In this series, we previously covered the power spectrum, the most basic and important function of an FFT analyzer as an electrical measuring instrument, in two parts, Part 1 and Part 2. In this third part, we will explain the calculation functions (overall, average function, etc.).
The power spectrum is the distribution of power (mean square) at each frequency contained in the time signal x(t) being analyzed. Therefore, its sum is equal to the total power (mean square) of the original signal x(t). In Ono Sokki 's FFT analyzers, this is called the overall (hereinafter abbreviated as OA) and is always displayed at the right end of the power spectrum graph (see Figure 1).
This will be equal to the mean squared (power) of the original time signal;
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Figure 1. Overall (OA) and All-Pass (AP) in frequency analysis.
In the field of frequency analysis, the mean squared value (power) of the original time signal before this analysis is sometimes specifically called the all-pass (hereinafter abbreviated as AP). This corresponds to the displayed value of general-purpose vibration meters and sound level meters in the field of acoustic vibration.
In actual FFT analyzers, the shape of the bandpass filter in Figure 1 changes depending on the type of time window explained previously, and as a result, the sum in equation (1) becomes larger. Therefore, the calculation is performed using the following equation (3), which corrects for the effect of the time window.
Here, P(k): the k-th power spectrum
H f: Correction factor
= 2/3 (Hanning)
= 1/3.671 (flat top)
=1 (other than the above)
It is important to note here that in equation (3), P(k) is the power value, so if the obtained power spectrum is a linear value (RMS value) or a dB value, it needs to be converted to a power value and summed up.
If the power spectrum of the linear value is L(k), then the linear value of its OA is:
If the power spectrum of the dB value is B(k), then the dB value of its OA is:
Next, I will explain about averages.
While averaging is generally unnecessary for deterministic signals such as periodic signals, it is essential for signals with many noise components or random signals to smooth out fluctuations and improve the accuracy of statistical estimation of random signal spectra.
The recorded continuous time signal is divided into FFT time windows to obtain the instantaneous power spectrum, and then the aggregate average is performed multiple times. Figure 2 shows an example where the FFT time window T is used, the recording time length is 5T, and the signal is divided into 5 time window blocks.
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Figure 2: Example of dividing a continuous-time signal into a 5-block time window.
Now, if we let S(k) be the average power spectrum over N trials:
This averaging method is called additive averaging (linear averaging, RMS averaging). Linear averaging is the name used in contrast to exponential averaging, which will be discussed later. RMS averaging means adding the power values and dividing by the number of times this is done (averaging the power values). It is important to note here, as mentioned in the overall calculation, that the average is always calculated using the power value (the square of the amplitude).
Without going into detail, the accuracy of power spectrum estimation using the FFT method depends on the number of times N is calculated as a ensemble average of spectra from data in independent time windows, and does not depend on the time window length T. For example, in Figure 2, the power spectrum obtained by equation (6) after performing a T time window FFT five times (N=5) has N1 times better estimation accuracy variability compared to the spectrum obtained from a single FFT with a 5T time window. The only advantage of a longer time window length is improved frequency resolution.
Figure 2 shows a rectangular window, but for continuous time signals, a Hanning window is usually used. To more accurately reflect the data in the spectrum, the time windows are overlapped as shown in the lower part (b) of Figure 3. This process is called overlapping. Overlapping slightly reduces the independence of the data, but it increases the number of averaging it can do for the same recording time, resulting in improved estimation accuracy. Figure 3 shows that with a Hanning time window T (and its sampling points M), a recording time of 5T (5M recording points) can be averaged 9 times with 50% overlapping, demonstrating that even with a Hanning window, the time data can be averaged with almost equal weighting.
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Figure 3: Example of applying a Hanning window and processing with 50% overlap.
Another averaging method is exponential averaging (Exp average). Unlike averaging, which is an equal-weighted average (linear average), exponential averaging is a weighted average as shown in equation (7) below. If the exponential average spectrum is Em(k) and the weight coefficients are N, then:
k = 0, 1, 2, ..., L L: Number of analysis lines
This type of average is used for time signals where spectral values fluctuate, and it can continuously average over a specified time constant that depends on a weight coefficient N. Furthermore, while the previously mentioned arithmetic average automatically terminates after N counts (or a certain averaging time), this exponential average continues averaging until stopped. Therefore, it is sometimes called a running average.
Finally, here's a summary.
- The sum of the power values of all frequency components in the power spectrum is called the overall power, and it is equal to the mean square of the original time signal.
- The mean square of the original time signal that was not analyzed is called the all-pass signal.
- Actual overall calculations require time window correction.
- If the obtained power spectrum results are linear or dB values, they need to be converted to power values for overall calculation.
- There are two main methods for averaging power spectra: averaging and exponential averaging.
- Averaging averages are ensemble averages with equal weights, while exponential averages are averages with certain weights.
- In averaging, overlap processing is usually performed, and with the Hanning window, optimal averaging can be achieved by applying approximately 50% overlap processing.
【keyword】
Overall, OA, All Pass, AP, Additive Average, Linear Average, RMS Average, Overlap Processing, Exponential Average, Exponential Average, Running Average
[Reference materials]
- "Signal Processing," by Iwao Morishita and Hidefumi Obata, Society of Instrument and Control Engineers (1982).
- "Spectral Analysis," by Mikio Hino, Asakura Shoten (1977)
(Excerpt from the email newsletter issued on May 22, 2014)
