Hi,

I'm having some troubles determining the amplitude/magnitude of the following equation.

\(A\cos(2\omega t+\beta_1)+B\cos(3\omega t+\beta_2)+C\cos(5\omega t+\beta_3)\)

Since each part is at a different frequency, i cannot sum the magnitudes of each part.

I have also thought about using variations of the double/triple angle formulae and some basic trigonometric identities, so that I can write the equation under a single frequency, but by doing so, I am introducing some higher order terms, which seem to negate the ability to sum the amplitudes together.

For example, the term with \(5\omega\) would look like
\(16\cos^5(\omega t)-20\cos^3(\omega t)+5\cos(\omega t)\)

I suppose another method would be to plot the first equation and then record the amplitude, but i would like a more generalised approach to solve this problem. All input is welcome and appreciated.

 

By the term magnitude or amplitude do you mean peak, rms, ....?

 

I believe I would like to determine the peak value.

I think the tricky part is that the coefficients A, B, C are governed by a filter, so their magnitudes change with the frequency, meaning I cannot simply plot my above equations and then determine a peak value.

 

I believe I would like to determine the peak value.

I think the tricky part is that the coefficients A, B, C are governed by a filter, so their magnitudes change with the frequency, meaning I cannot simply plot my above equations and then determine a peak value.

Keep in mind that filtering will likely change not only the amplitudes but also the phase relationships of the individual frequency components - which further complicates matters.

I'm not sure if I'm stating the obvious to you, but one mathematical approach in finding the peaks (maxima / minima) is to take the derivative(s) of the total function. Equating the function's first derivative to zero will give points of maxima or minima. Since the steady state function will be periodic there will be an infinity of max/min points but these will have recurrent equal values. The second derivative would indicate where one has a maximum or a minimum. It may not be necessary to take the second derivative as the peak max or min values may be identical [in terms of magnitude]. This of itself isn't a trivial process as one would still eventually be forced to reduce everything to common terms in "ωt" - albeit with higher order sine or cosine terms.

Not sure if this helps. There are probably others on the forum who have better suggestions.

 

Thanks for your suggestion t_n_k. I think your method would help to create a sketch of the waveform for sure. For my work, I suppose my goal is to find a closed form expression that can be extracted from my equations that, as you described, the variations in amplitude and phase.

I am sort of disappointed that my second method where I rewrote the equations all in terms of \(\omega t\) did not work. In my head, I had thought that since in the frequency domain, the terms would all be located in the same position, I could just sum up the resulting amplitudes and then somehow re-translate that back to the time domain. So I'm curious to know if there is something I might be missing here that is causing this to go wrong.

 

An issue for me is that you haven't explained exactly what you are hoping to achieve.

Do you actually want a plot of the composite waveform or simply the peak amplitude(s)...?? You added the issue of filtering which suggests you are delving into signal processing.

Perhaps if you gave a broader perspective on the problem it may be easier to offer suggestions.

 

Essentially, i am working on a problem related to envelope modulation, and I am trying to determine the amplitude of the envelope. I can pretty easily obtain a plot of the function in Matlab, but my end goal is to have a way to extract the amplitude numerically so that I can continue on with my analysis.

In my first trials, I had an equation that looked like \(A_o+ A\cos(\omega t)\), so I could easily pick out that the small signal amplitude was just equal to A. This method proved to be too inaccurate for my purposes, so I started to include harmonics in my equation, and ended up with the large equation stated in my first post. Unfortunately, in this case, the amplitude of the small signal envelope is not so obvious in this case, and thus I became stuck.

Hope that helps a little bit. Please let me know if I can provide more information.

 

No it didn't help. I'm still not sure what you are doing.

Are you looking at amplitude modulation (say) with multiple modulation terms at different modulation frequencies and indices?

What does filtering have to do with your problem?

Perhaps a diagram of the signal processing system might benefit.

 

A huge problem you are going to have is that the phase offsets of the harmonics drives the whole thing (unless one of the amplitudes is extremely dominant).

 

A huge problem you are going to have is that the phase offsets of the harmonics drives the whole thing (unless one of the amplitudes is extremely dominant).

This is correct. Actually my end goal is to determine what effects the harmonics play on the system, and how to model them appropriately for simulation.

 

Sorry to bump this up again, but I had an idea but I'm not sure if it would work.

Lets say we still have this equation

\(A\cos(2\omega t+\beta_1)+B\cos(3\omega t+\beta_2)+C\cos(5\omega t+\beta_3)\)

having this alone is difficult to come up with a value for the amplitude. What if I were to pass this signal into an envelope detector circuit? Could that possibly simplify the above equation into something that I can more easily deduce the amplitude values?