Error probability of coherent ASK

In ASK (on–off keying) a carrier is switched on for bit 1 and off for bit 0. Coherent detection uses a correlator with a locally generated carrier of the same frequency and phase.

ASK (on–off keying) signals over $0\le t\le T_b$:

$$s_1(t)=A\cos\omega_ct\ \ (\text{bit }1),\qquad s_2(t)=0\ \ (\text{bit }0)$$

Coherent receiver (correlator = matched filter):

 r(t) -->(x)--> integrate --> sample --> y > lam ? 1 : 0
          ^       0..Tb       t=Tb
          |
       cos(wc t)  (local carrier, in phase)

Correlator output at $t=T_b$, with $r(t)=s_i(t)+w(t)$:

$$y=\int_0^{T_b}r(t)\cos\omega_ct\,dt=a_i+n$$

• Bit 1: $a_1=\int_0^{T_b}A\cos^2\omega_ct\,dt=\frac{AT_b}{2}$ (for $f_c$ an integer multiple of $1/T_b$).
• Bit 0: $a_2=0$.
• Noise: $n=\int_0^{T_b}w(t)\cos\omega_ct\,dt$ is Gaussian, mean 0, variance

$$\sigma^2=\int_0^{T_b}\!\!\int_0^{T_b}\frac{N_0}{2}\delta(t-u)\cos\omega_ct\cos\omega_cu\,dt\,du=\frac{N_0}{2}\cdot\frac{T_b}{2}=\frac{N_0T_b}{4}$$

Threshold midway: $\lambda=\frac{AT_b}{4}$. Using the binary result $P_e=\frac12\operatorname{erfc}\left(\frac{a_1-a_2}{2\sqrt2\sigma}\right)$:

$$\begin{aligned} \frac{a_1-a_2}{2\sqrt2\,\sigma} &= \frac{AT_b/2}{2\sqrt2\sqrt{N_0T_b/4}}=\frac{A\sqrt{T_b}}{2\sqrt{2N_0}}=\sqrt{\frac{A^2T_b}{8N_0}} \end{aligned}$$

Energy of a "1" pulse is $E_1=\frac{A^2T_b}{2}$; the average energy per bit (half the bits are 0) is $E_b=\frac{E_1}{2}=\frac{A^2T_b}{4}$. Hence $\frac{A^2T_b}{8N_0}=\frac{E_b}{2N_0}$ and

$$P_e(\text{ASK})=\frac12\operatorname{erfc}\left(\sqrt{\frac{E_b}{2N_0}}\right)=Q\left(\sqrt{\frac{E_b}{N_0}}\right)$$

(In terms of peak energy $E_1$: $P_e=\frac12\operatorname{erfc}\sqrt{E_1/4N_0}$.)

Remarks: coherent ASK needs 3 dB more average power than BPSK for the same $P_e$, and the threshold depends on received amplitude, so ASK is sensitive to fading. Example: for $E_b/N_0=10$ (10 dB), $P_e=Q(\sqrt{10})=Q(3.162)\approx7.8\times10^{-4}$.