With the new year upon us, I'd like to start studying "Frequency Analysis from the Basics" again with renewed enthusiasm.
This time, we'll discuss the mathematical foundations needed to understand the Fourier transform, a fundamental and central technique in frequency analysis. The mathematics involved is roughly equivalent to high school level math plus a little extra.
As for basic mathematical knowledge:
- calculus
- Trigonometric functions (circular functions) and exponential functions
- Complex numbers (Cartesian coordinates, polar coordinates, complex plane)
- Complex exponential function (Euler's formula)
These will be necessary.
First, let me explain the units of angle.
A full rotation of a circle is 360 degrees. While expressing angles in degrees (deg) is called the degree system, there is also a method that uses pi (π) as the base value. This is called the radian system, and it is a common way of expressing angles in mathematics. Specifically, 360 degrees is 2π in radian measure, and its unit is radians (rad), although this unit is usually omitted.
Conversion between degrees and radians
1 degree = π/180 = 0.01745...
1 (rad) = 180/π = 57.2957...(degrees)
Why is radian measure used in mathematics? One reason is that it simplifies the notation of circumferences and areas of circles and sectors, but mathematically, it simplifies the calculus of trigonometric functions.
Let x be in radians;
.................................（1）
................................（2）
................................（3）
This makes it very simple. If we use the frequency method, a coefficient of π/180 is added, which makes it more complicated.
Next, I will talk about trigonometric functions (circular functions) that have been extended to general angles. Trigonometric functions such as sine and cosine are defined as ratios of the sides of a right triangle, but in mathematics, the x-coordinate of a point P on a unit circle of radius 1 is defined as the cosine and the y-coordinate as the point P rotates along that circle.
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Figure 1: Unit Circle and General Trigonometric Functions
In Figure 1, if we let θ be the angle between the vector OP formed by the origin 0 and point P and the x-axis;
.................................（4）
This is how it is defined. When a point P (or vector OP) rotates on a circle, if we define counterclockwise as a positive angle and clockwise as a negative angle, and draw a graph, we obtain the cosine and sine functions as shown in Figure 1. Also, since one revolution around the circle is 2π (= 360°), we can see that the cosine and sine functions are periodic functions with a period of 2π. Trigonometric functions are sometimes called circular functions by this definition.
Also, the radius of the circle (length of vector OP) is 1, so obviously;
.................................（5）
There is a relationship between them.
Next, let's consider Napier's number e, which is the base of the natural logarithm.
Napier's number e is defined as follows:
.................................（6）
If we define an exponential function using Napier's number e, and a natural logarithm logx with this value as base, the differentiation of these functions becomes incredibly simple (I'll omit the detailed explanation). That is:
.................................（7）
.................................（8）
Furthermore, if we expand the exponential function e x into a power series:
................................（9）
Substituting x = 1 into equation (9), we get:
...............................（10）
Therefore, calculating e from this equation gives us equation (6).
This is a bit of a digression, but around 2004, an advertisement billboard like the one below appeared on the streets of Silicon Valley. It was a job advertisement from the American IT company Google, and apparently it meant that you should access this URL and apply.
{ first 10-digit prime found in consecutive digits of e }.com
(The first 10-digit prime number found with consecutive digits of 'e'.)
Finding a 10-digit number that fits the above criteria from the value of the irrational number e would be difficult without programming.
Next, let's talk about complex numbers.
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Figure 2: Complex plane (Gaussian plane)
We define a number whose square is -1. This is called the imaginary unit i.
...............................（11）
A complex number z can be expressed using any real numbers a and b as follows:
z=a+ib ...............................（12）
It is defined as follows: On the complex plane in Figure 2, the absolute value r is:
...............................（13）
The argument θ is:
..............................（14）
Using the absolute value r and the argument θ, the polar form of z is:
z=r(cos θ+isin θ)
...............................（15）
Any complex number z can be expressed in two ways: in Cartesian coordinate form as shown in equation (12) and in polar coordinate form as shown in equation (15).
[Specific example]
As you can see, complex numbers are easier to understand when considered geometrically on the complex plane (Gaussian plane).
Now, let's consider the geometric meaning of the imaginary unit i. If we express i in polar form, it becomes:
...............................（16）
Multiplying equation (15) and equation (16) gives:
Thus, "multiplying a complex number z by i" is equivalent to advancing the phase angle by 90 degrees (= π/2) or rotating it by 90 degrees.
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Figure 3 When z is multiplied by i
Finally, let's talk about Euler's formula. Using the Napier's number e and the imaginary unit i that we've discussed so far:
..............................（17）
This equation (17) is called Euler's formula. Further transformation of this equation allows us to:
...............................（18）
This equation (18) is sometimes called Euler's formula.
Now, using equation (17), we can transform equation (15):
...............................（19）
Thus, any complex number can be expressed in a very simple form.
Thus, any complex number can be expressed in a very simple form.
The expression in equation (19) is indispensable for AC electrical circuits and wave equations such as sound and vibration, and is called the complex exponential function (complex sine function).
For a detailed explanation of complex numbers and Euler's formula, please refer to the previous measurement column (reference).
Finally, substituting x = π into equation (17) gives:
This yields the beautiful relationship between the constants (π, e, i) that we have discussed.
The following link will take you to our company's website.
[Reference materials]
Fundamentals of Digital Measurement - Part 16: "The Story of Complex Numbers"
Fundamentals of Digital Measurement - Part 17: "The Story of Complex Numbers (Part 2)"
(Excerpt from the email newsletter issued on January 26, 2012)
