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

23.6 Half-range series 745
EXERCISES23.5
1 Find the Fourier seriesrepresentation ofthe function
{
0 −5 <t < 0 period10
f(t)=
1 0<t<5
2 Find the Fourier seriesrepresentation ofthe function
{
−t −π <t < 0 period2π
f(t)=
0 0<t<π
3 Find the Fourier seriesrepresentation ofthe function
f(t)=t 2 +πt −π<t <π period2π
4 Find the Fourier seriesrepresentation ofthe function
{
−4 −π <t 0 period 2π
f(t)=
4 0<t<π
5 Find the Fourier seriesrepresentation ofthe function
{
2(1+t) −1<t0 period2
f(t)=
0 0<t<1
6 Find the Fourier seriesrepresentation ofthe function
with period2π given by
{
t 2 0t<π
f(t)=
0 πt<2π
7 Find the Fourier seriesrepresentation ofthe function
f(t)=2sint 0<t <2π
period2π
8 Forthe signal in Engineering application 23.1, show
that
a n = 2A
nπ sin(nπd)
Solutions
1
2
3
1
2 + 2 πt
sin
π 5 + 2 3πt
sin
3π 5 + 2 5πt
sin
5π 5
+ 2
4
7πt
sin
7π 5 · · ·
π
4 − 2 π cost−sint+1 2 sin2t− 2
5
9π cos3t−1 3 sin3t...
π 2
3 +2πsint−4cost−πsin2t+cos2t+ 6
2π
3 sin3t − 4 9 cos3t···
8
(2sint sin5t+···)
+ 3 2 sin3t + 5 2
π
1
{
2 +2 (2/π)cosπt−sinπt− 1 2 sin2πt
+(2/9π)cos3πt − 1 3 sin3πt+···}/π
π 2
6 + π2 −4
sint−2cost− π π 2 sin2t + 1 2 cos2t
+ 9π2 −4
27π
sin3t − 2 9 cos3t+···
7 2sint
23.6 HALF-RANGESERIES
Sometimes an engineering function is not periodic but is only defined over a finite
interval, 0 < t < T say, as shown in Figure 23.20. In cases like this Fourier analysis
can still be useful. Because the region of interest is only that between t = 0
2
and
t = T we may choose to define the function arbitrarily outside the interval. In particular,
we can make our choice so that the resulting function is periodic, with periodT.
2
Thereismorethanonewaytoproceed.Forexample,wecanreflecttheabovefunction
in the vertical axis and then repeat it periodically so that the result is the periodic even
function shown in Figure 23.21. We have performed what is called a periodic extension
of the given function. Note that within the interval of interest nothing has altered