Bandpass communication and the Complex Envelope EELE445-14 Lecture 25 1 Complex Envelope •Review sections 4-1 to 4-4 in the text •look at the complex envelope representations in table 4-1 •be able to calculate or find: •complex envelope of a waveform expression •calculate the signal power •calculate the PEP •define Modulation 1 Some Definitions A baseband waveform has a spectral magnitude that in nonzero for frequencies in the vicinity of the origin (i.e. f=0) and negligible elsewhere. A bandpass waveform has a spectral magnitude that is nonzero for frequencies in some band concentrated about a frequency f=+/- fc where fc is much greater than zero. Modulation is the process if imparting the source information onto a bandpass signal with a carrier frequency fc by the introduction of amplitude or phase perturbations or both. The resulting bandpass signal is called the modulated signal s(t), and the baseband source signal is called the modulating signal m(t) Figure 4–1 Bandpass Communication system. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 2 Complex Envelope Theorem: Any physical bandpass waveform can be represented by: s(t) = Re{g(t)e jωct } Re{•}denotes real part g(t) ≡ complex envelope of s(t) fc ≡ carrier frequency, wc ≡ 2πfc g(t) = x(t) + jy(t) = g(t) e∠g(t) = R(t)e jθ (t) Complex Envelope g(t) = x(t) + jy(t) = g(t) e∠g(t) = R(t)e jθ (t) x(t) = Re{}g(t) ≡ R(t)cos(θ (t)) y(t) = Im{}g(t) ≡ R(t)sin(θ (t)) R(t) = g(t) ≡ x(t)2 + y(t)2 −1⎛ y(t) ⎞ θ (t) = ∠g(t) ≡ tan ⎜ ⎟ ⎝ x(t) ⎠ jωct s(t) = x(t)cos(ωct) − y(t)sin(ωct) = Re{}g(t)e 3 Complex Envelope g(t) = x(t) + jy(t) x(t) is the “I” or In phase modulation y(t) is the “Q” or Quadrature modulation R(t) is >=0 and determines the signal power g(t) = g[]m(t) g[]• is the mapping operator for m(t) g[ ] function determines the bandpass modulation type Complex Envelope- Vector Modulator jωct s(t) = x(t)cos(ωct) − y(t)sin(ωct) = Re{g(t)e } cos(2πfct) π − 2 All modulation types may be generated using a vector modulator 4 Complex Envelope: time and frequency domain Bandpass waveform : v(t) = Re{}g(t)e jωct Spectrum : 1 V ( f ) = []G( f − f ) + G * (− f − f ) 2 c c PSD : 1 P ( f ) = []P ( f − f ) + P (− f − f ) v 4 g c g c Power : ∞ 1 2 P = v 2 (t) = P ( f )df = R (0) = g(t) v ∫ v v −∞ 2 Complex Envelope, of various modulations 5 Complex Envelope, (R θ), of various modulations DSB-SC and AM EELE445-14 Lecture 26 12 6 DSB-SC – section 5-1 in the text • This is AM without the Carrier • All of the power in the PSD is dependent on the information Æm(t) • The modulation efficiency is 100% • Synchronous demodulation is required s(t) = Acm(t)cos(2πfct) m(t) ≤ 1 g(t) = A m(t) pre - envelope m(t)2 ≤ 1 c cos(2πfct) DSB-SC – Spectrum s(t) = Acm(t) cos( 2πf ct) g(t) = Acm(t) pre - envelope m(t) ≤ 1, (normalize d m(t)) G( f ) = Ac M ( f ) A S( f ) = c []M ( f − f ) + M ( f + f ) 2 c c 7 DSB-SC – Signals Waveforms 1 1 ct() 0 Volts mt() − 1 1 05.10 4 0.001 0.0015 0.002 − 3 0 t 1.992× 10 time Carrier Voltage Modulation Voltage DSB-SC – Signal DSB-SC and Envelopes 1 1 s.dsb ()tn mt()n 0 Volts mt()n − mt()n − 1 1 4 05.10 0.001 0.0015 0.002 0 tn 1.992× 10− 3 Time s(t) m(t) positive envelope negative envelope 8 