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6.8 Linear Modulation Techniques

6.8 Linear Modulation Techniques. a(t) = . (1) Cartesian basis : . s(t) = s I (t)cos(2 π f c t) - s Q (t)sin(2 π f c t). (2) Polar basis : . s(t) = a(t)cos( 2 π f c t + θ (t) ). envelope of s(t) given as . θ (t) =. phase of s(t) given as . 6.8 Linear Modulation Techniques

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6.8 Linear Modulation Techniques

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  1. 6.8 Linear Modulation Techniques

  2. a(t) = (1) Cartesian basis: s(t) = sI(t)cos(2πfct) - sQ(t)sin(2πfct) (2) Polar basis: s(t) = a(t)cos( 2πfct + θ(t) ) envelope of s(t) given as θ(t) = phase of s(t) given as • 6.8 Linear Modulation Techniques • linear modulation: carrier’s amplitude carrier varies linearly with m(t) • bandwidth efficient • attractive for systems with limited spectrum (e.g. wireless) constructing signal constellations of linear modulated signals

  3. s(t) = Re[Am(t) exp(j2fct)] 6.65 s(t) = A[mR(t) cos(2fct) - m1(t) sin(2fct) ] Provides Basis for any transmitted signal A = signalamplitude fc = carrier frequency s(t) = transmitted signal m(t) = modulating digital signal mR + jm1(t) = complex envelope representation of m(t) • generally linear modulationdoesn’t have constant envelope • non-linear modulation has either linear or constant carrier • envelope

  4. 6.8.1 BPSK

  5. Ac = • 6.8.1 BPSK • phase of constant amplitudecarrier switched between 2 values • phase switches based on 2 possible symbols, m1 and m2 • normally phase separated by 180o p(t) = Assume rectangular pulse shape: For a carrier with frequency =fcandamplitude (volts) = Ac • carrier given by Accos(2πfct + θc) • bit energy Eb= ½ Ac2Tb • c = phase shift in carrier

  6. sBPSK(t)= sBPSK(t) = = BPSK constellation 1. Transmitted BPSK signal is binary 1 6.66 binary 0

  7. Basis Signal Set consists of one waveform (symbol), 1(t) 1(t) = 6.60 SBPSK = 6.61 generalize m1and m2as binary signal m(t) sBPSK(t)= m(t) 6.67 BPSK signal set expressed in terms of basis

  8. Spectrum and Bandwidth of BPSK • BPSK signal can be expressed in complex envelope form • use polar baseband data waveform of m(t), sBPSK = Re[gBPSK(t) exp(j2 fct)] 6.68 wheregBPSK(t)iscomplex envelope ofsBPSK(t)given by: gBPSK(t) = 6.69 • BPSK signal is a double sideband amplitude modulated waveform • suppressed carrier • applied carrier = Accos(2fct) • data signal = m(t) •  BPSK signal can be generated using balanced modulator

  9. PgBPSK(f) = 6.70 6.71 PgBPSK(f) = PSD of baseband complex envelope can be shown to be PSD for BPSK at RF passbandcan be evaluated by translating baseband spectrum according to 6.41. Ps(f) = ¼[Pg(f-fc) + Pg(-f-fc)]

  10. 1 1 0 0 1 1 1 0 -1 T 2T 3T 4T 5T 6T Peak PSD fc-5f0 fc-3f0 fc-f0 fc fc+f0 fc+3f0 fc+5f0 (occurs with 101010…pattern)

  11. 0 -10 -20 -30 -40 -50 -60 normalized PSD (dB) fc-3Rb fc-2Rb fc-Rb fc fc+Rb fc+2Rb fc+3Rb • PSD of BPSK signal with rectangular pulse and RC pulse shaping • null-to-null BW = 2Rb(Rb = bit rate) rectangular pulses RC shaping with  = 0.5

  12. 2 E b sBPSK(t)= p + q + q m ( t ) cos( 2 f t ) c c ch T b 6.72 2 E b p + q m ( t ) cos( 2 f t ) c T = b • 2. Received BPSK Signal • BPSK Receiver can be expressed as ch= phase shift from channel time delay c = phase shift in carrier fc = carrier frequency assumes no multipath impairments induced by channel

  13. BPSK Receiver uses coherent (synchronous) demodulation • receiver requires information about c & fc • options to recover fc and c include: • (1) send low levelpilot carrier signal & use PLL • (2) synthesize carrier phase & frequency • e.g. use Costas loop or - Squaringloop

