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

202 Chapter 6 Sequences and series
isthe sequence
x[0] = 0 x[1] = 3 2
x[2] = 3 x[3] = 9 2 ...
The values in the range of the function are the terms of the sequence. The independent
variableisk.FunctionsofthissortdifferfromthosedescribedinChapter2becausethe
independent variable is not selected from a continuous interval but rather isdiscrete. It
is,nevertheless,possibletorepresentx[k]graphicallyasillustratedinExamples6.1--6.3,
butinsteadofapiecewisecontinuouscurve,wenowhaveacollectionofisolatedpoints.
Example6.1 Graphthe sequences given by
{ 0 k <0
(a) x[k] = k=...,−3,−2,−1,0,1,2,...,thatisk∈Z
1 k0
{ 1 keven
(b) x[k] = k ∈ Z
−1 kodd
Solution (a) Fromthedefinitionofthissequence,thetermx[k]iszeroifk<0and1ifk 0.The
graph is obtained by plotting the terms of the sequence againstk(see Figure 6.1).
This sequenceisknown asthe unit stepsequence. We shall denotethisbyu[k].
(b) Thesequencex[k] isshown inFigure 6.2.
Example6.2 Graphthe sequencedefinedby
x[k] =
{ 1 k = 0
0 k≠0
k=...,−3,−2,−1,0,1,2,3,...
Solution From the definition, ifk = 0 thenx[k] = 1. Ifkis not equal to zero the corresponding
term in the sequence equals zero. Figure 6.3 shows the graph of this sequence which is
commonly called theKronecker delta sequence.
Example6.3 Thesequencex[k]isobtainedbymeasuringorsamplingthecontinuousfunction f (t) =
sint,t ∈ R, att = −2π, −3π/2, −π, −π/2, 0, π/2, π, 3π/2 and 2π. Write down the
termsofthissequenceand show themon a graph.
x[k]
1
x[k]
–4 –3 –2 –1 1
1
2 3 4 5
k
–4 –3 –2 –1 1
2 3 4 5
k
–1
Figure6.1
Theunitstepsequence.
Figure6.2 { 1 k even
Thesequencex[k] =
−1 k odd.