The Hanning Window

The Hanning window, after its inventor whose name was Von Hann, has the shape of one cycle of a cosine wave with 1 added to it so it is always positive. The sampled signal values are multiplied by the Hanning function, and the result is shown in the figure. Note that the ends of the time record are forced to zero regardless of what the input signal is doing.

While the Hanning window does a good job of forcing the ends to zero, it also adds distortion to the wave form being analyzed in the form of amplitude modulation; i.e., the variation in amplitude of the signal over the time record. Amplitude Modulation in a wave form results in sidebands in its spectrum, and in the case of the Hanning window, these sidebands, or side lobes as they are called, effectively reduce the frequency resolution of the analyzer by 50%. It is as if the analyzer frequency "lines" are made wider. In the illustration here, the curve is the actual filter shape that the FFT analyzer with Hanning weighting produces. Each line of the FFT analyzer has the shape of this curve -- only one is shown in the figure.

If a signal component is at the exact frequency of an FFT line, it will be read at its correct amplitude, but if it is at a frequency that is one half of delta F (One half the distance between lines), it will be read at an amplitude that is too low by 1.4 dB.

The illustration shows this effect, and also shows the side lobes created by the Hanning window. The highest-level side lobes are about 32 dB down from the main lobe.

The measured amplitude of the Hanning weighted signal is also incorrect because the weighting process removes essentially half of the signal level. This can be easily corrected, however, simply by multiplying the spectral levels by two, and the FFT analyzer does this job. This process assumes the amplitude of the signal is constant over the sampling interval. If it is not, as is the case with transient signal, the amplitude calculation will be in error, as shown in the figure below.

The Hanning window should always be used with continuous signals, but must never be used with transients. The reason is that the window shape will distort the shape of the transient, and the frequency and phase content of a transient is intimately connected with its shape.

The measured level will also be greatly distorted. Even if the transient were in the center of the Hanning window, the measured level would be twice as great as the actual level because of the amplitude correction the analyzer applies when using the Hanning weighting.

A Hanning weighted signal actually is only half there, the other half of it having been removed by the windowing. This is not a problem with a perfectly smooth and continuous signal like a sinusoid, but most signals we want to analyze, such as machine vibration signatures are not perfectly smooth. If a small change occurs in the signal near the beginning or end of the time record, it will either be analyzed at a much lower level than its true level, or it may be missed altogether. For this reason, it is a good idea to employ overlap processing. To do this, two time buffers are required in the analyzer. For 50% overlap, the sequence of events is as follows: When the first buffer is half full, i.e., it contains half the samples of a time record, the second buffer is connected to the data stream and also begins to collect samples. As soon as the first buffer is full, the FFT is calculated, and the buffer begins to take data again. When the second buffer is filled, the FFT is again calculated on its contents, and the result sent to the spectrum-averaging buffer. This process continues on until the desired number of averages is collected.

DLI Engineering provides products and services for Predictive Maintenance including vibration analysis instruments, monitoring and diagnostic software, and consulting for CBM programs.

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