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TELE3113 Analogue and Digital Communications Angle Modulation Wei Zhang w.zhang@unsw.edu.auSchool of Electrical Engineering and Telecommunications The University of New South Wales
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Last two weeks ...We have studied: Amplitude Modulation: s(t) = [1 + ka m(t)]c(t). Simple envelope detection, but low power/BW efﬁciency. DSB-SC Modulation: s(t) = m(t)c(t). High power efﬁciency, but low BW efﬁciency. SSB Modulation: s(t) = 1 Ac m(t) cos(2πfc t) 2 1 2 Ac m(t) sin(2πfc t). ˆ VSB Modulation: Tailored for transmission of TV signals. TELE3113 - Angle Modulation. August 18, 2009. – p.1/1
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Angle vs Amplitude Modulation Amplitude modulation: amplitude of a carrier wave varies in accordance with an information-bearing signal. Angle modulation: angle of the carrier changes according to the information-bearing signal. Angle modulation provides better robustness to noise and interference than amplitude modulation, but at the cost of increased transmission BW. TELE3113 - Angle Modulation. August 18, 2009. – p.2/1
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Deﬁnitions Let θi (t) denote the angle of a modulated sinusoidal carrier at time t. Assume θi (t) is a function of the information-bearing signal or message signal m(t). The angle-modulated wave is s(t) = Ac cos[θi (t)] Instantaneous frequency of s(t) is deﬁned as 1 dθi (t) fi (t) = 2π dt TELE3113 - Angle Modulation. August 18, 2009. – p.3/1
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PMTwo commonly used angle modulation: PM and FM. Phase modulation (PM): The instantaneous angle is varied linearly with m(t), as shown by θi (t) = 2πfc t + kp m(t), where kp denotes the phase-sensitivity factor. The phase-modulated wave is described by s(t) = Ac cos[2πfc t + kp m(t)]. TELE3113 - Angle Modulation. August 18, 2009. – p.4/1
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FM Frequency modulation (FM): The instantaneous frequency fi (t) is varied linearly with m(t), as shown by fi (t) = fc + kf m(t), where kf denotes the frequency-sensitivity factor. Integrating fi (t) with time and multiplying 2π, we get t t θi (t) = 2π fi (τ )dτ = 2πfc t + 2πkf m(τ )dτ. (1) 0 0 The frequency-modulated wave is therefore t s(t) = Ac cos 2πfc t + 2πkf m(τ )dτ . 0 TELE3113 - Angle Modulation. August 18, 2009. – p.5/1
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PM versus FM Phase Modulation Frequency Modulation tθi (t) 2πfc t + kp m(t) 2πfc t + 2πkf 0 m(τ )dτ kp dfi (t) fc + 2π dt m(t) fc + kf m(t) ts(t) Ac cos[2πfc t + kp m(t)] Ac cos 2πfc t + 2πkf 0 m(τ )dτ TELE3113 - Angle Modulation. August 18, 2009. – p.6/1
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PM/FM Relationship Modulating wave Phase FM wave Integrator modulator Ac cos( 2πf c t ) (a) Scheme for generating an FM wave by using a phase modulator. Modulating wave Frequency PM wave Differentiator modulator Ac cos( 2πf c t ) (b) Scheme for generating a PM wave by using a frequency modulator. TELE3113 - Angle Modulation. August 18, 2009. – p.7/1
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Properties of Angle Modulation Property 1 Constancy of transmitted power: The average power of angle-modulated waves is a constant, as shown by 1 2 Pav = Ac . 2 Property 2 Nonlinearity of the modulation process: Let s(t), s1 (t), and s2 (t) denote the PM waves produced by m(t), m1 (t) and m2 (t). If m(t) = m1 (t) + m2 (t), then s(t) = s1 (t) + s2 (t). TELE3113 - Angle Modulation. August 18, 2009. – p.9/1
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Properties of Angle Modulation Property 3 Irregularity of zero-crossings: A “zero-crossing” is a point where the sign of a function changes. PM and FM wave no longer have a perfect regularity in their spacing across the time-scale. Property 4 Visualization difﬁculty of message waveform: The message waveform cannot be visualized from PM and FM waves. TELE3113 - Angle Modulation. August 18, 2009. – p.10/1
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Example of Zero-crossings (1)Consider a modulating wave m(t) as shown by at, t ≥ 0 m(t) = 0, t < 0Determine the zero-crossings of the PM and FM waves producedby m(t) with carrier frequency fc and carrier amplitude Ac . TELE3113 - Angle Modulation. August 18, 2009. – p.11/1
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Example of Zero-crossings (2)The PM wave is given by A cos(2πf t + k at), t ≥ 0 c c p s(t) = Ac cos(2πfc t), t<0The PM wave experiences a zero-crossing when the angle is anodd multiple of π/2, i.e., π 2πfc tn + kp atn = + nπ, n = 0, 1, 2, · · · 2Then, we get 1/2 + n tn = , n = 0, 1, 2, · · · 2fc + kp a/π TELE3113 - Angle Modulation. August 18, 2009. – p.12/1
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Example of Zero-crossings (3)The FM wave is given by A cos(2πf t + πk at2 ), t ≥ 0 c c f s(t) = Ac cos(2πfc t), t<0To ﬁnd zero-crossings, we may set up π 2πfc tn + πkf at2 n = + nπ, n = 0, 1, 2, · · · 2The positive root of the above quadratic equation is 1 2 1 tn = −fc + fc + akf +n , n = 0, 1, 2, · · · akf 2 TELE3113 - Angle Modulation. August 18, 2009. – p.13/1
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Narrowband 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 Am cos(2πfm 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 - Angle Modulation. August 18, 2009. – p.15/1
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Narrowband FM (2)The FM wave is then given by s(t) = Ac cos[2πfc t + β sin(2πfm t)].Using cos(x + y) = cos x cos y − sin x sin y, we gets(t) = Ac cos(2πfc t) cos[β sin(2πfm t)]−Ac sin(2πfc t) sin[β sin(2πfm t)].For narrowband FM wave, β << 1. Then, cos[β sin(2πfm t)] ≈ 1and sin[β sin(2πfm t)] ≈ β sin(2πfm t). Therefore, s(t) ≈ Ac cos(2πfc t) − βAc sin(2πfc t) sin(2πfm t). TELE3113 - Angle Modulation. August 18, 2009. – p.16/1
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Generating Narrowband FMModulating __ wave Narrow- Product Integrator ∑ band Modulator FM wave Ac sin( 2πf c t ) + − 90 0 Phase-shifter Narrow-band phase modulator Carrier wave Ac cos(2πf c t ) TELE3113 - Angle Modulation. August 18, 2009. – p.17/1
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Narrowband FM vs. AMFor small β, the narrowband FM wave is given by s(t) ≈ Ac cos(2πfc t) − βAc sin(2πfc t) sin(2πfm t).Using sin x sin y = − 1 cos(x + y) cos(x − y), we get 2 1s(t) ≈ Ac cos(2πfc t) + βAc [cos[2π(fc + fm )t] − cos[2π(fc − fm )t]]. 2Recall the AM of the single-tone message signal is [p.11, Aug-4,TELE3113] 1sAM (t) = Ac cos(2πfc t)+ µAc [cos[2π(fc +fm )t]+cos[2π(fc −fm )t]]. 2The only difference between NB-FM and AM is the “sign”. TELE3113 - Angle Modulation. August 18, 2009. – p.18/1
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