Technical Report on FFT Analyzer 10

Before explaining the window function in the next chapter, let's take a look at the convolution integral and its properties, which form the basis for it. Please refer to the previous discussion on general Fourier transforms and inverse transforms. Also, please understand that there may be some repetition of similar content.

Given two time functions f ₁ (t) and f ₂ (t)

(Equation 6-1)

A function f defined by  (t) is called the convolution integral of f ₁ (t) and f ₂ (t). It is expressed symbolically as shown in equation 6-2. Furthermore, the commutative law holds here.

Equation 6-2

The Fourier transform of the convolution integral is:

Equation 6-3

Change the order of integration:

Equation 6-4

Here, we will consider the integral within the parentheses ( ). Let f ₁ (t) ←→ F ₁ (ω) and f ₂ (t) ←→ F ₂ (ω), and by setting t - τ = z, we have t = z + τ, dt = dz;

Equation 6-5

therefore

(Equation 6-6)

That is, the convolution integral of two functions f ₁ (t) and f ₂ (t) in the time domain becomes the product of their Fourier transforms F ₁ (ω) and F ₂ (ω) in the frequency domain. Similarly, the product of f ₁ (t) and f ₂ (t) becomes the convolution integral of their Fourier transforms F ₁ (ω) and F ₂ (ω) in the frequency domain. If we denote the Fourier transform and inverse transform by ←→, the above can be expressed in the following equation 6-7.

(Equation 6-7)

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

Consider a sample value series f(n) centered around a certain point, and calculate the average of the data within a time width b, using that average value as the value at that point. We then move to the next point in succession and take the average in the same way to form a new numerical series f _b (n). This method is called a moving average.

Taking a total of 5 points centered around point a ₀ with ±2 points, the moving average is shown in Figure 6-1. The newly created data series A ₀, A ₁...... is as follows;

(Equation 6-8)

The above formula can be summarized as follows.

Equation 6-9

Next, let us consider a rectangular wave w(t) with a width of b, a height of 1/b, and an area of 1, as shown in Figure 6-2.

(Equation 6-10)

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

(Equation 6-11)

Substituting this into 1/b of equation 6-9, the moving average is;

(Equation 6-12)

is expressed as a convolution integral formula.

Considering the previous figure 6-1, it has a window like that in figure 6-2, and nothing else is visible, meaning that everything visible through the window is averaged to represent the value at the center of the window, just like how a train window moves continuously. In this sense of observing the arrangement of data, this W(t) is called a window function.
Furthermore, the window in figure 6-2 has no distortion in the data like a flat glass and is uniformly averaged. Such a window is called a rectangular window. There are other window functions such as the Hanning window and flat-top window, but these all apply weighting to the data to perform averaging. The Fourier transform of the rectangular window function W(t) in figure 6-2 is:

(Equation 6-13)

This function is as shown in Figure 6-3, and in the FFT analyzer, it is set that b = T.

Next, we consider the moving average in the frequency domain. Let the spectrum of a certain waveform be G(f), and by taking the moving average while multiplying it by the frequency function W(f), we can derive the following equation 6-14, similar to the previous equation 6-12.

(Equation 6-14)

The term W (f) in this formula is called the spectral window.

By performing averaging, it is necessary to ensure that the power of the original waveform remains unchanged, and when seeking the average at a certain point, it is essential to maintain symmetry.

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

Considering the same thing as in Figure 6-2 in the frequency domain means taking a slice of W(f) over a certain frequency range and calculating a moving average of W(f) with a constant weighting.

(Equation 6-15)

We will calculate the variance of this formula. The variance σ ² is expressed as the square of the standard deviation in the following equation.

(Equation 6-16)

The rectangular window is represented by the following formula.

Equation 6-17

The shape of this function is similar to that of Figure 6-3, being large at the center and decreasing in weighting as one moves 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 bandwidth. The bandwidth for the rectangular pulse is clearly defined as b (-b/2 to b/2), but it is not clear where the bandwidth of the bandpass filter is for the rectangular window. Therefore, one method to determine this is to calculate the variance of each window function and use a rectangular pulse with the same variance to define the bandwidth of the window (referred to as the equivalent signal bandwidth).

When calculating the variance of a rectangular window;

(Equation 6-18)

The width of a rectangular window that has the same variance as a rectangular pulse is given by equations 6-16 and 6-18.

(Equation 6-19)

This situation is indicated by a blue line in the next figure 6-5.

Generally speaking,

Equation 6-20

will become.

Furthermore, to find the function W(f) obtained by performing the inverse Fourier transform of equation 6-17;

Equation 6-21

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

Multiplying the time waveform f(t) by the window function w(t) results in the product f(t)·w(t) in the time domain, and in the frequency domain, it corresponds to the convolution integral F(f) * W(f), which can be described as a weighted moving average of F(f) by W(f). The squared window function from Equation 6-13 and its logarithm are shown in the following Figure 6-7. In the figure, the range from -1/b to 1/b is referred to as the main lobe, while the higher frequency portions are called side lobes, and the values of each serve as indicators of the characteristics of the window function.