This document discusses wideband frequency modulation (FM) and its properties. It begins by introducing the concept of FM and defining terms like modulation index and frequency deviation. It then shows that the FM signal can be expressed as a complex Fourier series involving Bessel functions. The spectrum of the FM signal is derived and shown to consist of sidebands spaced at multiples of the modulating frequency away from the carrier frequency. For small modulation indices, the spectrum reduces to the carrier and first sidebands, corresponding to narrowband FM.

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
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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
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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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