Technical Report on FFT Analyzer 8

We will start by revisiting the Fourier transform and Fourier series, and examining the properties derived from them. Many equations will appear, but there is no need to memorize the equations themselves. The FFT analyzer will handle the equations internally. Here, please understand the meaning of the equations.
Furthermore, the content covered in this section and the next will form the foundation for understanding and mastering the window processing (Window function) described next.

First, as a review, let's consider Fourier series from the perspective of Fourier transforms.

The expressions representing the Fourier transform and its inverse have been presented in various ways depending on how the coefficient 1/T has been handled in the previous discussions; generally speaking,

(Equation 5-1)

is described in.

Here, if f(t) is a real function, then Equation 5-1 is;

Equation 5-2

It can be rewritten, with R(ω) and I(ω) representing the real part and the imaginary part, respectively.

Equation 5-3

can be placed.

Next, let us consider the functions f ₁ (t), f _e (t), and f ₀ (t) as shown in Figure 5-1. The second function f _e (t) in Figure 5-1 has a symmetrical relationship with f ₁ (t)/2 centered at 0;

Equation 5-4

can be established, and can also be expressed as follows.

(Equation 5-5)

Functions like this are called even functions, and cos ωt is a representative example. Also, the third function f ₀ (t) in Figure 5-1 has a point symmetry around 0;

(Equation 5-6)

is established, similarly:

Equation 5-7

Such a function is called an odd function, and sin ωt is its representative example.

Furthermore, from these two relationships;

Equation 5-8

I think it is also understandable that it can be established.

As can be predicted from the figure, the integral is;

　Equation 5-9

It shows how the integrals of even and odd functions behave. Note that the integral of an odd function becomes zero (which makes calculations easier).

Generally:

(Odd function) × (Odd function) = Even function

(Even function) × (Even function) = Even function

(Odd function) × (Even function) = Odd function

f _e (t) is an even function, so f _e (t) cos ωt is an even function, and f _e (t) sin ωt is an odd function. Therefore, this Fourier transform is given by equations 5-2 and 5-3;

Equation 5-10

Also, the inverse Fourier transform of R _e (ω) is similarly;

Equation 5-11

This means that, conversely, if the Fourier transform of the real function f _e (t) is a real function, then f _e (t) is an even function.

Similarly, for f ₀ (t) ;

(Equation 5-12)

If the Fourier transform of the real function f ₀ (t) is a purely imaginary function, then f ₀ (t) will be an odd function.

Next, considering f  (t), let the Fourier transforms of f _e (t) and f ₀ (t) be F _e (ω) and F ₀ (ω), respectively, then equation 5-3 is;

Equation 5-13

Next, from the relationship of the real functions of the above Fourier transform / the even functions of the Fourier inverse transform and the corresponding pure imaginary functions / the corresponding odd functions;

It can be understood that the above is summarized in the following equation 5-14. (The Fourier transform pairs are represented as ←→.)

Equation 5-14

This formula and the formula for finding the components of the Fourier series shown in the previous issue.

Equation 5-15

When we compare them, we can see that there is the following relationship.

Equation 5-16

In the FFT analyzer, a _n and b _n are displayed as discrete numerical sequences of the Fourier spectrum, while R (ω) and I (ω) are the mathematical expressions for a _n and b _n. R (ω) represents the real part and is an even function, while I (ω) represents the imaginary part and is an odd function. This relationship is illustrated schematically in Figure 5-2 below. As explained in the previous issue, C _n² in a _n² + b _n² = C _n² is displayed as a power spectrum after taking its logarithm, and φ（ω） is displayed as a phase spectrum.