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How to Select the Window Length When Extracting Orders with Order Analysis

Primary Software: LabVIEW Toolkits>>Order Analysis Toolset
Primary Software Version: 2.0
Primary Software Fixed Version: N/A
Secondary Software: N/A

Problem: I am trying to extract individual orders from my signal using the Order Analysis Toolkit and have noticed that the window length that I use will affect the shape of the order which is extracted. How do I know what window length I should use?

Solution: The window length in OAT Extract Order Waveforms VI largely determines the extracted order waveforms. To get accurate results of the specified order waveforms, you need to properly select the window length according to the characteristics of the signal and other parameters such as bandwidth.

OAT Extract Order Waveforms VI uses Gabor transform and Gabor expansion to extract order waveforms. It maps the signal to the time-frequency domain with the Gabor transform, locates the time-frequency components of the specified orders according to the speed information, and reconstructs the time signal with the time-frequency components of the specified orders through Gabor expansion. The reconstructed signal, or order waveforms are greatly affected by the time-frequency representation of the signal.

Ideally you would like to have good frequency resolution as well as good time resolution. Unfortunately, the bandwidth-time limitation, or uncertainty principle, shows us that the time duration T of a signal and bandwidth (usually equal to frequency increment) B has the following relationship:

This equation means that you can not get a good frequency resolution (requires a small B) and a good time resolution (requires a small T) at the same time. You have to balance the time and frequency resolution according to your input signal and other test specifications.

When choosing a large window length, the frequency resolution gets improved so that you can observe the slowly-varying signal in more detail. Whereas, with a small window block, you can observe the quickly-changing signal with a coarse frequency resolution. The following figure shows the time-frequency plot of a run-up vibration signal with small and large window length. The left plot uses a small window length to get a high time resolution while the right plot shows high frequency resolution with a large window length.

When extracting order waveforms of a certain order, you need to consider two things to get accurate results:

• The order waveform should not contain energy from adjacent orders so that you do not get any distorted order information.
  
• The energy of a certain order should be wholly extracted so that you do not lose any useful information of a certain order.
  

To ensure that the order waveforms do not get distorted, you need to make sure that the frequency increment (df) is less than the adjacent significant orders. That is:

where,
f_s is the sampling rate,
N is the window length,
n is the distance of adjacent significant orders that can cause distortion in unit of orders.
speed_min is the minimum speed value at which you need to separate close orders in unit of rpm.

To ensure the energy of the specified order does not get lost in extraction, you need to make sure that frequency change of the specified order is less than the frequency range of order bandwidth. That is:

where,
a_rpm is the acceleration rate in rpm/s.
N is the window length,
f_s is the sampling rate,
bandwidth is the bandwidth value specified in bandwidth in orders.
speed is the speed value at which you need to extract order waveforms in unit of rpm.
order is the specified order to extract.

When using the above equations, there are some guidelines you need consider:

1. The window length should be a power of 2.
   
2. If the significant orders are close to each other, you need to choose an appropriate small n in equation 1 to meet equation 1 strictly. Otherwise, you may choose a large n which will not cause much distortion.
   
3. When the speed changes rapidly, you should choose a speed value which will satisfy equation 2. This will ensure you to keep the significant vibrations in order waveforms.
   

However, in some run up or coast down tests, it is difficult to meet both requirements at the same time. Thus, you have to make a compromise. This compromise is mainly determined by the energy distribution of the signal at different speed. You can use the spectral map to see the energy distribution of your signal. For example, if the strong vibration is mostly at high speeds, you may consider use a large speed_min in equation 1 because distortion at low speed will have less impact. Also, when the speed changes quickly, you may consider reducing your bandwidth by setting it equal to 1.

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Report Date: 06/17/2005
Last Updated: 06/27/2005
Document ID: 3MGB8LNQ

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