Window Correction Factors
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.
While a window helps reduce leakage, the window itself distorts the data in two different ways:
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.
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.
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.
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.
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 LMS Test.Lab, 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 LMS Test.Lab displays like the FrontBack, Bode, UpperLower, etc. The 2D correction mode default is ‘Automatic’.
The selections available for 2D correction mode are:
Another important aspect in LMS Test.Lab of window correction modes is RMS calculations. When RMS calculations are performed in LMS Test.Lab, 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.
The RMS values between the cursors are identical. To calculate RMS consistently, the LMS Test.Lab 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.
RMS calculations can be used to get consistent values from windowed data.
A summary of the main points in this article:
Note: In this article, a periodic sine wave was used as an example. Window corrections are equally applicable to non-periodic signals.