4.10 Other Properties of the Fourier Series 285
and thus
0.5︸︷︷X︸k
Zk
=0.25
∑
m
XmXk−m ⇒ Xk=
1
2
∑
m
XmXk−m (4.48)
so thatα=0.5.
The Fourier series ofz(t)=0.5x(t)according to the results in Example 4.5 is
z(t)=0.5x(t)=
∑∞
k=−∞
sin(πk/ 2 )
πk
ejk^2 πt
If we define
S(k)=0.5Xk=
sin(kπ/ 2 )
kπ
⇒ Xk= 2 S(k)
we have from Equation (4.48) the interesting result
S(k)=
∑∞
m=−∞
S(m)S(k−m)
or the convolution sum of the discrete sinc functionS(k)with itself isS(k). n
4.10.4 Derivatives and Integrals of Periodic Signals
n Derivative: The derivativedx(t)/dtof a periodic signalx(t), of periodT 0 , is periodic of the same periodT 0.
If{Xk}are the coefficients of the Fourier series ofx(t), the Fourier coefficients ofdx(t)/dtare
jk 0 Xk (4.49)
where 0 is the fundamental frequency ofx(t).
n Integral: For a zero-mean, periodic signaly(t), of periodT 0 , the integral
z(t)=
∫t
−∞
y(τ)dτ
is periodic of the same period asy(t), with Fourier coefficients
Zk=
Yk
jk 0
k6= 0
Z 0 =−
∑
m6= 0
Ym
1
jm 0
0 =
2 π
T 0
(4.50)
These properties come naturally from the Fourier series representation of the periodic signal. Once
we find the Fourier series of a periodic signal, we can differentiate it or integrate it (only when the dc
