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Technical Report: About FFT Analyzers 14

8. Example of window function analysis using an FFT analyzer

Here, we present an example of analysis using an FFT analyzer to illustrate the differences caused by window functions. Please use it as a reference. This example should help you understand that when analyzing a signal with an FFT analyzer, you need to use different window functions depending on whether the value you want to obtain is the frequency or amplitude of the spectrum, and that you must set the sampling points appropriately to reduce leakage error and perform accurate analysis.

  • Figure 8-1

Figure 8-1 above shows the results of applying window functions to a rectangular waveform after cutting it over a certain time period. From top to bottom, a rectangular window, a Hanning window, and a flat-top window are applied to the same waveform.

 

  • Figure 8-2

The top image in Figure 8-2 above shows the time-domain waveform obtained by sampling an 800 Hz sine wave at 2048 points in the 6.4 kHz range. The subsequent images show the spectral waveforms obtained by applying rectangular, Hanning, and flat-top windows to the same waveform. A time length T = 125 ms can be extracted as 100 periods of 800 Hz (0.125 × 800 = 100), so a peak appears at 800 Hz in each spectrum.

  • Figure 8-3

Figure 8-3 above shows the results of an FFT analysis performed with 256 sampling points (time length T = 15.625 ms) and a rectangular window applied. From top to bottom, the first is the time-domain waveform in the 10 kHz range, followed by its FFT, then the time-domain waveform in the 6.4 kHz range, and finally its FFT. In the 10 kHz range, the clipping is synchronized to 800 Hz, resulting in a peak at 800 Hz. However, in the 6.4 kHz range, the clipping is not synchronized properly, causing leakage errors due to discontinuities, which manifest as a broadening of the waveform's tail. Furthermore, because the resolution is 6400/100 = 64 Hz, there is no frequency of 800 Hz. Instead, the spectral peaks appear at 768 Hz (= 64 Hz × 120 lines) and 832 Hz (= 64 Hz × 121 lines), and the Y-axis value at 768 Hz is smaller than that of the 10 kHz range. This attenuation corresponds to the window attenuation in Figure 7-14.

  • Figure 8-4

Figure 8-4 above shows the results of FFT analysis performed with a 6.8 kHz range and 256 sampling points, varying the window function. The top row shows the time-domain waveform, and the subsequent rows show the spectral waveforms with rectangular, Hanning, and flat-top windows applied. Looking at the amplitude value at 800 Hz (value B in the figure) for each spectral waveform, the value when the flat-top window is applied best represents the amplitude. However, with this flat-top window, the peak frequency width is widened, making it difficult to determine that the peak frequency is 800 Hz. Compare A in Figure 8-2 with B here.