Outline. Analog Communications. Lecture 03 Linear Modulation. Linear Modulation. Double Side Band (DSB) Modulation. Pierluigi SALVO ROSSI

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1 Outline Analog Communications Lecture 03 Linear Modulation Pierluigi SALVO ROSSI Department of Industrial and Information Engineering Second University of Naples Via Roma 29, Aversa (CE), Italy homepage: DSB 2 AM 3 SSB 4 QAM P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 03 1 / 33 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 03 2 / 33 Linear Modulation x(t) is the information low-pass signal or modulating signal p(t) = A cos(2πf 0 t + ϑ) is the carrier signal Double Side Band (DSB) Modulation The time-domain transmitted signal is z(t) = A x(t) cos(2πf 0 t) the transmitted signal or modulated signal has the form z(t) = z I (t) cos(2πf 0 t) + z Q (t) sin(2πf 0 t) where z I (t) and z Q (t) depend on x(t) through affine transformations The power efficiency η p is defined as the ratio between the transmitted power of the information signal component and the whole transmitted power The spectral efficiency η s is defined as the ratio between the bandwidth of the information signal (B x ) and the bandwidth of the transmitted signal (B z ) The frequency-domain transmitted signal is Z(f) = (X(f f 0) + X(f + f 0 )) P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 03 3 / 33 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 03 4 / 33

2 DSB Demodulation (1/2) DSB Demodulation (2/2) Assume an ideal LPF with unitary gain and cutoff frequency f t = B x ( ) f H(f) = rect y(t) = AB 2 x(t) cos(2π f t + ϑ) v(t) = ABx(t) cos(2πf 0 t) cos(2π(f 0 + f )t + ϑ) = AB 2 x(t) cos(2π f t + ϑ) + AB 2 x(t) cos(2π(2f 0 + f )t + ϑ) V (f) = AB ( ) e jϑ X(f f ) + e jϑ X(f + f ) 4 + AB ( ) e jϑ X(f (2f 0 + f )) + e jϑ X(f + (2f 0 + f )) 4 Y (f) = AB 4 Underlying assumption ( ) e jϑ X(f f ) + e jϑ X(f + f ) f 0 > B x P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 03 5 / 33 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 03 6 / 33 DSB Case 1: Perfect Synchronism If carrier frequency and phase are perfectly known at the receiver f = 0 and ϑ = 0 y(t) = AB 2 x(t) Y (f) = AB 2 X(f) the received signal is a scaled version of the transmitted signal the whole chain is non-distorting DSB Case 2: Phase Error If carrier frequency only is perfectly known at the receiver f = 0 and ϑ 0 y(t) = AB 2 cos(ϑ)x(t) Y (f) = AB 2 cos(ϑ)x(f) the received signal is a scaled version of the transmitted signal the whole chain is non-distorting critical only if the two oscillators are in quadrature (ϑ π/2) in this case y(t) 0 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 03 7 / 33 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 03 8 / 33

3 DSB Case 3: Frequency Error Correlation and PSD of DSB Signals If carrier phase only is perfectly known at the receiver f 0 and ϑ = 0 y(t) = AB 2 x(t) cos(2π f t) Y (f) = AB 4 (X(f f ) + X(f + f )) the received signal is a distorted version of the transmitted signal the whole chain is distorting non-linear distortion (residual modulation) If x(t) is a WSS signal, z(t) is cyclostationary with period 1/2f 0 R z (t, τ) = A2 2 R x(τ) cos(2πf 0 τ) + A2 2 R x(τ) cos(2πf 0 (2t τ)) R z (τ) = A2 2 R x(τ) cos(2πf 0 τ) P z (τ) = A2 4 (P x(f f 0 ) + P x (f + f 0 )) P z = A2 2 P x P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture 03 9 / 33 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 Complex Envelope of DSB Signals Power and Spectral Efficiencies for DSB Analytic signal and complex envelope are z(t) = Ax(t) exp(j2πf 0 t) z(t) = Ax(t) η p = 2 P x 2 P x The complex envelope is real-valued The complex envelope assumes both positive and negative values η s = B x /2 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33

Amplitude Modulation (AM) AM Coherent Demodulation (1/2) The time-domain transmitted signal is z(t) = A (1 + kx(t)) cos(2πf 0 t) The frequency-domain transmitted signal is Z(f) = (δ(f f 0) + δ(f + f

