Why don't we need frequency and phase correction in AM and FM?

Question

As far as I understand, double sideband AM with carrier and FM can be demodulated via SDR (Software Defined Radio) without frequency or phase correction. For AM, received signal is $$r(t) = A(t)\cos(\omega_c t) \tag{1}\label{received_am} $$

where $A(t) = 1 + m(t)$ and $-1<m(t)<1$. In ideal case we can just multiply \eqref{received_am} with $\cos(\omega_ct)$ and apply LPF to obtain $A(t)$. However, lets assume that both phase and frequency has error in local oscillator. Then, we will get the signal below when we apply first step of demodulation process

$$ A(t)\cos(\omega_ct)\cos((\omega_c+\omega_e)t + \phi_e) \\ = \tfrac{1}{2}A(t)\Big(\cos(\omega_et-\phi_e) + \cos\big((2\omega_c+\omega_e)t + \phi_e \big) \Big)$$

where $w_e$ and $\phi_e$ denotes small frequency and phase error respectively. After applying LPF we will get

$$\tfrac{1}{2}A(t)\cos(\omega_et-\phi_e)$$

but not $A(t)$ as in ideal case. Also, I can listen music via SDR without any distortion even if I tune frequency upper than original FM broadcast frequency. My question is simply, why don't we need frequency and phase correction in AM and FM?

Answer 1

With AM all information is encoded in the amplitude regardless of any phase modulation occurring. Therefore we can demodulate it with an amplitude discriminator (envelope detector) as long as the discriminator isn’t sensitive to phase modulation.

With FM all information is encoded in the phase regardless of any amplitude variation or modulation that may be occurring. Therefore we can demodulate it with a frequency discriminator as long as the discriminator isn’t sensitive to amplitude modulation.

The last points are important as there is often cross sensitivity. It is common in an FM receiver to hard limit the waveform prior to final demodulation to remove all amplitude variation first when the discriminator is sensitive to amplitude. Similarly if coherent demodulation is used for AM (as the OP has demonstrated) the result is sensitive to both AM and PM. (Envelope detection with large carrier AM is insensitive to PM so would be of no consequence). To counteract this for coherent AM demodulation, the phase modulation can be tracked alone with a hard-limited phase detector outside of the main demodulation path, and removed from the demodulated result (or the local oscillator can similarly be made to track the phase at a tracking bandwidth wider than the signal modulation).

Note the last point in the OP asking about a factor of 1/2. This has nothing to do with the phase offset but is due to half the signal being in the sum and half in the difference as given by the cosine product rule below:

$$A \cos(\alpha)\cos(\beta) = \frac{A}{2}\cos(\alpha+\beta)+\frac{A}{2}\cos(\alpha-\beta)$$

Thus we get a low frequency output (at DC when the frequencies match exactly) and a high frequency output, each scaled by $1/2$. The low pass filter removes the high frequency.