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Technical Report on FFT Analyzer 14

8. Example of Window Function Analysis Using FFT Analyzer

Here, I present an example of analysis using an FFT analyzer to illustrate the differences caused by window functions, so please use it as a reference. When analyzing signals with an FFT analyzer, it is necessary to choose different window functions depending on whether the desired value is the frequency of the spectrum or the amplitude. From this example, you should understand that to reduce leakage errors and analyze accurately, the number of sampling points must be set appropriately.

  • Figure 8-1

Figure 8-1 shows the result of applying different window functions to a square waveform over a certain duration. From top to bottom, the rectangular window, Hanning window, and flat-top window are applied to the same waveform.

 

  • Figure 8-2

The top part of Figure 8-2 shows a sine wave at 800 Hz sampled at 6.4 kHz over a time axis, with subsequent parts displaying the spectral waveforms after applying rectangular, Hanning, and flat-top windows to that waveform. The time length T = 125 ms can be cut to represent 100 cycles of the 800 Hz wave (0.125 × 800 = 100), which is why a peak appears at 800 Hz in each spectrum.

  • Figure 8-3

Figure 8-3 shows the results of FFT analysis with 256 sampling points (time duration T = 15.625 ms) using a rectangular window. From the top, the first is the time-domain waveform in the 10 kHz range, followed by its FFT, the third is the time-domain waveform in the 6.4 kHz range, and the fourth is its FFT. In the 10 kHz range, the cut-off is synchronized at 800 Hz, resulting in a peak at that frequency. However, in the 6.4 kHz range, synchronization could not be achieved effectively, leading to leakage errors due to discontinuities, which appear as spectral leakage in the waveform's tail. Additionally, due to the resolution of 6400/100 = 64 Hz, the frequency of 800 Hz does not exist; instead, spectral peaks appear at 768 Hz (= 64 Hz × 120 line) and 832 Hz (= 64 Hz × 121 line), with the Y-axis value at 768 Hz being smaller compared to the 10 kHz range. This attenuation corresponds to the window attenuation shown in the previous Figure 7-14.

  • Figure 8-4

Figure 8-4 shows the results of FFT analysis with a 6.8 kHz range and 256 sampling points, using different window functions. The top shows the time-domain waveform, and the subsequent ones display the spectral waveforms with Rectangular, Hanning, and Flat Top windows applied. Looking at the amplitude values at 800 Hz for each spectral waveform (the value labeled B in the figure), the value with the Flat Top window applied best represents that amplitude. However, with this Flat Top window, the peak frequency width is broadened, making it difficult to determine if the peak frequency is 800 Hz. Compare A in Figure 8-2 with B here.