modulation - How to soft decode DQPSK?

I'm successfully soft-decoding D-BPSK by taking the dot-product of the constellation-position of the symbol and of the previous symbol. If the result is >= 1, then the symbol phase hasn't changed and the bit is a zero. If the result is <= -1 then the phase has shifted and the result is a one. In between -1 and 1 the result is a soft 0 or soft 1.

I can't figure out how to do the same thing with D-QPSK. I can use just the phase, but this throws away a lot of information that could help the soft-decoder.

This paper explains how to do it and gives a formula (10):

$b_1 = \mathrm{Re}\{s_n s^*_{n-1}\}, b_2 = \mathrm{Im}\{s_n s^*_{n-1}\}$

But I don't understand the notation — what does a * floating above mean? I tried just multiplying the complex numbers and taking the real and imaginary parts but this didn't work.

Since the constellation can rotate, how can the two axes be teased apart?

Answer

Two successive symbols in the demodulator are $Z_1 = (X_1,Y_1)$ and $Z_2 =(X_2,Y_2)$ where $X$ is the output of the I branch and $Y$ the output of the Q branch of the receiver. The hard-decision DBPSK decision device considers the question:

Is the new symbol $Z_2$ closer to the old symbol $Z_1$ or to the negative $-Z_1$ of the old symbol?

and thus compares

$$(X_2-X_1)^2 + (Y_2-Y_1)^2 \gtrless (X_2+X_1)^2 + (Y_2+Y_1)^2$$

which can be simplified to a sign comparison on $\langle Z_1,Z_2\rangle = X_1X_2+Y_1Y_2$. Note that this is essentially asking

Are the two vectors $Z_1$ and $Z_2$ are pointing in roughly the same direction (in which case the inner product or dot product is positive) or in roughly opposite direction (in which case the dot product is negative)?

A third viewpoint thinks of $Z_1$ and $Z_2$ as complex numbers and asks

Is $\text{Re}(Z_1Z_2^*) = X_1X_2+Y_1Y_2$ positive or negative?

The soft decision decision device simply passes on the exact value of the dot product to the soft decision decoder which may opt to quantize dot products that are very large in magnitude into hard decisions and continue waffling on the rest. This is what the decision rule stated in the OP's question is, where large is taken as exceeding $1$ in magnitude.

In DQPSK, the encoding uses one of two conventions:

• 
  
  

  the signal phase is delayed by $0, \pi/2, \pi, 3\pi/2$ according as the dibit to be transmitted is $00, 01, 11, 10$

  
  


• 
  

  the signal phase is advanced by $0, \pi/2, \pi, 3\pi/2$ according as the dibit to be transmitted is $00, 01, 11, 10$

  
  

Note that a DQPSK signal is not the sum of two DBPSK signals modulated on phase-orthogonal carriers, but the I and Q bits jointly affect the net carrier phase.

For demodulating a DQPSK signal, the decision device needs to ask

Which of the four symbols $Z_1,\quad jZ_1 = (-Y_1,X_1),\quad -Z_1,\quad -jZ_1 = (Y_1,-X_1)$ is $Z_2$ closest to?

Thus, in addition to the comparison

$$(X_2-X_1)^2 + (Y_2-Y_1)^2 \gtrless (X_2+X_1)^2 + (Y_2+Y_1)^2$$

it is necessary to compare

$$(X_2+Y_1)^2 + (Y_2-X_1)^2 \gtrless (X_2-Y_1)^2 + (Y_2+X_1)^2$$

which works out to looking at $\text{Im}(Z_1Z_2^*)$ in addition to $\text{Re}(Z_1Z_2^*)$ and making the decision according as which quantity has the largest magnitude and the sign of the largest magnitude. The details of how the soft-decision decoder uses the decision statistic $Z_1Z_2^* = (\text{Re}(Z_1Z_2^*), \text{Im}(Z_1Z_2^*))$ will determine how these numbers are further massaged.