This measurement column addresses frequently asked questions received by our customer support center and provides answers to those questions.
In the 41st installment, we introduced the frequency resolution and time window (Hanning window) of the FFT. This time, we will discuss...
This section explains how the spectrum changes depending on the type of time window.
Please also refer to the back issues listed below.
Frequently Asked Questions about Measurement - Part 41: "FFT and Frequency Resolution"
Frequency Analysis from the Basics (14) - "DFT (FFT) and Time Windows"
4. Units of Noise Measurement
In FFT analysis (Fourier analysis), the time-domain waveform is cut to a certain length and then subjected to a Fourier transform.
In most cases, the time waveform becomes discontinuous at the start and end points of the clipped time waveform, resulting in errors in the frequency spectrum. The phenomenon is that the peak of the input frequency becomes blunted, and a gentle spread (tail) appears on both sides. The peak of the spectrum becomes smaller compared to the true value, and that amount of power leaks out into the spread on both sides. The spread is called the main lobe, and the tail is called the side lobe.
To minimize the errors described above, multiplying the extracted time waveform by a weighting function that causes both ends to converge to zero is called "applying a time window," and such a weighting function is called a "time window" or "time window function."
This time, we will introduce the characteristics of rectangular, Hanning, Humming, Blackman-Harris, and flat-top windows.
The time waveforms to be analyzed using FFT are sine waves with frequencies of 1000 Hz, 1002.5 Hz, and 1005 Hz.
These waveforms are analyzed using FFT analysis with a sample frequency of 20480 Hz (frequency range of 8000 Hz), with 2048 points per frame. The frequency step (resolution) is 10 Hz.
You can see that when the frequency of the time waveform shifts from the center of the frequency resolution, the shape of the spectrum changes, and the peak value becomes smaller. However, the overall power does not decrease, so the overall value does not decrease.
Figure 1 shows the processing results for a 1000 Hz sine wave. 1000 Hz perfectly matches the center frequency of the FFT analysis. The value at 1000 Hz is set to 0.0 dB.
From top to bottom, these are the results for rectangular, Hanning, Humming, Blackman-Harris, and flat-top time windows.
The graph on the right shows a magnified view of the frequency axis.
In the rectangular (unweighted) configuration, there is a single peak. The power is not spread to 990 Hz and 1010 Hz.
Hanning and Humming are almost identical, while Blackman Harris's version is more expansive compared to Hanning and Humming.
In the case of a flat top, it is significantly wider horizontally.
Since it perfectly matches the center frequency, there is no decrease in the peak value.
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Figure 1
Figure 2 shows the processing result for a 1002.5 Hz sine wave. The result is shifted from the center frequency of the FFT analysis. Since the frequency resolution is 10 Hz, the frequency is shifted by 1/4 of the resolution.
The beginning and end of the time waveform will be discontinuous.
From top to bottom, these are the results for rectangular, Hanning, Humming, Blackman-Harris, and flat-top time windows.
We can see that the peak value, which should originally be 0 dB, has decreased. In the rectangular shape, the widening of the tail is particularly pronounced. This causes other frequency components to be buried and hidden.
In Hanning, the spectral spread is slightly wider, but the level of the tail is lower.
In Humming, the spectral spread is narrow, but the level of the tail is high.
In Blackman Harris, he's somewhere between Hanning and Humming.
In flat-top patterns, the lateral spread is significant, but the peak value has hardly decreased.
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Figure 2
Figure 4 shows the processing result for a 1005 Hz sine wave. The frequency is shifted by half a Hz, which is the resolution of 10 Hz. Depending on the timing of waveform extraction, the discontinuity becomes clear, as shown in Figure 3.
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Figure 3
The decrease in the peak value is even greater compared to 1002.5 Hz.
From top to bottom, these are the results for rectangular, Hanning, Humming, Blackman-Harris, and flat-top time windows.
While the decrease in peak values is noticeable, the decrease is less pronounced in flat-top structures.
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Figure 4
summary
Rectangular:
Since no weighting is applied, this window is used for single-shot signals where the beginning and end of the original waveform are zero. It is an essential window for impact tests such as hammering tests and connector mating sound tests.
Hanning:
Compared to Humming and Blackman Harris, it has a better balance of flare and hem size.
Suitable for general noise and vibration measurements.
Flat top:
This method is used when you want to accurately observe peaks even if the signal frequency is off from the center frequency of the FFT resolution. It's useful for signal calibration with single-frequency signals, measuring harmonic distortion, and other situations where you need to accurately determine peak values.
(Excerpt from the email newsletter issued on December 16, 2020)
