Recently, I received a question asking, "What is the difference between a Fourier spectrum and a power spectrum?" So, this time, I will talk about Fourier spectra again.
Ono Ono Sokki 's FFT analyzers DS/CF series and time-series data analysis tool Oscope provide not only power spectra but also frequency functions called Fourier spectra.
While Fourier spectra have already been explained in detail in this series, "Frequency Analysis from the Basics (11) - Fourier Spectra," here we will explain the meaning and application of the function in the FFT analyzer mentioned above in more detail.
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Figure 1: Basic functions and processing flow of a 2-channel FFT analyzer.
Figure 1 shows the basic functions and processing flow of a typical FFT analyzer. As shown in this figure, the result of performing an FFT on a finite time series data segment up to a certain time is the Fourier spectrum, and the power spectrum is calculated from the obtained Fourier spectrum.
The finite Fourier series of the time series data x (n) (n = 0, 1, ..., N −1) of the N points that have been extracted is:
Here,
This is the result. Furthermore, since FFT performs operations on complex numbers;
If we assume this, the complex finite Fourier series is:
This is the result. Here, equation (5) is the discrete finite Fourier transform (i.e., FFT), and ck is called the complex Fourier coefficient. Ono Ono Sokki 's FFT analyzer calls this the Fourier spectrum. As explained in "Frequency Analysis from the Basics (11) - Fourier Spectrum," the usual definition of the Fourier spectrum X(f) is the value obtained by multiplying the complex Fourier coefficient ck by the time window length T, but here we assume they are equal.
Since we are using discrete notation, if we denote the Fourier spectrum as X(k) (=ck), then X(k) represents the magnitude (intensity) of the frequency components contained in the time series data x(n), and is a complex number;
Here, there is something to note regarding the amplitude. Equation (1) is a finite Fourier series expansion with only N/2 points, but equation (5) yields the complex Fourier coefficients of N points. This is because if we divide ck into two halves, the first and second halves, at the N/2 point, we obtain the same information from both. That is, the real part is symmetrical with respect to the line, and the odd part is symmetrical with respect to the point (complex conjugate) around N/2;
Therefore, considering X(k) as a one-sided spectrum, we use equation (3):
This can be calculated as follows: Equation (11) represents the RMS value of a constant frequency sine wave, and in amplitude (peak value) notation:
This is the result. |X (k)| is sometimes called the amplitude spectrum, and θ (k) is called the phase spectrum.
Thus, the Fourier spectrum is a complex number that contains amplitude and phase information for each frequency.
Next, what does phase represent?
The phase of the Fourier spectrum indicates where the sine wave at that frequency begins within the FFT time window, i.e., the initial phase φk in equation (2). However, a trigger function is necessary to obtain meaningful phase information.
(Example 1) The amplitude of x (t) = A cos (2π ft) is A, and the phase is 0 degrees.
(Example 2) The amplitude of x(t) = A sin(2π ft) is A, and the phase is -90 degrees (= -π/2).
(Example 3) The amplitude of x (t) = A cos (2π ft) + B sin (2π ft) is
When A = B = 1, the amplitude is √2 and the phase is -45 degrees (= -π/4).
The power spectrum is obtained as the square of the amplitude of the Fourier spectrum.
Expressed discretely, similar to the Fourier spectrum, from equation (11):
Thus, the power spectrum is the mean square of the signal (i.e., the square of the RMS value). Phase information is lost in the power spectrum.
Power spectra are not used for instantaneous spectra, but rather for smoothing out fluctuating components.
To improve the accuracy of spectral estimation for random signals, averaging is typically performed (averaging loop in Figure 1).
The most commonly used averaging method is additive averaging, as shown below.
In spectral analysis of sound and vibration signals, the primary objective is to measure the averaged power spectrum, as the goal is to determine the signal strength, and phase information is usually not necessary.
Table 1 Comparison of Fourier spectra and power spectra
| Fourier spectrum | Power Spectrum | |
| Amplitude information | - Signal strength at each frequency - Equivalent to an unaveraged power spectrum |
- Signal strength at each frequency - Equivalent to the mean square |
| phase | Required (trigger needed) | none |
| averaging | Normally, averaging is not possible. | The signal strength can be calculated by averaging it. |
| others | The original time series data can be reconstructed using IFFT. | Unable to revert to the original time-series data |
As shown in Table 1, conventional frequency analysis uses the power spectrum and not the Fourier spectrum. The greatest advantage of the Fourier spectrum is that it provides phase information, and it is typically used for the following purposes:
- Since amplitude and phase (real and imaginary parts) information is available, the original time-series data can be reconstructed using the inverse Fourier transform (IFFT) (see Figure 1). Furthermore, frequency bandwidth limiting and filtering are also possible.
- This technology can be applied to balancing measurements of rotating bodies. By inputting one pulse per rotation and using it as a trigger signal, meaningful phase information can be obtained, and positional information for correcting imbalances can be calculated.
- By performing phase tracking in tracking analysis, not only the amplitude but also the phase can be obtained for each rotational speed.
① The critical speed of a rotating body can be detected from the mode circle (polar diagram).
② By performing simultaneous tracking analysis at multiple points, data can be obtained for actual operation animation (ODS) at a specific rotational speed.
Finally, here's a summary.
- A Fourier spectrum is obtained by performing a Fourier transform on time-series data. It is a complex number and contains two pieces of information: amplitude and phase.
- The power spectrum is obtained by squaring the amplitude of the Fourier spectrum. If averaging is not performed, the amplitude information of both spectra will be the same. Also, phase information is lost in the power spectrum.
- Determining the spectrum of a time signal usually involves determining the average power spectrum, and Fourier spectra are not used.
- Applications of Fourier spectra include:
① Reconstructing time-series data using IFFT
② Balancing measurement
③ Phase tracking
These are some examples.
【keyword】
Power spectrum, Fourier spectrum, finite Fourier series, complex finite Fourier series, FFT, complex Fourier coefficients, time window length, complex conjugate, one-sided spectrum, RMS value, amplitude spectrum,
Phase spectrum, time window, initial phase, mean square, averaging, trigger, IFFT, balancing measurement, tracking analysis, phase tracking, mode circle, polar diagram, critical velocity, actual operation animation, ODS
【reference】
"Introduction to Spectral Analysis of Seismic Motion," by Yoshihiko Osaki, Kajima Publishing Co., Ltd. (1984)
"Digital Fourier Analysis (II) - Advanced Level -" edited by Kenichi Kido, Corona Publishing Co., Ltd. (2007)
(Excerpt from the email newsletter issued on November 25, 2016)
