In this measurement column, we address frequently asked questions received by our customer support center and provide answers. This time, we will introduce how to calculate overall (OA) and partial overall (POA) from the power spectrum obtained by FFT analysis.
The overall power (OA) is the sum of the power values (squared values) up to the analysis frequency range. If the frequency range for summing is limited, the value obtained by calculating the sum within that range is called the partial overall power (POA).
Calculation of overalls (OA) and partial overalls (POA)
The overall (OA) can be calculated using equation (1). The overall (OA) value obtained from this equation is the square of the physical value. If you want to display the overall value as the physical value instead of the square, take the square root of the result of equation (1).
Here:
PDC: Power value (squared value) of the DC component (0 Hz component)
P i: The value of the i-th power spectrum (effective value, squared value)
N: Number of power spectra (number of lines)
H f: Correction value for the window function.
Partial overall (POA) is an overall calculation that limits the range over which the sum is calculated. It can be calculated using formula (1) by changing the range of addition (within the brackets).
Table 1 shows the correction value Hf for each window (window function). Note that the correction value for flat-top (*) is for our DS-2000/DS-3000 series data stations and CF-5200/CF-7200/CF-9000/CF-4500/CF-4700 FFT analyzers. Our older products (CF-350/CF-360 FFT analyzers, etc.) have a different flat-top window function shape, so the correction value will also be different. In addition, other companies' products may use flat-top window functions with a different shape than ours, in which case the correction value will also be different.
Table 1. Correction values for the window function when calculating OA and POA.
| Window (window function) | Correction value |
| Rectangular (rectangular window) | 1 |
| Hanning | 2/3 = approximately 0.6667 |
| Flat top (*) | 1/3.6714416356 = approximately 0.2724 |
| force | 1 |
| index | 1 |
Overall calculation 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 10,000 Hz, and the number of sample points (cell B7) is 1,024, so the frequency resolution is 25 Hz. Cells A17 to A417 contain frequency values, arranged in 25 Hz increments from 0 Hz to 10 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 physical values (acceleration values) of each frequency component.
Entering the formulas shown in Table 2 into cells D17 to D417, D421, and B421 will display the overall (OA) physical quantity (acceleration value) in cell B421. Cells D17 to D417 contain formulas that calculate the square of the physical value of each frequency component. Cell B421 contains the formula "SQRT(D421 / 1.5)", where dividing by 1.5 corresponds to applying the correction value (2/3) of the Hanning window function. Cell D421 contains the formula "SUM(D17:D417)", and by changing the range for which this formula calculates the sum, you can calculate the partial overall (POA).
Since the Y-axis Magnitude (cell B16) is in RMS format, the values in cells B17 to B417 are RMS values. Therefore, the overall (OA) and partial overall (POA) values obtained using the method described above are also RMS values.
The data displayed in cells C16 to C17 as PSD, ESD, and V2 represent data measured using settings such as PSD (Power Spectral Density), ESD (Energy Spectral Density), and V2 (Display as the square of the physical value). Since the recorded data is different, the calculation method introduced in this section will need to be partially modified to perform the calculation.
Table 2: Example of calculating overall from acceleration power spectrum
| A | B | C | D | E | |
| 1 | Label: | CH2: Power Spectrum | |||
| 2 | DateTime: | Mon Jun 20 17:55:25 2016 | |||
| 3 | DataKind: | CH2 | PowerSpec | Mag | |
| 4 | DataPoints: | 402 | Filter: | FLAT | |
| 5 | DataCalc: | ||||
| 6 | Frequency: | 0 | 10000 | Hz | |
| 7 | Sample: | 1024 | Internal | ||
| 8 | Average: | 0 | 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 | |||
| 13 | X-AxisUnit: | Hz | |||
| 14 | Y-AxisScale: | Lin | |||
| 15 | Y-AxisUnit: | m/s2 | |||
| 16 | Y-AxisMagnitude: | rms | physical quantity squared value | Cell formulas | |
| 17 | 0.0 | 0.4217 | 0.177858442 | ＝B17*B17 | |
| 18 | 25.0 | 0.9046 | 0.818217277 | ＝B18*B18 | |
| 19 | 50.0 | 0.5663 | 0.320670754 | ＝B19*B19 | |
| 20 | 75.0 | 0.2481 | 0.06155948 | ＝B20*B20 | |
| … | … | ||||
| 414 | 9925.0 | 0.0489 | 0.002388137 | ＝B414*B414 | |
| 415 | 9950.0 | 0.0255 | 0.000652105 | ＝B415*B415 | |
| 416 | 9975.0 | 0.0547 | 0.002996523 | ＝B416*B416 | |
| 417 | 10000.0 | 0.0501 | 0.002509365 | ＝B417*B417 | |
| 418 | OVERALL | 17.9163 | |||
| 419 | |||||
| 420 | Overall value | Sum of the squared values of physical quantities | |||
| 421 | 17.91631297 | 481.4914056 | |||
| 422 | ＝SQRT（D421/1.5） | ＝SUM（D17:D417） | |||
The contents of Table 2 can be downloaded from the following link.
