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# Tele3113 wk5wed

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### Tele3113 wk5wed

1. 1. TELE3113 Analogue and Digital Communications Wideband FM Wei Zhang w.zhang@unsw.edu.auSchool of Electrical Engineering and Telecommunications The University of New South Wales
2. 2. Fourier SeriesLet gT0 (t) denote a periodic signal with period T0 . By using aFourier 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. 3. Bessel Function (1)The nth order Bessel function of the ﬁrst kind and argumentβ, 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 TELE3113 - Wideband FM. August 19, 2009. – p.2/1
4. 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 TELE3113 - Wideband FM. August 19, 2009. – p.3/1
5. 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. 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 thereal 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) TELE3113 - Wideband FM. August 19, 2009. – p.5/1
7. 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. TELE3113 - Wideband FM. August 19, 2009. – p.6/1
8. 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 ) TELE3113 - Wideband FM. August 19, 2009. – p.7/1
9. 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 −πUsing 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=−∞ TELE3113 - Wideband FM. August 19, 2009. – p.8/1
10. 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=−∞ TELE3113 - Wideband FM. August 19, 2009. – p.9/1
11. 11. Spectrum of Wideband FMThe 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). TELE3113 - Wideband FM. August 19, 2009. – p.10/1
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