In this measurement column, we address frequently asked questions from our customer service department and introduce the answers. This time, we will explain how to calculate the Power Spectral Density (PSD) from the power spectrum obtained through FFT analysis.
When performing FFT analysis on non-periodic signals such as random signals (signals with a continuous spectrum), the amplitude values of each component of the obtained power spectrum depend on the frequency resolution Δf of the FFT calculation, and changing the analysis conditions can alter these values. In such cases, Power Spectral Density (PSD) is used to express the power values per unit frequency width (1 Hz) so that the results do not depend on Δf.
The relationship between power spectrum and frequency resolution Δf.
There is a relationship between the frequency range, number of sample points, and frequency resolution when performing FFT analysis.
- Number of lines [points] = Number of sample points [points] ÷ 2.56
- Frequency resolution Δf [Hz] = Frequency range [Hz] ÷ Number of lines [points]
Figure 1 shows the power spectrum obtained from the FFT analysis of a signal with a continuous spectrum. The frequency range is 2 kHz for all cases, with sample points of 512, 1024, and 2048. The frequency resolution Δf is 10 Hz, 5 Hz, and 2.5 Hz, respectively.
The amplitude values of the 1 kHz component were 0.515 m/s 2, 0.337 m/s 2, and 0.234 m/s 2. For an ideal random signal, if the frequency resolution Δf is halved, the amplitude value becomes 1 over √2.
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Figure 1 Differences in power spectrum based on the number of sample points (Top: 512 points / Δf = 10 Hz, Middle: 1024 points / Δf = 5 Hz, Bottom: 2048 points / Δf = 2.5 Hz)
Frequency range: When performing FFT analysis with a frequency range of 2 kHz and a sample size of 512 points, the frequency resolution becomes 10 Hz, resulting in a power spectrum with 10 Hz intervals. The 1000 Hz component of the obtained power spectrum is the sum of the components from 995 Hz to 1005 Hz, while the 1010 Hz component is the sum of the components from 1005 Hz to 1015 Hz. Each value can be considered as the total value of a band with a width equal to the frequency resolution.
When performing FFT analysis with a frequency range of 2 kHz and a sample size of 1024 points, the frequency resolution becomes 5 Hz. Since the width of the band for which the total is calculated is halved, compared to an analysis with 512 points, the power value (the square of the amplitude) will be halved, resulting in the amplitude value being reduced to 1/√2.
Strictly speaking, the analysis result at 1000 Hz with 512 points does not only include components from 995 Hz to 1005 Hz; rather, it includes a slightly wider range of components due to the influence of window functions (Hanning window, flat-top window). However, since the influence of the window function remains the same regardless of changes in the number of sample points or frequency resolution, the relationship that when the frequency resolution Δf is halved, the amplitude value is reduced by a factor of √2 still holds.
Method for calculating Power Spectral Density (PSD)
The power spectrum obtained from FFT analysis shows that the power values (squared amplitude values) are proportional to the frequency resolution Δf. Therefore, to display the power values per unit frequency width (1 Hz), one should divide the power values of the power spectrum (squared amplitude values) by Δf.
In practice, simply dividing by Δf does not yield correct values due to the influence of window functions (Hanning window, flat-top window), so the power spectral density (PSD) is calculated using the following formula.
PSD=Power value÷(Δf×Wf)
=Amplitude value 2 ÷(Δf×Wf)
Here, Wf is the correction value for each window (window function). In the Measurement Column No. 179 (August 2016), we introduced the correction value Hf used when calculating overall (OA) and partial overall (POA). The reciprocal of Hf becomes the correction value Wf when calculating the power spectral density.
Table 1 Correction Values for Windows (Window Functions) When Calculating PSD
| Window (window function) | Correction value during calculation of OA (Hf) | Correction value (Wf) during PSD calculations |
| rectangular window | 1 | 1 |
| Hanning | 2/3 = approximately 0.6667 | 3/2 = 1.5 |
| Flat top | 1/3.6714416356 = approximately 0.2724 | 3.6714416356 |
| Force | 1 | 1 |
| index | 1 | 1 |
Power Spectral Density (PSD) is the power value (amplitude squared value) divided by the frequency resolution Δf, and its unit is [amplitude unit 2 / Hz]. If the amplitude unit is m/s 2, then the unit will be [(m/s 2) 2 / Hz], and if the amplitude unit is V, then the unit will be [V 2 / Hz].
The values of power spectral density (PSD) are often displayed in units of [amplitude value 2 / Hz], but they can also be presented as the square root of that value. In this case, the unit for the square root of the PSD value is expressed as [m/s 2 / √Hz], [V / √Hz], etc. When comparing past measurement results or specifications of the measured products, it is necessary to check whether the units of those PSD values are [amplitude value 2 / Hz] or [amplitude value/ √Hz] and to ensure that the units are consistent during measurement.
Our FFT analyzer has the capability to display power spectral density (PSD). At that time, you can choose the display value from V2 or V, and if you select V2, the PSD value is displayed in the unit of [amplitude value 2 / Hz]. If you select V, the square root of the PSD value is displayed in the unit of [amplitude value/ √Hz].
Calculation of power spectral density from acceleration power spectrum
An example of loading the acceleration power spectrum analyzed by the FFT analyzer into Excel is shown in Table 2. The frequency range (cell C6) is 2000 Hz, and the number of sample points (cell B7) is 1024, resulting in a frequency resolution of 5 Hz. Cells A17 to A417 contain frequency values ranging from 0 Hz to 2 kHz in increments of 5 Hz.
The Y-axis scale (cell B14) is set to Lin, so the values in cells B17 to B417 represent the physical values (acceleration values) of each frequency component measured with the Y-axis scale set to Lin. The data with the Y-axis scale set to MagLog represents the amplitude values (acceleration values) of each frequency component, with the same cells B17 to B417.
Entering formulas like those in Table 2 into cells D17 to D417 will display the power spectral density (PSD) values in the same cells. The unit of this value is [(m/s 2) 2 / Hz].
The Y-axis magnitude (cell B16) is in RMS, so the values from cells B17 to B417 are also in effective value (RMS value). Therefore, the power spectral density (PSD) values obtained using the above method are also in effective value (RMS value).
Additionally, the content of Table 2 can be downloaded from the link below.
Table 2 Example of calculating power spectral density from the acceleration power spectrum
Table 2 Example of calculating power spectral density from the acceleration power spectrum
Calculation of power spectral density from the power spectrum of a voltage signal.
An example of loading the power spectrum of a voltage signal, analyzed with an FFT analyzer and displayed in decibels, into Excel is shown in Table 3. The frequency range (cell C6) is 2000 Hz, and the number of sample points (cell B7) is 1024, resulting in a frequency resolution of 5 Hz. Cells A17 to A417 contain frequency values ranging from 0 Hz to 2 kHz in increments of 5 Hz.
The Y-axis scale (cell B14) is set to Log, so the data measured is based on this logarithmic Y-axis scale, and the values in cells B17 to B417 are the decibel values of each frequency component.
Table 3 Example of Overall Calculation from Voltage Signal Spectrum
When you enter a formula like the one in Table 3 into cells D17 through D417, the power spectral density (PSD) values are displayed in those cells. The unit for these values is [V 2 /Hz]. The formula for converting a decibel value to an amplitude value is 10^(decibel value/20), but in this case, we want to convert the decibel value to a power value (amplitude squared), so we used the formula 10^(decibel value/10).
The Y-axis magnitude (cell B16) is in RMS, so the values from cells B17 to B417 are also in effective value (RMS value). Therefore, the power spectral density (PSD) values obtained using the above method are also in effective value (RMS value).
Additionally, the content of Table 3 can be downloaded from the link below.
Table 3 Example of Overall Calculation from Voltage Signal Spectrum
(Excerpt from the newsletter published on October 20, 2016)
