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Signals and Spectra Chap. 2
Orthogonal Functions
Before the orthogonal series is studied, we define orthogonal functions.
DEFINITION. Functions w n 1t2 and w m 1t2 are said to be orthogonal with respect to each
other over the interval a 6 t 6 b if they satisfy the condition
w n (t)w m
*(t) dt = 0, where n Z m
(2–77)
La
Furthermore, if the functions in the set {w n 1t2} are orthogonal, then they also satisfy the
relation
where
b
b
w n (t)w* m (t) dt = e 0,
La
K n ,
n Z m
n = m f = K nd nm
(2–78)
d nm ! e 0, n Z m
(2–79)
1, n = m f
Here, d nm is called the Kronecker delta function. If the constants K n are all equal to 1,
the w are said to be orthonormal functions. †
n 1t2
Equation (2–77) is used to test pairs of functions to see if they are orthogonal. Any such
pair of functions is orthogonal over the interval (a, b) if the integral of the product of the functions
is zero. The zero result implies that these functions are “independent” or in “disagreement.”
If the result is not zero, they are not orthogonal, and consequently, the two functions
have some “dependence” on, or “alikeness” to each other.
Example 2–12 ORTHOGONAL COMPLEX EXPONENTIAL FUNCTIONS
Show that the set of complex exponential functions {e jnv0t } are orthogonal over the interval
a 6 t 6 b, where b = a + T 0 , T 0 = 1/f 0 , w 0 = 2pf 0 , and n is an integer.
Solution. Substituting w and w m 1t2 = e jnv 0t
n 1t2 = e jnv 0t
into Eq. (2–77), we get
b
a+T 0
a+T 0
w n (t)w m *(t) dt = e jnv0t e -jnv 0t
dt = e j(n-m)v 0t
dt
La
L a
L a
For m Z n, Eq. (2–80) becomes
a+T 0
(2–80)
e j(n-m)v 0t
dt = ej(n-m)v0a Ce j(n-m)2p - 1D
= 0
(2–81)
La
j(n - m)v 0
since e j(n-m)2p = cos [2p(n - m)] + j sin [2p(n - m)] = 1. Thus, Eq. (2–77) is satisfied, and
consequently, the complex exponential functions are orthogonal to each other over the interval
† To normalize a set of functions, we take each old w n 1t2 and divide it by 2K n to form the normalized w n 1t2.