Figure of merit of non-coherent FM; threshold effect

Figure of merit of non-coherent FM

FM signal and receiver. $s(t)=A_c\cos\left[\omega_ct+2\pi k_f\int m(t)dt\right]$, instantaneous frequency deviation $k_fm(t)$. Receiver: BPF ($B_T$) → limiter → discriminator (output $=\frac{1}{2\pi}\frac{d\phi}{dt}$) → LPF ($W$).

Noise in phasor form. Write the narrowband noise as $n(t)=r(t)\cos[\omega_ct+\psi(t)]$. At high carrier-to-noise ratio ($A_c\gg r$), the resultant phase is

$$\theta(t)\approx\phi(t)+\frac{r(t)}{A_c}\sin[\psi(t)-\phi(t)]$$

            r(t)
          .------>          resultant = carrier + noise
         /  ^               small noise phasor rotates
 Ac     /   | r sin(psi-phi) the carrier phasor slightly
 ------>----+

The noise term is statistically equivalent to $\frac{n_s(t)}{A_c}$ (the phase $\phi$ can be dropped for noise calculation). Discriminator output:

$$v(t)=\frac{1}{2\pi}\frac{d\theta}{dt}=k_fm(t)+\frac{1}{2\pi A_c}\frac{dn_s(t)}{dt}$$

Output signal power: $S_o=k_f^2P$.

Output noise power. $n_s$ has PSD $N_0$ for $\lvert f\rvert\le B_T/2$. Differentiation multiplies the spectrum by $j2\pi f$, so the noise PSD at discriminator output is

$$S_{n_d}(f)=\frac{(2\pi f)^2}{(2\pi A_c)^2}N_0=\frac{N_0f^2}{A_c^2},\qquad \lvert f\rvert\le\frac{B_T}{2}$$

This parabolic noise spectrum is the key property of FM. After the LPF ($\lvert f\rvert\le W$):

$$N_o=\int_{-W}^{W}\frac{N_0f^2}{A_c^2}df=\frac{2N_0W^3}{3A_c^2}$$

 S(f) |\                   /|
      | \                 / |   parabolic noise
      |  \               /  |   after discriminator
      |   \.           ./   |   (only |f|<W kept)
      +----+-----+-----+----+--> f
          -W     0     W

Output SNR:

$$\text{SNR}_o=\frac{k_f^2P}{2N_0W^3/3A_c^2}=\frac{3A_c^2k_f^2P}{2N_0W^3}$$

Input SNR. Received power $\frac{A_c^2}{2}$ (constant envelope); noise in $B_T$ is $N_0B_T$:

$$\text{SNR}_i=\frac{A_c^2}{2N_0B_T},\qquad \text{SNR}_{ref}=\frac{A_c^2}{2N_0W}$$

Gain and figure of merit:

$$\begin{aligned} \text{FOM} &= \frac{\text{SNR}_o}{\text{SNR}_{ref}}=\frac{3k_f^2P}{W^2} \\ \frac{\text{SNR}_o}{\text{SNR}_i} &= \frac{3k_f^2P\,B_T}{W^3} \end{aligned}$$

For a single tone $m(t)=A_m\cos\omega_mt$: $P=A_m^2/2$, $\Delta f=k_fA_m$, $\beta=\Delta f/W$, $B_T=2(\beta+1)W$ (Carson):

$$\text{FOM}=\frac{3}{2}\beta^2,\qquad \frac{\text{SNR}_o}{\text{SNR}_i}=3\beta^2(\beta+1)$$

So FM noise performance improves with the square of the deviation ratio: FM trades bandwidth for SNR. FM beats AM ($\text{FOM}=1/3$) when $\frac32\beta^2>\frac13$, i.e. $\beta>0.471$ (about 0.5).

Threshold effect

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)