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Sec. 2–1 Properties of Signals and Noise 41
expressing this concept is to say that the power is given on a per-ohm basis. In the signal-tonoise
power ratio calculations, R will automatically cancel out, so that normalized values of
power may be used to obtain the correct ratio. If the actual value for the power is needed, say,
at the end of a long set of calculations, it can always be obtained by “denormalization” of the
normalized value. From Eq. (2–12), it is also realized that the square root of the normalized
power is the RMS value of the voltage or the current.
DEFINITION. The average normalized power is
T/2
P = 8w 2 1
(t)9 = lim w 2 (t) dt
(2–13)
T: q T L-T/2
where w(t) represents a real voltage or current waveform.
Energy and Power Waveforms †
DEFINITION. w(t) is a power waveform if and only if the normalized average power P
is finite and nonzero (i.e., 0 6 P 6q).
DEFINITION.
The total normalized energy is
(2–14)
DEFINITION. w(t) is an energy waveform if and only if the total normalized energy is
finite and nonzero (i.e., 0 6 E 6q).
From these definitions, it is seen that if a waveform is classified as either one of these types,
it cannot be of the other type. That is, if w(t) has finite energy, the power averaged over infinite
time is zero, and if the power (averaged over infinite time) is finite, the energy is infinite. Moreover,
mathematical functions can be found that have both infinite energy and infinite power and, consequently,
cannot be classified into either of these two categories. One example is w(t) = e -t .
Physically realizable waveforms are of the energy type, but we will often model them by infiniteduration
waveforms of the power type. Laboratory instruments that measure average quantities—
such as the DC value, RMS value, and average power—average over a finite time interval. That is,
T of Eq. (2–1) remains finite instead of approaching some large number. Thus, nonzero average
quantities for finite energy (physical) signals can be obtained. For example, when the DC value is
measured with a conventional volt-ohm-milliamp meter containing a meter movement, the timeaveraging
interval is established by the mass of the meter movement that provides damping.
Hence, the average quantities calculated from a power-type mathematical model (averaged over
infinite time) will give the results that are measured in the laboratory (averaged over finite time).
Decibel
E = lim
T: q L-T/2
The decibel is a base 10 logarithmic measure of power ratios. For example, the ratio of the
power level at the output of a circuit compared with that at the input is often specified by the
decibel gain instead of the actual ratio.
T/2
w 2 (t) dt
† This concept is also called energy signals and power signals by some authors, but it applies to noise as well
as signal waveforms.