6.9 Constant Envelope Modulation

1. 6.9 Constant Envelope Modulation 6.9.1 BFSK 6.9.2 MSK 6.6.3 GMSK

2. 6.9 Constant Envelope Modulation • nonlinear modulation • constant carrier amplitude - regardless of variations in m(t) • improved power efficiency without degrading occupied spectrum • - use power efficient class Camplifiers (non-linear) • low out of band radiation (-60dB to -70dB) • use limiter-discriminator detection • - simplified receiver design • - high immunity against random FM noise & fluctuations from • Rayleigh Fading • larger occupied bandwidth than linear modulation

3. 6.9.1 BFSK

4. sBFSK(t)= vH(t) = binary 1 6.95 sBFSK(t)= vL(t) = binary 0 • 6.9.1 BFSK • constant amplitude carrier • carrier frequency switched between 2 values – 2 message states • implementation determines if phase between bits is continuous • or discontinuous General FSK signal • 2f = constant offset from nominal carrier frequency • Methods for Generating BFSK signals • (i) shift between 2 independent oscillators • (ii) modulate a single carrier

5. input data phase jumps cos w1t switch cos w2t = sBFSK(t)= vH(t) binary 1 6.96 binary 0 sBFSK(t)= vL(t) • (i) switching between 2 independent oscillators for binary 1 & 0 • results in phase discontinuities • discontinuities causes spectral spreading & spurious transmission • not suitedfor tightly designed systems =

6. sBFSK(t) = = where (t) = ii. single carrier that is frequency modulated using m(t) 6.97 • m(t) = discontinuous bit stream • (t) = continuous phase function proportional to integral of m(t)

7. x a0 a1 0 1 VCO 0 1 1 modulated composite signal cos wct

8. Spectrum & Bandwidth of BFSK Signals • complex envelope of BFSK is nonlinear function of m(t) • evaluation of spectrum is difficult • usually performed using actual time averaged measurements • PSD of BFSK consists of discretefrequency components at • fc • fc nf • PSD decay rate (inversely proportional to spectrum) • PSD decay rate for CP-BFSK  • PSD decay rate for non CP-BFSK  • f = frequency offset fromfc

9. Transmission Bandwidth of BFSK Signals (from Carson’s Rule) • B = bandwidth of digital baseband signal • BT = transmission bandwidth of BFSK signal • BT= 2f +2B 6.98 • assume 1st null bandwidth used for digital signal, B • - bandwidth for rectangular pulses is given by B = Rb • - bandwidth of BFSK using rectangular pulse becomes • BT = 2(f + Rb) 6.99 • if RC pulseshaping used, bandwidth reduced to: • BT = 2f +(1+) Rb6.100

10. cos wct output + - Decision Circuit  r(t) sin wct Probability of error in coherent FSK receiver given as: Pe,BFSK = 6.101 • Coherent BFSK Detector • 2 correlators fed with local coherent reference signals • difference in correlator outputs compared with threshold to • determine binary value • block diagram is optimum for coherent BFSK in AWGN channel Decision Circuit: if output > Threshold  ‘1’ else ‘0’

12. Matched Filter fH + - Envelope Detector r(t)  output Tb Decision Circuit Envelope Detector Matched Filter fL Pe,BFSK, NC = 6.102 • Non-coherent Detection of BFSK • operates in noisy channel without coherent carrier reference • pair of matched filters followed by envelope detector • - upper path filter matched to fH (binary 1) • - lower path filter matched to fL (binary 0) • envelope detector output sampled at kTb compared to threshold Average probability of error in non-coherent FSK receiver:

14. General FSK signal and orthogonality • Two FSK signals, VH(t) and VL(t) are orthogonal if 6.103 • interference between VH(t) and VL(t) will average to 0 during • demodulation and integration of received symbol received signal will contain VH(t) and VL(t) demodulation of VH(t) results in (VH(t) + VL(t))VH(t)

15. vH(t) = for 0 ≤ t ≤ Tb and vL(t) = then and vH(t)vL(t) = = = = General FSK signal vH(t)vL(t) areorthogonal if Δf sin(4πfcTb) = -fc(sin(4πΔf Tb)

