4.10 Other Properties of the Fourier Series 279
4.10 Other Properties of the Fourier Series
In this section we present additional properties of the Fourier series that will help us with its compu-
tation and with our understanding of the relation between time and frequency. We are in particular
interested in showing that even and odd signals have special representations, and that it is possible
to find the Fourier series of the sum, product, derivative, and integral of periodic signals without the
integration required by the definition of the series.
4.10.1 Reflection and Even and Odd Periodic Signals
If the Fourier series ofx(t), periodic with fundamental frequency 0 , is
x(t)=
∑
k
Xkejk^0 t
then the one for its reflected versionx(−t)is
x(−t)=
∑
m
Xme−jm^0 t=
∑
k
X−kejk^0 t (4.37)
so that the Fourier coefficients ofx(−t)areX−k(remember thatmandkare just dummy variables).
This can be used to simplify the computation of Fourier series of even and odd signals.
For an even signalx(t), we have thatx(t)=x(−t), and as suchXk=X−kand thereforex(t)is naturally
represented in terms of cosines and a dc term. Indeed, its Fourier series is
x(t)=X 0 +
∑−^1
k=−∞
Xkejkot+
∑∞
k= 1
Xkejkot
=X 0 +
∑∞
k= 1
Xk[ejkot+e−jkot]
=X 0 + 2
∑∞
k= 1
Xkcos(k 0 t) (4.38)
indicating thatXkare real-valued. This is also seen from
Xk=
1
T 0
∫
T 0
x(t)e−jk^0 tdt=
1
T 0
∫
T 0
x(t)[cos(k 0 t)−jsin(k 0 )]dt
=
1
T 0
∫
T 0
x(t)cos(k 0 t)dt
becausex(t)sin(k 0 t)is odd and their integral is zero. It will be similar for an odd function for which
x(t)=−x(−t), orXk=−X−k, in which case the Fourier series has a zero dc value and sine harmonics.
