Bit error probability of coherent BFSK

In binary FSK, bit 1 and bit 0 are sent as two different carrier frequencies $f_1$ and $f_2$ with the same amplitude.

Coherent BFSK signals over $0\le t\le T_b$:

$$s_1(t)=A\cos\omega_1t\ \ (1),\qquad s_2(t)=A\cos\omega_2t\ \ (0),\qquad E_b=\frac{A^2T_b}{2}$$

The tones are chosen orthogonal over $T_b$: $\int_0^{T_b}\cos\omega_1t\cos\omega_2t\,dt=0$ (e.g. $\lvert f_1-f_2\rvert=n/2T_b$).

Coherent receiver: two correlators, one per tone; their outputs are subtracted and compared with zero.

        +->(x)--> integ 0..Tb --> y1 --+
        |   ^ cos w1t                  |(+)
 r(t) --+                            (sum)--> l = y1 - y2
        |   v cos w2t                  |(-)   l > 0 ? 1 : 0
        +->(x)--> integ 0..Tb --> y2 --+

Signal part of $l$.

• 1 sent: $y_1=\frac{AT_b}{2}$, $y_2=0$ (orthogonality) so $l=+\frac{AT_b}{2}$.
• 0 sent: $y_1=0$, $y_2=\frac{AT_b}{2}$ so $l=-\frac{AT_b}{2}$.

Noise part. $n_1=\int w\cos\omega_1t\,dt$ and $n_2=\int w\cos\omega_2t\,dt$ each have variance $\frac{N_0T_b}{4}$ and are uncorrelated (orthogonal tones), so $n=n_1-n_2$ has

$$\sigma^2=\frac{N_0T_b}{4}+\frac{N_0T_b}{4}=\frac{N_0T_b}{2}$$

Error probability. $l$ is Gaussian with mean $\pm AT_b/2$ and variance $\sigma^2$; threshold 0. For 0 sent, error if $l>0$:

$$\begin{aligned} P_e &= \frac12\operatorname{erfc}\left(\frac{AT_b/2}{\sqrt2\,\sigma}\right)=\frac12\operatorname{erfc}\left(\frac{AT_b/2}{\sqrt{N_0T_b}}\right) \\ &= \frac12\operatorname{erfc}\left(\sqrt{\frac{A^2T_b}{4N_0}}\right) \end{aligned}$$

With $E_b=A^2T_b/2$:

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

Check with the general formula: $E_d=\int(s_1-s_2)^2dt=E_b+E_b=2E_b$, so $\frac12\operatorname{erfc}\sqrt{E_d/4N_0}=\frac12\operatorname{erfc}\sqrt{E_b/2N_0}$.

Comparison: BFSK has the same $P_e$ as ASK (for equal average energy) and needs 3 dB more $E_b/N_0$ than BPSK, because orthogonal signals are $\sqrt{2E_b}$ apart while antipodal ones are $2\sqrt{E_b}$ apart. Non-coherent FSK gives $P_e=\frac12e^{-E_b/2N_0}$, which is slightly worse but needs no carrier phase.