  14. (5) output of mixer sBPSK(t) m(t) 2fc 6.73 2 3 4 5 square law frequency  f/2 integrate & dump 1 6 cos(4fct+2) cos(2fct+) bit synch m2(t)A2ccos2(2fct+) m(t)Accos2(2fct+) sBPSK(t) = m(t)Accos(2fct+) (1) received signal is (2) sBPSK(t)2 generates dc signal & amplitude varying sinusoid at 4fc (3) dc signal is filtered using BPF with center frequency = 2fc (4) frequency divider () used to recreate cos(2fct +) BPSK Receiver with squaring loop

  15. W N ‘1’ S t ‘0’ Decision Boundary • (6) mixer output applied to integrate & dump • forms LPF segment of detector • if transmit & receive pulse shapes match  optimum detection • bit synch facilitates sampling of integrator output at end of Tb • at end of Tb  integrator switch closes & output dumped to • decision circuit • (7) decision circuit uses threshold to determine if bit is a 1 or 0 • threshold must be tuned to minimize error • if 1 or 0 are equally likely  use midpoint of detector • voltage output level

  16. for BPSK – the distance between points in constellation is • given by d12= Bit Error Probability for BPSK from substitution into 6.62 Pe,BPSK = 6.74 • Probability of Bit Error for BPSK • many modulation schemes in AWGN channel – use Q-function • of distance between signal points

  17. 6.8.2 DPSK

  18. dk= mk dk-1 • net effect also achieved by following rule: • if mk= 1  dk = dk-1 • if mk= 0  dk = dk-1  dk = kth differentially encoded output, generated from compliment of modulo 2 sum of mkandd k-1 6.8.2 DPSK • non-coherentPSK  receiver doesn’t need reference signal • simplified receiver - easy & cheap to build, widely used • Let {mk} = input binary sequence • {dk} = differentially encoded output sequence no symbol transition with mk = 1  possible synchronization issue

  19. k 0 1 2 3 4 5 6 7 8 {mk} - 1 0 0 1 0 1 0 1 {dk-1} - 1 1 0 1 1 0 0 1 {dk} 1 1 0 1 1 0 0 1 0 o o o = 1 ō = 0 • average probability of bit error: PeDPSK = 6.75 e.g. for given data stream: {mk} • less energy efficiency - about 3dB < coherent PSK

  20. dk DPSK signal mk  cos(2fct) Transmitter: DPSK obtained by passingdk to product modulator delay Tb dk-1 DPSK Signal threshold device integrate & dump logic circuit Delay Tb dk= mk dk-1  mk= dk  dk-1 • Receiver: • input signal demodulated • original sequence recovered by undoing differential encoding

  21. 6.8.3 QPSK

  22. QPSK signal sQPSK(t) can be expressed as: 6.76 sQPSK(t)= 0  t  Ts i = 1,2,3,4 is the symbol’s amplitude is phase of the symbol (0, 90 , 180 , 270) • 6.8.3 QPSK • 2 bits transmitted per symbol  2bandwidth efficiency of BPSK • symbol determined from 4 possible phases • Ts= 2Tb (one symbol time = two bit periods) • Es = 2Eb bit energy = ½ symbol energy

  23. rewrite equation 6.76over 0  t  Ts (cos(α + β) = cosαcosβ - sinα sinβ) define a Basis for S over interval 0  t  Ts: {1(t), 2(t)} 1(t) = 2(t) = define QPSK signal set: S = {s1(t), s2(t), s3(t), s4(t)} 6.77 sQPSK(t)= si(t)= 6.78 QPSK Basis

  24. m1(t) = 101101 m(t) = 110010100011 m2(t) = 100001 m2(t) = m1(t) = 1 for 0 ≤ t ≤ 2T 0 otherwise p(t) = • Alternate view of QPSK • parallel combination of 2 BPSK modulators operating in quadrature • phase to each other demultiplex binary stream m(t) into m1(t) and m2(t) bk,i= +1 for binary ‘1’ bk,i = -1 for binary ‘0’ p(t) = pulse shape, assume a rectangular pulse:

  25. phase shift key of binary streams • 0 and  phase shift mI(t) • /2and 3/2phase shift mQ(t) mI(t) = for i = 1, 3 Ac = mQ(t) = for i = 2,4 sI(t) = AcmI(t)cos(2fct) /2 sQ(t) = AcmQ(t)sin(2fct)  0 • sQPSK(t) = sI(t) + sQ(t) • = AcmI(t)cos(2fct) + AcmQ(t)sin(2fct) 3/2

