In one of our recent articles a question was asked regarding the practical use of real & imaginary type plots compared with modulus & phase type plots.
In general, noise or vibration signals are composed of one or more sinusoidal signals which can be quantified in terms of their magnitude (modulus) and phase components. These can be visualised as the result of a spectral transformation in the form of either a real & imaginary plot or a modulus & phase plot. The moduli & phases are directly related to the real & imaginary components. Real & imaginary is one way to visualise a complex number, modulus & phase is another way.
A sinusoidal vibration will have a magnitude which is the amount it is moving up and down. The phase component of the same signal is how much this sinusoid is delayed (in terms of an angle) compared with a reference sinusoid moving with the same frequency. This is an important point; the phase is only relevant if it is relative to a reference.
Say, for example, a simple, floating (free-free) beam has some form of sinusoidal excitation applied to one end of the beam and accelerometers are used to measure the responses at both ends. When both the phases are the same (at the excitation position and the response position) the ends of the beam will be moving up and down together as shown in Figures 1 and 2.
But when both the phases are not the same, the ends of the beam will be moving up and down at different times and the beam will pitch from side to side, as shown in Figures 3 and 4.
Now, to return to the original question, specifically about real & imaginary numbers:
A Real & Imaginary pair of numbers defines the position of the end point of a straight line drawn from the origin (0,0) of a two dimensional plot; one of the dimensions is the horizontal (real) part and the other dimension is the vertical (imaginary) part. This is known as rectangular complex data.
Real & imaginary data can also be expressed in the form of a pair of modulus & phase numbers, this is also known as polar complex. In this form the modulus is the distance from the origin and the phase is the angle that the line makes with the horizontal axis.
Essentially they are two different representations of the same data. Figures 7 and 8 show a frequency spectrum first in rectangular complex form and then in polar complex form respectively.
The output of a Fast Fourier Transform (FFT) analysis of a time signal is a spectrum of complex (real & imaginary) numbers. However, the human mind better understands and can visualise more easily a complex frequency spectrum when the data is displayed in the form of a modulus & phase plot as shown in Figure 8.
The same complex data can also be shown in two other ways: either by plotting the real spectrum versus the imaginary spectrum or the modulus spectrum versus the phase spectrum. This type of plot is commonly known as a Nyquist plot as shown in Figure 9 below. This combines the magnitude (modulus) and angle (phase) information into a single plot.
The data at each point of Nyquist plot corresponds to the complex amplitude at a particular frequency. Usually the data being displayed would be some sort of frequency response function which depicts how the response at a particular point varies in both amplitude and phase with changing frequency.
If a flexible structure is excited by a sinusoidal input force then in general the response in the time domain at any position will not be in phase with the excitation as shown in Figure 10 below.
From the time histories of the Excitation and Response the delay between the two signals can clearly be seen as both a time shift and a phase angle shift. In mathematical terms the response at a single frequency can be expressed as
The frequency response at all frequencies from zero to half the sampling rate can be represented by a modulus spectrum of values and a phase angle spectrum of values.
Normally, in noise and vibration work, we would use the modulus & phase form of presenting data. So what is the practical application of using the real & imaginary form? A good example would be modal analysis. Modal resonances usually appear as circles when a Frequency Response Function (FRF) is plotted in real versus imaginary form or as clearly defined peaks when the FRF is plotted as a modulus spectrum. However, in some situations when there are two or more resonance frequencies close together the two peaks can merge into a single peak thereby giving the impression that there is just a single modal resonance when in fact there are multiple resonances. By viewing an FRF as a Nyquist plot it is often easier to discern the presence of additional modes.
In our experience the modulus & phase form of display is by far the most widely used, and often the phase is not displayed at all. There are, however, cases where a real versus imaginary type of display shows phenomena that may be missed by the modulus & phase type of display.
Sound & Vibration Signal Processing Analyst at Prosig
Adrian Lincoln is Signal Processing Technology Manager at Prosig Ltd and has responsibilities for signal processing applications, training and consultancy. He was formerly a Research Fellow at the Institute of Sound & Vibration Research (ISVR) at Southampton University. He is a Chartered Engineer and member of the British Computer Society and Institute of Mechanical Engineers.