Technical Report: About FFT Analyzers 11

Based on the theory so far, let's consider FFT and window functions. We now sample a complex continuous waveform using an FFT analyzer, and extract a portion of the data—n = 1024 or 2048 points—and consider this region as having a period T. Assuming this waveform of period T repeats infinitely, the FFT calculation is performed. The value of the FFT will change depending on the position of the waveform being extracted.

Let's consider this using a sine wave.

Since Figure 7-1 is cropped to precisely match the period of the sin 2π f ₀ wave, the assumed waveform is also a continuous waveform, just like the original sine wave, and only the component with frequency f ₀ appears in its spectrum.

Next, Figure 7-2 below shows the case where the sine wave is cut off without matching its period. In this case, the assumed waveform has a discontinuity at period T where it abruptly changes from f (a - 0) to f(a + 0).

The waveform, thus clipped over a short time, is no longer the original waveform of sin 2π f ₀, but 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 look at Figure 7-2 from a different perspective.

A finite approximate spectrum can be obtained from a finite sample sequence of n = 2048 points. To what extent does the waveform obtained by inverse Fourier transforming this finite approximate spectrum approximate the original waveform? Looking at the original waveform that has been cut out in Figure 7-2, the discontinuous start and end points connect (converge) at the midpoint of f (a - 0) and f (a + 0), where they connect smoothly as shown by the dotted line. This is due to the following theorem, which is a property of integration for the integral formulas of the Fourier transform and inverse transform.

"When a function f(t) has a discontinuity at a, if we approach the discontinuity from the left and the value that converges to it is f(a - 0), and approaching from the right and the value that converges to it is f(a + 0), then the integral value converges to its mean value."

If we redraw the solid and dotted lines with each period representing a circle, we get Figure 7-3. The inverse Fourier transform completes one rotation with period T, and the resulting trajectory is the same.

It's problematic when the FFT results differ depending on how the same sin 2π f ₀ t waveform is cropped, even though it's the same waveform.

In an FFT analyzer, the period T is fixed, and since we are performing an FFT on a signal with an unknown period, it is always impossible to cut out the signal perfectly, as shown in Figure 7-1. Therefore, a method is needed to prevent discontinuities between the start and end points, as shown in Figure 7-2. One such method is to "apply a weighting function that gradually reduces the start and end points of the cut waveform until they become zero." One such function is called the Hanning window. When this weighting function is applied to the cut sine waveform, a repeating continuous waveform is assumed, as shown in Figure 7-4, and its spectrum appears mainly at frequency _f0. By applying the Hanning window, the influence of the cutting position can be reduced.

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

Next, let's look at some commonly used time windows, quoting from Kenichi Kido's "Introduction to Digital Signal Processing."

Let's look at the characteristics of a time window function w(t) that is actually used as an example. To do this, we consider how the spectrum of the original x(t) changes when a certain time window function w(t) is applied to the time-domain waveform x(t).

Let 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:

(Formula 7-1)

Equation 7-1 shows that the Fourier transform of the product of two functions f(t) and w(t) is the convolution integral of each function F(f) * W(f) in the frequency domain. An ideal window function in the frequency domain where W(f) is 1 when f = 0 and 0 otherwise becomes a window function in the time domain where t ranges from -∞ to +∞, which is not practical. Since the FFT uses a finite number of samples, the window function used should also be finite in length, and its spectrum should ideally have a narrow frequency spread.