4 Amplitude Modulation (AM) AM Coherent Demodulation (1/2) The time-domain transmitted signal is z(t) = A (1 + kx(t)) cos(2πf 0 t) The frequency-domain transmitted signal is Z(f) = (δ(f f 0) + δ(f + f 0 )) + k (X(f f 0) + X(f + f 0 )) The same scheme used for DSB would recover the information signal assuming that the DC component is absent (or irrelevant) and adding a system for DC removal v(t) = AB 2 (1 + kx(t)) and V (f) = AB 2 δ(f) + kab 2 X(f) y(t) = kab kab x(t) and Y (f) = 2 2 X(f) P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 AM Coherent Demodulation (2/2) AM Incoherent Demodulation (1/2) It is worth noticing that AM use part of the available power to transmit the unmodulated carrier That may appear as a waste of resources Using a coherent receiver DOES make that a waste of resources The main point of AM (i.e. of wasting transmission power) is the possibility to use an extremely-simple receiver v(t) = A 1 + kx(t) = A (1 + kx(t)) Last equality does not always hold, in the positive case it is apparent that y(t) = kax(t) P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33

5 AM Incoherent Demodulation (2/2) Correlation and PSD of AM Signals The condition is equivalent to i.e. 1 + kx(t) = (1 + kx(t)) 1 + kx(t) 0 t If x(t) is a WSS signal with null expected value, z(t) is cyclostationary with period 1/2f 0 R z (t, τ) = A2 2 cos(2πf 0τ) + A2 2 cos(2πf 0(2t τ)) + k2 2 R x(τ) cos(2πf 0 τ) + k2 2 R x(τ) cos(2πf 0 (2t τ)) R z (τ) = A2 2 cos(2πf 0τ) + k2 2 R x(τ) cos(2πf 0 τ) k x(t) 1 Finally, denoting x m = max t x(t) we get k 1 x m t P z (τ) = A2 4 (δ(f f 0) + δ(f + f 0 )) + k2 (P x (f f 0 ) + P x (f + f 0 )) 4 P z = A2 2 + k2 2 P x P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 Complex Envelope of AM Signals Power and Spectral Efficiencies for AM Analytic signal and complex envelope are z(t) = A(1 + kx(t)) exp(j2πf 0 t) z(t) = A(1 + kx(t)) η p = = k 2 2 P x 2 + k2 2 P x k 2 P x The complex envelope is real-valued The complex envelope assumes positive values only Also remember that k 2 P x 1 thus η p 1/2 η s = B x /2 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33

6 Single Side Band (SSB) Modulation (1/2) The time-domain transmitted signal is z(t) = A x(t) cos(2πf 0 t) A ˆx(t) sin(2πf 0 t) SSB Modulation (2/2) The main idea is to halve the bandwidth of the transmitted signal SSB-upper (SSB-U) selects the external frequencies and removes the internal frequencies SSB-lower (SSB-L) selects the internal frequencies and removes the external frequencies The frequency-domain transmitted signal is Z(f) = X(f f 0) (1 ± sign(f f 0 )) + X(f + f 0) (1 sign(f + f 0 )) Q(f) = (X(f f 0) + X(f + f 0 )) ( ) f H u (f) = 2 rect 2f 0 Z u (f) = Q(f)H u (f) ( ( )) f H l (f) = 2 1 rect 2f 0 Z l (f) = Q(f)H l (f) P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 SSB Demodulation (1/2) SSB Demodulation (2/2) Assume an ideal LPF with unitary gain and cutoff frequency f t = B x ( ) f H(f) = rect v(t) = ABx(t) cos 2 (2πf 0 t) ABˆx(t) sin(2πf 0 t) cos(2πf 0 t) = AB 2 x(t) + AB 2 x(t) cos(2π2f 0t) AB 2 ˆx(t) sin(2π2f 0t) y(t) = AB 2 x(t) Y (f) = AB 2 X(f) V (f) = AB 2 X(f) + AB 4 (X(f 2f 0) + X(f + 2f 0 )) Underlying assumption ± AB 4 (X(f 2f 0)sign(f 2f 0 ) X(f + 2f 0 )sign(f + 2f 0 )) f 0 > B x 2 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33

Correlation and PSD of SSB Signals Complex Envelope of SSB Signals If x(t) is a WSS signal, z(t) is WSS R z (τ) = R x (τ) cos(2πf 0 τ) ˆRx (τ) sin(2πf 0 τ) P z (τ) = A2 2 P

± ˆx(t)) The complex envelope is complex-valued Real and imaginary components have the same absolute value P. Salvo Rossi (SUN.

