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Basis:

u = exp( j2π nt) and u = exp( j − 2π nt) (7.71)
n n
Orthonormality of the basis vectors:

1 if

/
u u = ∫ 12 exp( j − 2π mt)exp( j2π nt dt =  m = n (7.72)
)
m n

− /12 0 if m ≠ n
Decomposition rule:
∞
∞
ϕ = ∑ cu n = ∑ c exp( j2 πnt) (7.73)
n
n
n =−∞ n =−∞
where

/
c = u ϕ = ∫ 12 exp( j − π2 nt ϕ) ( t dt) (7.74)
n n
− /12
Parseval’s identity:

∞
/
/
ϕ 2 = ϕ ϕ = ∫ 12 ϕ()t ϕ()t dt = ∫ 12 ϕ()t dt = ∑ c n 2 (7.75)
2
12
12
− / − /
n =−∞
Example 7.9
Derive the analytic expression for the potential difference across the capacitor
in the RLC circuit of Figure 4.5 if the temporal profile of the source potential
is a periodic function, of period 1, in some normalized units.
Solution:
1. Because the potential is periodic with period 1, it can be expanded using
Eq. (7.73) in a Fourier series with basis functions {e j2πnt }:


 nt 

˜
nj2π
Vt() = Re  ∑ V e  (7.76)
s s
 n 


where V ˜ n is the phasor associated with the frequency mode (2πn). (Note that
s
n in the expressions for the phasors is a superscript and not a power.)
© 2001 by CRC Press LLC