  26. sI(t) = mI(t)cos(2fct) bit = 1  sI(t) = cos(2fct) for 0 ≤ t ≤ 2T bit = 0  sI(t) = -cos(2fct) for 0 ≤ t ≤ 2T sQ(t) = mQ(t)sin(2fct) bit = 1  sQ(t) = sin(2fct) for 0 ≤ t ≤ 2T bit = 0  sQ(t) = -sin(2fct) for 0 ≤ t ≤ 2T normalize Ac =1 sQPSK(t) = cos(2fct) sin(2fct)

  27. Q Q /2 I I  0 3/4 /4 3/2 7/4 54 QPSK Constellation Diagram has four points • rotate constellation by /4obtain new QPSK signal set M1 = M2 = Es = 2Eb

  28. binary symbol grey coded QPSK signal si1 si2 binary symbol grey coded QPSK signal si1 si2 10 3π/2 0 /2 11 π 0 3/4 /4 10 7π/4 01 π/2 0  0 11 5π/4 00 0 0 01 3π/4 3/2 00 π/4 7/4 54 si(t) = si11(t) + si22(t) (3.36) Signal Space Characterization of QPSK Signal Constellations ithQPSK signal, based on message points (si1, si2) defined in tables for i = 1,2 and 0 ≤ t ≤ Ts

  29. Average probability of bit error: PeQPSK • assumes AWGN channel • since Ts= 2Tb Es = 2Eb PeQPSK = 6.79 • distance between adjacent points = = • ThusPeQPSK = PeBPSK • QPSK has 2 spectral efficiency of BPSK & same energy • efficiency • QPSK can be differentially encoded - allows non-coherent • detection

  30. I Q 1 1 0 1 1 0 -1 Ac 1 1 0 1 2T 4T 6T 8T 1 0 -1 2T 4T 6T 8T t • Baseband QPSK Signal in Time Domain • Ts= 0.1s • Tb = 0.05 Rb = 20bps S1 =

  31. Ac t • QPSK data stream • Ts= 0.1s • Tb = 0.05 Rb = 20bps

  32. PQPSK(f) = PQPSK(f) = Wnull= Null to Null Bandwidth • Wnull-QPSK= Rb • Wnull-BPSK = 2Rb • QPSK Spectrum & Bandwidth • PSD of QPSK using rectangular pulses given by PQPSK(f) • similar to PSD of BPSK, replace Tb with Ts 6.80

  33. PSD of QPSK signal 0 -10 -20 -30 -40 -50 -60 normalized PSD (dB) fc-Rb fc-½Rb fc fc+½Rb fc+Rb • rectangular pulse • RC pulse shaping with  = 0.5

  34. QPSK Transmitter - based on modulating 2 modulated BPSK signals mI(t) at ½ Rb QPSK output 1(t) m(t) at Rb LO Serial - Parallel  90o 2(t) mQ(t) at ½ Rb • (1) m(t) = bi-polar NRZ input with bit rate = Rb • (2) split m(t) into even and odd stream, mI(t) & mQ(t) eachwith ½ Rb • (3) modulate each stream with quadrature carriers 1(t), 2(t) • (4) sum two resultant BPSK signals to produce QPSK output • (5) band pass filter confines signal to allocated passband • *pulse shaping normally done at baseband, prior to modulation

  35. decision circuit received signal carrier recovery symbol timing recovery MUX recovered signal 90o decision circuit Coherent QPSK Receiver • (1) front end BPF removes out of band noise • (2) filtered output is split • (3) each part coherently demodulated using I & Q carriers • (4) revover carriers coherently from received signals with squaring • loop • (5) demodulated output passed to decision circuit which generates • I & Q streams • (6) I & Q streams are multiplexed to recover original binary stream

  36. 6.8.5 Offset QPSK

  37. 6.8.5 Offset QPSK (OQPSK) • QPSK is ideally constant envelope (e.g. amplitude is constant) • Pulse shaped (bandlimited) QPSK signals lose constant envelope • if phase shift =  signal envelope can momentarily pass • through 0 (zero crossing) • hardlimiting or non-linear amplification of zero crossings • brings back filtered side lobes • - fidelity of signal at small voltages is lost in transmission • - sidelobes  result in spectral widening • Use of linear amplifiers to amplify pulses will avoid this but • will result in inefficient power use

