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Turbo product code (TPC) decoder

`decoded = tpcdec(llr,N,K)`

`decoded = tpcdec(llr,N,K,S)`

`decoded = tpcdec(llr,N,K,S,maxnumiter)`

`decoded = tpcdec(llr,N,K,S,maxnumiter,earlyterm)`

`[decoded,actualnumiter] = tpcdec(___)`

performs 2-D TPC decoding on input log likelihood ratios, `decoded`

= tpcdec(`llr`

,`N`

,`K`

)`llr`

,
using two linear block codes specified by codeword length `N`

and
message length `K`

. For a description of 2-D TPC decoding, see
Algorithms.

performs 2-D TPC decoding and terminates early if the calculated syndrome or
parity-check of the component code evaluates to zero before
`decoded`

= tpcdec(`llr`

,`N`

,`K`

,`S`

,`maxnumiter`

,`earlyterm`

)`maxnumiter`

decoding iterations. To use
`maxnumiter`

and `earlyterm`

with
full-length messages, specify `S`

as empty, [ ].

`[`

also
returns the actual number of decoding iterations after performing 2-D TPC decoding
using any of the prior syntaxes.`decoded`

,`actualnumiter`

] = tpcdec(___)

Turbo product codes (TPC) are a form of concatenated codes used as forward error correcting (FEC) codes. Two or more component block codes, such as systematic linear block codes, are used to construct TPCs. The TPC decoder achieves near-optimum decoding of product codes using Chase decoding and the Pyndiah algorithm to perform iterative soft input, soft output decoding. For a detailed description, see [1] and [2]. This decoder implements an iterative soft input, soft output 2-D product code decoding, as described in [2], using two Linear Block Codes. The decoder expects the soft bit log likelihood ratios (LLRs) obtained from digital demodulation as the input signal.

The TPC decoder expects a positive bipolar mapped input, specifically –1 mapped to 0 and +1 mapped to 1. The output from demodulators in the Communications Toolbox™ is negative bipolar mapping, specifically +1 mapped to 0 and –1 mapped to +1. Therefore, the LLR output from demodulators must be negated to provide the positive bipolar mapped input expected by the TPC decoder.

**TPC Decoding Full-Length Messages**

TPC encoded full-length input messages are decoded using specified 2-D TPC code pairs.
Row-wise decoding uses the
(*N*_{C},*K*_{C})
code pair and column-wise decoding uses the
(*N*_{R},*K*_{R})
code pair. The input vector length must be *N*_{R} ×
*N*_{C}. To perform the 2-D TPC decoding, the
column vector of the input LLRs, composed of the message and parity bits, is arranged into
an
*N*_{R}-by-*N*_{C}
matrix.

The TPC decoder achieves near-optimum decoding of product codes using Chase decoding and the Pyndiah algorithm to perform iterative soft input, soft output decoding. Chase decoding forms a set of possible codewords for each row or column. The Pyndiah algorithm calculates soft information required for the next decoding step.

**Iterative Soft Input, Soft Output Decoder**

The iterative soft input, soft output decoding, as shown in the block diagram, carries out two decoding steps for each iteration.

The soft inputs for decoding are * R*(

Iteration loop counter

*i*increments from*i*= 1 to the specified number of iterations.*m*=`2`

*i*–`1`

is the decoding step index.is the received LLR matrix.**R**(**R***m*) is the soft input for the*m*th decoding step.(**W***m*) is the input extrinsic information for the*m*th decoding step.*α*(*m*) = [0,0.2,0.3,0.5,0.7,0.9,1,1, ...], where*α*is a weighting factor applied based on the decoding step index. For higher decoding steps,*α*= 1.*β*(*m*) = [0.2,0.4,0.6,0.8,1,1, ...], where*β*is a reliability factor applied based on the decoding step index. For higher decoding steps,*β*= 1.contains the decoded message bits. The output message bits are formed from**D**by mapping –1 to 0 and +1 to 1, then reshaping the message block into a column vector.**D**

The output message bits are formed after iterating through the specified number of iterations, or, if early termination is enabled, after code convergence.

**Early Termination of TPC Decoding**

If early termination is enabled, a code convergence check is performed on the hard decision of the soft input in each row-wise and column-wise decoding step. Early termination can be triggered after either the row-wise decoding or column-wise decoding converges.

The code is converged if, for all rows or all columns,

The syndrome evaluates to zero in the codes (Hamming codes, Extended Hamming codes, BCH codes, or Extended BCH codes).

The parity check is evaluated to zero in parity check codes.

The reported number of iterations evaluates to the iteration value that is currently in progress. For example, if the code convergence check is satisfied after row-wise decoding in the third iteration (after 2.5 decoding steps), then the number of iteration returned is 3.

**TPC Decoding Shortened Messages**

TPC encoded shortened input messages are decoded using specified 2-D TPC code pairs. Row-wise
decoding uses the (*N*_{C} –
*K*_{C} +
*S*_{C},
*S*_{C}) code pair and column-wise decoding uses
the (*N*_{R} –
*K*_{R} +
*S*_{R},
*S*_{R}) code pair. The input vector length must be
(*N*_{R} –
*K*_{R} +
*S*_{R}) ×
(*N*_{C}–
*K*_{C} +
*S*_{C}). To perform the 2-D TPC decoding of
shortened messages, the column vector of the input LLRs, composed of the shortened message
and parity bits, is arranged into an (*N*_{R} –
*K*_{R} +
*S*_{R})-by-(*N*_{C}
– *K*_{C} +
*S*_{C}) matrix.

The TPC decoder processes the received shortened message LLRs similar to full length codes, with these exceptions:

The shortened bit positions in the received codeword are set to –1.

The Chase algorithm does not consider the shortened bit positions while choosing the least reliable bits.

[1] Chase, D. "Class of Algorithms
for Decoding Block Codes with Channel Measurement Information." * IEEE
Transactions on Information Theory*, Volume 18, Number 1, January 1972,
pp. 170–182.

[2] Pyndiah, R. M. "Near-Optimum
Decoding of Product Codes: Block Turbo Codes." *IEEE Transactions on
Communications*. Volume 46, Number 8, August 1998, pp.
1003–1010.