B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

APPENDIX A
ORTHOGONALITY OF SOME SIGNAL SETS
A.1 Orthogonality of the Trigonometric and Exponential Signal Set
Consider an integral / defined by
I = f cos ncv o t cos mw o t dt
lr0
(A. la)
where } 11
stands for integration over any contiguous interval of T o = 2n / w 0 seconds. By
using a trigonometric identity (Appendix E), Eq. (A.la) can be expressed as
I= [ { cos (n + m)w o tdt + { cos (n - m)w o tdt]
2 1 1
(A. lb)
Since cos w o t executes one complete cycle during any interval of T o seconds, cos (n + m)w o t
executes (n + m) complete cycles during any interval of duration T o . Therefore, the first integral
in Eq. (A. I b ), which represents the area under (n + m) complete cycles of a sinusoid, equals
zero. The same argument shows that the second integral in Eq. (A.lb) is also zero, except when
n = m. Hence, / in Eq. (A.lb) is zero for all n -:f. m. When n = m, the first integral in Eq.
(A.1 b) is still zero, but the second integral yields
1 [ To
I=
2 } r o dt = 2
Thus,
f cos nw o t cos mw o t dt = [
lro 2
We can use similar arguments to show that
and
1. .
d
1
sm nw o t sm mw o t t =
To
2
f sin nw o t cos mw o t dt = 0
lr0
n -:f. m
m =n-:f.O
n -:f. m
n=m-:f.O
all n and m
(A.2a)
(A.2b)
(A.2c)
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