TELE3113 Analogue and DigitalCommunications

Wideband FM

Wei Zhang

w.zhang@unsw.edu.au

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

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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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