Technical Report on FFT Analyzer 11

Based on the theories up to this point, let us consider FFT and window functions. Now, we will sample a complex continuous waveform using an FFT analyzer, taking a segment of data consisting of n = 1024 or 2048 points, and consider this as one period T. It is assumed that the waveform of this period T repeats infinitely, and the FFT calculations will be performed. The values of the FFT will change depending on the position of the sampled waveform.

Let's consider this matter using a sine waveform.

Figure 7-1 is cut to exactly match the period of the sin 2π f ₀ wave, so the assumed waveform also becomes the same continuous waveform as the original sin waveform, and its spectrum only shows the component of frequency f ₀.

Next, the figure 7-2 below shows the case where the sine waveform is cut in a way that does not match its period. In this case, the hypothetical waveform will have a discontinuity at the point where it abruptly changes from f(a - 0) to f(a + 0) at period T.

The waveform cut out in such a short time is not the original waveform of sin 2π f ₀, but rather a different waveform. Therefore, it is expected that this waveform has many frequency components other than f ₀, as shown in Figure 7-2.

Let's take a look at Figure 7-2 from a different perspective.

A finite approximation spectrum can be obtained from a finite sample sequence of 2048 points. To what extent does the waveform obtained by inverse Fourier transforming this finite approximation spectrum approximate the original waveform? Regarding the truncated original waveform in Figure 7-2, the points where the discontinuous starting and ending points connect (converge) will smoothly connect like a dotted line at the midpoint of f(a - 0) and f(a + 0). This is due to the following theorem regarding the properties of integrals in the integral formulas of Fourier transform and inverse transform.

"When a function f(t) has a discontinuity at a, the value that converges when approaching the discontinuity from the left is denoted as f(a - 0), and the value that converges when approaching from the right is denoted as f(a + 0). The integral value converges to the average of these two values."

When the solid and dotted lines are redrawn as one period in a circle, it becomes like Figure 7-3, and in the Fourier inverse transform, it rotates once over the period T, resulting in the same trajectory.

It is troubling that even though it is the FFT of the same sin 2π f ₀ t sine waveform, the results differ depending on how it is truncated.

In the FFT analyzer, since the period T is fixed and we are performing FFT on signals with unknown periods, it is not always possible to cut them well as shown in Figure 7-1. Therefore, it is necessary to devise a method to ensure that the starting and ending points do not become discontinuous, as shown in Figure 7-2. One approach could be to apply a weighting function that gradually reduces the starting and ending points of the cut waveform to zero. One such function is called the Hanning window, and when this weighting function is applied to the cut sine waveform, it is assumed that a continuous waveform is repeated, and its spectrum appears mainly at frequency f ₀. By applying the Hanning window, we can minimize the influence of the cutting position.

A weight function based on such reasons is called a time window (time window function).

Next, let's look at commonly used time windows, citing Kenichi Kido's 'Introduction to Digital Signal Processing'.

We will examine the characteristics of a time window function w(t) that is actually used. This involves considering how the spectrum of the original waveform x(t) changes when the time window function w(t) is applied to it.

If we assume f(t) ↔ F(f) and w(t) ↔ W(f)

The Fourier transform X(f) of the product x(t) of f(t) and w(t) is:

(Equation 7-1)

This equation 7-1 represents that the Fourier transform of the product of two functions f(t) and w(t) corresponds to the convolution integral of their respective functions F(f) and W(f) in the frequency domain. The ideal window function W(f), which is 1 at f = 0 and 0 elsewhere, translates to a window function in the time domain that takes the value of 1 for t from -∞ to +∞, which is impractical. Since FFT involves a finite number of samples, the window function used must also be of finite length, and its spectrum should ideally have a narrow frequency spread.