Table 2: Example of calculating overall from acceleration power spectrum
Calculation of the overall sound pressure level from the sound pressure level spectrum.
Table 3 shows an example of importing the power spectrum of sound pressure levels analyzed with an FFT analyzer into Excel. The frequency range (cell C6) is 10,000 Hz, and the number of sample points (cell B7) is 1,024, so the frequency resolution is 25 Hz. Cells A17 to A417 contain frequency values, arranged in 25 Hz increments from 0 Hz to 10 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 (sound pressure level values) of each frequency component.
If you enter the formulas shown in Table 3 into cells D17 to D417, D421, and B421, the overall (OA) decibel value (sound pressure level value) will be displayed in cell B421. Cells D17 to D417 contain formulas that calculate the squared sound pressure value from the decibel values of each frequency component. Cell B421 contains the formula "10*LOG10(D421/1.5)", where dividing by 1.5 is the process of applying the correction value (2/3) of the Hanning window function, and "10*LOG10" is the process of converting the squared sound pressure value to a decibel value. Cell D421 contains the formula "SUM(D17:D417)", and by changing the range for which this formula is summed, you can calculate the partial overall (POA).
Since the Y-axis Magnitude (cell B16) is in RMS format, the values in cells B17 to B417 are RMS values. Therefore, the overall (OA) and partial overall (POA) values obtained using the method described above are also RMS values.
The data displayed in cells C16 to C17 as PSD, ESD, and V2 represents data measured with settings such as PSD (Power Spectral Density) and ESD (Energy Spectral Density). Since the recorded data is different, the calculation method introduced in this section will need to be partially modified to perform the calculation.
The contents of Table 3 can be downloaded from the following link.
Table 3: Example of calculating the overall value from the sound pressure level spectrum.
Table 3: Example of calculating the overall value from the sound pressure level spectrum.
| A | B | C | D | E | |
| 1 | Label: | CH1: Power Spectrum | |||
| 2 | DateTime: | Mon Jun 20 17:55:25 2016 | |||
| 3 | DataKind: | CH1 | PowerSpec | Mag | |
| 4 | DataPoints: | 402 | Filter: | FLAT | |
| 5 | DataCalc: | ||||
| 6 | Frequency: | 0 | 10000 | Hz | |
| 7 | Sample: | 1024 | Internal | ||
| 8 | Average: | 0 | Power/Sum | ||
| 9 | Voltage（CH1）: | -30 | dBVrms | ||
| 10 | EU/V（CH1）: | 3.98E+01 | 0dBRef.（CH1）: | 2.00E-05 | |
| 11 | Window（CH1）: | Hann | |||
| 12 | X-AxisScale: | Lin | |||
| 13 | X-AxisUnit: | Hz | |||
| 14 | Y-AxisScale: | Log | |||
| 15 | Y-AxisUnit: | Pa | |||
| 16 | Y-AxisMagnitude: | rms | physical quantity squared value | Cell formulas | |
| 17 | 0.0 | 52.256 | 168096.9835 | ＝10^（B17/10） | |
| 18 | 25.0 | 50.504 | 112293.3427 | ＝10^（B18/10） | |
| 19 | 50.0 | 37.941 | 6224.698059 | ＝10^（B19/10） | |
| 20 | 75.0 | 35.992 | 3973.889804 | ＝10^（B20/10） | |
| … | … | ||||
| 414 | 9925.0 | 29.817 | 958.7666301 | ＝10^（B414/10） | |
| 415 | 9950.0 | 21.824 | 152.1842996 | ＝10^（B415/10） | |
| 416 | 9975.0 | 27.408 | 550.5741158 | ＝10^（B416/10） | |
| 417 | 10000.0 | 35.021 | 3177.875289 | ＝10^（B417/10） | |
| 418 | OVERALL | 75.045 | |||
| 419 | |||||
| 420 | Overall value | Sum of the squared values of physical quantities | |||
| 421 | 75.04507623 | 47929056.75 | |||
| 422 | ＝10*LOG10（D421/1.5） | ＝SUM（D17:D417） | |||
Regarding the Hanning window function correction value
When performing FFT analysis using the Hanning window function, the values of each frequency component and the overall (OA) change due to the influence of the window function. Therefore, FFT analyzers correct for this influence and display the values of each frequency component and the overall (OA).
