# Finding the amplitude of a multi-tone signal

How do you find the amplitude of a signal consisting of multiple sinusoids with different frequencies such as
$$f(t) = A\cos(\omega_1t)+B\sin(\omega_2t)\text?$$

I'm thinking that you take the derivative and set it equal to zero (finding the maximum), but I can't really say why this works. This is for communication systems when finding the amplitude of a message signal.

• For what purpose are you finding the amplitude? If you need the worst case peak amplitude e.g. to ensure you never clip the signal, that's trivial. A+B. – user_1818839 May 9 '20 at 20:15
• I'm ultimately trying to find the modulation index Am/Ac. I should also mention this is for academia. Does A+B work for signals with different frequencies? – Aaron May 9 '20 at 20:25
• the instantaneous amplitude is your $f(t)$, and there's no other way than evaluating the cosine and the sine, multiply them with A and B, respectively and sum them up. And your question has little to do with the modulation index... so I'm not quite sure what you're asking about! – Marcus Müller May 9 '20 at 20:41
• The definition of the modulation index is Message Amplitude/Carrier Amplitude right? I'm trying to find the Message Amplitude. – Aaron May 9 '20 at 21:06
• i.e. f(t) is the message of an amplitude modulated signal? – Ocanath May 9 '20 at 21:14

In a comment you clarified that you're trying to find the modulation index of an amplitude modulated signal. I assume this means that the signal provided takes the form of your message signal, and that is the reason you're trying to find the maximum. If you're unable to predict the exact frequencies, amplitudes, and phases of your sinusoids, you should assume the worst case upper bound is $$\ |A| + |B| \$$.
If all of your signals are cosines and have zero phase shift, then if the amplitudes are all positive, obviously the maximum is the sum $$\(t = 0)\$$. Interestingly, if there are only two cosines with no phase shift, and if the amplitudes are not both positive, you can still assume the maximum is $$\ |A| + |B| \$$ if the ratio $$\r = \frac{\omega_1}{\omega_2}\$$ is not rational.