SNR gain of non-coherent FM detector

A non-coherent FM detector (limiter + frequency discriminator) responds to the instantaneous frequency of the received signal; no local carrier is needed. The gain parameter $\text{SNR}_o/\text{SNR}_i$ is found assuming high carrier-to-noise ratio (above threshold).

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).