This time we'll be looking at window functions. This is a continuation of our previous discussion on bandpass filters.
Please refer to Figure 7-6 from the previous section.
Furthermore, the above is mentioned in Chapter 7, "FFT and Time Windows," of Ono Sokki Technical Report, "About FFT Analyzers."
Please refer to the reference materials cited from the bibliography.
7-5. Window Functions
The time length T of the data processed by the FFT is determined by the frequency range and the number of sample points.
The various frequency components included in the observed waveform are unrelated to this T, so when the waveform is cut out...
The beginning and end will not be continuous. The data was sampled and cut into 2048 points.
The "ta" is just like the scenery framed by a car window. In FFT, the framed waveform is periodic.
We assume this is repeated. When extracting 2048 points, the first of the 2048 sampling points
By weighting the parts so that the first and last parts gradually become zero, the discontinuity is eliminated.
By creating a smooth connection, leakage errors can be minimized.
Various weighting methods have been proposed for this purpose, and they are known as window functions.
This image is shown in Figure 7-7. This weighting effect is the same as the bandpass filter we discussed in the previous issue.
In terms of form, or terminology, it appears in the form of the main lobe and side lobes of a window function.
Reference materials
The equivalent signal bandwidth is the calculated frequency bandwidth obtained when performing an FFT, and represents the frequency separation.
In the previous issue, we explained that it is a series of bandpass filters.
The width is represented by the equivalent signal bandwidth and corresponds to the power of the bandpass filter.
Furthermore, fo (frequency resolution) is given by fo = 1/T, and the X-axis is represented by frequency discrete points n fo (n = 0 to N).
I previously explained that in a rectangular window, it becomes 1/T, the same as fo.
This time, we will focus on frequency separation and bandwidth, and for the sake of explanation, we will refer to "equivalent signal bandwidth" as "bandwidth".
We will explain this using the notation "Δf".
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Figure 7-7 Window function
DS-0221 General-purpose FFT analysis software offers the following window functions: [Rectangular window], [H
[Rising Window] and [Flat Top Window] are available.
(1) Rectangular window
The rectangular window will have flat weights (as in the sample data).
The window width Δf is the same as the frequency resolution fo, and is narrower than other window functions, improving frequency separation.
This makes a good window function. It's suitable for impact waveforms such as those used in impact tests, which start and end at zero.
Alternatively, in tests using the signal output built into the FFT analyzer, the length of the FFT time T is equal to the signal period.
Because they are synchronized (the period is an integer multiple), they are used in cases like this.
If sampling cannot be performed for a time length synchronized with the period, the side lobes will be large as shown in Figure 7-10.
This will happen. In the example of 502Hz in Figure 7-10, if the analysis signal contains 502Hz and other frequency components...
Additionally, if other frequency components are small, they may be buried within these side lobes.
(2) Hanning window
The Hanning window assigns weights to the sampled data in the form shown in Figure 7-8.
This weighting will reduce the power. Our FFT analyzer
In RIZER, this attenuation is normalized and displayed so that the amplitude of the sine wave can be read directly.
The normalized main lobe becomes the bandpass filter shown in Figure 7-6 from the previous issue.
Furthermore, by normalizing from the reference materials, the bandwidth (power) becomes 3/2 times Δf (total power)
This is 3/2 times the size of a rectangular window. It's a bit confusing, but...
Think of bandwidth as a difference in the shape of the bandpass filter. (Hanning wind)
In U, the side lobes are smaller compared to the rectangular window.
This window function is commonly used for signals and general continuous waveforms.
In the case of the hammer's striking waveform, when the impact waveform reaches the initial position, the impact waveform becomes a hammer.
Because of the weighting of the ndou, it is attenuated, so the rectangular shape which is unaffected by the weighting
A window will be used. The Hanning window is the next flat top window.
A window function that has properties intermediate between an indwelling window and a rectangular window.
Yes, I can.
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Figure 7-8-1 Hanning window function and signal
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Figure 7-8-2 Signal with Hanning window function applied
(3) Flat-top windows
The flat-top window is a window proposed to read amplitude more accurately.
The area where the beginning and end of the sample are zeroed out is wider than the Hanning window.
This acts as a weight, and the bandwidth becomes approximately 3.67 times Δf (Figure 7-9). Frequency separation is
Other windows are better, but the signal is at frequencies between the frequency resolution limits.
The window displays a smaller amplitude than the actual amplitude, but in the flat-top window...
Because it has a wider bandwidth, it can be read more accurately. This was suggested by Professor Kido, the author of the reference.
This is a window function.
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Figure 7-9-1 Flat-top window and signal
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Figure 7-9-2 Signal with flat-top window function applied
The shape of the bandpass filter changes depending on the window type, so frequency separation
This results in differences in spectral values. The degree of this difference varies depending on the frequency distribution of the input signal.
However, to read the amplitude of the peak value more accurately, it is measured using a flat-top.
For waveforms that fit within a time length T, such as impulse waveforms, a rectangular window is used.
For waveforms that don't fit this description, we use the Hanning window. Also, when focusing on amplitude, we use the flat window.
Depending on the measurement purpose, you can use different window types for analysis, such as using the top-of-the-clock window.
Generally, using the Hanning window allows for reliable measurements in terms of both frequency separation and amplitude.
You can.
(4) Main robe and side robes
Figure 7-10 shows the signal and the sample count set to 1 kHz, with a coarse frequency resolution of 256.
Input a sine wave with an amplitude of 1Vrms (0dBVr) and frequencies of 500Hz and 502Hz, and then apply a window function.
I've lined up the results of performing FFT with different numbers.
In the case of 500Hz, a sample can be taken with a time length T that matches this period, so the side lobes
Although it is small, 502Hz cannot be sampled in the same way, so the side lobes are large.
It has become so. Also, the reading of the amplitude value at this time differs depending on the window function.
They will be.
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Figure 7-10 Window Features
Left column: 500Hz sine signal Right column: 502Hz sine signal
From top to bottom: Time waveform, Rectangular window, Hanning window, Flat-top window
<Key Points>
In an FFT analyzer, an anti-aliasing filter is applied during sampling.
When analyzing Wahr spectra, we take into account that a window function is applied during processing.
Generally, a rectangular window is used for impulse waveforms, and a Hanning window is used for continuous signals.
A flat-top window is used to accurately measure the amplitude level of the peak value.
(Excerpt from the email newsletter issued on August 22, 2007)
