PSK31 Convolution Encoding

Convolutional Codes are often charactarized by three aspects:

For QPSK31 we have [n,k,K] = [1,2,5]. We give the encoder 1-bit of input, out comes 2-bits, and 5-bits of input are stored in our shift register. For the 2-bits of output 1-bit is In-Phase and the other is Quadrature. Let’s look at a diagram that allows use to envision [n,k,K].

x0 x1 x2 x3 x4 Σ g0 Σ g1

The convolutional encoder is effectively a 5-bit shift register with bits [x0,x1,x2,x3,x4] where x0 is the new incoming bit and x4 is the oldest bit in the register that is shifted out on each cycle. For PSK31 each bit will come in at 31.25Hz.

Generator functions

From the block diagram shown we can see 2 output bits g0 and g1. These are 1-bit outputs that come from the Σ block. This Σ block is a modulo-2 operation and is 1 when the output is odd and 0 when the output is even. These generator functions are defined as follows:

int g0 = (x4+x2+x1+x0)%2;
int g1 = (x4+x3+x0)%2;

Convolutional Encoder Output

To summarize we have a K=5 bit convolutional encoder giving 32 states starting at 0 and ending at 31. Each time a new bit x0 comes into our encoder we will output 2-bits giving and output symbol rate of k=2 for a base code rate of n=1.

Convention

As a matter of convention a state of say 1 will be [x0,x1,x2,x3,x4]=[1,0,0,0,0] and a state of 16 will be [x0,x1,x2,x3,x4]=[0,0,0,0,1]. However, we will store this as variables in C as state 1 = 0x01 and state 16 = 0x10.

The output to the encoder is can have 4 possible states 0b00, 0b01, 0b10, and 0b11. The state 0b10 will correspond to and output of [g0,g1]=0b10. This notation can be a bit backwards as g0 is the most significant bit. This is kept this way as the examples I’ve seen online and in textbooks. I assume this is because time is typically from left to right.

Example PSK31 Encoding

Let’s take an example input sequence and find display the output of the PSK31 convolutional encoder. We will also display the current and next state of the shift register.

Here is the key to explain the encoder output table:

t  each sample interval in time in PSK31 we have 31.25Hz
i  input to the encoder, we will have 0101110...
cs current state of the encoder
ns next state of the encoder
ob the output in binary where we have x0,x1
o  the output in decimal which gives 0321001...

Below shows the input, current state, next state, and output of the PSK31 Convolutional Encoder.

t  i cs ns  ob  o
-----------------
0  0  0  0 0b00 0
1  1  1  3 0b11 3
2  0  2  6 0b10 2
3  1  5 15 0b01 1
4  1 11 31 0b00 0
5  1 23 31 0b00 0
6  0 14 30 0b01 1
7  0 28 28 0b00 0
8  1 25 27 0b01 1
9  0 18 22 0b01 1
10 1  5 15 0b01 1
11 0 10 30 0b11 3
12 0 20 28 0b01 1
13 0  8 24 0b01 1
14 1 17 19 0b00 0
15 0  2  6 0b10 2
16 0  4 12 0b10 2
17 0  8 24 0b01 1
18 0 16 16 0b11 3
19 0  0  0 0b00 0

Implementation of a PSK 31 Encoder in C

Below is an example written in C of the convolutional encoder for PSK31. The filename is psk31_enc.c.

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

// convolutional encoder function
// arr is an array of length len
// output will be an array of length len with the encoded bits
// where the output 2 bits where the msb is g0 and the lsb is g1
void cnv_enc(unsigned int len, unsigned int arr[], unsigned int *output[]);
int cnv_enc_next(int i);

// generator functions
int g0(int x4, int x3, int x2, int x1, int x0);
int g1(int x4, int x3, int x2, int x1, int x0);

int main(int argc, char *argv[])
{
  // print to stdout unless a file name is provided
  FILE* fdo = stdout;
  if(argc > 2)
  {
    printf("%s [filename]\n",argv[0]);
    return 1;
  }
  else if(argc == 2 && (strcmp("--help",argv[1])==0 || strcmp("-h",argv[1])==0))
  {
    printf("%s [filename]\n",argv[0]);
    return 1;
  }
  else if(argc == 2)
  {
    fdo = fopen(argv[1],"w");
  }
  // our sample input stream
  unsigned int len = 20;
  unsigned int input[] = {0,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0,0,0,0,0};

  // the 2-bit output of our encoder
  // note the bit0 is g1 and bit1 is g0
  unsigned int *output;

  cnv_enc(len,input,&output);
  for(int i=0;i<len;i++) {
    fprintf(fdo,"%d",output[i]);
  }
  if(fdo != stdout) {
    fclose(fdo);
  }
  free(output);
  return EXIT_SUCCESS;
}

void cnv_enc(unsigned int len, unsigned int input[], unsigned int *output[])
{
  unsigned int *arr = calloc(len,sizeof(int));
  *output = arr;

  for(int i=0;i<len;i++)
  {
    int enc = cnv_enc_next(input[i]);
    arr[i] = enc;
  }
}

int cnv_enc_next(int x0)
{
  static int x1=0,x2=0,x3=0,x4=0;
  int r0,r1;
  r0 = g0(x4,x3,x2,x1,x0);
  r1 = g1(x4,x3,x2,x1,x0);

  x4 = x3;
  x3 = x2;
  x2 = x1;
  x1 = x0;

  return (r0<<1)|r1;
}

int g0(int x4, int x3, int x2, int x1, int x0)
{
  int g0 = (x4+x2+x1+x0)%2;
  return g0;
}

int g1(int x4, int x3, int x2, int x1, int x0)
{
  int g1 = (x4+x3+x0)%2;
  return g1;
}

A simple Makefile to build this project is as follows. Running a make followed by a ./psk31_enc.o encode the 1-bit input in the example and output the 2-bit encoding using the PSK31 generator functions.

all: psk31_enc.o

%.o: %.c
	cc -o $@ $<

clean:
	rm *.o