Tele3113 wk5wed
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TELE3113 Analogue and DigitalCommunications
Wideband FM
Wei Zhang
School of Electrical Engineering and Telecommunications
The University of New South Wales
Fourier Series
Let gT0(t) denote a periodic signal with period T0. By using a
Fourier series expansion of this signal, we have
gT0(t) =
∞∑
n=−∞
cn exp(j2πnf0t)
where
f0 is the fundamental frequency: f0 = 1/T0,
nf0 represents the nth harmonic of f0,
cn represents the complex Fourier coefficient,
cn =1
T0
∫ T0/2
−T0/2gT0
(t) exp(−j2πnf0t)dt, n = 0,±1,±2, · · ·
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Bessel Function (1)
The nth order Bessel function of the first kind and argument
β, denoted by Jn(β), is given by
Jn(β) =1
2π
∫ π
−πexp [j(β sin x − nx)] dx. (1)
0 2 4 6 8 10 12 14 16 18−0.5
0
0.5
1
J0(β)
J1(β)
J2(β)
J4(β)
J3(β)
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Bessel Function (2)
Some properties:
For different integer values of n,
Jn(β) =
J−n(β), for n even
−J−n(β), for n odd
For small values of β,
Jn(β) ≈
1, n = 0
β2 , n = 1
0, n ≥ 2
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Wideband FM (1)
Consider a sinusoidal modulating wave defined by
m(t) = Am cos(2πfmt).
The instantaneous frequency of the FM wave is
fi(t) = fc + kfm(t) = fc + ∆f cos(2πfmt)
where ∆f = kfAm is called the frequency deviation.
The angle of the FM wave is
θi(t) = 2πfct + β sin(2πfmt)
where β = ∆ffm
is called the modulation index of the FM
wave. TELE3113 - Wideband FM. August 19, 2009. – p.4/10
Wideband FM (2)
The FM wave is then given by
s(t) = Ac cos[θi(t)] = Ac cos[2πfct + β sin(2πfmt)].
Using cos θ = <[exp(jθ)], where the operator <[x] denotes the
real part of x, we get
s(t) = <[Ac exp(j2πfct + jβ sin(2πfmt))]
= <[s̃(t) exp(j2πfct)], (2)
where
s̃(t) = Ac exp [jβ sin(2πfmt)] . (3)
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Wideband FM (3)
Theorem 1: s̃(t) in Eq. (3) is a periodic function of time t with a
fundamental frequency equal to fm.
Proof: Replacing time t in s̃(t) with t + k/fm for any integer k,
we have
s̃(t + k/fm) = Ac exp [jβ sin(2πfm(t + k/fm))]
= Ac exp [jβ sin(2πfmt + 2πk)]
= Ac exp [jβ sin(2πfmt)]
= s̃(t).
It completes the proof.
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Wideband FM (4)
Since s̃(t) is a periodic signal with period 1/fm (see Theorem 1),
we may expand s̃(t) in the form of a complex Fourier series as
follows:
s̃(t) =∞
∑
n=−∞
cn exp(j2πnfmt), (4)
where the complex Fourier coefficient
cn = fm
∫ 1/(2fm)
−/(2fm)s̃(t) exp(−j2πnfmt)dt
= fmAc
∫ 1/(2fm)
−/(2fm)exp [jβ sin(2πfmt)] exp(−j2πnfmt)dt.(5)
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Wideband FM (5)
Define x = 2πfmt. Hence, we may express cn in Eq. (5) as
cn =Ac
2π
∫ π
−πexp [j(β sin x − nx)] dx.
Using Bessel function Jn(β) in Eq. (1), we therefore have
cn = AcJn(β).
Then, Eq. (4) can be written as
s̃(t) = Ac
∞∑
n=−∞
Jn(β) exp(j2πnfmt). (6)
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Wideband FM (6)
Substituting Eq. (6) into Eq. (2), we get
s(t) = <
[
Ac
∞∑
n=−∞
Jn(β) exp[j2π(fc + nfm)t]
]
= Ac
∞∑
n=−∞
Jn(β)< [exp(j2π(fc + nfm)t)]
= Ac
∞∑
n=−∞
Jn(β) cos[2π(fc + nfm)t]. (7)
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Spectrum of Wideband FM
The spectrum of s(t) is given by
S(f) =Ac
2
∞∑
n=−∞
Jn(β)[δ(f − fc − nfm) + δ(f + fc + nfm)].
S(f) contains an infinite set of side frequencies ±fc,
±fc ± fm, ±fc ± 2fm, · · ·
For small values of β, S(f) is effectively composed of ±fc
and ±fc ± fm. This case corresponds to the narrow-band
FM.
The amplitude of the carrier component varies with β
according to J0(β).
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