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

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 $$-1. In ideal case we can just multiply \eqref{received_am} with $$\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?

• What's "SDR"? Feb 26, 2023 at 20:11
• I guess my post is not specific to software defined radio implementation but I wanted to state that I was using SDR while I was experimenting to demodulate FM Broadcast with incorrect tuning setting. Feb 26, 2023 at 20:27
• Well, now at least I know what you mean. I thought it could have been "Single Diode Rectifier" or something. Feb 27, 2023 at 1:08
• I edited the post. Thank you for making my question more understandable :) Feb 27, 2023 at 9:07

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.

• Thank you Dan. Your answer is very clear and intuitive for me. However, I don't understand why frequency error $w_e$ appears to change amplitude in the last equation in my post? Is something wrong with my equation or my demodulation process? Feb 26, 2023 at 13:04
• @unique frequency is the time derivative of phase: a change of phase over a change in time is frequency. So frequency error causes a time varying phase error-- does that help? You will always have the $1/2$ factor regardless of no error. Feb 26, 2023 at 13:07
• Yes, definetly. Your first edit made me realize that I use coherent demodulation. Now, it is much more clear. Thanks. Feb 26, 2023 at 13:12
• So you either specify the transmitter LO and your receiver LO to be of sufficient spectral purity (this is typically the case), or when that is outside of your control there is something more exotic you can do as I described (since we can easily determine the phase alone even in the presence of AM by hard-limiting the waveform first). Feb 26, 2023 at 13:15
• If you're building an SDR for AM or FM, you should try to understand how they did it (synchronizing the local oscillator to the IF carrier frequency). There are some super easy ways (like a single diode rectifier), as long as $|m(t)|<1$. Because there is no phase reversal of the modulated carrier, you can always sync up your local oscillator. Feb 27, 2023 at 18:56