# distspec

Compute distance spectrum of convolutional code

## Syntax

```spect = distspec(trellis,n) spect = distspec(trellis) ```

## Description

`spect = distspec(trellis,n)` computes the free distance and the first `n` components of the weight and distance spectra of a linear convolutional code. Because convolutional codes do not have block boundaries, the weight spectrum and distance spectrum are semi-infinite and are most often approximated by the first few components. The input `trellis` is a valid MATLAB trellis structure, as described in Trellis Description of a Convolutional Code. The output, `spect`, is a structure with these fields:

FieldMeaning
`spect.dfree`Free distance of the code. This is the minimum number of errors in the encoded sequence required to create an error event.
`spect.weight`A length-`n` vector that lists the total number of information bit errors in the error events enumerated in `spect.event`.
`spect.event`A length-`n` vector that lists the number of error events for each distance between `spect.dfree` and `spect.dfree+n-1`. The vector represents the first `n` components of the distance spectrum.

`spect = distspec(trellis)` is the same as `spect = distspec(trellis,1)`.

## Examples

The example below performs these tasks:

• Computes the distance spectrum for the rate 2/3 convolutional code that is depicted on the reference page for the `poly2trellis` function

• Uses the output of `distspec` as an input to the `bercoding` function, to find a theoretical upper bound on the bit error rate for a system that uses this code with coherent BPSK modulation

• Plots the upper bound using the `berfit` function

```trellis = poly2trellis([5 4],[23 35 0; 0 5 13]) spect = distspec(trellis,4) berub = bercoding(1:10,'conv','hard',2/3,spect); % BER bound berfit(1:10,berub); ylabel('Upper Bound on BER'); % Plot.```

The output and plot are below.

```trellis = numInputSymbols: 4 numOutputSymbols: 8 numStates: 128 nextStates: [128x4 double] outputs: [128x4 double] spect = dfree: 5 weight: [1 6 28 142] event: [1 2 8 25]``` ## Algorithms

The function uses a tree search algorithm implemented with a stack, as described in .

## References

 Bocharova, I. E., and B. D. Kudryashov, “Rational Rate Punctured Convolutional Codes for Soft-Decision Viterbi Decoding,” IEEE Transactions on Information Theory, Vol. 43, No. 4, July 1997, pp. 1305–1313.

 Cedervall, M., and R. Johannesson, “A Fast Algorithm for Computing Distance Spectrum of Convolutional Codes,” IEEE Transactions on Information Theory, Vol. 35, No. 6, Nov. 1989, pp. 1146–1159.

 Chang, J., D. Hwang, and M. Lin, “Some Extended Results on the Search for Good Convolutional Codes,” IEEE Transactions on Information Theory, Vol. 43, No. 5, Sep. 1997, pp. 1682–1697.

 Frenger, P., P. Orten, and T. Ottosson, “Comments and Additions to Recent Papers on New Convolutional Codes,” IEEE Transactions on Information Theory, Vol. 47, No. 3, March 2001, pp. 1199–1201.