Technical Report: About FFT Analyzers 8

Let's return to the Fourier transform and Fourier series, and begin by examining the properties derived from them. Many mathematical formulas will appear, but you don't need to memorize them. The FFT analyzer will process them internally. Here, just understand the meaning of the formulas.
The content covered in this section and the next section forms the foundation for understanding and mastering window processing (Window functions), which will be discussed next.

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

The formulas for the Fourier transform and its inverse transform have been expressed in various ways depending on how the coefficient 1/T is treated, but generally:

(Formula 5-1)

It is described as follows:

Here, if f(t) is a real function, then equation 5-1 becomes:

(Formula 5-2)

It can be rewritten as follows, and if R (ω) and I (ω) are the real and imaginary parts, respectively:

(Formula 5-3)

It can be placed like this.

Next, let's consider the functions _f₁ (t), _f₀ (t), and _f₀ (t) as shown in Figure 5-1 below. The second function _f₀ (t) in Figure 5-1 has a symmetric relationship with _f₁ (t)/2 around 0;

(Formula 5-4)

The following holds true, and can also be expressed as follows:

(Formula 5-5)

Such functions are called even functions, and cos ωt is a typical example. Furthermore, the third function f ₀ (t) in Figure 5-1 is point-symmetric about 0;

(Formula 5-6)

The following holds true, and similarly:

(Formula 5-7)

Such functions are called odd functions, and sin ωt is a typical example.

Furthermore, from this relationship between the two:

(Formula 5-8)

I think you can understand why that is true.

As can be expected from the figure, the integral is:

　(Formula 5-9)

This shows how the integrals of even and odd functions work. Note that the integral of an odd function is 0 (which makes it easier to calculate).

in general:

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

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

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

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

(Formula 5-10)

And similarly, the inverse Fourier transform of _Re (ω) is:

(Formula 5-11)

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

Similarly, for f ₀ (t):

(Formula 5-12)

Therefore, if the Fourier transform of a real function _f₀ (t) is a purely imaginary function, then _f₀ (t) is an odd function.

Next, considering f(t), if we 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 becomes:

(Formula 5-13)

Therefore, from the relationship between the real function of the Fourier transform and the even function of the inverse Fourier transform, and the same pure imaginary function and the same odd function,

This is what we can see. The above is summarized in the following equation 5-14. (Fourier transform pairs are represented by ←→)

(Formula 5-14)

This formula, along with the formula for finding the Fourier series and its components shown in the previous issue;

(Formula 5-15)

When we compare them, we can see that they have the following relationship.

(Formula 5-16)

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