563489578934

Sec. 2–4 Orthogonal Series Representation of Signals and Noise 67
A 2
Weight is
––– 4 A 2
––– 4
–f 0 f 0
f
Figure 2–9 Power spectrum of a sinusoid.
This value, A 2 /2, checks with the known result for the normalized power of a sinusoid:
P = 8w 2 2
(t)9 = W rms = 1A/122 2 = A 2 /2
(2–74)
It is also realized that A sin v 0 t and A cos v 0 t have exactly the same PSD (and autocorrelation
function) because the phase has no effect on the PSD. This can be verified by evaluating the PSD for
A cos v 0 t, using the same procedure that was used earlier to evaluate the PSD for A sin v 0 t.
Thus far, we have studied properties of signals and noise, such as the spectrum, average
power, and RMS value, but how do we represent the signal or noise waveform itself? The
direct approach is to write a closed-form mathematical equation for the waveform itself.
Other equivalent ways of modeling the waveform are often found to be useful as well. One,
which the reader has already studied in calculus, is to represent the waveform by the use of a
Taylor series (i.e., power series) expansion about a point a; that is,
w(t) = a
q
n=0
w (n) (a)
n!
(t - a) n
(2–75)
where
w (n) (a) = ` dn w(t)
dt n `
t=a
(2–76)
From Eq. (2–76), if the derivatives at t = a are known, Eq. (2–75) can be used to reconstruct
the waveform. Another type of series representation that is especially useful in
communication problems is the orthogonal series expansion. This is discussed in the next
section.
2–4 ORTHOGONAL SERIES REPRESENTATION OF SIGNALS
AND NOISE
An orthogonal series representation of signals and noise has many significant applications in
communication problems, such as Fourier series, sampling function series, and the representation
of digital signals. Because these specific cases are so important, they will be studied in
some detail in sections that follow.