16. consider binary CPFSK signal defined over the interval 0 ≤ t ≤ T s(t) = • θ(t) = phaseof CPFSK signal • θ(t) is continuous s(t) is continuous at bit switching times • θ(t) increases/decreases linearly with t during T θ(t) = θ(0) ± ‘+’ corresponds to ‘1’ symbol ‘-’ corresponds to ‘0’ symbol h = deviation ratio of CPFSK CPFSK Modulation • elimination of phase discontinuity improves spectral efficiency & • noise performance 0 ≤ t ≤ T

17. 2πfct +θ(0) + = 2πf2 t+θ(0) f1 = 2πfct +θ(0) - = 2πf1t+θ(0) fc= f2 = yields and thus h = T(f2 – f1) determine fc and h by substitution • nominal fc= mean of f1 and f2 • h≡f2 – f1 normalized by T

18. at t = T θ(T) = θ(0) ± πh kFSK= symbol‘1’  θ(T) - θ(0) = πh symbol‘0’  θ(T) - θ(0) = -πh • peak frequency deviation F = |fc-fi | = ‘1’ sent  increases phase of s(t) by πh ‘0’ sent  decreases phase of s(t) by πh • variation ofθ(t)with t follows a path consisting of straight lines • slope of lines represent changes in frequency FSK modulation index =kFSK (similar to FM modulation index)

19. θ(t) - (0) rads 3πh 2πh πh 0 -πh -2πh -3πh 0 T 2T 3T 4T 5T 6T t • Phase Tree depicted from t = 0 • clarifies phase transitions across interval boundaries of incoming • bit sequence • θ(t) - (0) = phase of CPFSK signal is even or even multiple of πh • at even or even multiples of T

20. θ(t) - (0) 3π 2π π 0 -π -2π -3π fi=  f1= 2/T andf2= 3/T fc= = 5/2T 0 T 2T 3T 4T 5T 6T t h = T(f2 – f1) = 1 θ(t) = θ(0) ± 0 ≤ t ≤ T Phase Tree is a manifestation of phase continuity – an inherent characteristic of CPFSK consider Sunde FSK 1 0 0 0 0 1 1 • thus change in phase over T is either πor -π • change in phase of π = change in phase of -π • Sunde FSK has no memory • e.g. knowing value of bit i doesn’t help to find the value of bit i+1

21. fi= nc = fixed integer assume fi given by as si(t) = 0 ≤ t ≤ T for i = 1, 2 si(t) = 0 ≤ t ≤ T for i = 1, 2 = 0 otherwise = 0 otherwise • Sunde FSK • CPFSK = continuous phase FSK • phase continuity during inter-bit switching times

22. 2(t) 0 for i = 1, 2 i(t) = 0 ≤ t ≤ T 1 1(t) = 0 otherwise Sunde BFSK constellation: define two coordinates as let nc = 2 and T = 1us (1Mbps) then f1= 3MHz,f2 = 4MHz 1(t) = 0 ≤ t ≤ T = 0 otherwise 2(t) = 0 ≤ t ≤ T = 0 otherwise

23. 0 ≤ t ≤ T s1(t) = = 0 otherwise 0 ≤ t ≤ T = = s2(t) = = 0 otherwise

24. Sunde FSK Tb = 0.01, Rb= 100bps f1= 300Hz, f2= 400Hz two data carriers baseband time domain baseband frequency domain

25. Demodulation of Sunde FSK using Correlators Autocorrelation Cross Correlation

26. 6.9.2 MSK

27. 6.9.2 MSK (aka fast FSK) FSK modulation index • type of continuous phase FSK (CPFSK) • spectrally efficient • constant envelope • good BER performance • self-synchronizing capability • requires coherent detection • minimum frequency spacing (bandwidth) for 2 FSK signals to • be coherently orthogonal • minimum bandwidth that allows orthogonal detection kFSK= MSK modulation index is kMSK= 0.5  FMSK=

28. MSK can be thought of as special case of OQPSK • uses half-sinusoidal pulses instead of baseband rectangular pulses • arch shaped pulse during period = 2Tb • modify OQPSK equations for half-sine pulses for N-bit stream • several variations of MSK exist with different basic pulse shapes • e.g. • - use only positive ½ sinusoids • - use alternating negative & positive ½ sinusoids • all variations are CPFSK that use different techniques to achieve • spectral efficiency

29. sMSK(t) = • mI(t) & mQ(t)are bipolar bit streams (1) that feedI& Q • arms of the modulator - each arm fed at Rb/2 6.104 • mIi(t) = ith bit of mI(t), the even bits of m(t) • mQi(t) = ith bit of mQ(t), the odd bits of m(t) ½ sine pulse given by p(t) = 6.105 p(t – 2iTb)cos(2πfct) m(t) = ±1 bipolar bit stream p(t – 2iTb-Tb)sin(2πfct) TransmittedMSK signal (OQPSK variant)

