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70
Signals and Spectra Chap. 2
obtaining
b
b
[w(t)]w m
*(t) dt = c an
a n w n (t) dw m
*(t)dt
La
L a
b
= an
a n w n (t)w m
*(t)dt = an
a n K n d nm
L
= a m K m
(2–86)
Thus, Eq. (2–84) follows.
The orthogonal series is very useful in representing a signal, noise, or signal–noise
combination. The orthogonal functions w j 1t2 are deterministic. Furthermore, if the waveform
w(t) is deterministic, the constants {a j } are also deterministic and may be evaluated using Eq.
(2–84). In Chapter 6, we will see that if w(t) is stochastic (e.g., in a noise problem), the {a j }
are a set of random variables that give the desired random process w(t).
It is also possible to use Eq. (2–83) to generate w(t) from the wj(t) functions and the
coefficients a j . In this case, w(t) is approximated by using a reasonable number of the wj(t)
functions. As shown in Fig. 2–10, for the case of real values for a j and real functions for wj(t),
w(t) can be synthesized by adding up weighted versions of wj(t), where the weighting factors
are given by {a j }. The summing-and-gain weighting operation may be conveniently realized
by using an operational amplifier with multiple inputs.
a
Function
generator 1
1 (t)
a 1
Function
generator 2
2 (t)
a 2
Clock
•
•
•
•
•
•
•
•
•
•
•
•
w(t)
Function
generator N
N (t)
a N
Figure 2–10
Waveform synthesis using orthogonal functions.