5—Fourier Series 133
This time average of the power is (up to that constant factor that I’m ignoring)
〈
f^2
〉
= lim
T→∞
1
2 T
∫+T
−T
dtf(t)^2
Now put the Fourier series representation of the sound into the integral to get
lim
T→∞
1
2 T
∫+T
−T
dt
[∞
∑
−∞
aneinω^0 t
] 2
The soundf(t)is real, so by problem 11 ,a−n=a*n. Also, using the result of problem 18 the time average of
eiωtis zero unlessω= 0; then it’s one.
〈
f^2
〉
= lim
T→∞
1
2 T
∫+T
−T
dt
[
∑
n
aneinω^0 t
][
∑
m
ameimω^0 t
]
= lim
T→∞
1
2 T
∫+T
−T
dt
∑
n
∑
m
aneinω^0 tameimω^0 t
=
∑
n
∑
m
anam lim
T→∞
1
2 T
∫+T
−T
dtei(n+m)ω^0 t
=
∑
n
ana−n
=
∑
n
|an|^2 (31)
Put this into words and it says that the time-average power received is the sum of many terms, each one of
which I can interpret as the amount of power coming inat that frequencynω 0. The Fourier coefficients squared
(absolute-squared really) are then proportional to the part of the power at a particular frequency. The “power
spectrum.”
Other Applications
In section10.2Fourier series will be used to solve partial differential equations, leading to equations such as
Eq. (10.14).