8.2 Discrete-Time Signals 473The representation of any signalx[n] in terms ofδ[n] results from thesifting propertyof the unit-sample
signal:
x[n]δ[n−n 0 ]=x[n 0 ]δ[n−n 0 ]which is due to
δ[n−n 0 ]={
1 n=n 0
0 otherwiseThus, consideringx[n] a sequence of samples
...x[−1]x[0]x[1]...at times... −1, 0, 1,..., we can writex[n] as
x[n]=···+x[−1]δ[n+1]+x[0]δ[n]+x[1]δ[n−1]+···=
∑∞
k=−∞x[k]δ[n−k]The generic representation (Eq. 8.25) of any signalx[n] will be useful in finding the output of a
discrete-time linear time-invariant system.
nExample 8.15
Use the generic representations in terms of unit-sample signals to represent theramp signal r[n]
defined asr[n]=nu[n]and from it show thatr[n]=∑nm= 0(n−m)−∑nm= 1(n−m)