2.4.3 Spectrum of Discrete Parameter Process, Aliasing 
For discrete process {X;}, 
1. R(r) are defined only for integer values of r, ry = 0, +1,+2,.... 
Therefore 
sw) == FIO) 
r=-@ 
2 [oe] 
Onell 
=— +— R(r) cos ro. DST 
= SRO) (2.57) 
r=1 
Here Ar, the time interval of the discrete process, is assumed to be Ar = 1. If 
Ar = 1, changew into w' =w/Ar; then 
s(w') = s(w) At. (2.58) 
From now on, in this lecture note, At = 1 is assumed. 
2. s(@) are continuous functions but are defined only for frequency —z <= 
@ = x, and are affected by aliasing. 
Roy = |e ave) r= 0, (file 2g 
(2n+1) ae EA 
= \, | e"dvw)= > | exp [i(w + 2nx)r] dV(w + 2nz) 
nae (2n-1)x ee ge 
n=—@& 
ll 
Mis 
48 
c/4 co 
exp (iwr) dV(w +2nx) = | ee” S$” dV + 2nn) = | e” dS(w) 
i —-o 
(2.59) 
Thus 
S@)= > Vw +2nn), (2.60) 
n=—o 
20 
