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Chap 3
3.1 A continuous-time periodic signal x(t) is real value and has a fundamental period T=8. The nonzero Fourier series coefficients for x(t) are
*a1?a?1?2,a3?a?3?4j.
Express x(t) in the form
x(t)??Akcos(?kt??k)
k?0?Solution:
Fundamental period T?8.?0?2?/8??/4
x(t)?k?????akej?0kt?a1ej?0t?a?1e?j?0t?a3ej3?0t?a?3e?j3?0t
?2ej?0t?2e?j?0t?4jej3?0t?4je?j3?0t?3??4cos(t)?8sin(t)443.2 A discrete-time periodic signal x[n] is real valued and has a fundamental period N=5.The nonzero Fourier series coefficients for x[n] are
*j?/3a0?1,a?2?e?j?/4,a2?ej?/4,a4?a? 4?2eExpress x[n] in the form
x[n]?A0??Aksin(?kn??k)
k?1?Solution:
?j?/4j?/4for, a0?1, a?2?e , a2?e ,
a?4?2e?j?/3,
a4?2ej?/3
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x[n]?k??N??akejk(2?/N)n
?a0?a2ej(4?/5)n?a?2e?j(4?/5)n?a4ej(8?/5)n?a?4e?j(8?/5)n
?1?ej?/4ej(4?/5)n?e?j?/4e?j(4?/5)n?2ej?/3ej(8?/5)n?2e?j?/3e?j(8?/5)n4?8??1?2cos(?n?)?4cos(?n?)
545343?85??1?2sin(?n?)?4sin(?n?)
5456
3.3 For the continuous-time periodic signal
2?5?x(t)?2?cos(t)?4sin(t)
33Determine the fundamental frequency ?0 and the Fourier series coefficients ak such that
x(t)?Solution:
for the period ofcos(ofsin(k???jk?0tae?k.
?2?t)is T?3, the period 35?t)is T?6 3so the period ofx(t)is 6, i.e. w0?2?/6??/3
2?5?x(t)?2?cos(t)?4sin(t)
331 ?2?cos(2?0t)?4sin(5?0t)
21 ?2?(ej2?0t?e?j2?0t)?2j(ej5?0t?e?j5?0t)
21then, a0?2, a?2?a2? , a?5?2j, a5??2j
2
3.5 Let x1(t) be a continuous-time periodic signal with fundamental frequency ?1 and Fourier coefficients ak. Given that
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x2(t)?x1(1?t)?x1(t?1)
How is the fundamental frequency ?2 of x2(t) related to? Also, find a relationship between the Fourier series coefficients bk of x2(t) and the coefficients ak You may use the properties listed in Table 3.1. Solution:
(1). Because x2(t)?x1(1?t)?x1(t?1), then x2(t) has the same period as x1(t),
that is T2?T1?T, w2?w1 (2). bk??1T?T1?jkw2t?jkw1tx(t)edt?(x(1?t)?x(t?1))edt 211??TTT1x1(1?t)e?jkw1tdt??x1(t?1)e?jkw1tdt
TT?a?ke?jkw1?ake?jkw1?(a?k?ak)e?jkw1
3.8 Suppose given the following information about a signal x(t):
1. x(t) is real and odd.
2. x(t) is periodic with period T=2 and has Fourier coefficients ak. 3. ak?0 for |k|?1. 4
122|x(t)|dt?1. ?02Specify two different signals that satisfy these conditions. Solution:
x(t)?k?????akej?0kt
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while: x(t) is real and odd, then ak is purely imaginary and odd， a0?0, ak??a?k,.
and so
for
T?2, then ?0?2?/2??
ak?0 for k?1
x(t)?
k?????akej?0kt?a0?a?1e?j?0t?a1ej?0t
j?t ?a1(e?e?j?t)?2a1sin(?t)
1222222x(t)dt?a?a?a?2a?1 0?1112?0? ?
a1??2/2j x(t)??2sin(?t)
3.13 Consider a continuous-time LTI system whose frequency response is
H(j?)??h(t)e?j?tdt????sin(4?)?
If the input to this system is a periodic signal
?1,0?t?4 x(t)????1,4?t?8With period T=8,determine the corresponding system output y(t). Solution:
Fundamental period T?8.?0?2?/8??/4
x(t)?k??????akej?0kt
? y(t)?k????akH(jk?0)ejk?0t
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H(jk?0)?sin(4k?0)?4,.......k?0 ??k?0?0,.......k?0? y(t)?k?????akH(jk?0)ejkw0t?4a0
11418 Because a0??x(t)dt??1dt??(?1)dt?0
TT8084另：x(t)为实奇信号，则ak为纯虚奇函数，也可以得到a0为0。 So y(t)?0.
3.15 Consider a continuous-time ideal lowpass filter S whose frequency response is
??1,.......??100H(j?)??
??0,.......??100When the input to this filter is a signal x(t) with fundamental
period
T??/6and
Fourier series
coefficientsak, it is found that
x(t)?y(t)?x(t).
For what values of k is it guaranteed that ak?0? Solution:
for
Sx(t)?k??????akej?0kt
?
y(t)?k????akH(jk?0)ejk?0t
即对于所有的k，H(jk?0)?1
??1,.......??100H(j?)?for ?0,.......??100??也就是说k?0?100, T??/6??0?12
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信号与系统第二版课后习题解答(3-4)奥本海姆
