## Order Cuts And Overall Level

Order cuts are taken from a set of FFTs, each one at a different rpm. The rms level is then found as the Square root of the Sum of the squares of each of the FFT values. Mathematically, if $x_{ks}$ is the modulus (magnitude) of the $k^{th}$ value of the FFT at speed s for $k = 1,\dots,N-1$ then the rms value at that speed is given by

$rms_s = \sqrt{\sum_{k=0}^{N-1}{x_{ks} ^2}}$

This takes into account the entire energy at that speed both the order and the non order components, including any noise.

Graphically, suppose we have a waterfall such as shown below.

Extracting the FFT at, say, 4200 rpm gives the graph shown below.

The rms considers all of the energy in the entire spectrum. That is, in the graph below, it considers all of the blue shaded area. All the values are squared, summed and the result is square rooted, which gives the rms value.

When one is finding the order values we know that because the speed is changing the actual order will have been skewed over several frequencies. Also if is extremely unlikely that the frequency values at which the FFT was evaluated will exactly match the frequency of the order at that speed. The FFT spectrum thus suffers from leakage as well as smearing. The net effect of both of these artefacts is that the spectrum is “broadened and flattened”. That in the energy is now distributed over several frequencies so that once again one has to use a square root of sum of squares approach to determine the rms level centred over the frequency range of the order. This frequency range is usually referred to as the order bandwidth. Taking the peak valve will be grossly inaccurate.

Mathematically then we have that the component of Order j at speed s is given by

$Order_{js} = \sqrt{\sum_{k=j-b}^{j+b}{x_{ks} ^2}}$

where b is the half bandwidth. The actual implementation is a little more sophisticated than above, but the signal processing essence is as shown.

Again, graphically, we now consider only part of the spectrum centered at the order frequency and over a finite bandwidth.

For a specific order we would only consider the blue part as illustrated above. Once again all the “blue” values are squared, summed and the result is square rooted, which now gives the rms value of that specific order at the selected speed.

Now if we take all of the significant orders, square, sum and square root them, then we will have an estimate of the rms value. If the orders are well spaced then we will miss some of the noise and the very small orders. So we would expect the rms calculated from the orders to be smaller than the overall rms value. Note that the correct way of collecting the orders together is the square root of the sum of squares. Adding the orders together directly will give an incorrect value.

This is not the end of the story because sometimes the bandwidth of the individual orders overlap so that some energy is counted twice. In this situation the rms calculated will be greater than the true rms. If this occurs then we know we have had order bandwidth overlap.

For reference, one may be used to seeing the rms value calculated from time series. When computing the value from the time series, a division by the number of values in the summation occurs. This division does not occur if we are computing the rms from the frequency spectrum. Mathematically, if $x_j$ is the time series and $x_k$ is its Fourier transform then we have

$rms = \sqrt{\frac{\sum_{j=0}^{N-1}{x_j ^2}}{N}} = \sqrt{\sum_{k=0}^{N-1}{x_k^2}}$

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#### Dr Colin Mercer

Chief Signal Processing Analyst at Prosig
Dr Colin Mercer was formerly at the Institute of Sound and Vibration Research (ISVR), University of Southampton where he founded the Data Analysis Centre. He then went on to found Prosig in 1977. Colin retired as Chief Signal Processing Analyst at Prosig in December 2016. He is a Chartered Engineer and a Fellow of the British Computer Society.

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