App Note #8
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2/25/2014
ME’scope Application Note #8 Peak & Power Values in Auto Spectrum & PSD Measurements INTRODUCTION NOTE: The steps in this Application Note can be duplicated using any Package that includes the VES-3000 Signal Processing option. The Auto Spectrum shows how the power of a signal is distributed in the frequency domain. The Auto Spectrum is calculated by multiplying the Fourier spectrum (FFT) of a time waveform by its complex conjugate. The Power Spectral Density (PSD) is the Auto Spectrum normalized to a 1 Hz bandwidth. A PSD is calculated by dividing a Auto Spectrum by its frequency resolution ().
In ME’scope, Fourier spectrum values can be calculated using either a two sided FFT or a one sided FFT. Spectrum values of a two sided FFT are 1/2 of the values of a one sided FFT. Consequently, Power spectrum values from a two sided FFT must be multiplied by 4 to compare them with Power spectrum values from a one sided FFT. NOTE: The Power calculation in ME’scope corrects for one sided versus two sided FFTs. Power can also be displayed as a Linear quantity by taking the square root of its Power value. This is the Linear (or RMS) Power.
POWER
EXAMPLE #1 - Periodic Sine Wave
ME’scope has a command for displaying the power in a Trace, or the power in a band if the Band cursor is displayed. The power in a time domain waveform is calculated as,
In this example, we will measure the power in a sine wave and in its Auto spectrum and PSD. To eliminate the effects of leakage, we will synthesize a sine wave that is periodic in its sampling window. (See to Application Note #1 for details on leakage.)
T
Power
1 f ( t ) 2 dt T t 0 1 N
Real N
i 1
2
Open the App Note 08-Power Measurements.VTprj from the ME’copeVES\Application Notes folder.
Click on BLK: 3125 in the Project panel to open its window
i
where: N = the number of samples of Trace data. Real i = the time waveform data for the ith sample.
This window contains a 0.3125 Hz sine wave with a peak magnitude of 1.0. Notice that the Trace (shown in Figure 1) has exactly 4 cycles of the sine wave in it.
In a Linear frequency spectrum (also called Root Mean Squared or RMS spectrum), Power is calculated as,
Power
1 2
Real
1 2
Mag
N
i 1
2 i
N
i 1
Imagi
2
2
i
where: Real i = the Real part of the ith sample Imag i = the Imaginary part of the ith sample Mag i = the magnitude of the ith sample. In a Power frequency spectrum (also called Mean Squared or MS spectrum), Power is calculated as,
1 N Power Mag i 2 i 1
Figure 1. Periodic Sine Wave. Since an integer number of cycles are contained within its window, this signal is said to be periodic in the sampling window. Since it is periodic in the window, the FFT will calculate its frequency spectrum without leakage.
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App Note #8
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Power in the Time Waveform
Right click in the Trace graphics area, and execute Tools | Statistics
Notice that the waveform statistics are displayed to the right of the Trace. The Power = 0.50 (g)2, and the Linear Power = 0.707 g’s. Auto Spectrum of a Periodic Sine Wave
2/25/2014
Display the Line cursor, and place it on the peak in the Auto spectrum.
Notice that the peak value at 0.3125 Hz is 0.25 (g)2 or 0.5 g’s, only half of the original sine wave peak value. Half of this signal (0.5 g’s) is represented by a negative frequency peak at -0.3125 Hz and half by its positive frequency peak at 0.3125 Hz. One Sided FFT
Now we will calculate the Auto spectrum of this periodic sine wave and display its peak and power values.
Right click in the Trace graphics area, and execute Data Block Options.
On the Show/Hide tab in the dialog box that opens, check the FFT Trace column, and click on OK
In the FFT column of the Traces spreadsheet, select Two Sided
Execute Transform | Spectra. A dialog box will open.
Make sure that Auto spectrum is selected in the list, and click on Calculate
Click on OK in the next dialog box. Another dialog box will open
Click on the New File button, enter “Auto Spectrum” into the dialog box, and click on OK.
In the FFT column of the Traces spreadsheet, select One Sided
Calculate the Auto Spectrum over again using the steps above.
Now, when you place the Line cursor on the peak at 0.3125 Hz, the cursor value will be the expected value of “1”. Power in the Auto Spectrum Notice that the Power and Lin Power in the signal are also the same as the time domain values. Hence, the power in a signal is the same in both the time & frequency domains, and the power is not changed by using a one sided versus two sided FFT.
The resulting Auto spectrum of the sine wave is shown in Figure 2. As expected, it has a single peak at 0.3125 Hz, the frequency of the sine wave.
Figure 3. One Sided Auto Spectrum of Periodic Sine. A Linear Auto spectrum (with linear units) is simply the square root of a Power Auto spectrum (with power units). Power units are squared units. Notice Auto spectrum units in Figure 3 that (g)^2. To convert the Auto spectrum from Power to Linear units,
Figure 2. Two Sided Auto Spectrum of Periodic Sine. Notice that the Power and Lin Power in the signal are the same as it time domain values. With a two sided FFT half the power is distributed to the positive frequencies and half to the negative frequencies in the spectrum.
