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Signal Power, Energy, and Frequency

Module by: Carlos E. Davila. E-mail the author

Signals can be characterized in several different ways. Audio signals (music, speech, and really, any kind of sound we can hear) are particularly useful because we can use our existing notion of “loudness" and “pitch" which we normally associate with an audio signal to develop ways of characterizing any kind of signal. In terms of audio signals, we use “power" to characterize the loudness of a sound. Audio signals which have greater power sound “louder" than signals which have lower power (assuming the pitch of the sounds are within the range of human hearing). Of course, power is related to the amplitude, or size of the signal. We can develop a more precise definition of power. The signal power is defined as:

p x = lim T 1 T - T / 2 T / 2 x 2 ( t ) d t p x = lim T 1 T - T / 2 T / 2 x 2 ( t ) d t
(1)

The energy of this signal is similarly defined

e x = - x 2 ( t ) d t e x = - x 2 ( t ) d t
(2)

We can see that power has units of energy per unit time. Strictly speaking, the units for energy depend on the units assigned to the signal. If x(t)x(t) is a voltage, than the units for exex would be volts22-seconds. Notice also that some signals may not have finite energy. As we will see shortly, periodic signals do not have finite energy. Signals having a finite energy are sometimes called energy signals. Some signals that have infinite energy however can have finite power. Such signals are sometimes called power signals.

We use the concept of “frequency" to characterize the pitch of audio signals. The frequency of a signal is closely related to the variation of the signal with time. Signals which change rapidly with time have higher frequencies than signals which are changing slowly with time as seen Figure 1. As we shall see, signals can also be represented in terms of their frequencies, X(jΩ)X(jΩ), where ΩΩ is a frequency variable. Devices which enable us to view the frequency content of a signal in real-time are called spectrum analyzers.

Figure 1: The signal y(t)y(t) contains a greater amount of high frequencies than x(t)x(t).
Figure 1 (pitch.png)

Something to keep in mind is that the signals shown in Figures (Reference), (Reference), and (Reference) each have different units (degrees Fahrenheit, pressure, and voltage, respectively). So while we can compare relative frequencies between these signals, it doesn't make much sense to compare their power since each signal has different units. We will take a more formal look at the frequency of signals starting in Chapter 2.

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