# APP Decoder

Decode convolutional code using a posteriori probability (APP) method

## Library

Convolutional sublibrary of Error Detection and Correction

## Description

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

### Input Signals and Output Signals

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 (Simulink) block to the input
*L*(*u*) and a Simulink^{®}
Terminator (Simulink) 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.

### Specifying the Encoder

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.

### Specifying Details of the Algorithm

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.

## Parameters

**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).

## References

[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.

## Extended Capabilities

**Introduced before R2006a**