Right click in the Trace graphics area, and execute Data Block Options.
On the Show/Hide tab in the dialog box that opens, check the Linear Power column, and click on OK
In the Linear Power column of the Traces spreadsheet, select linear engineering units (EU).
Peak Value in the Auto Spectrum
Zoom the Trace display to show the peak at 0.3125 Hz more clearly.
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App Note #8
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2/25/2014
Right click in the Trace graphics area, and execute Data Block Properties to open the Properties dialog box displaying the frequency resolution.
Notice in Figure 5 that the power in the PSD is 0.5 (g)2, as expected since it is a power quantity. Linear PSD A Linear PSD (with linear units) is the square root of the PSD (with squared or power units). To convert the PSD to linear units,
In the Linear Power column of the Traces spreadsheet, select linear engineering units (EU).
Place the Line cursor on the peak in the PSD.
Figure 4. Linear Auto spectrum. Calculating a PSD To calculate a PSD for the sine wave in BLK: 3125,
Execute Transform | Spectra in the BLK: 3125 window.
In the dialog box that opens, choose BLK: 3125 as the Source File, choose Power Spectral Density from the spectrum pick list, and click on the Calculate button.
Click on OK in the Spectrum Averaging dialog box
Click on the New File button in the next dialog box that opens, enter “PSD”, and click on OK.
The BLK: PSD Data Block window will open. The PSD of the sine wave also has a single peak at 0.3125 Hz, the frequency of the sine wave.
Use Zoom to display the peak clearly.
Display the Line cursor, and position it on the peak in the PSD.
Figure 6. Linear PSD Showing Power = 0.707 g’s. Notice that the magnitude at 0.3125 Hz is now 3.578 (g/(Hz)1/2), and that the power has remained the same; 0.707 g’s or 0.50 (g)2. EXAMPLE #2 - Non-Periodic Sine Wave Non-periodic time waveforms yield different results than periodic signals when transformed from the time to the frequency domain. This is due to leakage effects. The spectrum of a non-periodic sine wave is “smeared” over several frequencies, instead of being a single peak only at the frequency of the sine wave. Also, leakage drastically affects the peak value of a sine wave signal in its frequency spectrum.
It is evident that this Data Block contains a non-periodic sine wave, since it completes 4 ½ cycles.
Figure 5. PSD of the Periodic Sine Wave. Notice that the cursor value of the PSD at 0.3125 Hz is 12.8 (g)2/Hz. This value is equal to the peak value of the Auto spectrum (1.0) divided by the frequency resolution of the spectrum, = 1/T = 1/(12.8 sec) = 0.078125 Hz
Open BLK: 3515 from the Project panel.
Power in the Time Waveform
Right click in the Trace graphics area, and execute Tools | Statistics
Notice in Figure 7 that the power is the same as the periodic sine wave; Linear Power = 0.707 g’s, Power = 0.50 (g)2.
To check the frequency resolution,
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App Note #8
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2/25/2014
Right click in the Trace graphics area, and execute Tools | Statistics
Notice that the power has remained the same (0.50 (g)2) in spite of the leakage. Leakage has merely spread the power to other frequencies in the frequency spectrum. CONCLUSIONS From these two examples, we can conclude the following,
Figure 7. Non-Periodic Sine Wave.
If a sine wave is periodic in its sampling window and a One Sided FFT is used, the peak value of its Auto spectrum is equal to its time domain peak value.
If a sine wave is periodic in its sampling window and a Two Sided FFT is used, the peak value of its Auto Spectrum is one half of its time domain peak value.
The power in a Linear (RMS) frequency spectrum is equal to the square root of the power in a Power (MS) frequency spectrum.
Auto Spectrum of a Non-Periodic Sine Wave Now we will calculate the Auto spectrum of the nonperiodic sine wave and display its peak and power values.
Execute Transform | Spectra
Choose BLK: 3515 in the dialog box that opens, choose Auto spectrum from the spectrum list, and click the Calculate button.
If a sine wave is non-periodic in its sampling window, the peak value of its Auto Spectrum or PSD will always be less than its time domain peak value, due to leakage.
Click on OK in the Spectrum Averaging dialog box.
Click on the New File button in the next dialog box, enter “Non-Periodic Auto spectrum”, and click on OK.
The total power in the Auto spectrum of a signal is the same as the power in its time domain signal, independent of whether the time domain signal is periodic or non-periodic in its sampling window.
Figure 8. Auto Spectrum of Non-Periodic Sine Wave. A new Data Block window will open with the Auto spectrum in it, as shown in Figure 8. The Auto spectrum of has a peak at approximately 0.3515 Hz, the frequency of the sine wave.
Use Zoom to display the peak clearly.
Display the Line cursor, and position it on the peak in the Auto spectrum.
Notice that the peak amplitude of the Auto spectrum is only 0.114 (g)2 instead of 1.0 (g)2. The sine wave frequency is 0.3515 Hz, but there is no sample at that frequency. Instead, the peak is at 0.3525 Hz. This is due to leakage.
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