Amplitude Modulation Triplets

Understanding of “beats” and amplitude modulation requires going back to basics to distinguish two similar treatments that are often confused with one another.

Acoustic Beats
The following link shows the modulation which leads to the acoustic beats phenomenon and contains audio examples to listen to:
Frequencies in amplitude modulation:

The basic trigonometrical identity [1] that is used to relate modulation to interference patterns is this:

cos a * cos b = 0.5 * ( cos (a+b) + cos (a-b) )

If a physical system contains modulation as expressed by the multiplication on the left of the equation, spectral analysis will split things into a series of additive components and we will have to interpret what we find in the spectrum as being the sum and the difference of the physical frequencies that are modulating each other.

Superposition (beats).
In the other direction the superposition (addition) of two signals can be converted back to modulation :
cos (f1t) + cos ( f2t) = 2 cos ((f1t + f2t)/2) * cos ((f1t – f2t)/2)
which if we rename the variables using a and b
cos (a) + cos ( b) = 2 cos ((a + b)/2) * cos ((a – b)/2)

So the presence of two frequencies seen in a Fourier spectrum is equivalent to physical modulation of their average frequency by half the difference of their frequencies. This is a mathematical identity, the two interpretations are equivalent and interchangeable, thus this is totally general and independent of any physical system where this sort of pattern may be observed. If these kind of patterns are found, the cause could be either a modulation or superposition.

In the presence of perfect sampling, the two forms are mathematically identical and again what would be found in the spectra would be the left-hand side: the two additive signals. However, what happens to the modulating envelop on the right in the climate system may well mean that the faster one get smoothed out, or the sampling interval and all the averaging and data processing breaks it up . The longer, lower frequency signal may be all that is left and then that is what will show up in the spectrum.

This is similar to what is called “beats” in an acoustic or musical context, except that the ear perceives twice the real physical frequency since human audition senses the amplitude variation: the change in volume, NOT the frequency of the modulation. The amplitude peaks twice per cycle and what we hear as two “beats” per second is a modulation of 1 hertz. Care must be taken when applying this musical analogy to non-acoustic cycles such as those in climate variables.

Also, if one part ( usually the faster one ) gets attenuated or phase delayed by other things in climate it may still be visible but the mathematical equivalence is gone and the two, now separate frequencies are detected.


Since the ‘side-lobe’ frequencies are symmetrically placed about the central frequency, this creates a symmetric pair of frequencies of equal magnitude whose frequencies are the sum and the difference of the originals. This is sometimes referred to as a ‘doublet’.

If the two cosine terms are equal as shown above, neither of the original signal frequencies remain. However, if the higher frequency is of larger amplitude, a residual amount of it will remain giving rise to a ‘triplet’ of frequencies. This is what is usually done in radio transmission of an amplitude modulated signal (AM radio). In this case the central peak is usually at least twice the magnitude of the each side bands.

It can be seem mathematically from the equations given above, that if both inputs are of equal amplitude the central frequency will disappear, leaving just a pair side frequencies. It may also be so small as to no longer be distinguishable from background noise in real measurements.

All of this can confound detection of the underlying cycles in a complex system because the periods of the causative phenomena may be shifted or no longer visible in a frequency analysis.

There are many non-linear effects, distortions and feedbacks that will deform any pure oscillation and thus introduce higher harmonics. Indeed such distortions will be the norm rather than a pure oscillation and so many harmonics would be expected to be found.

As a result, even identifying the basic cause of a signal can be challenging in a complex system with many interacting physical variables.

The triplet is useful pattern to look for, being suggested by the presence equally spaced frequencies, although the side peaks may attenuated by other phenomena and are not always of equal height as in the abstract example.

Examples of these kind of patterns can be found in variations of Arctic ice coverage.


Sum and Product of Sine and Cosine: