In this measurement column, we address frequently asked questions received by our customer support center and provide answers. This time, we will introduce a method for calculating power spectral density (PSD) from the power spectrum obtained by FFT analysis.
When non-periodic signals, such as random signals (signals with a continuous spectrum), are analyzed using FFT, the amplitude values of each component in the resulting power spectrum depend on the frequency resolution Δf of the FFT calculation, and these values change when the analysis conditions are altered. In such cases, power spectral density (PSD) is used to express the power value per unit frequency width (1 Hz) so that the analysis result does not depend on Δf.
Relationship between power spectrum and frequency resolution Δf
The following relationship exists between the frequency range, number of samples, 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 spectra obtained by FFT analysis of signals with a continuous spectrum. The frequency range is 2 kHz in all cases, and the number of sample points is 512, 1024, and 2048. The frequency resolution Δf is 10 Hz, 5 Hz, and 2.5 Hz, respectively.
The amplitude values for the 1 kHz component were 0.515 m/ s², 0.337 m/ s², and 0.234 m/ s², respectively. For an ideal random signal, if the frequency resolution Δf is halved, the amplitude value becomes √2.
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Figure 1. Differences in power spectrum based on the number of sample points (Top panel: 512 points/Δf = 10 Hz, Middle panel: 1024 points/Δf = 5 Hz, Bottom panel: 2048 points/Δf = 2.5 Hz)
When FFT analysis is performed with a frequency range of 2 kHz and 512 sample points, the frequency resolution becomes 10 Hz, and a power spectrum is obtained in 10 Hz increments. The 1000 Hz component of the obtained power spectrum is the sum of the components from 995 Hz to 1005 Hz, the 1010 Hz component is the sum of the components from 1005 Hz to 1015 Hz, and so on. Each value can be considered the sum of a bandwidth with a frequency resolution width.
When performing an FFT analysis with a frequency range of 2 kHz and 1024 sample points, the frequency resolution becomes 5 Hz, and the bandwidth for calculating the total is halved. Therefore, compared to an analysis with 512 points, the power value (square of the amplitude) is halved, and the amplitude value becomes √2.
Strictly speaking, the 1000 Hz value in the analysis results from 512 points does not only include components from 995 Hz to 1005 Hz, but also includes components in a slightly wider range due to the influence of window functions (Hanning window, flat-top window). However, since the effect of the window function remains the same even if the number of sample points and frequency resolution change, the relationship that the amplitude value becomes 1/√2 when the frequency resolution Δf is halved still holds true.
How to calculate Power Spectral Density (PSD)
The power spectrum obtained from FFT analysis has a power value (amplitude squared) that is proportional to the frequency resolution Δf. Therefore, to display the power value per unit frequency width (1 Hz), you simply need to divide the power value (amplitude squared) of the power spectrum by Δf.
In reality, simply dividing by Δf will not yield the correct value due to the influence of window functions (Hanning window, flat-top window). Therefore, 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 Measurement Column No. 179 (August 2016), we introduced the correction value Hf used when calculating overall (OA) and partial overall (POA), and the reciprocal of Hf is the correction value Wf used when calculating power spectral density.
Table 1. Correction values for the window (window function) when calculating the PSD.
| Window (window function) | Correction value (Hf) during OA calculation | Correction value (Wf) during PSD calculation |
| Rectangular (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 calculated by dividing the power value (amplitude squared) by the frequency resolution Δf, and its unit is [amplitude unit 2 / Hz]. If the amplitude unit is m/ s², the unit is [(m/ s²) ² / Hz], and if the amplitude unit is V, the unit is [V² / Hz].
Power spectral density (PSD) values are often expressed in units of [amplitude value in units of 2 /Hz], but they can also be expressed as the square root of that value. In this case, the units of the square root of the PSD are expressed as [m/ s² /√Hz], [V/√Hz], etc. When comparing results from past measurements or the specifications of the product being measured, it is necessary to check whether the units of those PSD values are [amplitude value in units of 2 /Hz] or [amplitude value in units of √Hz] and to match the units when measuring.
Our FFT analyzer has a function to display power spectral density (PSD). When doing so, you can choose between V2 and V as the display value. If you select V2, the PSD value will be displayed in units of [amplitude value in units of 2 / Hz]. If you select V, the square root value of the PSD will be displayed in units of [amplitude value in units of / √Hz].
Calculation of power spectral density from acceleration power spectrum
Table 2 shows an example of importing an acceleration power spectrum analyzed with an FFT analyzer into Excel. The frequency range (cell C6) is 2000 Hz, and the number of sample points (cell B7) is 1024, so the frequency resolution is 5 Hz. Cells A17 to A417 contain frequency values, arranged in 5 Hz increments from 0 Hz to 2 kHz.
The Y-axis scale (cell B14) is set to Lin, so this is data measured with the Y-axis scale set to Lin, and the values in cells B17 to B417 are the physical values (acceleration values) of each frequency component. The data with the Y-axis scale set to MagLog is data measured with the Y-axis scale set to Log/MagLog, and in this case as well, the values in cells B17 to B417 are the amplitude values (acceleration values) of each frequency component.
Entering the formulas shown in Table 2 into cells D17 through D417 will display the Power Spectral Density (PSD) values in those cells. The unit of this value is [(m/ s²) ² / Hz].
Since the Y-axis Magnitude (cell B16) is in RMS, the values in cells B17 to B417 are RMS values. Therefore, the Power Spectral Density (PSD) values obtained using the method described above are also RMS values.
The contents of Table 2 can be downloaded from the following link.
Table 2. Example of calculating power spectral density from acceleration power spectrum.
Table 2. Example of calculating power spectral density from acceleration power spectrum.
| A | B | C | D | E | |
| 1 | Label: | CH2: Power Spectrum | |||
| 2 | DateTime: | Tue Oct 18 20:35:50 2016 | |||
| 3 | DataKind: | CH2 | PowerSpec | Mag | |
| 4 | DataPoints: | 402 | Filter: | FLAT | : |
| 5 | DataCalc: | ||||
| 6 | Frequency: | 0 | 2000 | Hz | |
| 7 | Sample: | 1024 | Internal | ||
| 8 | Average: | 681 | Power/Sum | ||
| 9 | Voltage（CH2）: | -10 | dBVrms | ||
| 10 | EU/V（CH2）: | 1.00E+03 | 0dBRef.（CH2）: | 1.00E+00 | |
| 11 | Window（CH2）: | Hann | |||
| 12 | X-AxisScale: | Lin | Frequency resolution [Hz] | ||
| 13 | X-AxisUnit: | Hz | 5 | =C6/（B7/2.56） | |
| 14 | Y-AxisScale: | Lin | |||
| 15 | Y-AxisUnit: | m/s2 | PSD value | Cell formulas | |
| 16 | Y-AxisMagnitude: | rms | （m/s2）^2/Hz | ||
| 17 | 0.0 | 0.211 | 0.005917 | =（B17*B17）/（$D$13*1.5） | |
| 18 | 5.0 | 0.151 | 0.003042 | =（B18*B18）/（$D$13*1.5） | |
| 19 | 10.0 | 0.077 | 0.000800 | =（B19*B19）/（$D$13*1.5） | |
| 20 | 15.0 | 0.173 | 0.003980 | =（B20*B20）/（$D$13*1.5） | |
| 21 | 20.0 | 0.201 | 0.005400 | =（B21*B21）/（$D$13*1.5） | |
| 22 | 25.0 | 0.222 | 0.006581 | =（B22*B22）/（$D$13*1.5） | |
| 23 | 30.0 | 0.245 | 0.008021 | =（B23*B23）/（$D$13*1.5） | |
| … | |||||
Calculation of power spectral density from voltage signal power spectrum
Table 3 shows an example of importing the power spectrum of a voltage signal, analyzed with an FFT analyzer and displayed in decibels, into Excel. The frequency range (cell C6) is 2000 Hz, and the number of sample points (cell B7) is 1024, so the frequency resolution is 5 Hz. Cells A17 to A417 contain frequency values, arranged in 5 Hz increments from 0 Hz to 2 kHz.
