Outline Analog Communications. Lecture 05 Angle Modulation. Instantaneous Frequency and Frequency Deviation. Angle Modulation. Pierluigi SALVO ROSSI

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1 Outline Analog Communications Lecture 05 Angle Modulation 1 PM and FM Pierluigi SALVO ROSSI Department of Industrial and Information Engineering Second University of Naples Via Roma 9, Aversa (CE), Italy Narrowband Approximation 3 Wideband Modulation homepage: pierluigi.salvorossi@unina.it P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 05 1 / P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 05 / Angle Modulation Instantaneous Frequency and Frequency Deviation x(t) is the information low-pass signal or modulating signal p(t) = A cos(f 0 t + ϑ) is the carrier signal the transmitted signal or modulated signal has the form z(t) = A cos(f 0 t + ψ x (t)) where ψ x (t) depends on x(t) through linear transformations Phase Modulation (PM) ψ x (t) = s x(t) Frequency Modulation (FM) ψ x (t) = s t t 0 x(τ)dτ PM FM f z (t) = 1 f z (t) = 1 f z (t) = f 0 + s f z (t) = s t (f 0t + ψ x (t)) = f t ψ x(t), t x(t), t x(t) f z (t) = f 0 + s x(t) f z (t) = s x(t), t ψ x(t) f max = max t R f z(t) f max = s t x(t) f max = s x(t) max max P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 05 3 / P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 05 4 /

2 Signal Plots Narrowband Approximation (1/) Using series expansion of exponential function z(t) = A cos(f 0 t + ψ x (t)) = AR {exp (jf 0 t) exp (jψ x (t))} { } j n ψ n = AR exp (jf 0 t) x(t) n! n=0 The bandwidth of ψ n x(t) is n times the bandwidth of ψ x (t) The bandwidth of z(t) is infinite Assuming small phase variations, i.e. ψ x (t) 1 z(t) AR {exp (jf 0 t) (1 + jψ x (t))} = A cos(jf 0 t) Aψ x (t) sin(jf 0 t) Looks like AM modulation: the modulating signal is in quadrature w.r.t. the carrier signal P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 05 5 / P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 05 6 / Correlation and PSD of Narrowband Angle Modulation If ψ x (t) is a WSS signal with null expected value, then z(t) is cyclostationary with period 1/f 0 R z (t, τ) = A cos(f 0τ) + A cos(f 0(t τ)) + A R ψ x (τ) cos(f 0 τ) A R ψ x (τ) cos(f 0 (t τ)) R z (τ) = A cos(f 0τ) + A R ψ x (τ) cos(f 0 τ) P z (τ) = A 4 (δ(f f 0) + δ(f + f 0 )) + A 4 (P ψ x (f f 0 ) + P ψx (f + f 0 )) P z = A + A P ψ x B z = B x P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 05 7 / Complex Envelope of Narrowband Angle Modulation Analytic signal and complex envelope are z(t) = A(1 + jψ x (t)) exp(jf 0 t) z(t) = A(1 + jψ x (t)) The complex envelope is complex-valued Real (resp. imaginary) component has arbitrary (resp. arbitrary small) value P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 05 8 /

3 Wideband Modulation: One Sinusoid Modulating Wideband Modulation: Bessel Expansion (1/) Focus on the FM case with modulating signal x(t) = A m cos(f m t + θ m ) then ψ x (t) = sa m f m sin(f m t + θ m ) and the transmitted signal is ( z(t) = A cos f 0 t + sa ) m sin(f m t + θ m ) f m = A cos (f 0 t + β m sin(f m t + θ m )) z(t) = A cos (f 0 t + β m sin(f m t + θ m )) = AR {exp(jf 0 t) exp(jβ m sin(f m t + θ m ))} { = AR exp(jf 0 t) = A + n= + n= J n (β m ) exp(jnθ m ) exp(jnf m t) J n (β m ) cos ((f 0 + nf m )t + nθ m ) } with f max = sa m and β m = f max f m Infinite spectral components with frequency spacing equals to f m P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 05 9 / P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / Wideband Modulation: Bessel Expansion (/) Wideband Modulation: Bandwidth J n (x) = 1 +π exp(j(x sin(α) nα))dα = 1 +π cos(x sin(α) nα)dα π π 0 J 0 (0) = 1 and J n (0) = 0 J n ( x) = J n (x) and J n (x) = J n (x) J n 1 ( x) = J n 1 (x) and J n+1 (x) = J n 1 (x) J n (x) 0 if n x + n= J n(x) = 1 z(t) A +β m n= β m J n (β m ) cos ((f 0 + nf m )t + nθ m ) B z = β m f m P z = A P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 05 1 /

