4.3 Java1484-spectrum analysis using java, complex spectrum and  (Page 4/14)

I also explained that the Fourier transform of the input time series at that frequency can be viewed as a complex number having a real part and an imaginary part as in the third expression in Figure 3 .

(The amplitude of the spectrum at that frequency can be determined by computing the square root of the sum of the squares of the real and imaginaryparts at that frequency but that isn't of major interest in this module.)

The Fourier transform of an input time series can be computed by performing these calculations across a range of frequencies.

The results of performing a spectral analysis on the pulse

The bottom four plots in Figure 2 show the results of performing a Fourier transform on the pulse in the top plot.

(In this display format, which was produced by the program named Graph06, each sample value is represented by a vertical bar whose height isproportional to the value of the sample.)

These four plots show the values of the Fourier transform output at a set of uniformly spaced frequencies ranging from zero to 0.25 times the samplingfrequency.

The amplitude spectrum

The second plot from the top in Figure 2 shows the value of the amplitude spectrum. This is the Fourier transform output that we have been using in theprevious modules in this series.

(Those modules ignored the complex spectrum and the phase angle.)

As you can see, the amplitude spectrum peaks at a frequency equal to 0.0625 times the sampling frequency. The reason for this will become clear when weexamine the code that produced the pulse shown in the first plot.

The real and imaginary parts of the transform

The real part of the transform is shown in the third plot and the imaginary part of the transform is shown in the fourth plot. (I believe that this is the first time that I have presented the real and imaginary parts of the spectrum in this series ofmodules.)

The phase angle in degrees

The phase angle in degrees is shown in the bottom plot. There are a variety of different ways to display phase angles. This program displays the phase angleas values that range from -180 degrees to +180 degrees.

(It is also possible to display the phase angle as ranging from 0 to 360 degrees, or any combination that equates to 360 degrees or one fullrotation. It is also possible to display the phase angle in radians instead of degrees.)

How is the phase angle computed?

Basically, the phase angle is the angle that you get when you compute the arc tangent of the ratio of the imaginary part to the real part of the complexspectrum at a particular frequency. However, beyond computing the arc tangent, you must do some additional work to take the quadrant into account.

How should we interpret the phase angle?

To begin with, you should ignore the result of phase angle computations at those frequencies for which there is insignificant energy. It is always possibleto form a ratio of the values of the real and imaginary parts of the complex Fourier transform at any frequency. However, if the real and imaginary valuesproduced by the Fourier transform at that frequency are both very small, the phase angle resulting from that ratio is of no practical significance. In fact,the angle can be corrupted by arithmetic errors resulting from performing arithmetic on very small values.