8.1. Maximum Likelihood Non-coherent Demodulation#

  • Let us begin with the simplifying model that the TX signal is \(x(t) = s_m(t)\), where \(s_m(t)\) is one signal from the \(M\)-ary signal constellation \(\{s_0(t), s_1(t), \ldots, s_{M-1}(t)\}\). The RX signal is modeled as

    \[\begin{equation*} r(t) = x(t) e^{j\theta} + n(t) \end{equation*}\]

    where, as before, the uniform random variable \(\theta\) models the unknown carrier phase and \(n(t)\) is AWGN. We have also assumed that the long-term (averaged over many symbols) RX signal gain is normalized to \(1\) by the AGC. Since timing synchronization has been achieved, there is no need to model the transmission delay.

  • If all the signals in the constellation are equally likely to be transmitted, then the ML non-coherent demodulator minimizes the average symbol (demodulation) error probability. It is not hard to show (see [1] Ch. 4) that the ML non-coherent demodulator is given as below:

    (8.1)#\[\begin{equation} \hat m = \arg\hspace{-15pt}\max_{m \in \{0,1,\ldots,M-1\}} \ln I_0 \left( \frac{1}{N_0} \left| \int_{-\infty}^{\infty} r(t) s^*_m (t) dt \right| \right) -\frac{E_m}{2 N_0} \end{equation}\]

    where \(E_m \triangleq \int_{-\infty}^{\infty} |s_m(t)|^2 dt\) is the energy of the \(m\)th signal and \(I_0(\cdot)\) is the \(0\)th order modified Bessel function of the first kind.

  • In general, we need to know the noise spectral density \(N_0\) (after the AGC normalization) in order to implement the ML non-coherent demodulator.

  • However, if all the signals in the constellation are of equal energy, the ML non-coherent demodulator reduces to

    (8.2)#\[\begin{equation} \hat m = \arg\hspace{-15pt}\max_{m \in \{0,1,\ldots,M-1\}} \left| \int_{-\infty}^{\infty} r(t) s^*_m (t) dt \right| = \arg\hspace{-15pt}\max_{m \in \{0,1,\ldots,M-1\}} \left| r(t) * s^*_m(-t) \big|_{t=0} \right| \end{equation}\]

    because of the strictly increasing nature of \(\ln I_0(x)\) for non-negative \(x\).

  • For linear modulation, the signal constellation is \(\{ b_{0} p(t), b_{1} p(t), \ldots, b_{M-1} p(t)\}\), where the pulse shape \(p(t)\) has (normalized) unit energy. Thus, the ML non-coherent demodulator becomes

    (8.3)#\[\begin{equation} \hat m = \arg\hspace{-15pt}\max_{m \in \{0,1,\ldots,M-1\}} \ln I_0 \left( \frac{|b_m|}{N_0} \left| \tilde r[0] \right| \right) -\frac{|b_m|^2}{2 N_0} \end{equation}\]

    where \(\tilde r[n] \triangleq r(t) * p^*(-t) \big|_{t=nT}\) is the MF output sampled at time \(t=nT\).