'s output symbol is X given that the previous encoder state is m'"'"' and the present encoder state is m;

forming said vectors α

_t, β

_t, and λ

_t, respectively, by;

(i.a) starting with an initial α

₀ equal to (1/M, . . . , 1/M), calculating L times the forward recursion;

space="preserve" listing-type="equation">α

.sub.t =α

.sub.t-1 γ

.sub.t, t=1, . . . , L;

and normalizing the results so that the elements of each α

_t sum to unity, retaining all L α

_t vectors;

(i.b) letting α

₀ equal α

_L from step (i.a) and, starting at t =1, calculating

space="preserve" listing-type="equation">α

.sub.t =α

.sub.t-1 γ

.sub.t, for t=1, 2, . . . , L.sub.w.sbsb.min

where L_w.sbsb.min is a predetermined minimum number of trellis stages;

normalizing the results so that the elements of each α

_t sum to unity, retaining only the most recent set of L α

_t '"'"'s found by the recursion in steps (i.a) and (i.b) and the ##EQU18## found in step (i.a);

(i.c) comparing ##EQU19## from step (i.b) to the ##EQU20## found in step (i.a) and proceeding to step (ii) if within a tolerance range;

otherwise, continuing to step (i.d);

(i.d) letting t=t+1 and calculating α

_t =α

_t-1 γ

_t, normalizing the results of the iteration so that the elements of each α

_t sum to unity, retaining only the most recent set of L α

'"'"'s calculated and this α

_t found previously in step (i.a).

(i.e) comparing the α

_t to the most recent, previously-calculated α

_t from steps (i.a), (i.b) and (i.d) and proceeding to step (ii) if within a tolerance range, continuing with step (i.d) if the two most recent vectors do not agree to within said tolerance range and if the number of recursions does not exceed a predetermined maximum;

proceeding to step (ii) otherwise;

(ii) initializing β

_L =(1, 1, 1, . . . ,

     1)^T and forming the preceding β

_t by a backward recursion as follows;

space="preserve" listing-type="equation">β

.sub.t =γ

.sub.t+1 β

.sub.t+1, t=L-1, . . . , 1;

and normalizing the results of the recursion so that the elements of each β

_t sum to unity, retaining all L β

_t vectors;

(iii) forming the joint probability λ

_t vectors, the elements of which are said joint probabilities λ

_t (i, j) by multiplying the elements of said row vectors by the elements of said column vectors as follows;

space="preserve" listing-type="equation">λ

.sub.t (i)=α

.sub.t (i) β

.sub.t (i), all i, t=1, . . . , L; and

determining from λ

_t the probability that a given data bit inputted to the encoder at time=t is equal to zero, the data bit being the m^th of k data bits, and providing a soft output as a function of said probability.

19. The method of claim 18 further comprising the step of implementing a decision rule to provide decoded output bits from the probability that the dam bit inputted to the encoder at time=t is equal to zero.

20. The method of claim 18 wherein said tail-biting trellis code comprises a convolutional code.

21. The method of claim 18 wherein the encoder encodes a block of data bits, said dam bits being grouped into k-bit symbols for encoding, the predetermined maximum number of recursions comprising two times the number of k-bit input symbols in a block of data bits.

22. A method for decoding a tail-biting trellis code generated by an encoder by determining the joint probability that the state of the encoder at time t, S_t, is m and that a sequence of L channel outputs having values Y₁^L ={y₁, . . . , y_L } is observed, as represented by λ

_t (m)=P{S_t =m;

Y₁^L }, said trellis code having M encoder states, said method comprising steps for determining L conditional probability matrices γ

_t, one at each of L trellis levels, the elements of said probability matrices being defined by;

space="preserve" listing-type="equation">γ

.sub.t (ij)=P{state j at time t;

y.sub.t /state i at time t-1};

and for determining row vectors α

_t having M joint probability elements defined by;

space="preserve" listing-type="equation">α

.sub.t (j)=P{state j at time t;

y.sub.1, . . . , y.sub.t};

and for determining column vectors β

_t having M conditional probability elements defined by;

space="preserve" listing-type="equation">β

.sub.t (j)=P{y.sub.t+1, . . . y.sub.L /state j at time t}

for j=0, 1, . . . , (M-1), the steps of said method comprising;

determining the scalar elements of said probability matrices γ

_t from said channel outputs, the channel transition probability R(Y_t, X), the probability p_t (m/m'"'"') that the encoder makes a transition from state m'"'"' to m at time t, and the probability q_t (X/m'"'"', m) that the encoder'"'"'s output symbol is X given that the previous encoder state is m'"'"' and the present encoder state is m;

forming said vectors α

_t, β

_t, and λ

_t, respectively, by;

(i.a) starting with an initial α

₀ equal to (1/M, . . . , 1/M), calculating L times the forward recursion;

space="preserve" listing-type="equation">α

.sub.t =α

.sub.t-1 γ

.sub.t, t=1, . . . , L;

and normalizing the results so that the elements of each α

_t sum to unity, retaining all L α

_t vectors;

(i.b) letting α

₀ equal α

_L from step (i.a) and, starting at t =1, calculating

space="preserve" listing-type="equation">α

.sub.t =α

.sub.t-1 γ

.sub.t, for t=1, . . . L.sub.w,

where the wrap depth L_w is a predetermined number of trellis stages;

normalizing the results so that the elements of each α

_t sum to unity, replacing the α

_t calculated in step (i.a) with the α

_t calculated in step (i.b) for t=1, 2, . . . L_w ;

(ii.a) initializing β

_L =(1, 1, 1, . . . ,

     1)^T and forming the preceding β

_t by a backward recursion as follows;

space="preserve" listing-type="equation">β

.sub.t =γ

.sub.t+1 β

.sub.t+1, t=L-1, . . . , 1;

and normalizing the results of the recursion so that the elements of each β

_t sum to unity, retaining all L β

_t vectors;

(ii.b) letting β

_L+1 equal β

₁ from step (ii.a) and letting γ

_L+1 =γ

₁, starting at t=L, and calculating

space="preserve" listing-type="equation">β

.sub.t =γ

.sub.t+1 β

.sub.t+1, for t=L, (L-1), . . . (L-L.sub.w);

where the wrap depth L_w is a predetermined number of trellis stages;

normalizing the results so that the elements of each β

_t sum to unity, replacing the β

_t calculated in step (ii.a) with the β

_t calculated in step (ii.b) for t=L, (L-1), . . . , L-(L_w +1);

(iii) forming the joint probability λ

_t vectors, the elements of which are said joint probabilities y_t, by multiplying the elements of said row vectors by the elements of said column vectors as follows;

space="preserve" listing-type="equation">λ

.sub.(i)=α

.sub.t (i) β

.sub.t (i), all i, t=1, . . . , L; and

determining from λ

_t the probability that a given data bit inputted to the encoder at time=t is equal to zero, the data bit being the m^th of k data bits, and providing a soft output as a function of said probability.

23. The method of claim 22, further comprising the step of implementing a decision rule to provide decoded output bits from the probability that the data bit inputted to the encoder at time =t is equal to zero.

24. The method of claim 22 wherein said tail-biting trellis code comprises a convolutional code.

25. The method of claim 22 wherein the encoder encodes a block of data bits, said data bits being grouped into k-bit symbols for encoding, the predetermined maximum number of recursions comprising two times the number of k-bit input symbols in a block of data bits.