Technical Report: About FFT Analyzers 10

Before explaining window functions in the next chapter, let's look at the convolution integral, which forms its basis, and its properties. Please refer to the general Fourier transform and inverse transform from the previous chapter. Note that this will be a repetition of similar content.

Given two time functions f1(t) and f2(t)

(Formula 6-1)

The function f(t) defined by is called the convolution integral (sum product) of f1(t) and f2(t). Equation 6-1 is symbolically expressed as Equation 6-2. Also, the commutative law holds here.

(Formula 6-2)

The Fourier transform of a convolution integral is:

(Formula 6-3)

Reverse the order of integration:

(Formula 6-4)

Now let's consider the integral inside the square brackets ( ). Let _f1 (t) ←→ _F1 (ω) and _f2 (t) ←→ _F2 (ω), and set t - τ = z. Then t = z + τ and dt = dz;

(Formula 6-5)

Therefore;

(Formula 6-6)

In other words, the convolution integral of two time-domain functions _f₁ (t) and _f₂ (t) is the product (frequency domain) of their respective Fourier transforms _F₁ (ω) and _F₂ (ω). Similarly, the product of _f₁ (t) and _f₂ (t) is the frequency domain convolution integral of their respective Fourier transforms _F₁ (ω) and _F₂ (ω). If we represent the Fourier transform and inverse transform with ←→, the above can be expressed by the following equation 6-7.

(Formula 6-7)

Let's consider the content of section 6-1 using a sample value sequence.

We take a sample value sequence f(n), find a point in it, and calculate the average of the data over a time interval b. This average is then taken as the value of a given point. We then move the point to the next point and repeat the same process to obtain a new numerical sequence _fb (n). This method is called a moving average.

As an example, we take a total of 5 points centered on a point _a0, with ±2 points, and the moving average of these points is shown in Figure 6-1. The newly created data series _A0, _A1, etc. are:

(Formula 6-8)

The above equation can be expressed as follows:

(Formula 6-9)

Next, let's consider a square wave w(t) with width b, height 1/b, and area = 1, as shown in Figure 6-2.

(Formula 6-10)

Consider the case where equation 6-10 is shifted by τ;

(Formula 6-11)

Substituting this into 1/b in equation 6-9, the moving average is:

(Formula 6-12)

This is expressed as follows, and it becomes the formula for the convolution integral.

Considering Figure 6-1, it can be likened to a window, as shown in Figure 6-2, where nothing is visible outside the window. In other words, everything visible through the window is averaged to the center value of the window, and the data moves along it like a train window. This W(t) is called a window function because it allows us to view the sequence of data in this way.
Furthermore, the window in Figure 6-2 is like a flat glass pane, with no distortion in the data, and is uniformly averaged. Such a window is called a rectangular window. Other window functions include the Hanning window and the flat-top window, but these all average by assigning weights to the data. The Fourier transform of the rectangular window function W(t) in Figure 6-2 is:

(Formula 6-13)

This function is as shown in Figure 6-3, and in an FFT analyzer, b = T.

Next, let's consider the moving average in the frequency domain. If we let G (f) be the spectrum of a certain waveform, and take a moving average while multiplying it by the frequency function W (f), we can derive equation 6-14 in the same way as equation 6-12 above.

(Formula 6-14)

In this equation, W (f) is called the spectral window.

Averaging is necessary to preserve the power of the original waveform, and when calculating the average at a certain point, it is necessary to maintain symmetry.

Next, let's consider an example of W (f).

If we consider the same thing as in Figure 6-2 in the frequency domain, this means that we cut out W(f) over a certain frequency range and take a moving average of W(f) with a certain weighting.

(Formula 6-15)

We will now calculate the variance of this equation. The variance ^σ² is the square of the standard deviation and is expressed by the following formula:

(Formula 6-16)

A rectangular window is represented by the following formula:

(Formula 6-17)

The shape of this function is similar to that shown in Figure 6-3, with a large weight at the center and a smaller weight as you move away from the center.

Both the rectangular pulse in (a) and the rectangular window in (b) act as bandpass filters in the sense that they allow frequency components within a certain width to pass through. In the case of a rectangular pulse, the bandwidth b (-b/2 to b/2) is clear, but in this rectangular window, it is not clear how far the bandpass filter bandwidth extends. Therefore, one approach is to calculate the variance of each window function and use the width of a rectangular pulse with an equal variance as the bandwidth of that window (called the equivalent signal bandwidth).

Calculating the distribution of the rectangular window:

(Formula 6-18)

The width of a rectangular window with equal dispersion to a rectangular pulse is given by equations 6-16 and 6-18;

(Formula 6-19)

This is illustrated by the blue line in Figure 6-5.

Generally speaking;

(Formula 6-20)

It will be.

Furthermore, if we obtain the function W(f) by inverse Fourier transforming equation 6-17, we get:

(Formula 6-21)

This results in a similar rectangle to the one in Figure 6-2, where u = b/2 is substituted. Equations 6-17 and 6-21 are shown in Figures 6-5 and 6-6.

Multiplying a time waveform f(t) by a window function w(t) results in the product of f(t) and w(t) in the time domain, and the convolution integral of F(f) * W(f) in the frequency domain, or in other words, a moving average of F(f) weighted by W(f). The square of the window function in equation 6-13 and its logarithm are shown in Figure 6-7. In the figure, the range from -1/b to 1/b is called the main lobe, and the higher frequency portion is called the side lobe, and their respective values are indicators that represent the characteristics of the window function.