# Why is a precoder necessary for DQPSK and what does it accomplish?

I've implemented a soft-decoder for DQPSK using the wonderful answers I received here:

How to soft decode DQPSK?

To get the soft-decoder working properly I needed to precode the data I was sending out. I implemented the precoder mentioned in this paper:

$I_k=\overline{u_k \oplus v_k}*(u_k \oplus I_{k-1})+(u_k \oplus v_k)*(v_k \oplus Q_{k-1})$ $Q_k=\overline{u_k \oplus v_k}*(v_k \oplus Q_{k-1})+(u_k \oplus v_k)*(u_k \oplus I_{k-1})$

I'd like to know why this precoder is necessary -- what does that complicated expression of XORs actually accomplish?

Here's a table showing what the equation yields. If "to_encode" is 00, the to_send symbol is the same as the previous ("prev") symbol. If the "to_encode" is 11, the to_send symbol is the previous symbol xor 11. What is the meaning in other cases?

to_encode prev  to send
[ 0 0 ] [ 0 0 ] [ 0 0 ]
[ 0 1 ] [ 0 0 ] [ 1 0 ]
[ 1 0 ] [ 0 0 ] [ 0 1 ]
[ 1 1 ] [ 0 0 ] [ 1 1 ]
[ 0 0 ] [ 0 1 ] [ 0 1 ]
[ 0 1 ] [ 0 1 ] [ 0 0 ]
[ 1 0 ] [ 0 1 ] [ 1 1 ]
[ 1 1 ] [ 0 1 ] [ 1 0 ]
[ 0 0 ] [ 1 0 ] [ 1 0 ]
[ 0 1 ] [ 1 0 ] [ 1 1 ]
[ 1 0 ] [ 1 0 ] [ 0 0 ]
[ 1 1 ] [ 1 0 ] [ 0 1 ]
[ 0 0 ] [ 1 1 ] [ 1 1 ]
[ 0 1 ] [ 1 1 ] [ 0 1 ]
[ 1 0 ] [ 1 1 ] [ 1 0 ]
[ 1 1 ] [ 1 1 ] [ 0 0 ]

• This question is unanswerable. You implemented a precoder which you do not describe very much, and which might, or possibly might not, be the same as the one mentioned in the link you provide. Now you ask "Why it is necessary?" You need to tell us why you thought it was necessary to have a precoder in your system. How can we tell why a precoder that you designed and about which we have no knowledge is necessary for your system? – Dilip Sarwate Mar 8 '13 at 0:12
• @DilipSarwate, I've made the question clearer. I've checked and what I implemented was the same as the precoder suggested in the paper. – Dan Sandberg Mar 8 '13 at 19:25

Regard the dibits as Gray code representations of the integers $0,1,2,3$, more specifically,

$$[0, 0] \leftrightarrow 0, ~~ [0, 1] \leftrightarrow 1, ~~ [1, 1] \leftrightarrow 2, ~~ [1, 0] \leftrightarrow 3.$$

Then, the precoding scheme is nothing but differential encoding for QPSK with

to send = prev - to_encode modulo $4$.

For example the line [ 1 0 ] [ 0 1 ] [ 1 1 ] in the question says that to encode $3 =$ [1 0] when the previous symbol was $1 =$ [0 1], we send $1-3 = -2 \equiv 2 \bmod 4$ where $2 =$ [1 1].