Determine the amplitude and phase of the various frequency components of a PM signal with k_p = 1 and with m(t) a periodic signal given by

in one period.

Question

Determine the amplitude and phase of the various frequency components of a PM signal with kp = 1 and with m(t) a periodic signal given by    in one period.

David · Accepted Answer

u(t) = Ac cos(2πfct + kpm(t)) = AcRe
[
ej2πfctejkpm(t)
]
= AcRe
[
ej2πfctejm(t)
]
The signal ejm(t) is periodic with period Tm = 1fm and Fourier series expansion
cn =
1
Tm
∫ Tm
0
ejm(t)e−j2πnfmtdt
= 1
Tm
∫ Tm
2
0
eje−j2πnfmtdt + 1
Tm
∫ Tm
Tm
2
e−je−j2πnfmtdt
= − e
j
Tmj2πnfm
e−j2πnfmt
∣∣∣∣ Tm2
0
− e
−j
Tmj2πnfm
e−j2πnfmt
∣∣∣∣TmTm
2
= (−1)
n − 1
2πn
j(ej − e−j) =
 0 n = 2l2
π(2l+1) sin(1) n = 2l+ 1
Hence,
ejm(t) =
∞∑
l=−∞
2
π(2l+ 1) sin(1)e
j2πlfmt
and
u(t) = AcRe
[
ej2πfctejm(t)
]
= AcRe
ej2πfct ∞∑
l=−∞
2
π(2l+ 1) sin(1)e
j2πlfmt

= Ac
∞∑
l=−∞
∣∣∣∣ 2 sin(1)π(2l+ 1)
∣∣∣∣ cos(2π(fc + lfm)t +φl)
where φl = 0 for l ≥ 0 and φl = π for negative values of l.
The modulated PM signal is given by
acer aspire
Text Box
Determine the amplitude and phase of the various frequency components of a PM signal with kp = 1 and with m(t) a periodic signal given by
acer aspire
Text Box
in one period.
acer aspire
Text Box
Solution:

Determine the amplitude and phase of the various frequency components of a PM signal with k p = 1 and with m(t) a periodic signal given by in one period.

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