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

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Review exercises 23 755
The values of the first few coefficients are
b 1
= 2 π (1−cosπ)= 4 π
b 2
= 2
2π (1−cos2π)=0
b 3
= 2 4
(1 −cos3π) =
3π 3π
The next stage is toevaluate the gain and phase changes of the Fourier components.
Using Equations (23.13) and (23.14):
n = 1
ω 1
=1
|G(jω 1
)| =
1
√
1+0.09×1
= 0.96
n = 3
n = 5
G(jω 1
) = −tan −1 0.3 = −16.7 ◦
ω 3
=3
|G(jω 3
)| =
1
√
1+0.09×9
= 0.74
G(jω 3
) = −tan −1 0.9 = −42.0 ◦
ω 5
=5
|G(jω 5
)| =
1
√
1+0.09×25
= 0.55
G(jω 5
) = −tan −1 1.5 = −56.3 ◦
It is clear that high-frequency Fourier components are attenuated and phase shifted
more than low-frequency Fourier components. The effect is to produce a rounding
of the rising and falling edges of the square wave input signal. This is illustrated
in Figure 23.29(b). The output signal has been obtained by adding together the
attenuated and phase-shifted output Fourier components. This is possible because
the systemislinear.
REVIEWEXERCISES23
1 Find the half-range Fourier sineseries representation
off(t)=tsint,0tπ.
2 Find the half-range sineseries representationof
f(t)=cos2t,0tπ.
3 Find (a) the half-range sineseries,and(b)the
half-rangecosine seriesrepresentation ofthe function
defined in the interval [0, τ]by
⎧
4t
⎪⎨ τ
f(t)= ( )
4
⎪⎩ 1 − t 3 τ
0t τ 4
τ
4 tτ
4 Find the Fourier series representation ofthe function
with periodT defined by
{
V(constant) |t| <T/6
f(t)=
0 T/6|t|T/2