Using a window to combat the evil effects of leakage on data? That is great!
But hold on! The window itself also distorts your data as shown in Figure 1. It reduces the both the amplitude and energy of the signal.
Figure 1: Multiplying a signal (blue) by a Hanning window (green) reduces the amplitude and energy in the signal (red).
While a window helps reduce leakage, the window itself distorts the data in two different ways:
Amplitude – The amplitude of the signal is reduced
Energy – The area under the curve, or energy of the signal, is reduced
Window correction factors are used to try and compensate for the effects of applying a window to data. There are both amplitude and energy correction factors.
Amplitude and Energy Correction Factors
To correct for amplitude or energy distortion, each spectral line of a windowed frequency spectrum are multiplied by a fixed factor. This factor is determined by the window type that was applied.
For different types of windows, different correction factors are used, as summarized in Figure 2. Only the Uniform window, which is equivalent to no window, has the same amplitude and energy correction factors.
Figure 2: Table of Window Correction Factors
For example, for a Hanning window, the amplitude correction factor is 2.00, while the energy correction factor is 1.63. Correction factors are applied in the frequency domain by multiplication of all the amplitude values across the spectrum.
Hanning Window Example
Take an example of a Hanning window applied to a periodic sine wave. The amplitude of a periodic sine wave is reduced by a factor of two by the window as shown in Figure 3.
Figure 3: Applying a Hanning window to a sine wave (blue) reduces the amplitude by a factor of 2 (red).
It is possible to correct both the amplitude and energy content of the windowed signal to equal the original signal. However, both corrections cannot be applied simultaneously.
To correct the amplitude, for a Hanning window, all the values in the frequency spectrum are multiplied by two as shown in Figure 4.
Figure 4: With amplitude correction, by multiplying by two, the peak value of both the original and corrected spectrum match. However the energy content is not the same.
Notice that the energy in the two signals shown in Figure 3 are now different after the multiplication by a factor of two. The amplitude corrected signal (red) appears to have more energy, or area under the curve, than the original signal (blue).
A different correction value has to be used to adjust the energy content of the signal. To correct the energy content for a Hanning window, the spectral values must be multiplied by 1.63 factor, instead of 2 as shown in Figure 5.
Figure 5: Multiplying the values in the spectrum by 1.63, rather than 2, makes the area under the curve the same for both the original signal (blue) and energy corrected signal (red).
Now the peak amplitude values of the two spectrums do not match. Only one type of window correction can be applied at a time from a visualization point of view.
This Hanning window example used a periodic sine wave, where the effects of the window can be plainly seen. However, correction factors are applicable to non-periodic signals as well.
In Simcenter Testlab, under ‘Tools -> Options -> General’, the window correction behavior can be set with ‘2D Correction Mode’ as shown in Figure 6. This governs which correction mode is used in 2D Simcenter Testlab displays like the FrontBack, Bode, UpperLower, etc. The 2D correction mode default is ‘Automatic’.
The selections available for 2D correction mode are:
Automatic – Correction mode is set based on type of measurement:
Spectrum, Autopower, and Orders – Amplitude Correction mode
Power Spectral Density – Energy Correction mode
Fixed Amplitude – All type of spectrums are displayed with Amplitude Correction mode
Fixed Energy – All type of spectrums are displayed with Energy Correction mode
Not Corrected – No window correction is applied to the spectrums. Expect the amplitude to be the lowest of all the possible ‘2D Correction modes’
Original – If the data was acquired with a specific correction mode active, will be displayed with this mode
Another important aspect in Simcenter Testlab of window correction modes is RMS calculations. When RMS calculations are performed in Simcenter Testlab, energy correction is automatically applied to get the correct RMS value. This conversion to energy correction values is done even if the spectrum is being displayed with amplitude correction.
A RMS value can be calculated by right clicking in the display and selecting ‘Add Double Cursor -> X’. Right click on the cursor and select ‘Calculations -> RMS’. For more information, see the "Double X Cursor Knowledge Base article";.
The amplitude corrected spectrums in Figure 7 appear to have different ‘areas under the curve’, but the RMS calculation values are the same.
Figure 7: RMS is identical for original signal (blue) and amplitude corrected signal (red)
The RMS values between the cursors are identical. To calculate RMS consistently, the Simcenter Testlab software changes the spectral data to energy corrected values during the calculation, even if they are displayed with amplitude correction.
Naturally, if the spectral data is energy corrected, the RMS values are also identical as shown in Figure 8.
Figure 8: RMS is identical for original signal (blue) and energy corrected signal (red)
RMS calculations can be used to get consistent values from windowed data.
A summary of the main points in this article:
Applying a window to a signal reduces the amplitude and energy of the signal
In the frequency domain, either the amplitude or energy can be corrected by multiplying by the appropriate correction factor
The RMS calculation in Simcenter Testlab will always use energy corrected values, even if amplitude corrected values are being displayed, to get the correct RMS value
Note: In this article, a periodic sine wave was used as an example. Window corrections are equally applicable to non-periodic signals.