Apple Sanity School > Fourier Series
Signal Transformations, Trigonometric Formulas, Complex Notation
Welcome to the school for Apple Sanity, Sunshine Ecstasy. This page describes the process for Fourier Series: why they are necessary and essential for taking any periodic signal and converting them into a trigonetric formula. Explains the Nyquist Frequency, sampling rates, power spectral densities, and complex notation, as well as shortcuts for handling even and odd functions.
NOTICE FOR MECHANICAL ENGINEERS: the following is all you need to understand the math behind labs that require usage of DAQ cards.
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Introduction
Given any periodic signal under certain conditions, we can represent the signal as a series of trigonometric formulas. Often, these signals are recorded on a sampling device. Here are a few examples of signals that have wave-like characteristics, but cannot be expressed as a continuous function. Consequently, a Fourier transform is needed.
Example 1 - odd signal
Example 2 - odd signal
Example 3 - even signal
Sampling Rates
In order to determine a useful sampling rate, note that:
where fh is the highest frequency recorded and fcrit is the maximum sampling rate or frequency that will record all of the components of our signals. To take into account the two unknown coefficients associated with analog signals of the form,
we determined that the frequency should be measured at two times fh to account for the entire wave forms. This particular frequency is the Nyquist Frequency :
where fDAQ is the sampling rate or frequency at which data is acquired by the the sampling device. Without using this Nyquist frequency the phenomenon of aliasing occurs and our digital readings would be inaccurate.
Fourier Transformations
Signals recorded by a sampling device are approximated in the form of a trigonometric Fourier series. Any periodic signal can be expressed as a series of sines and cosines if 1) the signal has a finite number of discontinuities period, 2) if the signal has a finite average value, and 3) if the signal has a finite number of relative maxima and minima. Then this finite signal can be expressed as
where n = the number of iterations, t = the time, and T = period or wavelength. Furthermore,
and,
Obviously, y(t) cannot always express the entire signal; otherwise, Fourier transforms would be unnecessary. In the three introduction examples, it is clear that y(t) becomes a piecemeal function. For Example 3, one would have to express y(t) = -mx from -T / 2 to 0, and y(t) = mx from 0 to T / 2, as opposed to expressing y(t) over the range of -T / 2 to T / 2.
The goal is to perform the summation until n reaches a fairly large number. The higher the number, the closer the Fourier transform is to an approximation to the given signal.
Even & Odd Functions
In addition, calculating the Fourier series for a periodic wave signal can be simplified by examining whether if the signal itself is an even or odd function. If the signal is an even function (Example 3), then it is symmetric about the y-axis. Consequently,
and
Moreover, if the signal is an odd function (Examples 1 and 2), then
Consequently,
and
Complex Notation
Let us now approximate the following signal, where the wavelength T = 2, and y(t) is defined as y(t) = -1 from -nT / 2 to 0 and y(t) = 1 from 0 to nT / 2. The first thing we should notice is that this signal is an odd function, so
Furthermore, this trigonometric Fourier Series can be simplified further using complex number notation. Given, already, that:
and,
then we have:
where
Rearranging the terms give:
If we define:
then
where
PSD: the Power Spectral Density
With these last two equations, the power spectral density, or PSD, of the waveform can be found. The PSD of a periodic waveform, or signal, displays the series of frequencies and their corresponding amplitudes derived from the original signal. With this display, we can find the highest frequency in each signal, which will prove crucial in determining sampling rates. The time average of the square of the y(t) is:
where
if y(t) is given in the form of complex number notation. Consequently,
Nyquist Frequency
These equations are how the sampling device and computer find the Fourier series and PSD of the signals it receives. One crucial consideration for the sampling device is the sampling rate. As with any analog-to-digital process, low sampling rates translate into aliasing, or even worse, amplitude ambiguity or falseness. Hence or otherwise, if there are two few sampling points, then the periodic waveform of the signal that the sampling device receives cannot be correctly interpreted. Obviously, the highest frequency found in the signal, or fh, must be smaller than the sampling rate, or fcrit:
The Nyquist frequency rule is that we must choose a sampling frequency fcrit such that it is equal to or at least twice the highest frequency fh in the signal.
where fN is the Nyquist Frequency, and fDAQ is the sampling rate of the DAQ card. We can find the highest frequency, and consequently, the minimum Nyquist Frequecy by analyzing the PSD of the periodic waveform of each signal.