082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

6.2 Sequences 203
sin t
1
x[k]
1
– 3p
–2p –2 –p – p –2 p
0 –2 p
3p –2
2p
t
–4 –3 –2 –1 1
Figure6.3
TheKronecker delta sequence.
2 3 4 5
k
–1
Figure6.4
Thefunction f (t) = sint with sampled points
shown.
x[k]
1
–4 –3 –2 –1 1 2 3 4 k Figure6.5
Sequence formed from sampling
–1
f(t) =sint.
Solution The function f (t) = sint, for −2π t 2π, is shown in Figure 6.4. We sample the
continuousfunctionattherequiredpoints.Thesamplevaluesareshownas •.Fromthe
graph we see that
x[k]=0,1,0,−1,0,1,0,−1,0
The graph ofx[k] isshown inFigure 6.5.
k=−4,−3,...,3,4
Sometimes it is possible to describe a sequence by a rule giving the kth term. For example,
the sequence for whichx[k] = 2 k ,k = 0,1,2,..., is given by 1,2,4,8.... On
occasions,arulegivesx[k]intermsofearliermembersofthesequence.Forexample,the
previoussequencecouldhavebeendefinedbyx[k] = 2x[k−1],x[0] = 1.Thesequence
isthensaidtobedefinedrecursivelyandthedefiningformulaiscalledarecurrencerelationordifferenceequation.Differenceequationsareparticularlyimportantindigital
signalprocessingand are dealtwith inChapter22.
Example6.4 Write down the termsx[k] fork = 0,...,7 ofthe sequencedefined recursively as
x[k] =x[k −2] +x[k −1]
wherex[0] = 1 andx[1] = 1.
Solution Thevalues ofx[0]andx[1]aregiven. Usingthe given recurrencerelationwefind
x[2] =x[0] +x[1] = 2
x[3] =x[1] +x[2] = 3
Continuing inthisfashion wefind the first eight termsofthe sequenceare
1,1,2,3,5,8,13,21
This sequenceis known asthe Fibonaccisequence.