# Trouble in understanding AM modulation

Amplitude modulation is in fact, superimposing the low frequency transmission signal into a high frequency carrier signal, right?

So, if the transmission signal can be represented as $c(t)=A_c \sin (\omega_ct )$ and the carrier wave can be represented as $c(t)=A_m \sin (\omega_mt)$, then upon superimposing and simplifying the equation using trigonometric identities we will have $$c_m(t)=A_c \sin (\omega_ct )+\frac{\mu A_c}{2}\cos(\omega_c-\omega_m)t-\frac{\mu A_c}{2}\cos(\omega_c+\omega_m)t,$$ but this equation has the carrier wave and two sinusoidal waves with frequencies $\omega_c-\omega_m$ and $\omega_c+\omega_m$. So where is the signal which is transmitted? What we've got is a corrupted signal with uneven frequencies i.e. $\omega_c-\omega_m$ and $\omega_c+\omega_m$

Correct me if I'm wrong at any point.

• I'm not sure if this helps: the transmitted waveform is now somewhat complicated, but the usefulness is that the low frequency signal can be extracted on the receiving end. Multiply $c_m(t)$ by $\sin(\omega_c t)$. Feb 20, 2018 at 19:05

You are correct in your assertion that the AM procedure, when applied to a monochromatic signal, will generate a radio emission that, spectrally, has a spike at the carrier and then two sidebands on either side, separated from the carrier frequency $\omega_c$ by the modulation frequency $\omega_m$.

More specifically:

• The strength of the signal is encoded in the depth of the modulation and therefore on the strength of the sidebands when compared to the carrier.
• The frequency of the signal is encoded in the separation of the sidebands with respect to the carrier.
• The phase of the signal is encoded in the phase of the sidebands with respect to the carrier.

This means that the amplitude-modulation procedure has successfully encoded all the relevant information of your signal into a higher-frequency radio beam that can be easily transmitted. It is then the job of the detector (i.e. your consumer radio) to decode that information into audio signals that can be played by a speaker; how the decoder does that is up to the device, but the important thing is that the full information is there to be decoded.

The decoding procedure itself can be done by forgetting about this sideband business and just looking at the signal in the time domain with an envelope detector, or you can explicitly use this carrier-sideband structure with a locally-generated carrier that you then use to extract information from the sidebands in a synchronous detector.

• There will be no "spike at the carrier frequency" (i.e., the carrier frequency will not be present at all) if the modulation index is 100%. Feb 20, 2018 at 15:08
• @Emilio Pisanty, How does the transmission signal correctly became the envelope enclosing the carrier wave? Can it be seen mathematically? Feb 20, 2018 at 15:34
• @SlayerDiAngelo, An AM radio signal is the instantaneous product of the audio signal and the carrier wave. The product can be "computed" by a simple analog circuit. Feb 20, 2018 at 15:54

You are right that by superimposing a constant modulation on the carrier wave you cannot transmit a signal because you get a constant single frequency carrier wave plus two constant sideband frequencies. For the transmission of a signal, you need a modulation changing in time (frequency and/or amplitude) like in speech transmission in AM radio or in Morse signals transmission in amateur shortwave radio.

The low frequency transmission signal means slowly changing $A_c(t)$, not fixed $A_c$ with fixed $\omega$ which does not carry any information.

• I don't understand, can you say it clearly? Feb 20, 2018 at 12:51
• An amplitude modulated signal is: $c_m(t)=A_c(t)*sin(\omega_m t)$ and not a summation of two fixed $sin$ functions. The amplitude changes in time slowly in comparison with the periodicity of $\omega_m$. This is the meaning of superimposing a low frequency signal on the carrier. Feb 20, 2018 at 12:56
• @SlayerDiAngelo I think he means the same as I in my answer below. Please look at that. With a modulation of constant frequency and constant amplitude you cannot transmit any information! Only with changes in modulation you can transfer information, e.g. like with changing frequencies of the amplitude modulation. Feb 21, 2018 at 16:52
• That's right @freecharly. The discussions here somehow missed the fact that AM is a term from Communication Engineering rather than a mathematical equation. Feb 21, 2018 at 17:10
• @npojo- You are exactly right! Feb 21, 2018 at 17:14