DIII) Analog Communications - Lecture 03 26 / 33 Power and Spectral Efficiencies for SSB Quadrature Amplitude Modulation (QAM) The time-domain transmitted signal is z(t) = A

7 Correlation and PSD of SSB Signals Complex Envelope of SSB Signals If x(t) is a WSS signal, z(t) is WSS R z (τ) = R x (τ) cos(2πf 0 τ) ˆRx (τ) sin(2πf 0 τ) P z (τ) = A2 2 P x(f f 0 ) (1 ± sign(f f 0 )) P z = P x + A2 2 P x(f + f 0 ) (1 sign(f + f 0 )) Analytic signal and complex envelope are z(t) = A (x(t) ± ˆx(t)) exp(j2πf 0 t) z(t) = A (x(t) ± ˆx(t)) The complex envelope is complex-valued Real and imaginary components have the same absolute value P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 Power and Spectral Efficiencies for SSB Quadrature Amplitude Modulation (QAM) The time-domain transmitted signal is z(t) = A x 1 (t) cos(2πf 0 t) + A x 2 (t) sin(2πf 0 t) η p = A2 P x P x η s = B x B x The frequency-domain transmitted signal is Z(f) = (X 1(f f 0 ) + X 1 (f + f 0 )) + j (X 2(f f 0 ) X 2 (f + f 0 )) P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33

8 QAM Demodulation (1/2) QAM Demodulation (2/2) Assume an ideal LPF with unitary gain and cutoff frequency f t = B x ( ) f H(f) = rect y 1 (t) = AB 2 x 1(t) y 2 (t) = AB 2 x 2(t) v 1 (t) = ABx 1 (t) cos 2 (2πf 0 t) + ABx 2 (t) sin(2πf 0 t) cos(2πf 0 t) = AB 2 x 1(t) + AB 2 x 1(t) cos(2π2f 0 t) + AB 2 x 2(t) sin(2π2f 0 t) V 1 (f) = AB 2 X 1(f) + AB 4 (X 1(f 2f 0 ) + X 1 (f + 2f 0 )) + AB 4j (X 2(f 2f 0 ) X 2 (f + 2f 0 )) P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 Y 1 (f) = AB 2 X 1(f) Y 2 (f) = AB 2 X 2(f) Underlying assumption f 0 > B x P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 Correlation and PSD of QAM Signals Complex Envelope of QAM Signals If x(t) is a WSS signal, z(t) is cyclostationary with period 1/2f 0 R z (τ) = A2 2 (R x 1 (τ) + R x2 (τ)) (τ) cos(2πf 0 τ) Usually R x1 x 2 (τ) = 0, thus A2 2 (R x 1 x 2 (τ) R x2 x 1 (τ)) (τ) sin(2πf 0 τ) R z (τ) = A2 2 (R x 1 (τ) + R x2 (τ)) (τ) cos(2πf 0 τ) P z (τ) = A2 4 (P x 1 (f f 0 ) + P x1 (f + f 0 ) + P x2 (f f 0 ) + P x2 (f + f 0 )) P z = A2 2 P x 1 + A2 2 P x 2 Analytic signal and complex envelope are z(t) = A (x 1 (t) x 2 (t)) exp(j2πf 0 t) z(t) = A (x 1 (t) x 2 (t)) The complex envelope is complex-valued Real and imaginary components have arbitrary values P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33 P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33

9 Power and Spectral Efficiencies for QAM η p = 2 P x 1 + A2 2 P x 2 2 P x 1 + A2 2 P x 2 η s = P. Salvo Rossi (SUN.DIII) Analog Communications - Lecture / 33

Chapter 3: Analog Modulation Cengage Learning Engineering. All Rights Reserved.

Chapter 3: Analog Modulation Cengage Learning Engineering. All Rights Reserved. Contemporary Communication Systems using MATLAB Chapter 3: Analog Modulation 2013 Cengage Learning Engineering. All Rights Reserved. 3.1 Preview In this chapter we study analog modulation & demodulation,