TELE3113 Analogue and Digital      Communications                   Wideband FM                      Wei Zhang            ...
Fourier SeriesLet gT0 (t) denote a periodic signal with period T0 . By using aFourier series expansion of this signal, we ...
Bessel Function (1)The nth order Bessel function of the first kind and argumentβ, denoted by Jn (β), is given by           ...
Bessel Function (2)Some properties:    For different integer values of n,                                                ...
Wideband FM (1) Consider a sinusoidal modulating wave defined by                     m(t) = Am cos(2πfm t). The instantaneo...
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 = [e...
Wideband FM (3)Theorem 1: s(t) in Eq. (3) is a periodic function of time t with a            ˜fundamental frequency equal ...
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 ...
Wideband FM (5)Define x = 2πfm t. Hence, we may express cn in Eq. (5) as                          π                   Ac   ...
Wideband FM (6)Substituting Eq. (6) into Eq. (2), we get                          ∞        s(t) =       Ac        Jn (β) e...
Spectrum of Wideband FMThe spectrum of s(t) is given by                  ∞            Ac  S(f ) =               Jn (β)[δ(f...
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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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