8.2. \(M\)-ary Amplitude Shift Keying (\(M\)-ASK)

• From the form of the ML non-coherent demodulator in (8.3), we see that any linear modulation that can be non-coherently demodulated must have distinct \(|b_m|\) in the constellation. If all \(b_m\)s are real, this results in ASK.

• We may assume \(0 \leq b_0 < b_1 < \cdots < b_{M-1}\). For high SNR (\(N_0 \ll 1\)), the ML non-coherent demodulator in (8.3) may be approximated by

  \[\begin{equation*} \hat m = \arg\!\max_{m \in \{0,1,\ldots,M-1\}} b_m \left| \tilde r[0] \right| - \frac{|b_m|^2}{2} \end{equation*}\]

  which is equivalent to comparing the statistic \(\left| \tilde r[0] \right|\) to a sequence of thresholds \(0 = a_0 < a_1 < a_2 < \cdots < a_{M-1} < a_{M} = \infty\) to get \(\hat m = m\) if \(a_m \leq \left| \tilde r[0] \right| < a_{m+1}\). It is easy to see that \(a_m = \frac{b_{m-1}+b_m}{2}\) for \(m=1,2,\ldots,M-1\).

• Fixing the average signal energy in the constellation, we may obtain the optimal ASK constellation by minimizing the average symbol error probability

  \[\begin{equation*} \frac{1}{M} \sum_{m=0}^{M-1} \Pr \left\{ \left| \tilde r[0] \right| \notin [a_m,a_{m+1}) \big| m \right\}. \end{equation*}\]

  The optimal ASK constellation is one that gives equal conditional symbol error probabilities. In general, this optimization problem is rather hard to solve. For the special case of \(M=2\) (also called OOK), the optimal choice is clearly \(b_0 =0\) and \(b_1 = \sqrt{2}\) if the average symbol energy fixed at \(1\).

• If the RRC pulse shape is used, it can be shown (see [1] Ch. 9) that \(p_{RRC}(t-nT) * p^*_{RRC}(-t) \big|_{t=0} = 0\) for all \(n \neq 0\). This result, which is usually referred to as the ISI-free property of the RRC pulse, implies that adjacent symbols play no role in the decision statistic of the current symbol when fine timing synchronization is achieved. Hence the ML non-coherent demodulator discussed above directly applies to the practical ASK signal containing multiple symbols.