Technical Report: About FFT Analyzers 13

The Hanning window is a practical window function that is so frequently used that it's practically synonymous with FFT.

The Hanning window is given by the following equation:

(Formula 7-8)

This Fourier transform is:

If we express W (f) using DFT, then:

When n / T (n = 0): 0.5

When n = ±1: 0.25

n = Other cases: 0

Power attenuation rate = -4.26 dB

Equation 7-9 is shown in Figure 7-10. As you can see from the figure, the main part of the spectrum is broadened to about twice the size of the rectangular window, but the secondary part is rapidly attenuated.

The Hanning window has the advantage of providing clearer separation of _f0 than a square wave window due to less leakage on the high-frequency side. To obtain the same frequency resolution as the rectangular window mentioned earlier, the window length must be doubled to 2T. To set the window length to 2T in an FFT analyzer, you would need to either lower the frequency range or use the data length change function.

With a rectangular window, the extracted waveform does not have amplitude distortion, but when a Hanning window is applied, the amplitude of the waveform is distorted, thus reducing its power.

The normalized power, after scaling by 1/0.5T so that the amplitude of _f0 can be read directly, is:

(Formula 7-10)

b = 3/2 T, meaning the Hanning window is 3/2 times wider than the filter width of the rectangular window.

The power of w(t) before normalization is:

(Formula 7-11)

This results in a power reduction of -4.26 dB compared to the rectangular window. This value is indicated as the power reduction percentage after each window formula.

When calculating the overall value of the spectrum, this would result in a value that is 3/2 times larger than the rectangular window. Therefore, FFT analyzers perform a window correction during the overall calculation and display the result accordingly.

The Hamming window is a window used to further reduce the spectral size of the secondary part adjacent to the primary part of the Hanning window. In the Hamming window, the secondary part becomes less than 1/100th the size of the primary part.

The Hamming window and its spectrum are given by the following equations:

(Formula 7-12)

(Formula 7-13)

In the DFT of W _m (f):

n/T when n = 0: 0.54

When n = ±1: 0.23

n = Other cases: 0

Power reduction rate: −4.0 dB

Equation 7-13 is shown in Figure 7-11.

The spectra of rectangular, Hanning, and Hamming windows do not have flat tops. Therefore, for waveforms with line spectra such as cosine waves, sine waves, and square pulse waves, when DFT is performed, the magnitude of the spectrum corresponding to each frequency will be represented as a smaller value than the actual value unless the time window length is an exact integer multiple of the wave period. To address this, the flat-top window was devised as a time window that does not cause much change in the magnitude of frequency components even if the time window length is not an integer multiple of the fundamental period. The flat-top window is obtained by multiplying the Hamming window by a waveform sin(4π t/T)/4π t/T, which has a flat spectrum from -2/T to +2/T, and this spectrum has a flat top as shown in Figure 7-12.

The flat-top window is a window function devised by Kenichi Kido, the author of "Introduction to Digital Signal Processing," which I am using as a reference book for this explanation.

A flat-top window can be expressed as follows:

(Formula 7-14)

Furthermore, this Fourier transform is given by the following equation as the convolution integral of equations 7-13 and 7-16.

(Formula 7-15)

Here:

(Formula 7-16)

Power reduction rate: -7.0 dB

Equations 7-14 and 7-15 are shown in Figure 7-12.

Since the representation in DFT is not straightforward, here, as practice for convolution integrals, we will use the main part of the Hanning window and the discrete spectral values of equations 7-13 and 7-16, and perform the convolution integral of equation 7-15.

(Formula 7-17)

(Formula 7-18)

As mentioned earlier, the convolution integral is a moving average, so let's calculate W f (n) by considering that W _m (f) is at stations x (1) to x (12), and P _2/T (_f) is a train with y (1) to y (5) from the front, and the train passes through the stations. The calculation process is shown in Table 1, and a simplified diagram of the obtained W _f (f) is shown in Figure 7-13.

Table 1. Approximate calculation of flat-top window (convolution integral)

For example, W _f (4), W _f The formula for finding (5), omitting the multiplication with 0, is as follows:

W (4) = x (8) y (1) + x (7) y (2) + x (6) y (3) + x (5) y (4) + x (4) y (5)

W (5) = x (9) y (1) + x (8) y (2) + x (7) y (3) + x (6) y (4) + x (5) y (5)

It will be.

While a flat-top window is not suitable for separating closely spaced line spectra, it is a very effective window function for accurately determining the amplitude of those line spectra.

Figure 7-14 shows one side of the main part of each window overlaid, which should clearly illustrate the differences caused by the windows (filters).

Generally, when analyzing periodic or random signals, a Hanning window is used when frequency is important, a flat-top window when amplitude is important, and a rectangular window is used when measuring transfer functions in impact tests. You can see that this is due to the characteristics of each window. Also, the longer the time length T, the smaller the power of leakage error becomes, and the less impact it has. There are many other window functions that have been proposed, but we will omit them here.

Table 2 shows the features of the windows discussed so far for your reference.

Table 2 Window Characteristics