Fourier transform

[1] In math, the Fourier transform (FT) is an integral transform taking a function as input and outputs another function depicting the extent to which various frequencies are present in the original function. I.e., it breaks down complex wave/signals into their fundamental sine/cosine frequencies or in short, "transforming" signals from their time domain to their frequency domains.

The transform's output is a complex valued frequency function.

[1] The term Fourier transform refers to both the mathematical operation and to this complex-valued function. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The FT is analogous to decomposing a musical chord's sound into its constituent pitches' intensities.

When each note plays, each have a unique wave at some frequency that can be graphed through measurements.

When the notes are played together, the measured wave is a result of the sum of the notes' waves being added together.

[2.1] E.g., the wave of A440 (the standard or concert pitch), the 'A' note's usual 440 Hz frequency is graphed as shown in yellow.

[2.1] The wave of D294, the 'D4' note's usual frequency of 293.66 Hz is graphed as shown in pink.

[2.1] If both notes are played together, at 1 point in time of the resulting signal, that point's amplitude (pressure difference) is the sum of the amplitude of both signals at that corresponding time.

[2.1] Adding more notes makes the resulting wave more complex.

[2.1] To decompose the resulting signal back into the individual signals of each notes, we use a mathematical tool to filter out some signals.

E.g., Take a 3 beats/second interval and use a finite part of the wave (x = 0 to x = 4.5 in this case).

The key idea is to take the graph and wrap it around a circle.

[2.1] Imagine a vertical vector that goes across the wave (on the horizontal graph) and a rotating vector (on the circle graph). where each time point, its length is the graph's height at that time.

I.e., high points = farther from origin and low points = closer to origin.

Moving forward in 2 s corresponds to a full rotation below or 1 s is half a rotation.

[2.1] So 2 frequencies are present here:

Frequency of "3 beats/second" on the horizontal graph.
Frequency of "0.5 cycles/second" on the circle graph.

We can adjust the 2nd frequency by wrap it around faster or slower and the choice of winding frequency determines what the wound up graph look slike

[2.1] But if the winding frequency matches the signal's 3 beats or cycles/second (by separating axis to have verticale lines distanced to be 1/3 = 0.33 from each other), the circle graph will be graphed in a peach-like circle, with all high points of the signal corresponds to the circle's right side and all low points to the left side

[2.1] To make a frequency decomposer machine using full advantage of this, imagine the circle graph to have mass by placing a dot at its center, the center of mass. In the process of changing the frequency (by moving the dotted vertical lines from left to right, while more are moved into the graph from the left, which rotates each of their circle graph), the center mass will wobble a bit around a bit.

But as we approach the 3 cycles/second frequency, the circle graph becomes the peach shaped graph and the center of mass got farther to the left.

Note: The center of mass isn't located on any part of the yellow line; it's the average position of all point on the whole "wound-up" graph at any given moment.

If we graph out the center of mass' x-axis value, in the circle graph, with respect (as the y-axis) to the frequency (as the x-axis), we'd get the frequency domain.

As we start increasing the frequency (from 0), the red graph would start from (0, 1), since the frequency is currently at 0 (x-axis) and the x-value the center of mass is on is 1.

When frequency approaches 3, the red graph will go up to y = 0.5 (since the center of mass, on the circle graph is at x = 0.5) before going down to its typical oscilation .

References

Page updated

Google Sites

Report abuse

Fourier transform

References

Site navigation

Main Site

Science Site

Math Site

Computer Science Site

Usual headlines

Definition structures

Headlines for references

Webcull - to store all links

Legend: