Wiley signals and systems e book TLFe BO 543

528

Appendix A. Solutiorii to the Exercisrs

Solution 14.2
a) shift-vasiaiit, because

S { S [ k - :V]}= x[2(k- N)]# y[k

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N]

s [ 2 k - N]

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b) likewise shihvariant, see a,).
C)

'L'lir systciii sets to zero every sample with odd k :

Exaiuplc for N = 2: shift-rnvar.iarit.

The syst,rni is iiivariailt, whcn shift,ed by an even iiiirriber of cyclcs, 11ut shifting: x [ k ]by an odd niimber of cycles does iiol deliver a shifted vrrsiori of' ;q[k].
Note: Systcms t h a t , a,re invariant for a cert,ain shift m d its multiples are
called perzod,ic ,sii:i~fl-ziivnrznn.t.
Solution 14.3
a ) lJ[-1] = 0
y'O] = Y[O] = 1
U [ I j = . L [ 1 ] 2y[k - 11 = 0 - 2 ' 1 = -2

y[2; = x[2] 2y[k - I] - ?/[A, - = 0 2 . (-2) - 1 = 3
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