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Windowing Functions
Windowing functions are an important component for spectrum measurements.The CoCo includes all commonly used windowing functions including force/exponential for impact testing.
Windowing functions modify a nonperiodic signal and reduce the leakage effects in spectrum analysis.The Fourier Transform assumes that the time signal is periodic and infinite in duration. When only a portion of a record is analyzed the record must be truncated by a data window to preserve the frequency characteristics. A window can be expressed in either the time domain or in the frequency domain, although the former is more common. To reduce the edge effects, which cause leakage, a window is often given a shape or weighting function. For example, a window can be defined as
w(t) = g(t) -T/2 < t < T/2
= 0 elsewhere
where g(t) is the window weighting function and T is the window duration.
The data analyzed, x(t) are then given by
x(t) = w(t) x(t)’
where x(t)’ is the original data and x(t) is the data used for spectral analysis.
A window in the time domain is represented by a multiplication and hence, is a convolution in the frequency domain. A convolution can be thought of as a smoothing function. This smoothing can be represented by an effective filter shape of the window; i.e., energy at a frequency in the original data will appear at other frequencies as given by the filter shape. Since time domain windows can be represented as a filter in the frequency domain, the time domain windowing can be accomplished directly in the frequency domain.
In most DSA products, rectangular, Hann, Flattop and several other data windows are used;
Rectangular Window
w(k) = 1 0 £ k £ N-1
Hann Window
w(k) = 0.5 * (1 - cos (2pk /(N-1) ) 0 £ k £ N-1
Because creating data window attenuates a portion of the original data, a certain amount of correction has to be made in order to get an un-biased estimation of the spectra. In linear spectral analysis, an Amplitude Correction is applied; in power spectral measurements, an Energy Correction is applied. See the sections below for details.
Leakage Effect
Windowing of a simple signal, like a sine wave may cause its Fourier transform to have non-zero values (commonly called leakage) at frequencies other than the frequency of this sine. This leakage effect tends to be worst (highest) near sine frequency and least at frequencies farthest from sine frequency. The effect of leakage can easily be depicted in the time domain when a signal is truncated. As shown in the picture, after data windowing, truncation distorted the time signal significantly, hence causing a distortion in its frequency domain.
Illustration of a non-periodic signal resulting from sampling.If there are two sinusoids, with different frequencies, leakage can interfere with the ability to distinguish them spectrally. If their frequencies are dissimilar, then the leakage interferes when one sinusoid is much smaller in amplitude than the other. That is, its spectral component can be hidden or masked by the leakage from the larger component. But when the frequencies are near each other, the leakage can be sufficient to interfere even when the sinusoids are equal strength; that is, they become undetectable.
There are two possible scenarios that leakage does not occur. The first is that when the whole time capture is long enough to cover the complete duration of the signals. This can occur with short transient signals. For example in a hammer test, if the time capture is long enough it may extend to the point where the signal decays to zero. In this case, data window is not needed.
The second case is when a periodic signal is sampled at such a sampling rate that is perfectly synchronized with the signal period, so that with a block of capture, an integer number of cycles of the signal are always acquired. For example, if a sine wave has a frequency of 1000Hz and the sampling rate is set to 8000Hz. Each sine cycle would have 8 integer points. If 1024 data points are acquired then 128 complete cycles of the signal are captured. In this case, with no window applied you still can get a leakage-free spectrum.
The figure below shows a sine signal at 1000 Hz with no leakage resulting in a sharp spike. The figure below shows the spectrum of a 1010 Hz signal with significant leakage resulting in a wide peak. The spectrum has significant energy outside the narrow 1010 Hz frequency. It is said that the energy leaks out into the surrounding frequencies.
Sine spectrum with no leakage.
Sine spectrum with significant leakage.Several windowing functions have been developed to reduce the leakage effect. The picture below shows a Flattop window applied to the same sine signal with frequency 1010Hz:
Sine spectrum with Flattop windowing function.When Flattop window is used, the leakage effect is reduced. Both the sine peak and noise floor can be seen now. However, such data windowing operation also makes the spectrum peak “fatter” and less accurate. In the rest of the sections we will discuss how to choose different data windows.
Guidelines of Choosing Data Windows
If a measurement can be made so that no leakage effect will occur, then do not apply any window (in the software, select Uniform.). As discussed before, this only occurs when the time capture is long enough to cover the whole transient range, or when the signal is exactly periodic in the time frame.
If the goal of the analysis is to discriminate two or multiple sine waves in the frequency domain, spectral resolution is very critical. For such application, choose a data window with very narrow main slope. Hanning is a good choice.
If the goal of the analysis is to determine the amplitude reading of a periodic signal, i.e., to read EUpk, EUpkpk, EUrms or EUrms2, the amplitude accuracy of a single frequency component is more important than the exact location of the component in a given frequency bin, choose a window with a wide main lobe. Flattop window is often used.
If you are analyzing transient signals such as impact and response signals, it is better not to use the spectral windows because these windows attenuate important information at the beginning of the sample block. Instead, use the Force and Exponential windows. A Force window is useful in analyzing shock stimuli because it removes stray signals at the end of the signal. The Exponential window is useful for analyzing transient response signals because it damps the end of the signal, ensuring that the signal fully decays by the end of the sample block.
If the nature of the data is has a random nature or unknown, choose Hanning window.
