Threshold effect in FM; error probability of coherent PSK

Threshold effect in FM detection

Threshold effect in FM. The FM improvement $\text{SNR}_o\propto\beta^2$ holds only when the carrier is much stronger than the noise at the discriminator input. When the carrier-to-noise ratio (CNR) falls below a threshold, about 10 dB (typically 10–13 dB), the noise phasor sometimes becomes larger than the carrier and the resultant phasor encircles the origin, causing a sudden $2\pi$ phase jump.

  high CNR: noise wobbles     low CNR: resultant can swing
  the tip slightly            around origin -> 2pi jump
        .-.                         .---.
  0 -->( * )                   0 --( *   )
        '-'                         '---'

Each $2\pi$ jump produces an impulse (spike) in the discriminator output, heard as clicks. The click rate rises rapidly as CNR drops, so the output SNR falls much faster than linearly and the signal is mutilated. Above threshold, FM is excellent; below it, FM is worse than AM. Wider deviation (larger $\beta$) means larger $B_T$, more noise and a higher threshold.

 SNRo(dB)            . FM above threshold
     |             .   (slope 1, offset 3/2 beta^2)
     |           .
     |         .
     |       . <- threshold knee (CNR ~ 10 dB)
     |     .:
     |   . :     AM / baseband
     |  .  :
     +------------------> SNRi (dB)

The threshold can be lowered (threshold extension) by using FM feedback (FMFB) or PLL demodulators. Pre-emphasis/de-emphasis improves the output SNR above threshold but does not lower the threshold.

Error probability of coherent PSK

BPSK signals over $0\le t\le T_b$ (antipodal):

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

Coherent receiver: multiply by the locally generated $\cos\omega_ct$, integrate over $T_b$, sample, compare with threshold $\lambda=0$.

 r(t) -->(x)--> integrate --> sample --> y > 0 ? 1 : 0
          ^       0..Tb       t=Tb
          |
       cos(wc t)  (from carrier recovery)

Correlator output $y=a_i+n$ with

$$a_1=+\frac{AT_b}{2},\qquad a_2=-\frac{AT_b}{2},\qquad \sigma^2=\frac{N_0T_b}{4}$$

The pdfs of $y$ are Gaussian centred at $\pm AT_b/2$. Error when 0 is sent ($y>0$):

$$\begin{aligned} P_e &= \int_0^{\infty}\frac{1}{\sqrt{2\pi}\sigma}e^{-(y+AT_b/2)^2/2\sigma^2}dy=\frac12\operatorname{erfc}\left(\frac{AT_b/2}{\sqrt2\,\sigma}\right) \\ &= \frac12\operatorname{erfc}\left(\frac{AT_b/2}{\sqrt{N_0T_b/2}}\right)=\frac12\operatorname{erfc}\left(\sqrt{\frac{A^2T_b}{2N_0}}\right) \end{aligned}$$

By symmetry the error for a sent 1 is the same, so

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

BPSK is the best binary scheme: it needs 3 dB less $E_b/N_0$ than coherent ASK or FSK for the same $P_e$. Example: $E_b/N_0=10$ dB gives $P_e=Q(\sqrt{20})\approx3.9\times10^{-6}$.