According to Lathi, the AM signal can be demodulated coherently (demodulation synchronous) or noncoherently (demodulation assynchronous). In practice, two demodulation noncoherent methods are used: (1) rectifier detection and (2) envelope detection.
(1) Rectifier Detector: If an AM signal is applied to a diode and resistor circuit, the negative part of the AM wave will be supressed. The rectified output, $v_r(t)$, is:
$$v_r(t) = [A+m(t)]\cos \omega_ct [1/2 + 2/\pi(\cos\omega_ct- 1/3 \cos3\omega_ct + \ldots)] = 1/\pi[A + m(t)] + hft$$
where $hft$ are high frequency terms, $m(t) = B \cos\omega_mt$ is the information (low frequency signal), $\omega_m$, information frequency, $\omega_c$, carrier frequency, $A$ and $B$, amplitude of carrier and low frequency signals, respectively.
When $v_r(t)$ is applied to a low-pass filter of cutoff $B$ Hz, the output is $[A +m(t)]/\pi$, and all the other terms in $v_r$ of frequencies higher than $B$ Hz are supressed. The dc term $a/\pi$ may be blocked by a capacitor to give the desired output $m(t)/\pi$.
(2) Envelope Detector: the output of the detector follows the envelop of the modulated signal. The circuit is a diode followed by a RC-filter. Mathematical details in Lathi.
The information (low frequency signal) in FM resides in the instantaneous frequency $\omega_i = \omega_c + k_f m(t)$, $k_f$ is a modulation index. A frequency-selective network with a transfer function $$|H(\omega)| = a\omega + b$$over the FM band would yield an output proportional to the instantaneous frequency, $\omega_i$. There are several possible networks with such characteristics, the simplest is an ideal differentiator with transfer function $j\omega$. The mathematical details can be seen in Lathi.
