1. TELE3113 Analogue and Digital
Communications
Wideband FM
Wei Zhang
w.zhang@unsw.edu.au
School of Electrical Engineering and Telecommunications
The University of New South Wales

2. 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) = cn exp(j2πnf0 t)
n=−∞
where
f0 is the fundamental frequency: f0 = 1/T0 ,
nf0 represents the nth harmonic of f0 ,
cn represents the complex Fourier coefficient,
T0 /2
1
cn = gT0 (t) exp(−j2πnf0 t)dt, n = 0, ±1, ±2, · · ·
T0 −T0 /2
TELE3113 - Wideband FM. August 19, 2009. – p.1/1

3. Bessel Function (1)
The nth order Bessel function of the first kind and argument
β, denoted by Jn (β), is given by
π
1
Jn (β) = exp [j(β sin x − nx)] dx. (1)
2π −π
1
J0(β)
J1(β)
J2(β)
0.5
J (β)
3 J (β)
4
0
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4. Bessel Function (2)
Some properties:
For different integer values of n,

 J (β), for n even
−n
Jn (β) =
 −J−n (β), for n odd
For small values of β,

 1,
 n=0


Jn (β) ≈ β
 2, n=1

 0,

n≥2
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5. Wideband FM (1)
Consider a sinusoidal modulating wave defined by
m(t) = Am cos(2πfm t).
The instantaneous frequency of the FM wave is
fi (t) = fc + kf m(t) = fc + ∆f cos(2πfm t)
where ∆f = kf Am is called the frequency deviation.
The angle of the FM wave is
θi (t) = 2πfc t + β sin(2πfm t)
∆f
where β = fm is called the modulation index of the FM
wave. TELE3113 - Wideband FM. August 19, 2009. – p.4/1

6. Wideband FM (2)
The FM wave is then given by
s(t) = Ac cos[θi (t)] = Ac cos[2πfc t + β sin(2πfm t)].
Using cos θ = [exp(jθ)], where the operator [x] denotes the
real part of x, we get
s(t) = [Ac exp(j2πfc t + jβ sin(2πfm t))]
= [˜(t) exp(j2πfc t)],
s (2)
where
s(t) = Ac exp [jβ sin(2πfm t)] .
˜ (3)
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7. Wideband FM (3)
Theorem 1: s(t) in Eq. (3) is a periodic function of time t with a
˜
fundamental frequency equal to fm .
Proof: Replacing time t in s(t) with t + k/fm for any integer k,
˜
we have
s(t + k/fm ) = Ac exp [jβ sin(2πfm (t + k/fm ))]
˜
= Ac exp [jβ sin(2πfm t + 2πk)]
= Ac exp [jβ sin(2πfm t)]
= s(t).
˜
It completes the proof.
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8. Wideband FM (4)
Since s(t) is a periodic signal with period 1/fm (see Theorem 1),
˜
we may expand s(t) in the form of a complex Fourier series as
˜
follows:
∞
s(t) =
˜ cn exp(j2πnfm t), (4)
n=−∞
where the complex Fourier coefficient
1/(2fm )
cn = f m s(t) exp(−j2πnfm t)dt
˜
−/(2fm )
1/(2fm )
= f m Ac exp [jβ sin(2πfm t)] exp(−j2πnfm t)dt. (5)
−/(2fm )
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9. Wideband FM (5)
Define x = 2πfm t. Hence, we may express cn in Eq. (5) as
π
Ac
cn = exp [j(β sin x − nx)] dx.
2π −π
Using Bessel function Jn (β) in Eq. (1), we therefore have
cn = Ac Jn (β).
Then, Eq. (4) can be written as
∞
s(t) = Ac
˜ Jn (β) exp(j2πnfm t). (6)
n=−∞
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10. Wideband FM (6)
Substituting Eq. (6) into Eq. (2), we get
∞
s(t) = Ac Jn (β) exp[j2π(fc + nfm )t]
n=−∞
∞
= Ac Jn (β) [exp(j2π(fc + nfm )t)]
n=−∞
∞
= Ac Jn (β) cos[2π(fc + nfm )t]. (7)
n=−∞
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11. Spectrum of Wideband FM
The spectrum of s(t) is given by
∞
Ac
S(f ) = Jn (β)[δ(f − fc − nfm ) + δ(f + fc + nfm )].
2 n=−∞
S(f ) contains an infinite set of side frequencies ±f c ,
±fc ± fm , ±fc ± 2fm , · · ·
For small values of β, S(f ) is effectively composed of ±f c
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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