4.3 Java1484-spectrum analysis using java, complex spectrum and  (Page 13/14)

(I did make one other change. This change was to add a tiny spike to one of the samples near the center of the pulse. This creates a tiny amountof wide-band energy and tends to stabilize the computation of the phase angle. It prevents the imaginary part of the spectrum from switching backand forth between very small positive and negative values due to arithmetic errors.)

The spectral analysis output

The output from the spectral analysis is shown in Figure 13 . The magnitude spectrum hasn't changed. The real part of the spectrum has changedsignificantly. It is now a true (sin x)/x curve with both positive and negative lobes.

Figure 13. Spectral analysis of 11-sample boxcar centered in time.

The imaginary part of the spectrum is zero or nearly zero at every frequency. (It would be zero in the absence of arithmetic errors.)

The phase angle is zero across the entire main energy lobe of the spectrum. It is -180 degrees in those frequency areas where the real part of the spectrumis negative, and is zero in those frequency areas where the real part of the spectrum is positive. There is no linear phase shift because the boxcar pulse iscentered on the time origin.

Probably more than you ever wanted to know

And that is probably more than you ever wanted to know about the complex spectrum, phase angles, and time shifts. I will stop writing and leave it atthat.

Run the program

I encourage you to copy, compile, and run the program provided in this module. Experiment with it, making changes and observing the results of yourchanges.

Create more complex experiments. For example, you could create pulses of different lengths with complex shapes and examine the complex spectra and phaseangles for those pulses.

If you really want to get fancy, you could create a pulse consisting of a sinusoid whose frequency changes with time from the beginning to the end of thepulse. (A pulse of this type is often referred to as a frequency modulated sweep signal.) See what you can conclude from doing spectral analysis on a pulse of this type. Pay particular attention to the phase angle across thefrequency band containing most of the energy.

Most of all enjoy yourself and learn something in the process.

Summary

The default pulse for the Dsp034 program is a damped sinusoid. This is a pulse whose shape is commonly found in mechanical andelectronic systems in the real world. The phase angle in the complex spectrum for a pulse of this shape is nonlinear. Among other things, nonlinear phaseangles introduce phase distortion into audio systems.

The simplest pulse of all is a single impulse. A pulse of this type has an infinite bandwidth (theoretically) and a linear phase angle. The slope of the phase angle depends on the location of the pulse relative to the timeorigin.