  38. QPSK: • phase (bit) transitions of mI(t) & mQ(t) occur at same time instants • phase transitions occur every Ts= 2Tb • maximum phase transition = 180o () both mI(t) & mQ(t) change • non-linear amplification results in spectral widening • OQPSK (offset QPSK) • phase (bit) transition instants of mI(t) & mQ(t) are offset by Tb • phase transitions occur every Tb= ½Ts • max phase shift = 90o (/2) only one bit stream value changes • ensures smaller phase transitions applied to RF amplifier •  reduces spectral growth after amplification

  39. m0 m2 m4 m6 m8 m10 m12 mI(t) m1 m3 m5 m7 m9 m11 m13 mQ(t) -Tb0 Tb2Tb3Tb4Tb5Tb6Tb7Tb8Tb9Tb 10Tb11Tb12Tb13Tb • OQPSK (offset QPSK) • bit transitions of mI(t) & mQ(t) are offset by Tb in relative alignment • - phase transistions occur every Tb= ½ Ts • - at any time, only one bit stream can change values •  maximum phase shift of transmitted signal limited to 90°

  40. /2 /2   0 0 3/2 3/2 OQPSK possible phase shifts OQPSK possible phase shifts

  41. s2(t)and s2_offset(t) s1(t)and s2_offset(t) Offset QPSK

  42. sI(t) = AcmI(t)cos(2fct) sQ(t) = AcmQ(t-Tb)sin(2fct) • sOQPSK(t) = sI(t) + sQ(t) • = AcmI(t)cos(2fct) + AcmQ(t-Tb)sin(2fct) main differences from QPSK is time alignment of mI(t) & mQ(t) mI(t)= even bit stream mQ(t-Tb) = odd bit streams, offset by Tb Ts = symbol period Tb= bit period

  43. OQPSK vs QPSK • OQPSK switching occurs at Tb vs 2Tbfor QPSK • OQPSK eliminates 180° phase transition • spectrum of OQPSK = spectrum of QPSK – unaffected by offset • alignment of bit streams • OQPSK retains bandlimited nature even with non-linear • amplification • - critical for low power operations • OQPSK appears to perform better than QPSK with phase jitter • from noisy reference signals

  44. 6.8.6 /4 QPSK

  45. 6.8.6 /4 QPSK • compromise between OQPSK & QPSK • either coherent or non-coherent demodulation • maximum phase change limited to 135o • - 180o for QPSK • - 90o for OQPSK • preservation of constant BW property in between 2 variants • performs better than both in multipath spread & fading • /4 DQPSK differential encoded version • facilitate differential detection or coherent modulation

  46. Q = possible states forkfork-1= n/4 = possible states for kfork-1= n/2 I all possible signal transitions • /4 QPSKmodulation • modulated signal selected from 2 QPSK constellations shifted • by /4 • for each symbol  switch between constellations –total of 8 • symbols states 4 used alternately • phase shift between each symbol =nk= /4 , n = 1,2,3 • - ensures minimal phase shift, k= /4 between successive symbols • - enables timing recovery & synchronization

  47. 6.8.7 /4 Transmitter

  48. 6.8.7 /4 QPSK Transmitter • (i) partition input mk into symbol stream mIk, mQk • (ii) produce pulses Ikand Qk by signal mapping during [kT,(k+1)T] • (iii) form I(t) & Q(t) from p(t), Ik, Qk & modulateby quadrature • carriers • (iv) pre-modulationpulse shaping /4QPSK Transmitter cos wct /4QPSK output mIk Ik I(t) mk Serial - Parallel Signal Mapping  amplifier Q(t) mQk Qk sin wct

  49. (i) input bitstream m(t) partitioned into symbol streams mIk, mQk • by serial-parallel conversion • each symbol stream with symbol rate, Rs = ½Rb • for symbol at k+1, phase shift = kis a function ofmIk, mQk inputs mIk, mQk phase shift k 11 /4 01 3/4 10 -3/4 00 -/4

  50. Ik= cosk • = Ik-1cos k - Qk-1sin k 6.81 k = k-1 + k • Qk= sink • = Ik-1sin k - Qk-1cos k 6.83 6.82 • (ii) signal mapping circuit produces Ik& Qk during kT  t  (k+1)T • Ik= kthin-phase pulse • Qk = kthquadrature pulse • k= phase of kthsymbol  is a function of k

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