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38
Signals and Spectra Chap. 2
For any physical waveform, we are actually interested in evaluating the DC value only over a
finite interval of interest, say, from t 1 to t 2 , so that the DC value is
However, if we use a mathematical model with a steady-state waveform of infinite extent,
we will obtain the correct result by using our definition, Eq. (2–5), which involves the limit as
T : q. Moreover, as will be shown in Chapter 6, for the case of ergodic stochastic waveforms,
the time average operator 8[ # ]9 may be replaced by the ensemble average operator
¬
.
Power
t
1
2
w(t) dt
t 2 - t 1 L
In communication systems, if the received (average) signal power is sufficiently large compared
to the (average) noise power, information may be recovered. This concept was demonstrated
by the Shannon channel capacity formula, Eq. (1–10). Consequently, average power is
an important concept that needs to be clearly understood. From physics, it is known that
power is defined as work per unit time, voltage is work per unit charge, and current is charge
per unit time. This is the basis for the definition of power in terms of electrical quantities.
t 1
DEFINITION. Let v(t) denote the voltage across a set of circuit terminals, and let i(t)
denote the current into the terminal, as shown in Fig. 2–2. The instantaneous power
(incremental work divided by incremental time) associated with the circuit is given by
p(t) = v(t)i(t)
(2–6)
where the instantaneous power flows into the circuit when p(t) is positive and flows out
of the circuit when p(t) is negative.
[# ]
Example 2–1 INSTANTANEOUS POWER
Plot the instantaneous power as a function of time for a circuit that has a sinusoidal current with a
peak value of 5 A and a sinusoidal voltage with a peak value of 10 V. For the solution, see and run
the MATLAB file Example2_01.m.
DEFINITION.
The average power is
P = 8p(t)9 = 8v(t)i(t)9
(2–7)
i(t)
+
v(t)
Circuit
Figure 2–2
Polarity convention used for voltage and current.