4.2 Java1483-spectrum analysis using java, frequency resolution  (Page 5/19)

Spectrum of an ideal sinusoid

That having been said, different people will probably interpret these results in different ways. Let's begin by stating that the theoretical spectrum for asinusoid of infinite length in the absence of noise is a single vertical line having zero width and infinite height.

In the real world of measurements, however, there is no such thing as a sinusoid of infinite length. Rather, every measurement that we make musttruncate the sinusoid at some point in time. For a theoretical signal of infinite length, every spectral analysis that we can perform is an imperfectestimate of the spectrum.

Two viewpoints

There are at least two ways to think of the pulses shown in Figure 1 .

1. Each pulse is a truncated section of an ideal sinusoid of infinite length.
2. Each pulse is a signal having a definite planned start and stop time.

The way that you interpret the results shown in Figure 2 depends on your viewpoint regarding the pulses.

The first viewpoint

If your viewpoint is that each pulse is a truncated section of an ideal sinusoid of infinite length, then the width of each of the peaks (beyond zero width) is the result of measurement error introduced by the truncation process.

The second viewpoint

If your viewpoint is that each pulse is a signal having a definite planned start and stop time, then the widths and the shape of each of the peaksdescribes the full range of frequency components required to physically generate such a pulse. This is the viewpoint that is consistent with the hypothetical situation involving a device on a submarine that I described earlier in this module.

A simplified hypothetical situation

Assume for the moment that the hypothetical device on the submarine contains only one rotating machine and that this device is turned on and off occasionallyin short bursts. Because of the rotating machine, when the device is turned on, it will emit acoustic energy whose frequency matches the rotating speed of themachine.

(In reality, it will probably also emit acoustic energy at other frequencies as well, but we will consider it to be a very ideal machine. Wewill also assume the complete absence of any other acoustic noise in the environment.)

Assume that you have a recording window of 400 samples, and that you are able to record five such bursts within each of five separate recording windows.Further assume that the lengths of the individual bursts match the time periods indicated by the pulses in Figure 1 .

The spectra of the bursts

If you perform spectral analysis on each of the five individual 400-sample windows containing the bursts, and if you normalize the peak values for plottingpurposes, you should get results similar to those shown in Figure 2 .

The spectral bandwidth of the signal

The frequency range over which energy is distributed is referred to as the bandwidth of the signal. As you can see in Figure 2 , shorter pulses require wider bandwidth.

For example, considerably more bandwidth is required of a communication system that is required to reliably transmit a series of short truncatedsinusoids than one that is only required to reliably transmit a continuous tone at a single frequency.