8.3. \(M\)-ary Differential Phase Shift Keying (\(M\)-DPSK)

• Consider the \(M\)-PSK constellation \(\left\{ e^{j 2\pi m/M}: m=0,1,\ldots,M-1 \right\}\). Let \(c[n]\) be the information symbol sequence chosen from the \(M\)-PSK constellation, and \(b[n] = c[n] b[n-1]\) is the differentially encoded symbol sequence. Then the linearly modulated signal \(x(t) = \sum_n b[n] p(t-nT)\) gives us \(M\)-ary DPSK.

• Practically, a known value needs to be set for the initial symbol, say, \(b[-1] = 1\).

• Consider the ML non-coherent demodulation of \(c[0]\). We may treat

  \[\begin{equation*} s_m(t) = b[-1] p(t+T) + b[0] p(t) = b[-1] p(t+T) + e^{j 2\pi m/M} b[-1] p(t) \end{equation*}\]

  in this case. If \(p(t)\) has most of its energy concentrated to within a symbol period, the signals \(s_m(t)\)’s have approximately equal energies. Hence the ML non-coherent demodulator is given in (8.2):

  (8.4)\[\begin{split}\begin{align} \hat c[0] & = \arg\hspace{-15pt}\max_{m \in \{0,1,\ldots,M-1\}} \left| \tilde r[-1] + e^{-j 2\pi m/M} \tilde r[0] \right| \notag \\ & = \arg\hspace{-15pt}\max_{m \in \{0,1,\ldots,M-1\}} \text{Re} \left(e^{-j 2\pi m/M} \tilde r[0] \tilde r^*[-1] \right) \notag \\ & = \arg\hspace{-15pt}\max_{m \in \{0,1,\ldots,M-1\}} \cos \left(\angle \tilde r[0] - \angle \tilde r[-1] - \frac{2\pi m}{M} \right) \end{align}\end{split}\]

  where \(\tilde r[n] = r(t) * p^*(-t) \big|_{t=nT}\).

• The demodulation rule in (8.4) is simply finding the \(M\)-PSK signal point on the unit circle that is closest to the point with angle \(\angle \tilde r[0] - \angle \tilde r[-1]\).

• The ISI-free property of the RRC pulse discussed before guarantees that the demodulation rule in (8.4) is applicable to the case of multiple TX symbols.