Frequency Analysis from the Basics (3) - "Fourier Series Expansion (Part 2)"

Following on from last time, I will continue to discuss Fourier series expansions.

Let's rewrite the definition of the Fourier series expansion. First, let T be the period of the periodic function x(t);

................................(1)

Here, ω0 is the fundamental angular frequency and f0 is the fundamental frequency.

(1) Series expansion using standard trigonometric functions

.......(2)

.......(3)

.......(4)

.......(5)

.......(6)

(2) Complex Fourier series expansion

.......(7)

.......(8)

The relationship between an and bn above is;

.......(9)

Let's consider the meaning of these equations.

Equation (2) represents Fourier's fundamental idea that any complex period-time function can be expressed as a sum of trigonometric functions (sine and cosine waves) at its fundamental frequency and its integer multiples of that frequency. Equation (3) is a transformation of equation (2) using the synthesis of simple harmonic motion, expressed as a sum of cosine waves (frequency , amplitude , phase ). The Fourier spectrum displayed in an FFT analyzer usually shows the corresponding numerical values.

So, what is the physical meaning of equation (7)?

The complex exponential function (complex sine wave) is represented by e ^jnω0t, which is a vector that rotates counterclockwise on the unit circle with time at an angular frequency nω0, and e^-jnω0t, which is a vector that rotates clockwise on the unit circle with time at an angular frequency nω0. As shown in Figure 1, the two vectors are always symmetric with respect to the real axis (they are complex conjugates), so their sum is a real number.

Next, let's consider the complex coefficients c and n in equation (7);

.....(10)

Since we can set it as follows, Xn represents the amplitude of the rotating vector, and represents the initial phase of the rotating vector (the phase at t = 0 in the time signal, the starting position of the rotating vector).
Here, if we add the counterclockwise and clockwise vectors at angular frequency , we get:

..(11)

This is equivalent to equation (3) of the Fourier series expansion of a trigonometric function.

Rewriting equation (7):

...............................(12)

Equation (12) shows that the amplitude is

Initial phase is πn, angular frequency nω0 (period)

This means that the sum of vectors rotating counterclockwise (n > 0) and clockwise (n < 0) is calculated, and then the sum of these vectors for each angular frequency is taken.
For example, the vector n=1, Xn=10, and -π/30 (-60 degrees) is shown in Figure 2.

Figure 2 shows the vector in the initial phase state at time t = 0.

【Note】
The sign of the angular frequency ω indicates the direction of rotation: positive (plus) means counterclockwise, and negative (minus) means clockwise. Regarding the phase sign, relative to the counterclockwise rotation direction, positive (plus) means the phase is leading, and negative (minus) means the phase is lagging. As shown in Figure 2, the vector obtained by combining both positive and negative vectors always lies on the real axis, and the locus of that point is...

This results in a cosine waveform that is delayed by a certain amount.

Since Fourier series expansion expresses a time signal in the frequency domain, that is, it obtains the frequency spectrum, in equation (8), cn is called the complex Fourier coefficient (complex Fourier spectrum), Xn is called the amplitude spectrum, and πn is called the phase spectrum.

Now, let's calculate the Fourier series expansion of a periodic function.

[Specific Example 1]

A sawtooth wave with period T, as shown in the figure above;

...............................(13)

We will expand this into a series.

Since the sawtooth wave in Figure 3 is clearly an odd function, we only need to calculate the sine term of the Fourier coefficient;

Therefore, the Fourier series expansion of the sawtooth wave equation (13) is:

.(14)

It will be.

Now, substituting T/4 into both sides of equation (13), the left side becomes 1/2;

from now;

...............................(15)

And we can derive Leibniz's formula.

[Specific Example 2]

A parabola as shown in the diagram above;

...............................(16)

We will expand this into a series.

In this example, since it is an even function, we only need to calculate the Fourier coefficient of the cosine term, an;

Also;

Therefore, the Fourier series expansion of the parabola (16) is:

..(17)

By substituting t = 0 into both sides of equation (17);

...............................(18)

The following relationship is obtained. Here, if we let the right-hand side of equation (18) be S;

The value of S is ^π² /12, so ultimately;

...............................(19)

It will be.

The problem of finding the sum of the squares of the reciprocals of natural numbers in equation (19) has long been called the "Basel problem."
It was solved by Euler in the 18th century. This problem was generalized in the 19th century.
This is Riemann's zeta function.

That is, if s is a complex number;

...............................(20)

From equation (19), the value when s = 2 is;

It will be.

Finally, here is a summary of the Fourier series expansion (Figure 5).

1. Applicable to continuous-time signals with period T.
2. The frequency spectrum consists of infinitely many discrete frequencies, f₀ (= 1/T), and integer multiples thereof.
3. In a complex Fourier series expansion, there are positive and negative frequencies, and their values are related by a complex conjugate relationship. That is, Xn = X - n and φn = - -n.

------------------------------------------

[Reference materials]

1. "Signal Processing," co-authored by Iwao Morishita and Hidefumi Obata, Society of Instrument and Control Engineers.
2. "Spectral Analysis" by Mikio Hino, published by Asakura Shoten.

(Excerpt from the email newsletter issued on May 24, 2012)