Summary: This is a general overview of how to solve simple electrical engineering problems.

A periodic signal, such as the half-wave rectified sinusoid,
consists of a sum of elemental sinusoids. A plot of the Fourier
coefficients as a function of the frequency index, such as shown
in
Figure 1,
displays the signal's spectrum. The word "spectrum"
implies that the independent variable, here

Fourier Series spectrum of a half-wave rectified sine wave |
---|

A subtle, but very important, aspect of the Fourier spectrum is
its *uniqueness*: You can unambiguously find
the spectrum from the signal
(decomposition)
and the signal from the spectrum
(composition).
Thus, any aspect of the signal can be found from the spectrum
and vice versa. *A signal's frequency domain
expression is its spectrum*. A periodic signal can be
defined either in the time domain (as a function) or in the
frequency domain (as a spectrum).

A fundamental aspect of solving electrical engineering problems is whether the time or frequency domain provides the most understanding of a signal's properties and the simplest way of manipulating it. The uniqueness property says that either domain can provide the right answer. As a simple example, suppose we want to know the (periodic) signal's maximum value. Clearly the time domain provides the answer directly. To use a frequency domain approach would require us to find the spectrum, form the signal from the spectrum and calculate the maximum; we're back in the time domain!

Another feature of a signal is its average power. A signal's instantaneous power is defined to be its square. The average power is the average of the instantaneous power over some time interval. For a periodic signal, the natural time interval is clearly its period; for nonperiodic signals, a better choice would be entire time or time from onset. For a periodic signal, the average power is the square of its root-mean-squared (rms) value. We define the rms value of a periodic signal to be

What is the rms value of the half-wave rectified sinusoid?

The rms value of a sinusoid equals its amplitude divided by

To find the average power in the frequency domain, we need to
substitute the spectral representation of the signal into this
expression.

Power Spectrum of a Half-Wave Rectified Sinusoid |
---|

It could well be that computing this sum is easier than
integrating the signal's square. Furthermore, the contribution
of each term in the Fourier series toward representing the
signal can be measured by its contribution to the signal's
average power. Thus, the power contained in a signal at its

In high-end audio, deviation of a sine wave from the ideal is measured by the total harmonic distortion, which equals the total power in the harmonics higher than the first compared to power in the fundamental. Find an expression for the total harmonic distortion for any periodic signal. Is this calculation most easily performed in the time or frequency domain?

Total harmonic distortion equals

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