Mixed-Frequency Signals

As an interesting footnote, this is precisely why differentiation is rarely performed on real-world signals. Since the frequency of noise often exceeds the frequency of the signal, differentiating a “noisy” signal will only lead to a decreased signal-to-noise ratio.

For a practical example of this, tell your students about vibration measurement, where it is more common to calculate velocity based on time-integration of an acceleration signal than it is to calculate acceleration based on time-differentiation of a velocity signal.

Although the sequence of images presented in the question by no means constitutes a formal proof, it should lead students to observe a trend: that odd harmonics do not make a waveform unsymmetrical about the horizontal axis, whereas even harmonics do. Given these two facts, we may make qualitative judgments about the harmonic content of a waveform simply by visually checking for symmetry about the horizontal axis.

Incidentally, some students have a difficult time grasping the concept of symmetry about the horizontal axis of a waveform. Take this simple example, which is symmetrical about its horizontal centerline:

Some students will protest that this waveform is not symmetrical about its centerline, because it does not look exactly the same as before after flipping. They must bear in mind, though, that this is just one cycle of a continuous waveform. In reality, the waveform looks like this before and after flipping:

All one needs to do to see that these two waveforms are indeed identical is to do a 180 degree phase shift (shifting either to the left or to the right):

By contrast, a waveform without symmetry about the horizontal axis cannot be made to look the same after flipping, no matter what subsequent phase shift is given to it:

Another way to describe this asymmetry is in terms of the waveform’s departure from the centerline, compared to its return to the centerline. Is the rate-of-change ([dv/dt] for a voltage waveform) equal in magnitude and opposite in sign at each of these points, or is there a difference in magnitude as well? Discuss ways to identify this type of asymmetry, and what it means in terms of harmonic content.

Mathematically, this symmetry is defined as such:

Where,

f(t) = Function of waveform with time as the independent variable

t = Time

T = Period of waveform, in same units of time as t

I’ve had electrical power system experts confidently tell me that even-numbered harmonics cannot exist in AC power systems, due to some deep mathematical principle mysteriously beyond their ability to describe or explain. Rubbish! Even-numbered harmonics can and do appear in AC power systems, although they are typically much lower in amplitude than the odd-numbered harmonics due to the nature of most nonlinear loads.

If you ever wish to prove the existence of even-numbered harmonics in a power system, all you have to do is analyze the input current waveform of a half-wave rectifier!