DSB-SC – Signal DSB-SC Voltage Spectrum 1 1 2 DSB k 0.5 Volts peak 0 0 0 5000 1 .104 1.5 .104 2 .104 0 1 20KHz ⋅k t.max Frequency Hz AM – section 5-1 g(t) = Ac [1 + m(t)] pre - envelope s(t) = Ac [1 + m(t)]cos( 2πfct) | m(t) |≤ 1 m(t)2 ≤ 1 A − A %modulation = max min 100 2 Ac max[ m(t)] − min[m(t) = 100 2 9 AM – section 5-1 in the text 1 2 1 s(t) = g(t) = A2 []1 + m(t) 2 2 2 c 1 1 = A2 + A2 m(t) + A2 m(t)2 2 c c 2 c = Carrier + Carrier shift + Sideband for 0 mean, m(t) = 0 : s(t)2 = Carrier power + sideband power 1 1 = A2 + A2 m(t)2 2 c 2 c Figure 5–1 AM signal waveform. s(t) = Ac [1 + m(t)]cos(ωt) physical wavform g(t) = Ac []1 + m(t) envelope determines the power Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 10 Mathcad AM signal waveform, u=50% Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Mathcad AM signal waveform μ=0.5 AM Modulation Voltage Spectrum 1.5 1.25 1 0.75 Volts peak Volts 0.5 0.25 0 6 8 10 12 14 Frequency KHz AM Power 0.5 0.45 0.4 0.35 0.3 0.25 Watts 0.2 0.15 0.1 0.05 .0312 0 6 8 10 12 14 Frequency KHz Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 11 Mathcad AM signal waveform μ=1 AM and Envelopes AM Modulation Voltage Spectrum 2 1.5 1.25 1 1 0.75 0 peak Volts Volts 0.5 0.25 1 0 6 8 10 12 14 Frequency KHz 2 . 4 0510 0.001 0.0015 0.002 AM Power 0.5 Time s(t) 0.45 Positive Envelop e 0.4 Negative Envelope 0.35 0.3 0.25 Watts 0.2 0.15 .125 0.1 0.05 0 6 8 10 12 14 Frequency KHz Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 AM – Power and modulation efficiency 1 A2 s 2 (t) = g 2 (t) = c []1 + m(t) 2 2 2 A2 = c 1 + 2m(t) + m 2 (t) 2 A2 A2 = c + A2 m(t) + c m 2 (t) 2 c 2 = Carrier power + power shift in carrier + sideband power 12 AM – section 5-1 s(t) = Ac [1 + m(t)]cos( 2πfct) real envelope : R(t) = g(t) = Ac []1 + m(t) , | m(t) |≤ 1 m(t)2 ≤ 1 A − A %modulation = max min 100 2 Ac max[ m(t)] − min[m(t) = 100 2 Figure 4–2 Spectrum of AM signal. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 13 AM – section 5-1 in the text • use your class notes and review section 5-1 in the text. • also review chapter 4 for AM detection and the superheterodyne receiver. • AM is used for commercial broadcasting in order that low cost envelope detection may be used. Bandpass Digital Signal Examples on board… EELE445-14 14 Asymmetric Sideband Signals USSB, LSSB, VSB EELE445-13 Lecture 27 Figure 5–4 Spectrum for a USSB signal. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 15 Figure 5–5 Generation of SSB. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Figure P5–12 Weaver’s method for generating SSB. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 16 Single Side Band- Phasing Method m(t) g(t) = m(t) + mˆ (t) mˆ (t) Single Side Band- Phasing Method M ( f ) ˆ M ( f ) = H h ( f )M ( f ) 1 1 f f 0 0 -1 G( f ) = M ( f ) + Mˆ ( f ) 17 Single Side Band U(-f) Single Side Band 18 Single Side Band Single Side Band 19 Single Side Band Single Side Band 20 Single Side Band Single Side Band 21 Figure 5–5 Generation of SSB. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Single Side Band 22 Single Side Band Single Side Band 23 Figure 5–6 VSB transmitter and spectra. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 24.