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TELE3113 Analogue and Digital Communications Pulse Modulation Wei Zhang w.zhang@unsw.edu.auSchool of Electrical Engineering and Telecommunications The University of New South Wales
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What did we studyIn previous lectures, we studied continuous-wave (CW)modulation: Some parameter of a sinusoidal carrier wave is varied continuously in accordance with the message signal. Amplitude Modulation (AM, DSB-SC, SSB, VSB) Angle Modulation (PM, FM) TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.1/1
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What will we studyNext, we will study Pulse Modulation: Some parameter of a pulse train is varied in accordance with the message signal. Analogue pulse modulation: some feature of the pulse (e.g. amplitude, duration, or position) is varied continuously in accordance with the sample value of the message signal. Digital pulse modulation: the message signal is discrete in both time and amplitude, thereby transmitting a sequence of coded pulses. TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.2/1
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Sampling Process 1Let ga (t) be a continuous-time (CT) signal that is sampleduniformly at t = nT , generating the sequence g[n], g[n] = ga (nT ), −∞ < n < ∞ (1)where T is the sampling period and n is an integer. TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.3/1
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Sampling Process 2 Sampling ga(t) g[n] ga(t) gp(t) p(t) ∞p(t) is a periodic impulse train: p(t) = n=−∞ δ(t − nT ). TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.4/1
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Sampling Process 3p(t) can be expressed as a Fourier series as (see page 18 fordetails) ∞ 1 2π p(t) = exp(j( )kt). (2) T T k=−∞The sampling operation is a multiplication of the continuous-timesignal ga (t) by a period impulse train p(t): ∞ 1 2π gp (t) = ga (t) · p(t) = ga (t) · exp(j( )kt) . (3) T T k=−∞ TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.5/1
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FT of Sampled SignalAssume Ga (jω) ⇔ ga (t), i.e., Ga (jω) = F [ga (t)]. From thefrequency-shifting property of the FT, we have 2π 2π F [ga (t) · exp(j( )kt)] = Ga (j(ω − k )). (4) T TNext, taking FT on both sides of (3) and using (4), we get ∞ 1Gp (jω) = F [gp (t)] = Ga (j(ω − kωT )), −∞ < k < ∞ (5) T k=−∞ 2πwhere ωT = T denotes the angular sampling frequency. TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.6/1
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Sampling TheoremSampling theorem: Let ga (t) be a bandlimited signal withGa (jω) = 0 for |ω| > ωm . Then ga (t) is uniquely determined byits samples ga (nT ), −∞ < n < ∞, if ωT ≥ 2ωm , (6)where 2π ωT = . (7) T TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.7/1
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Signal Recovery 1Question: Suppose that g[n] is obtained by uniformly samplinga bandlimited analog signal ga (t) with a highest frequency ωm ata sampling rate ωT = 2π satisfying (6), can the original analog Tsignal ga (t) be fully recovered from the given sequence g[n]?Answer: YES, ga (t) can be fully recovered by generating animpulse train gp (t) and then passing gp (t) through an ideal lowpass ﬁlter (LPF) H(jω) with a gain T and a cutoff frequency ω csatisfying ωm < ωc < ωT − ωm . ∧ g[n] gp(t) LPF g a(t) TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.8/1
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Representation in Spectrum G a ( jω ) ω − ωm ωm Sampling Recovery LPF G p ( jω ) ••• ••• ω − ωT ωT 2ωT ωm ωc TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.9/1
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Signal Recovery 2Taking the inverse FT of the frequency response of the ideal LPFH(jω): T, |ω| ≤ ω c H(jω) = (8) 0, |ω| > ωcThen, the impulse response h(t) of the LPF is given by ωc 1 ∞ jωt T h(t) = H(jω)e dω = ejωt dω 2π −∞ 2π −ωc sin(ωc t) = . (9) πt/T TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.10/1
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Signal Recovery 3Consider the impulse train gp (t) be expressed as ∞ gp (t) = ga (t) · p(t) = ga (t) · δ(t − nT ) n=−∞ ∞ ∞ = ga (nT )δ(t − nT ) = g[n]δ(t − nT ). (10) n=−∞ n=−∞Therefore, the output of the LPF is given by the convolution ofgp (t) with the impulse response h(t): ∞ ga (t) = ˆ g[n]h(t − nT ). (11) n=−∞ TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.11/1
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Signal Recovery 4Substituting h(t) from (9) in (11) and assuming for simplicityωc = ωT /2 = π/T , we arrive at ∞ sin[π(t − nT )/T ] ga (t) = ˆ g[n] n=−∞ π(t − nT )/T ∞ t − nT = g[n] · sinc( ), (12) n=−∞ Twhere sinc(x) is deﬁned as sinc(x) = sin(πx)/(πx).The reconstructed analog signal ga (t) is obtained by shifting in ˆtime the impulse response of the LPF h(t) by an amount nT andscaling it an amplitude by the factor g[n] for −∞ < n < ∞ andthen summing up all shifted versions. TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.12/1
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PAMPulse-amplitude modulation (PAM): The amplitudes ofregularly spaced pulses are varied in proportion to thecorresponding sample values of a continuous message signal.Generation of PAM: Natural Sampling: easy to generate, only an analog switch required. Flat-Top Sampling: generated by using a sample-and-hold (S/H) type of electronic circuit. TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.13/1
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PDM and PPM Pulse-duration modulation (PDM): The duration of the pulses are varied according to the sample values of the message signal. Also referred to as pulse-width modulation or pulse-length modulation. Pulse-position modulation (PPM): The leading or trailing edge of each pulse is varied in accordance with the message signal. TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.16/1
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PDM and PPM Message Signal Pulse train PDM PPM TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.17/1
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Derivation of Eq. (2)Using Fourier series, p(t) can be expressed as ∞ k p(t) = ck exp(j2π t), T k=−∞where T /2 1 k ck = p(t) exp(−j2π t)dt T −T /2 T T /2 ∞ 1 k = δ(t − nT ) exp(−j2π t)dt T −T /2 n=−∞ T T /2 1 k = δ(t) exp(−j2π t)dt T −T /2 T 1 = . (13) T TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.18/1
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