The Y-axis scale (cell B14) is set to Log, so this is data measured with the Y-axis scale set to Log. The values in cells B17 to B417 are the decibel values of each frequency component.
Table 3: Example of calculating the overall value from a voltage signal spectrum
| A | B | C | D | E | |
| 1 | Label: | CH2: Power Spectrum | |||
| 2 | DateTime: | Tue Oct 18 20:37:00 2016 | |||
| 3 | DataKind: | CH2 | PowerSpec | Mag | |
| 4 | DataPoints: | 402 | Filter: | FLAT | AnalogFilter（CH2）: |
| 5 | DataCalc: | ||||
| 6 | Frequency: | 0 | 2000 | Hz | |
| 7 | Sample: | 1024 | Internal | ||
| 8 | Average: | 681 | Power/Sum | ||
| 9 | Voltage（CH2）: | -10 | dBVrms | ||
| 10 | EU/V（CH2）: | 1.00E+00 | 0dBRef.（CH2）: | 1.00E+00 | ADOverHold（CH2）: |
| 11 | Window（CH2）: | Hann | |||
| 12 | X-AxisScale: | Lin | Frequency resolution [Hz] | ||
| 13 | X-AxisUnit: | Hz | 5 | =C6/（B7/2.56） | |
| 14 | Y-AxisScale: | Log | |||
| 15 | Y-AxisUnit: | V | PSD value | Cell formulas | |
| 16 | Y-AxisMagnitude: | rms | V2/Hz | ||
| 17 | 0.0 | -73.53 | 5.9171E-09 | =（10^（B17/10））/（$D$13*1.5） | |
| 18 | 5.0 | -76.42 | 3.0421E-09 | =（10^（B18/10））/（$D$13*1.5） | |
| 19 | 10.0 | -82.22 | 7.9964E-10 | =（10^（B19/10））/（$D$13*1.5） | |
| 20 | 15.0 | -75.25 | 3.9802E-09 | =（10^（B20/10））/（$D$13*1.5） | |
| 21 | 20.0 | -73.93 | 5.3996E-09 | =（10^（B21/10））/（$D$13*1.5） | |
| 22 | 25.0 | -73.07 | 6.5810E-09 | =（10^（B22/10））/（$D$13*1.5） | |
| 23 | 30.0 | -72.21 | 8.0211E-09 | =（10^（B23/10））/（$D$13*1.5） | |
| … | |||||
Entering the formulas shown in Table 3 into cells D17 to D417 will display the Power Spectral Density (PSD) values in those cells. The unit of this value is [V² / Hz]. The formula for converting decibel values to amplitude values is 10^(decibels/20), but in this case, we want to convert the decibel values to power values (amplitude squared values), so we used the formula 10^(decibels/10).
Since the Y-axis Magnitude (cell B16) is in RMS, the values in cells B17 to B417 are RMS values. Therefore, the Power Spectral Density (PSD) values obtained using the method described above are also RMS values.
The contents of Table 3 can be downloaded from the following link.
Table 3: Example of calculating the overall value from a voltage signal spectrum
summary
This time, we introduced a method for calculating the power spectral density (PSD) from the power spectrum obtained by FFT analysis.
Our FFT analyzer has a function to calculate and display power spectral density (PSD), but if you save the data without using that function, you can calculate it using the method described here.
Power spectral density (PSD) is expressed as the power value per Hz (the square of the amplitude value), and its unit is [amplitude unit 2 /Hz], i.e., [(m/ s²) ² /Hz], [V² /Hz], etc.
When performing FFT analysis and comparing it with past measurement results or the specifications of the product being measured, please confirm whether the past results or specifications are PSD or simply power spectra, and if PSD, whether the unit of the value is [amplitude value unit 2 / Hz] or [amplitude value unit / √Hz], and then perform the measurement with the same units and measurement conditions.
(Excerpt from the email newsletter issued on October 20, 2016)