4 Wideband Modulation: Two Sinusoids Modulating (1/) Focus on the FM case with modulating signal with f 1 > f then x(t) = A 1 cos(f 1 t + θ 1 ) + A cos(f t + θ ) ψ x (t) = sa 1 f 1 sin(f 1 t + θ 1 ) + sa f sin(f t + θ ) Wideband Modulation: Two Sinusoids Modulating (/) Using twice the Bessel expansion z(t) A +β 1 +β J n1 (β 1 )J n (β ) n 1 = β 1 n = β cos ((f 0 + n 1 f m1 + n f m )t + n 1 θ 1 + n θ ) and the transmitted signal is with z(t) = A cos (f 0 t + β 1 sin(f 1 t + θ 1 ) + β sin(f t + θ )) β 1 = sa 1 f 1 and β = sa f P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / B z = β 1 f 1 and P z = A P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / Wideband Modulation: N Sinusoids Modulating Focus on the FM case with modulating signal N x(t) = A l cos(f l t + θ l ) with β l = sa l f l then l=1 z(t) A +β 1 +β N J n1 (β 1 ) J nn (β N ) n 1 = β 1 n N = β N ( ( ) ) N N cos f 0 + n l f l t + n l θ l m = arg max l=1,...,n f l l=1 l=1 B z = β m f m and P z = A Carson s Bandwidth Rule Assume a low-pass modulating signal x(t) with B x is the bandwidth of x(t) A x = max t x(t) is the maximum amplitude = fmax B x β = sax B x then extending the previous considerations we get B z = βb x = f max Merging results for narrowband modulation (β 1) and wideband modulation (β 1), we get B z = (1 + β)b x P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture /

5 Wideband Modulation and Woodward s Theorem (1/3) Consider a II-order stationary random process ψ x (t) as modulating signal R z (t, τ) = A E {cos(f 0 t + ψ x (t)) cos(f 0 (t τ) + ψ x (t τ))} = A E {cos(f 0(t τ) + ψ x (t) + ψ x (t τ))} + A E {cos(f 0(τ) + ψ x (t) ψ x (t τ))} = A [ R e jf0t e jf0τ E{e jψx(t) e jψx(t τ) } + A [ R e jf0τ E{e jψx(t) e jψx(t τ) } = A [ R e jf0t e jf0τ E{y(t)y(t τ)} + A [ R e jf0τ E{y(t)y (t τ)} where y(t) = exp(jψ x (t)) P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / Wideband Modulation and Woodward s Theorem (/3) The II-order stationarity assumption makes both E{y(t)y(t τ)} and E{y(t)y (t τ)} = R y (τ) independent on t z(t) is cyclostationary with period 1/f 0 and R z (τ) = A [ R e jf0τ R y (τ) = A 4 ( ) e jf0τ R y (τ) + e jf0τ R y ( τ) P z (f) = A 4 (P y(f f 0 ) + P y ( f f 0 )) P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / Wideband Modulation and Woodward s Theorem (3/3) Assume ψ x (t) slowly changing, then ψ x (t) ψ x (t τ) τ ψ x (t) and { } R y (τ) = E e jτ ψ x(t) = e jτξ f ψ x (ξ)dξ { R } = F 1 f ψ x (f) (τ) then P y (f) = f ψ x (f), giving P z (f) = πa ( ) f ψ x ((f f 0 )) + f ψ x ( (f + f 0 )), and P z = A Bandwidth The shape of the PSD of the modulated signal z(t) follows the shape of the pdf of the modulating signal x(t) The bandwidth B z is approximatively given by the support of the pdf f x ( ) The support is proportional to the standard deviation σ x In the case of FM we have the Woodward s Theorem ( ( ) ( P z (f) = πa f x s s (f f 0) + f x )) s (f + f 0) The bandwidth is proportional to the amplitude of the modulating signal P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 05 0 /

6 Modulation and Demodulation Frequency Discriminator FM modulation and demodulaton PM modulation and demodulation ψ x (t) = s t x(τ)dτ t 0 z(t) = A cos (f 0 t + ψ x (t)) ż(t) = A(f 0 + ψ x (t)) sin (f 0 t + ψ x (t)) Assuming f 0 s x(t) max we get v(t) = A(f 0 + sx(t)) y(t) = Asx(t) P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 05 1 / P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 05 /

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