30. sMSK(t) = 6-106 • k= 0 or depending on whether mI(t) = +1 to -1 • sMSK(t) has constant amplitude • to ensure phase continuity at bit interval  selectfc = • - n = integer • fc - and • fc + rewrite 6.104  view MSK waveform as a special case of CPFSK MSK is FSK signal with binary signaling frequencies given by • phase of MSK varies linearly over Tb

31. Phase Continuity of MSK h = ½ θ(t) = θ(0) ± 0 ≤ t ≤ T phase, θ(t) can take on only 2 values at odd or even multiples of T t =even multiple of T θ(T) - θ(0)= πor 0 t = odd multiple of Tθ(T) - θ(0)= ± π/2 assume = θ(0) 0

32. π π/2 0 -π/2 -π θ(t) - (0) 1 0 0 1 1 1 0 0 2T 4T 6T t Phase Trellis: path depicts θ(t) corresponding to a binary sequence • for h = ½ ΔF = Rb/4 • minimum ΔF for two binary FSK signals to be coherently • orthogonal • e.g. if Rb = 100Mbps  = ΔF = 25MHz

33. and f1 = f2 = then fc= = 1/T h = T(f1 - f2) = 1/2 θ(t) = θ(0) ± 0 ≤ t ≤ T MSK signal can be expressed as = s(t) = assume θ(0)= 0, then if bit = ‘1’ then s(t) = if bit = ‘0’ then s(t) = e.g. assume MSK modulation with

34. define orthonormal basis for MSK as 1(t) = 2(t) = 0 ≤ t ≤ T 0 ≤ t ≤ T bi θ(0) θ(T) s1 s2 then s(t) = s1(t)1(t) + s2(t)2(t) with ‘0’ 0 -π/2 ‘1’ π -π/2 s1= -T ≤ t ≤ T ‘0’ π π/2 ‘1’ 0 π/2 s2= = 0 ≤ t ≤ 2T =

35. RF power spectrum obtained by frequency shifting |F{p(t)}|2 • F{} = fourier transform • p(t)= MSK baseband pulse shaping function (1/2 sin wave) p(t) = 6.107 Normalized PSD for MSK is given as PMSK(f) = 6.108 MSK Power Spectrum

36. PSD of MSK & QPSK signals 10 0 -10 -20 -30 -40 -50 -60 QPSK, OQPSK MSK normalized PSD (dB) fc fc+0.5Rb fc+Rb fc+1.5Rb fc+2Rb • MSK spectrum • (1) has lower side lobes than QPSK (amplitude) • (2) has wider side lobes than QPSK (frequency) • 99% MSK power is within bandwidthB = 1.2/Tb • 99% QPSK power is within bandwidth B = 8/Tb

37. MSK has faster roll-off due to smoother pulse function • Spectrum of MSK main lobe > QPSK main lobe • - using 1st null bandwidth  MSK is spectrally less efficient • MSK has no abrupt phase shifts at bit transitions • - ok to bandlimit MSK signal to meet specified bandwidths • - bandlimiting doesn’t cause envelop to cross 0 • - envelope is  constant after bandlimiting • small variations in envelope removed using hardlimiting • - does not raise out of band radiation levels • constant amplitude  non-linear amplifiers can be used • continuous phase is desirable for highly reactive loads • simple modulation and demodulation circuits

38. mI(t) _ + SMSK(t) + + mQ(t)  + + cos(2fct)   cos(t/2T) MSK Transmitter (i) cos(2fct)cos(t/2T) 2 phase coherent signals atfc ¼R (ii) Separate 2 signals with narrow bandpass filters (iii) Combined to formI & Q carrier components x(t), y(t) (iv) Mix and sum to yield SMSK(t) = x(t)mI(t) + y(t)mQ(t) x(t) y(t)

