The fundamental technique in signal processing is the Fourier transform, and this time we will discuss the Fourier series expansion of periodic functions, which forms the basis of the Fourier transform.
The Fourier series was first conceived by Joseph Fourier, a French mathematician and physicist from the Napoleonic era, as his name suggests. It is said that he devised the Fourier series in the early 19th century to solve the differential equation for the heat conduction equation in solids.
Fourier's fundamental idea was a bold one: "Any time-period function, no matter how complex, can be expressed as a sum of trigonometric functions (sine and cosine waves) at its fundamental frequency and its integer multiples of that frequency." Even Lagrange, a leading figure in the French mathematical society at the time, opposed this idea and refused to accept the paper. Today, it has become an indispensable fundamental technology not only in physics and engineering fields such as electromagnetism, optics, acoustics, communications, image processing, and quantum mechanics, but also in economics.
If x(t) is a periodic function with period T, then its fundamental frequency f0 and fundamental angular frequency ω0 are;
.................................(1)
The Fourier series expansion of x(t) is;
.................................(2)
.................................(3)
This is how it is expressed. Here, there is a question as to whether both sides of equation (2) are truly equal, but it has been proven that in the ordinary time signals we deal with, we can make it coincide with x(t) by letting n go up to infinity.
Now, next is the formula (2) The coefficient of the right-hand side a n and b n The question then becomes how to find this, but this can be easily calculated using the orthogonality of trigonometric functions.
The orthogonality of trigonometric functions is that, for any integers n and m ,;
.................................(4)
.................................(5)
This means that the following holds true.
Multiply both sides of equation (2) by cosn ω₀ t and sinn ω₀ t, and integrate each with respect to period T;
.................................(6)
.................................(7)
.................................(8)
Equation (2) (or equation (3)) is called the Fourier series expansion of the periodic function x(t), and a n and b n are called its Fourier coefficients.
Now, equation (2) includes both sine and cosine waves, and the trigonometric calculations are quite cumbersome, so it is often transformed into complex exponential notation before use.
From Euler's formula, which was explained last time;
.................................(9)
Substituting this into equation (2), we get:
....(10)
Here, although it's somewhat formal;
...............................(11)
Then, equation (10) is;
...............................(12)
It can be expressed very concisely like that.
Furthermore, since e jn ω0 t has the same orthogonal properties as trigonometric functions, by multiplying both sides of equation (12) by e-jn ω0 t and integrating with respect to period T, we can:
n=0、±1、±2、
...............................(13)
Here, equation (12) is called the complex Fourier series expansion, and c n in equation (13) is called the complex Fourier coefficient. Also, c n and c- n are complex conjugates of each other.
The meaning of the Fourier series expansion is that equation (2) can be transformed into equation (3), so it is clearly a superposition of cosine waves consisting of the fundamental angular frequency ω0 and its integer multiples, nω0, but equation (12) is not so intuitive.
In equation (12), the complex exponential function e jn ω0 t represents a vector rotating in the positive direction (counterclockwise) around the origin with angular frequency nω 0, and similarly, e-jn ω0 t represents a vector rotating in the negative direction (clockwise) around the origin with angular frequency nω 0. Furthermore, c n and c- n represent the initial vectors (time axis t = 0) of the vector rotating in the positive direction and the vector rotating in the negative direction, respectively.
Since these positive and negative vectors are symmetric with respect to the real axis, their composite vector is always real.
These are trigonometric functions that exist only on the number axis, i.e., real numbers. The right-hand side of equation (12), x(t), is its equivalent.
It represents the sum of vectors. A negative rotation vector represents a negative frequency component.
By allowing the existence of negative frequency components, real-valued time waveforms are represented using complex numbers.
This is true. This is because Euler's formula (9), which expresses sine and cosine waves that are real numbers, is used in this equation.
That should also be clear.
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Figure 1: The geometric meaning of the complex Fourier series
Fourier series expansion separates a regular time function into frequency components, that is, it transforms the time-domain representation into frequency components.
Since this involves converting to a numerical domain representation, both representations contain the same information. Here, equation (2)
Squaring both sides gives the period. T If we take the average:
（14）
As before, due to the orthogonality property, the right-hand side can be easily calculated;
................................... （15）
This is the relationship between the time axis and the frequency axis, and it is generally known as Parseval's theorem.
The physical meaning of equation (15) is that the left side is the mean square, and the right side is the square of the coefficient for each frequency component.
This is a composite sum of (which we call power). In an FFT analyzer, the right-hand side is the analyzed power
This refers to the overall value of the spectrum (total power value) (of course, the value of n is
(Although it will be finite), the overall value of the power spectrum is the mean square of the original time waveform.
This indicates that it is equal to the value (the square of the effective value).
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[Reference materials]
- "Signal Processing," co-authored by Iwao Morishita and Hidefumi Obata, Society of Instrument and Control Engineers.
- "Spectral Analysis" by Mikio Hino, published by Asakura Shoten.
(Excerpt from the email newsletter issued on March 23, 2012)
