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Sec. 2–3 Power Spectral Density and Autocorrelation Function 65
By the use of Parseval’s theorem, Eq. (2–41), the average normalized power becomes
q
q
1
P = lim
(2–65)
T: q T L
ƒ W T(f)ƒ 2 ƒW T (f)ƒ 2
df = a lim bdf
- q L - q T: q T
where W T 1f2 = [w T (t)]. The integrand of the right-hand integral has units of
watts/hertz (or, equivalently, volts 2 /hertz or amperes 2 /hertz, as appropriate) and can be
defined as the PSD.
DEFINITION. The power spectral density (PSD) for a deterministic power waveform is †
w (f) ! lim a ƒW r(f)ƒ 2
b
(2–66)
T: q T
where w T 1t2 4 W T 1f2 and w 1f2 has units of watts per hertz.
Note that the PSD is always a real nonnegative function of frequency. In addition, the PSD is
not sensitive to the phase spectrum of w(t), because that is lost due to the absolute value operation
used in Eq. (2–66). If the PSD is plotted in dB units, the plot of the PSD is identical to
the plot of the Magnitude Spectrum in dB units. This is the result of the definition of dB as
given by Eqs. (2–15) and (2–18). Identical plots for the Magnitude Spectrum and the PSD are
illustrated by file Example2_17.m.
From Eq. (2–65), the normalized average power is †
q
P = 8w 2 (t)9 = P w (f) df
(2–67)
L
That is, the area under the PSD function is the normalized average power.
Autocorrelation Function
A related function called the autocorrelation, R( t), can also be defined. ‡
DEFINITION. The autocorrelation of a real (physical) waveform is §
T/2
1
R w (t) ! 8w(t)w(t + t)9 = lim w(t)w(t + t) dt (2–68)
T: q T L-T/2
Furthermore, it can be shown that the PSD and the autocorrelation function are Fourier transform
pairs; that is,
-q
R w (t) 4 w (f)
(2–69)
† Equations (2–66) and (2–67) give normalized (with respect to 1 Ω) PSD and average power, respectively.
For unnormalized (i.e., actual) values, w 1f2 is replaced by the appropriate expression as follows: If w(t) is a voltage
waveform that appears across a resistive load of R ohms, the unnormalized PSD is w 1f2/R W/Hz, where w( f) has
units of volts 2 /Hz. Similarly, if w(t) is a current waveform passing through a resistive load of R ohms, the unnormalized
PSD is w (f)R W/Hz, where w (f) has units of amperes 2 /Hz.
‡ Here a time average is used in the definition of the autocorrelation function. In Chapter 6, an ensemble
(statistical) average is used in the definition of R(τ), and as shown there, these two definitions are equivalent if w(t) is
ergodic.
§ The autocorrelation of a complex waveform is R w (t) ! 8w*(t)w(t + t)9.