39. (i) multiplycos(2fct)cos(t/2T) • cos(2fct) =carrier • cos(t/2T) =baseband pulseshaping function •  produces two phase coherent signals atfc ¼R • (ii) Separate 2 FSK signals with narrow bandpass filters • (iii) Combined to form • I carrier component: x(t) = (fc - 1/4Tb) – (fc + 1/4Tb) = -1/2Tb • Q carrier component y(t) = (fc + 1/4Tb) + (fc-1/4Tb) = 2fc if fc = 1/4Tb 2fc = 1/2Tb MSK Transmitter • (iv) SMSK(t) = x(t)mI(t) + y(t)mQ(t) • mI(t) & mQ(t) = even & odd bit streams

40. Threshold Device mI(t) t = 2(k+1)T x(t) SMSK(t) y(t) Threshold Device mQ(t) t = 2(k+1)T Coherent MSK Receiver • (i) SMSK(t) split & multiplied by local x(t) & y(t) (I & Q carriers) • (ii) mixer outputs are integrated over 2T & dumped • (iii) integrate & dump output fed to decision circuit every 2T • input signal level compared to threshold  decide 1 or 0 • output data streams correspond to mI(t) & mQ(t) • mI(t) & mQ(t) are offset & combined to obtain demodulated signal • *assumes ideal channel – no noise, interference

41. MSK Tb = 0.001, Rb= 1kbps f1=750Hz, f2= 1250Hz two data carriers 1 1 1 0 0 1 0 0 1 baseband time domain baseband frequency domain

42. Orthogonality Property of MSK Autocorrelation Cross Correlation

43. 6.9.3 GMSK 6.9 Constant Envelope Modulation 6.9.1 BFSK 6.9.2 MSK 6.6.3 GMSK

44. 6.9.3 GMSK Modulation • simple to apply Gaussian pulse shaping MSK • - smooths phase trajectory of MSK signal  over time, stabilizes • instantaneous frequency variations • - results in significant additional reduction of sidelobe levels • premodulation pulse shaping filter used to filter NRZ data • - converts full response message signal into partial response • scheme • full response baseband symbols occupy Tb • partial responsetransmitted symbols span several Tb • - pulse shaping doesn’t cause pattern’s averaged phase trajectory • to deviate from simple MSK trajectory • GMSK detection can be coherent (like MSK) or noncoherent • (like FSK)

45. GMSKs main advantages are • power efficiency - from constant envelope (non-linear amplifiers) • excellent spectral efficiency • pre-modulation filtering introduces ISI into transmitted signal • if B3dbTb > 0.5  degradation is not severe • B3dB= 3dB bandwidth of Gaussian Pulse Shaping Filter • Tb= bit duration = baseband symbol duration • irreducible BER caused by partial response signaling is the • cost for spectral efficiency & constant envelope • GMSK filter can be completely defined form B3dB Tb • - customary to define GMSK by B3dBTb

46. impulse responseof pre-modulation Gaussian filter given by hG(t) = 6-109 transfer function of pre-modulation Gaussian Filter given by 6-110 HG(f) =  is related toB3dB by  = 6.111

47. ReducingB3dBTb • (i) spectrum becomes more compact (spectral efficiency) • causes sidelobes of GMSK to fall off rapidly • B3dBTb = 0.5  2nd lobe peak is 30dB below main lobe • MSK 2nd peak lobe is 20dB below main lobe • MSK  GMSK with B3dBTb =  • (ii) increases irreducible error rate (IER)due to ISI • ISI degradation caused by pulse shaping increases • however - mobile channels induceIERdue to mobile’s velocity • mobile channels induce IER due to mobile’s velocity • if GMSK IER < mobile channel IER  no penalty for • using GMSK

48. 0 -10 -20 -30 -40 -50 -60 0 0.5 1.0 1.5 2.0 (f-fc)T PSD of GMSK signals BTb =  (MSK) BTb = 1.0 BTb = 0.5 BTb = 0.2 • increasing BTb • reduces signal spectrum • results in temporal spreading and distortion

49. RF bandwidth containing % power as fraction of Rb e.g. for BT = 0.2  99.99% of the power is in the bandwidth of 1.22Rb • [Ish81] BER degradation from ISI caused by GMSK filtering is • minimal at B3dBTb= 0.5887 • degradation in required Eb/N0 = 0.14dB compared to case of no ISI

50. 6.112 Pe = • GMSK BER for AWGN channel • [Mur81] shown to perform within 1dB of optimal MSK with • B3dBTb = 0.25 • since pulse shaping causes ISI  Peis function of B3dBTb Pe= bit error probability  is constant related to B3dBTb • B3dBTb = 0.25   = 0.68 • B3dBTb =    = 0.85 (MSK)