I am studying DAPSK(Differential amplitude phase shift keying) at the moment. I have a 64-DAPSK. We are using 4 concentric rings and 6 bits per symbol. The first two bits contain the information of amplitude difference and the rest contain the information of the phase difference. I am having trouble understanding how the table on the bottom left of this example is filled out:



Each circle is identified by its amplitude, either 1, $a$, $a^2$, or $a^3$. The table shows the "amplitude transition factor"; that is, you need to multiply the previous amplitude by this factor to find the next amplitude. In other words, it is not the amplitude difference that is encoded, but the amplitude ratio.

Let's follow the example above the table. You start with $d_{n-1}$ with amplitude $a^3$. The next symbol, $d_n$, has amplitude 1, so the factor is $1/a^3$, which corresponds to bits $01$ (indicated also as the first two bits under "info bits" in the figure, top right).

Now, $d_n$ has amplitude 1 and the next symbol, $d_{n+1}$, has amplitude $a^2$. The factor is $a^2$ and the corresponding bit sequence is $11$.

Notice how the table is designed to allow transitions from any of the four amplitudes to any other amplitude.


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