Fourier Transform's Impression

Fourier series represents that any signal shapes can be approximated by a composition of trigonometric
functions.
This program calculates frequency compositions(spectrum=amplitue&phase) by Discrete Fourier transform,
and execute compositions(Fourier series) of them. Let's experience Fourier transform as itself without
mathmatical formulas.

Source Signal Selection

Constant Peak
2 Peaks 3 Peaks
Trapezoid Ramp
Random

Execute Discrete Fourier Transform


DC(0th)
1st 2nd
3rd 4th

Composite Spectrums

Rectangle
0

Explanation

This program executes Discrete Fourier transform of 8 points. Here, there is an implicit assumption.
Although source signals of this program are data series of \(y\left(0\right)\)`\(y\left(7\right)\), they are regarded as the periodic
waveform which returns to a starting point as \(y\left(0\right)\)=\(y\left(8\right)\).
In this point, it seems that many users of numerical calculation library do not acknowledge. In
addition, it also can be conceivable that staring and ending point have expansions of 0.5 for the
integer discrete space(refer to this site" Range Mapping").
Next, the spectrum gotten by Discrete Fourier transform is that the horizontal axis is frequencies,
and the vertical axis is spectrum intensities(complex number). For this program, spectrum(amplitude&
phase) is shown as signal component before the transform.
The frequencies are heading towards fundamental frequency, 2 times of fundamental, 3 times... from
DC(Direct Current component) as the lowest degree. For the data series of 8 points, maximum frequency
becomes 4 times. This is for 2 points which are minimum number of sample points and they can represent
1 period's sine or cosine wave. Generally, the lower the degree is, the bigger the spectrum's amplitude
is, and the higher the degree is, the smaller the amplitude becomes. By also this program, you can see
that whole envelope gathers in low degrees and high degree components contribute to only details.
In addition, about the discontinuous point, you can check the overshoot(Gibbs phenomenon). Both ends
of data seriies often becomes discontinuous, there is a method which can ease it by multiplying a
window function in such a situation.
For this program, an aspect of the composition which replace the trigonometric functions to rectangle
waves can be checked. The impression of composition seems to be clear for "Peak" or "Ramp". As an
orthogonal transform of rectangles, there is Walsh-Hadamard transform.
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Now, this program is represented by (1) as Fourier series.

\(\large{y\left(t\right) = DC + \displaystyle \sum_{i=1}^{n/2} \left(u_i \cos \left(2\pi\frac{it}{n}\right) + v_i \sin \left(2\pi\frac{it}{n}\right)\right)} \qquad \left(n=8\right) \qquad (1)\)
\(DC\): Average of all data(Direct Current Component)
\(u_i\): Spectrum's real component \(v_i\): Spectrum's imaginary component

Although this is a formula generally expressed, it can be tranformed to (2) by a composition formula of
trigonometric functions.

\(\large{y\left(t\right) = DC + \displaystyle \sum_{i=1}^{n/2} A_i \sin \left(2\pi\frac{it}{n} + \varphi_i \right)} \qquad (2)\)
\(A_i=\sqrt{u_i^2 + v_i^2}\): Power spectrum \(\varphi_i = \arctan {\left(v_i, u_i\right)}\): Phase spectrum

As the author, this way of expression is more clear for the amplitude and phase, and seems to be easy
to understand.
As the orthogonal basis like Fourier transform and Walsh Hadamard transform which have been formulated,
you can also generate the basis. As generating the orthogonal system by principal component analysis
(uncorrelation), please see this site"Whitening". Regarding Fourier transform, lower frequency's amplitude
tends to grow bigger as already mentioned, this also means that the principal components are concentrated
in lower orders.