The Hanning window function is defined by equation (2).
The mean value w and mean squares 2w of the Hanning window function can be calculated using equations (3) and (4), respectively, and their values are 1/2 and 3/8. When a window function is applied to a stationary time-domain waveform, the mean value of that time-domain waveform becomes 1/2, and the mean squares value becomes 3/8.
If we directly perform an FFT (Fourier Transform) on the time-domain waveform after applying the window function, the amplitude of the resulting power spectrum will be halved. Therefore, we correct the amplitude by doubling the amplitude ratio (quadrupling the power ratio) of the power spectrum.
The overall (OA) is the mean square, so it ends up being 3/8. Since it is multiplied by 4 during amplitude correction, the overall (OA) value obtained from the amplitude-corrected power spectrum will be 3/8 × 4 = 3/2 times the original value.
Therefore, when calculating the overall power (OA) from the power spectrum, a correction is applied by multiplying the sum of each component (power value) of the spectrum by a correction value Hf = 2/3. The overall correction value Hf can be calculated from the mean value-w and mean square value-w² of the Hanning window function using equation (5).
If you calculate the overall (OA) and partial overall (POA) values without applying this correction, the resulting values will be 1.5 times larger in power ratio and approximately 1.225 times larger in amplitude ratio than the original values. In terms of decibels, this increases by approximately 1.761 dB.
Regarding the flat-top window function correction value
When performing FFT analysis using a flat-top window function, the values of each frequency component and the overall (OA) change due to the influence of the window function. Therefore, FFT analyzers correct for this influence and display the values of each frequency component and the overall (OA) accordingly.
The flat-top window function used in our products is defined by equation (6).
(0≤t≤1)
The mean value-w and mean squares value-w² of the flat-top window function can be calculated using equations (7) and (8), respectively, and their values are 1/4.6 and 3.6714416356 /21.16. When the window function is applied to a time-domain waveform, the mean value of that time-domain waveform becomes 1/4.6, and the mean squares value becomes 3.6714416356/21.16.
If we directly perform an FFT (Fourier Transform) on the time-domain waveform after applying the window function, the amplitude of the resulting power spectrum will be reduced to 1/4.6. Therefore, we correct the amplitude by multiplying the power spectrum by 4.6 in amplitude ratio (21.16 in power ratio).
The overall (OA) value obtained from the amplitude-corrected power spectrum will deviate from the original value, so a correction is applied by multiplying it by the correction value H f = 1/3.6714416356.
The overall correction value Hf can be calculated using equation (9) from the mean value - w and the mean square value -w² of the flat-top window function.
If you calculate the overall (OA) and partial overall (POA) without applying this correction, the resulting values will be approximately 3.667 times larger in power ratio and approximately 1.918 times larger in amplitude ratio than the original values. In terms of decibels, this increases by approximately 5.655 dB.
summary
This time, we introduced a method for calculating the overall power (OA) and partial overall power (POA) from the power spectrum obtained by FFT analysis.
Overall (OA) is the sum of power values (squared values) up to the analysis frequency range, while Partial Overall (POA) is the sum of power values (squared values) within a limited frequency range. However, if you calculate it directly, the value will be larger due to the effect of the window function applied during FFT analysis, so it is necessary to apply a correction value determined by the shape of the window function.
(Excerpt from the email newsletter issued on August 25, 2016)
