A look at Fourier from a High School Math Perspective: Fourier and AC Signal Processing

History

No discussion of AC Signal analysis is complete without mention of Jean Baptiste Joseph Fourier, the 18^th century mathematician whose work has been honored by giving his name to the mathematical methods based on conjectures he made in his study of the conduction of heat: The Fourier Series and Fourier Transform^[1]. Fourier's conjecture was that virtually any function could be represented as a summed series of sines and cosines, something that, with suitable restrictions (see Johan Dirichlet), is assumed as a given by engineers and mathematicians of today. The validity of this conjecture was by no means obvious to mathematicians of Fourier's time. There was considerable doubt as to whether the series would actually converge.^[2] However, in 1900, a young (19 years old) mathematician, [Lipót Fejér], in writing his doctoral thesis, realized that if the Fourier Series was cast in the form of the means of the sine and cosine functions, then the series could be shown to converge. Certain conditions still applied of course (limitations on discontinuities, the requirement of periodicity - or at least the ability to pretend periodicity by means of repetition), but for the most part, almost all functions describing natural phenomena are valid candidates for analysis using Fourier methods.

The Fourier Transform for the Mathematically Gifted

One who is interested in the Fourier methods would typically pick up any one of the excellent textbooks on the subject and begin to read up on the mathematics, only to be frustrated by encountering something like this

$X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}\, dt\,\!$ and $x(t)=\int_{-\infty}^{\infty}X(f)e^{-j\omega t}\, d\omega\,\!$

on the first page^[3]. This is the classic Fourier Forward/Reverse Transform pair. The accompanying explanation is often in mathematical terms that are difficult for non-mathematicians to deal with. Gifted mathematicians can stop here - they already have the thing figured out.

The Fourier Transform for the Not so Mathematically Gifted

For the rest of us, the truth is that it takes a great deal of study and intellectual effort to gain an intuitive feel for the subtle meaning of these two integrals. While they may be elegant and complete, they often leave the inexperienced with a feeling of wondering exactly why, or how, the whole thing works. There are very few (if any) sources that convey, at least to amateurs, a deeper sense of exactly why Fourier analysis works. For this, one needs to do a bit of homework on their own. Fortunately, a great deal of insight can be had using only high school level math: algebra, trigonometry and a bit of introductory calculus.

While the discussion that follows is mathematically correct in essence, it is by no means rigorous. It's purpose after all, is to clarify the workings of the Fourier Transform, in particular as it applies to the analysis of AC waveforms, not to replace formal textbooks on Fourier Methods^[4].

Necessary Mathematical Background

There are only a few well known and relatively simple bits of math that will be required. These are:

Basic Algebra. This really requires no additional comments.
The Sine^[5] and Cosine^[5] Functions. Since, for all practical purposes, Fourier deals with functions that are periodic^[6], it should come as no surprise that sine and cosine functions would be involved. Here is a one period plot of these functions:

The idea here is to remind us that the only difference between the sine and cosine functions is a simple shift of $- π / 2$ along the abscissa^[7]. The axes have no labels at this point - on purpose, since the function itself is dimensionless. Later on we'll scale the axes to put time on the abscissa and some measured quantity like voltage or current on the ordinate. We'll also be using a few common relationships between these functions:

1. The Tangent function. While typically thought of as a trigonometric function in it's own right, it is far more useful to define it in terms of the sine and cosine functions:

$tan(\alpha)=\frac{sin (\alpha)}{cos(\alpha)}\,\!$

2. A Fundamental Identity. Everyone knows this one:

$sin^2(\alpha)+cos^2(\alpha)=1\,\!$

3. Sum of angles formulas. Everyone has to learn these in high school and probably forgets them almost immediately. They are stated here for reference:

$sin(\alpha \pm \beta)=sin(\alpha)\, cos(\beta) \pm cos(\alpha)\, sin(\beta)\,\!$

$cos(\alpha \pm \beta)=cos(\alpha)\, cos(\beta) \mp sin(\alpha)\, sin(\beta)\,\!$

4. The forms in which α and β are equal are also useful:

$sin(2\alpha)=2 sin(\alpha) cos(\alpha)\,\!$

$cos(2\alpha)=cos^2(\alpha)-sin^2(\alpha)=2 cos^2(\alpha)-1=1-2 Sin^2(\alpha)\,\!$

Simple Integrals. While some may consider integral calculus beyond high school math, in fact almost all high school curricula include some elementary calculus, at least for some of the students. We're only going to be using one integral that may go a bit beyond this minimal treatment, that being the integral defining the mean of an arbitrary function, $f (u)$ :

$\overline{f(u)}=\frac{1}{b-a}\int_{a}^{b}f(u)\, du\,\!$

All this integral says is that we can get the average height by dividing the area under some bounded segment of a curve by its width. This is intuitively obvious for shapes like squares, rectangles and triangles. It's really cool that it's also true for any arbitrary function that can be integrated.
The Concept of Orthogonality. The concept of Orthogonal functions is tossed about in many math texts as though everyone on the planet is born with an inherent knowledge of orthogonality. Yet the vast majority of people I encounter have never even heard of the concept - some have never even heard the word. Geometrically, it refers to right angles and has been abstracted to functions in the sense that the product of two functions that are orthogonal to each other will always be exactly zero. Working from the geometrical concept, the simplest pair of orthogonal functions are the x and y axes of the Cartesian coordinate system, x=0 and y=0, the product of which is clearly zero. While x=0 and y=0 are really boring orthogonal functions, it turns out that sin(x) and cos(x) are also orthogonal functions under the right circumstances. This is much more interesting and will be critical to understanding the operation of the Fourier Transform.

Next: The Fourier Series >>>>

Notes

↑ [Fourier, J.B.J. The Analytical Theory of Heat. 1822.] (Trans., 1878, by Alexander Freeman from Théorie analytique de la chaleur.)
↑ That is, whether the series of sines could actually be made to equal the original function.
↑ A small apology is in order here. For the sake of making a point, I have presented the Fourier integrals. However, the FFT and its relatives are really based on the Fourier Series. More about this shortly.
↑ If rigor is what you need, there are many fine texts that discuss Fourier methods in more rigorous mathematical terms. In my opinion, for sheer understandability alone, the best of these is the Ramirez book: The FFT: Fundamentals and Concepts, Robert W. Ramirez. Prentice-Hall, Inc., Englewood Cliffs, N.J. 1985.
↑ ^5.1 ^5.2 See these references for a more complete mathematical definition of the [sine] and [cosine] functions.
↑ Meaning, specifically, repeating over and over again in time.
↑ The conventional x-axis. See [Cartesian Coordinate System] for a more complete discussion.

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