AM vs FM efficiency and complexity; coherent ASK error

AM vs FM

Error probability of coherent ASK

Signals over a bit: $s_1(t) = A\cos\omega_c t$ for "1", $s_0(t) = 0$ for "0"; AWGN of PSD $N_0/2$; equal priors.

Correlator output $y = \int_0^{T_b} r(t)\cos\omega_c t\,dt$ has mean $AT_b/2$ for "1" and 0 for "0", with Gaussian noise variance

$$\sigma^2 = \frac{N_0}{2}\cdot\frac{T_b}{2} = \frac{N_0T_b}{4}$$

Optimum threshold is midway, $AT_b/4$, so

$$\begin{aligned} P_e &= Q\!\left(\frac{AT_b/4}{\sqrt{N_0T_b/4}}\right) = Q\!\left(\sqrt{\frac{A^2T_b}{4N_0}}\right) \end{aligned}$$

With average bit energy $E_b = \tfrac12\cdot\tfrac{A^2T_b}{2} = \tfrac{A^2T_b}{4}$:

$$P_e = Q\!\left(\sqrt{\frac{E_b}{N_0}}\right)$$

This is 3 dB worse than coherent BPSK, $Q(\sqrt{2E_b/N_0})$.