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Decode convolutional code using a posteriori probability (APP) method

Convolutional sublibrary of Error Detection and Correction

The APP Decoder block performs a posteriori probability (APP) decoding of a convolutional code.

The input *L*(*u*) represents
the sequence of log-likelihoods of encoder input bits, while the input *L*(*c*)
represents the sequence of log-likelihoods of code bits. The outputs *L*(*u*)
and *L*(*c*) are updated versions
of these sequences, based on information about the encoder.

If the convolutional code uses an alphabet of 2^{n} possible
symbols, this block's *L*(*c*) vectors
have length *Q***n* for some positive
integer *Q*. Similarly, if the decoded data uses
an alphabet of 2^{k} possible
output symbols, then this block's *L*(u) vectors
have length *Q***k*.

This block accepts a column vector input signal with any positive
integer for *Q*.

If you only need the input *L*(*c*)
and output *L*(*u*), you can attach
a Simulink Ground block to the
input *L*(*u*) and a Simulink^{®} Terminator block to the output *L*(*c*).

This block accepts `single`

and `double`

data
types. Both inputs, however, must be of the same type. The output
data type is the same as the input data type.

To define the convolutional encoder that produced the coded input, use the **Trellis
structure** parameter. This parameter is a MATLAB^{®} structure whose format is described in Trellis Description of a Convolutional Code. You can use this
parameter field in two ways:

If you have a variable in the MATLAB workspace that contains the trellis structure, enter its name as the

**Trellis structure**parameter. This way is preferable because it causes Simulink to spend less time updating the diagram at the beginning of each simulation, compared to the usage described next.If you want to specify the encoder using its constraint length, generator polynomials, and possibly feedback connection polynomials, use a

`poly2trellis`

command within the**Trellis structure**field. For example, to use an encoder with a constraint length of 7, code generator polynomials of 171 and 133 (in octal numbers), and a feedback connection of 171 (in octal), set the**Trellis structure**parameter to`poly2trellis(7,[171 133],171)`

To indicate how the encoder treats the trellis at the beginning
and end of each frame, set the **Termination method** parameter
to either `Truncated`

or `Terminated`

.
The `Truncated`

option indicates that the
encoder resets to the all-zeros state at the beginning of each frame.
The `Terminated`

option indicates that the
encoder forces the trellis to end each frame in the all-zeros state.
If you use the Convolutional Encoder block
with the **Operation mode** parameter set to ```
Truncated
(reset every frame)
```

, use the `Truncated`

option
in this block. If you use the Convolutional
Encoder block with the **Operation mode** parameter
set to `Terminate trellis by appending bits`

,
use the `Terminated`

option in this block.

You can control part of the decoding algorithm using the **Algorithm** parameter.
The `True APP`

option implements a posteriori
probability decoding as per equations 20–23 in section V of [1].
To gain speed, both the `Max*`

and `Max`

options
approximate expressions like

$$\mathrm{log}{\displaystyle \sum _{i}\mathrm{exp}({a}_{i})}$$

by other quantities. The `Max`

option
uses max(*a*_{i}) as the approximation,
while the `Max*`

option uses max(*a*_{i})
plus a correction term given by $$\mathrm{ln}(1+\mathrm{exp}(-\left|{a}_{i-1}-{a}_{i}\right|))$$ [3].

The `Max*`

option enables the **Scaling
bits** parameter in the dialog box. This parameter is the
number of bits by which the block scales the data it processes internally
(multiplies the input by (2^`numScalingBits`

) and
divides the pre-output by the same factor). Use this parameter to
avoid losing precision during the computations.

**Trellis structure**MATLAB structure that contains the trellis description of the convolutional encoder.

**Termination method**Either

`Truncated`

or`Terminated`

. This parameter indicates how the convolutional encoder treats the trellis at the beginning and end of frames.**Algorithm**Either

`True APP`

,`Max*`

, or`Max`

.**Number of scaling bits**An integer between 0 and 8 that indicates by how many bits the decoder scales data in order to avoid losing precision. This field is active only when

**Algorithm**is set to`Max*`

.**Disable L(c) output port**Select this check box to disable the secondary block output, L(c).

For an example using this block, see the Iterative Decoding of a Serially Concatenated Convolutional Code example.

[1] Benedetto, S., G. Montorsi, D. Divsalar,
and F. Pollara, “A Soft-Input Soft-Output Maximum A Posterior
(MAP) Module to Decode Parallel and Serial Concatenated Codes,” *JPL
TDA Progress Report*, Vol. 42-127, November 1996.

[2] Benedetto, Sergio and Guido Montorsi, “Performance
of Continuous and Blockwise Decoded Turbo Codes.” *IEEE
Communications Letters*, Vol. 1, May 1997, 77–79.

[3] Viterbi, Andrew J., “An Intuitive
Justification and a Simplified Implementation of the MAP Decoder
for Convolutional Codes,” *IEEE Journal on Selected
Areas in Communications*, Vol. 16, February 1998, 260–264.

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