# Mapping soft symbols in DQPSK

I'm trying to soft decode DQPSK. According to this paper the bits are:

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

If I put 0.707 as a 100% probability of 0:

$$\begin{eqnarray} s_n=(0.707 + 0.707i) \nonumber \\ s_{n-1}=(0.707 + 0.707i) \nonumber \end{eqnarray}$$

This gives:

$$\begin{eqnarray} b_1=0.707 * 0.707 - (0.707 * -0.707)=0.999 \nonumber \\ b_2=0.707 * -0.707 + 0.707 * 0.707 = 0.0 \nonumber \end{eqnarray}$$

And the result doesn't make sense:

1. The real part (0.999) is fine. Original data corresponds to 100% probability of 0 and real part now from -0.999 to 0.999.
2. The imag part (0.000) is weird. It can span from -0.999 to 0.999. Being 0.000 means bit 0 has 50% probability. Which is wrong, because original $$s_n$$ and $$s_{n-1}$$ correspond to 100% probability.

So it looks like the result constellation is no longer the same as original DQPSK: (0,0) bits now mapped to (1,0) point.

Questions are:

1. Am I right in assumptions?
2. How can I get soft symbols from the result constellation?

The paper actually deals with $$\pi/4$$-QPSK, not with plain DQPSK. In $$\pi/4$$-QPSK, the symbols are always rotated by a minimum of $$\pm \pi/4$$. Rappaport's Wireless Communications textbook has this table:
\begin{align*} 11 \rightarrow & \, \pi/4 \\ 01 \rightarrow & \, 3\pi/4 \\ 00 \rightarrow & \, -3\pi/4 \\ 10 \rightarrow & \, -\pi/4 \end{align*}
Unfortunately, the paper you point to does not specify what mapping it's using; the only clue is equation (2), where we see that the rotations are $$n\pi/4$$ for $$n=1,3,5,7$$.
Let's assume that $$\text{00} \rightarrow 7\pi/4$$. Then, in your case you would have $$s_{n-1} = \sqrt{2}/2 + i\sqrt{2}/2$$, and $$s_n = e^{i7\pi/4}s_{n-1} = 1$$. Then, according to the paper,
\begin{align*} b_1 = & \text{Re}(s_n s_{n-1}^*) = \sqrt{2}/2 \\ b_2 = & \text{Im}(s_n s_{n-1}^*) = \sqrt{2}/2 \end{align*}
I worked out $$\text{00} \rightarrow 7\pi/4$$ as the only rotation that produces results that make sense. However, it's not often that I see a paper with so many typos and so poorly typeset, and I wouldn't trust it without working out the details by myself first.