# Relation of carrier frequency and signal frequency

#### curious_mind

##### Member level 5 In amplitude modulation, what is the relation between carrier and signal frequency. In theory how close can they be. I am doing an experiment in lab to understand this phenomenon. After brief survey I found that demodulation becomes difficult if signal frequency were to be same as carrier frequency. Could any body explain this?

#### FvM

##### Super Moderator
Staff member #### c_mitra Do this gedanken (an experiment in mind only):

Consider a simple carrier frequency. Say this is denoted as A*sin(omega*t). Here omega is the angular frequency and A is the amplitude.

Now modulate this carrier with another frequency: B*sin(phi*t). B is the amplitude and phi is the frequency of the modulation.

What is the result? The two waves will be added point by point. The result will be A*sin(omega*t) + B*sin(phi*t).

We want to know what happens when phi gets close to omega. So we keep omega constant and change phi from omega-alpha to omega+alpha where alpha is a number of the order or greater than omega. That means we scan over the frequency like a radio tuner.

Just assume (else it will be messy to write) A=B=1; then sin(omega*t)+sin(phi*t)=2*sin((omega+phi)*t/2)*cos((omega-phi)*t/2)

Now you see that you get product of two frequencies: one sum and the other difference.

What happens if phi is close to omega? Look at the cosine term. The cosine term becomes large (close to 1)- very low frequency.

Hence the modulation frequency should be much less than the carrier (it is possible to reverse the carrier and the signal frequency in principle).

#### curious_mind

##### Member level 5 Just wanted to know what is the ratio we need to maintain between Fc and Fs?

#### albbg Now modulate this carrier with another frequency: B*sin(phi*t). B is the amplitude and phi is the frequency of the modulation.

What is the result? The two waves will be added point by point. The result will be A*sin(omega*t) + B*sin(phi*t).

We want to know what happens when phi gets close to omega. So we keep omega constant and change phi from omega-alpha to omega+alpha where alpha is a number of the order or greater than omega. That means we scan over the frequency like a radio tuner.

Just assume (else it will be messy to write) A=B=1; then sin(omega*t)+sin(phi*t)=2*sin((omega+phi)*t/2)*cos((omega-phi)*t/2)
Sorry, but this way you don't modulate the signal, just add up two sinusoidal signal then the spectrum will shown just two frequencies: omega and phi.
In order to amplitude modulate (since I think this was you purpose) you have to multiply two sinusoidal signals as, for instance:
A*sin(omega*t)*[1+B*sin(phi*t)].
This is a very simple case, of course there are many other types of modulation.

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#### FvM

##### Super Moderator
Staff member Just wanted to know what is the ratio we need to maintain between Fc and Fs?
Depends on demodulator properties and selectivity of low pass filter that extracts the message signal. Fs = 0.9 Fc is possible with envelope detector.

##### Super Moderator
Staff member A reasonable choice for AM carrier frequency is 4x or greater than the signal. This allows easy detection of the signal envelope.
Whereas, if the two frequencies are nearly the same then it creates a beat (or subtractive or difference) frequency. Beat frequencies can also occur if the signal is approximately a subharmonic of the carrier.

The waveforms below are AM modulation.
#1 output has a ratio which allows the initial signal to be observed and detected.
However #2 output contains a 3kHz difference frequency, hiding evidence of the initial signal. #### c_mitra The waveforms below are AM modulation.
#1 output has a ratio which allows the initial signal to be observed and detected.
However #2 output contains a 3kHz difference frequency, hiding evidence of the initial signal.
Very interesting graphs. The question next is whether the input signal can be recovered with reasonable fidelity from the modulated output?

The answer is not simple; but some answers can be obtained from the sampling theory (faltung). How the convolutions work in real life.

There is another problem pointed out by Nyquist. Without going into the detailed mess, it is best to stick to half the carrier frequency (for the max signal frequency).

#### FvM

##### Super Moderator
Staff member There's no relation to Nyquist. The bandwidth of the modulated signal isn't restricted to fc. A baseband of 0 to fc can be recovered without aliasing, but not with a simple envelope detector. The example of fs = 0.9 fc mentioned above still works with a full wave envelope detector and a steep low pass.

##### Super Moderator
Staff member To get an idea of the entire process, here's a demo showing recovery of a signal which is 1/4 the frequency of the carrier.

Diode rectification detects the positive half of the AM. Followed by a peak detector in the form of a smoothing capacitor.

It's a sample-and-hold. Notice 4 or 5 steps are sufficient to recover the quasi-sine waveform.

However there's more that needs to be added in simulation. What worked is to discharge the cap partially just before a peak arrives. Although I tried various filter networks with a capacitor and inductor, the rise and fall slopes could not be made smooth enough to reveal the initial waveform clearly. 