4.2 Java1483-spectrum analysis using java, frequency resolution  (Page 14/19)

Create more complex experiments. For example, you could create pulses containing three or more sinusoids at closely spaced frequencies, and you couldcause the amplitudes of the sinusoids to be different. See what it takes to cause the peaks in the spectra of those pulses to be separable and identifiable.

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.

Try using the random number generator of the Math class to add some random noise to every value in the 400-sample time series. See whatthis does to your spectral analysis results.

Move the center frequency up and down the frequency axis. See if you can explain what happens as the center frequency approaches zero and as the centerfrequency approaches the folding frequency.

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

Summary

This program provides the code for three spectral analysis experiments of increasing complexity.

Bandwidth versus pulse length

The first experiment performs spectral analyses on five simple pulses consisting of truncated sinusoids. This experiment shows:

• Shorter pulses require greater bandwidth.
• The bandwidth of a truncated sinusoidal pulse is inversely proportional to the length of the pulse.

Peak resolution versus pulse length and frequency separation

The second experiment performs spectral analyses on five more complex pulses consisting of the sum of two truncated sinusoids having closely spacedfrequencies. The purpose is to determine the required length of the pulse in order to use spectral analysis to resolve spectral peaks attributable to the twosinusoids. The experiment shows that the peaks are barely resolvable when the length of the pulse is the reciprocal of the frequency separation between thetwo sinusoids.

Five pulses with barely resolvable spectral peaks

The third experiment also performs spectral analyses on five pulses consisting of the sum of two truncated sinusoids having closely spacedfrequencies. In this case, the frequency separation for each pulse is the reciprocal of the length of the pulse. The results of the spectral analysisreinforce the conclusions drawn in the second experiment.

What's next?

So far, the modules in this series have ignored the complex nature of the results of spectral analysis. The complex results have been converted into realresults by computing the square root of the sum of the squares of the real and imaginary parts.

The next module in the series will meet the issue of complex spectral results head on and will explain the concept of phase angle. In addition, the modulewill explain the behavior of the phase angle with respect to time shifts in the input time series.

Complete program listings

Complete listings of the main programs discussed in this module are provided in this section. Listings for other programs mentioned in the module, such as Graph03 and Graph06 , are provided in other modules. Those modules are identified in the text of this module.