# How to plot Full spectrum with negative frequencies

I'm trying to do frequency analysis on signal of a rotating shaft. The signal is generated in two measurement planes i,e X-position and Y-Position. I was able to use fft and transform into frequency domain for individual planes. The plot I get for each place is only Half spectra. What I'm looking for is a full spectra which could combine both the planes. I did some research and found the following description online.

Full spectrum: The full spectrum is an additional diagnostic tool and is also called the spectrum of an orbit. It shows the same information as an orbit but in a different format. It helps to determine the degree of ellipticity (or flattening) associated with the various machinery conditions along with the precessional direction for all the frequency components present. To obtain the full spectrum, the orthogonal X and Y transducer signals are fed into the direct and quadrature parts of the FFT input. The positive and negative vibration components for each frequency are obtained. Positive is defined to be the forward precession and the negative component as the reverse precession. These components yield the following ellipticity and precessional information for a given orbit of any particular frequency (1× or 2× or …):

• The sum of two components, forward and reverse, is the length of the orbit major axis. • The difference between the two components is the length of the orbit minor axis. • The larger of the two components, positive or negative, determines the direction of precession that is forward or reverse.

One of the possible applications of full spectrum is analysis of the rotor runout caused by mechanical, electrical or magnetic irregularities. Depending on the periodicity of such irregularities observed by the X–Y proximity probes, different combinations of forward and reverse components are observed. The method forms the basis for many useful machinery diagnostics.

The full spectrum (just like the normal FFT) can be obtained in a steady-state analysis (a single FFT or waterfall) and even in transient analysis, which would then be called the full spectrum cascade (Figure attached).

Could anyone help how the 2 half spectra is converted into full spectra plot. Does anyone have any idea about this? Any hints or suggestions please!!

• don't you just take the fft of x+i*y? – endolith Feb 15 '15 at 19:00
• @endolith did that but the question now is how to derive the phase angle? drive.google.com/file/d/0Byib6yrCQ9vnRUM2Yi1sMklXeFk/… – Agni Feb 15 '15 at 19:44
• the same way you derive the phase angle for any complex number: en.wikipedia.org/wiki/… the FFT gives you a bunch of complex numbers, and typically you work with the magnitude and phase of each one. – endolith Feb 15 '15 at 20:56
• @endolith But that would give only one complex number So now I can only derive one phase angle. But if you can see in the figure I have shared in my previous comment. There are 2 phase angles alpha and beta. So How to deal with the other phase angle? Or since the X and Y sensors are orthogonal positioned, so assume that one angle is whatever I get from the complex number and the other is (90-phase angle)? – Agni Feb 16 '15 at 6:02
• I don't see any alpha and beta in your figures. As you said in your post "The positive and negative vibration components for each frequency are obtained. Positive is defined to be the forward precession and the negative component as the reverse precession." I'm not sure what the question is. – endolith Feb 17 '15 at 17:03

## 1 Answer

Take the fft of x+i*y.

The phase angles in the picture you linked https://drive.google.com/file/d/0Byib6yrCQ9vnRUM2Yi1sMklXeFk/view are from the result of this fft.

The image linked have an error, from the article it came from, where the phases of the negative frequencies of the fft result are Beta. The image uses Alpha in the negative and positive results. In the end, the image just shows mathematical prove that the amplitudes of the fft are derived from the amplitudes of the individual signals.

The fullspectrum process is simple as the fft of x+i*y. The majoraxis of an ellipsis of one frequency will be the sum of the R+ and R-, and the direction will be the (Alpha+Beta)/2.

This paper could clarify it more "Use of directional spectra of vibration signals for diagnosis of misalignment in rotating machinery"