Combined Linear & Constant Envelope Modulation M-ary modulation: digital baseband data sent by varying RF carrier’s (i) envelope ( eg. MASK) (ii) phase /frequency ( eg. MPSK, MFSK) (iii) envelope & phase offer 2 degrees of freedom ( eg. MQAM) (i) n bits encoded into 1 of M symbols, M 2n (ii) each symbol mapped to signal si(t), M possible signals: s1(t),…,sM(t) (iii) a signal, si(t) , sent during each symbol period, Ts = n.Tb 1 Combined Linear & Constant Envelope Modulation M-ary modulation is useful in bandlimited channels • greater B log2M • significantly higher BER - smaller distances in constellation - sensitive to timing jitter MPSK MQAM MFSK OFDM 2 Mary Phase Shift Keying Carrier phase takes 1 of M possible values – amplitude constant i = 2(i-1)/M, i = 1,2,…M Modulated waveform: si(t) = 2 Es 2 cos 2f c t (i 1) Ts M Es = log2MEb Ts = log2MTb 2 Ts 2 Es cos (i 1) M for i = 1,2,…M i = 1,2,…M energy per symbol symbol period written in quadrature form as: si(t) = 0 t Ts, Basis Signal ? 2 cos2f c t Es sin (i 1) M sin 2f c t 3 Mary Phase Shift Keying Orthogonal basis signals 1(t) = 2(t) = 2 cos2f ct Ts 2 sin 2f ct Ts defined over 0 t Ts MPSK signal can be expressed as sMPSK(t) = Es cos (i 1) 1 (t ) Es sin (i 1) 2 (t ) 2 2 i = 1,2,…M 4 Mary Phase Shift Keying • MPSK basis has 2 signals 2 dimensional constellation • M-ary message points equally spaced on circle with radius Es • MPSK is constant envelope when no pulse shaping is used 2(t) MPSK signal can be • coherently detected • = Arctan(Y/X) •Minimum | I - | • non-coherent detected with differential encoding 1(t) Es 2 Es sin M 5 Mary Phase Shift Keying Probability of symbol error in AWGN channel – using distance between adjacent symbols as 2 Es sin M Pe = average symbol error probability in AWGN channel Pe 2 Eb log 2 M 2Q sin N0 M When differentially encoded & non-coherently detected, Pe estimated for M 4 as: Pe 2Q 4 Es sin 2M No 6 Power Spectrum of MPSK Ts = Tblog2M - Ts = symbol duration - Tb = bit duration Ps(f) = ¼ { Pg(f-fc) + Pg( -f-fc) } PMPSK(f) = PMPSK(f) = Es 2 sin ( f f )T c s ( f f c )Ts Eb log 2 M 2 2 sin ( f f c )Ts ( f f c )Ts sin 2 ( f f )T log M c b 2 2 ( f f c )Tb log 2 M 2 2 sin 2 ( f f c )Tb log 2 M 2 ( f f c )Tb log 2 M 2 7 PSD for M = 8 & M = 16 0 rect pulses RCF -10 normalized PSD (dB) -20 Increase in M with Rb held constant -30 • Bnull decreases B increases • denser constellation higher BER -40 -50 -60 fc-½Rb fc-⅔Rb fc-¼Rb fc-⅓Rb fc fc+¼Rb fc+⅓Rb fc+½Rb fc+⅔Rb 8 MPSK Bandwidth Efficiency vs Power Efficiency M 2 4 8 16 2 B = Rb/Bnull 0.5 1.0 1.5 Eb/N0 (dB) 10.5 10.5 14.0 18.5 32 2.5 23.4 64 3 28.5 • B = bandwidth efficiency • Rb = bit rate • Bnull = 1st null bandwidth • Eb/N0 for BER = 10-6 bandwidth efficiency & power efficiency assume • Ideal Nyquist Pulse Shaping (RC filters) • AWGN channel without timing jitter or fading 9 Mary Phase Shift Keying Advantages: Bandwidth efficiency increases with M Drawbacks: Jitter & fading cause large increase in BER as M increases EMI & multipath alter instantaneous phase of signal – cause error at detector Receiver design also impacts BER Power efficiency reduces for higher M MPSK in mobile channels require Pilot Symbols or Equalization 10 Mary- Quadrature Amplitude Modulation • allows amplitude & phase to vary • general form of M-ary QAM signal given by si(t) = 2Emin 2Emin ai cos 2f ct bi sin 2f ct Ts Ts 0 t Ts i = 1,2,…M Emin = energy of signal with lowest amplitude ai, bi = independent integers related to location of signal point Ts = symbol period • energy per symbol / distance between adj. symbols isn’t constant probability of correct symbol detection is not same for all symbols • Pilot tones used to estimate channel effects 11 Mary- Quadrature Amplitude Modulation Assuming rectangular pulses - basis functions given by 1(t) = 2 cos2f ct Ts 2(t) = 2 sin 2f ct Ts 0 t Ts 0 t Ts QAM signal given by: si(t) = Emin ai1(t) + Emin bi2(t) 0 t Ts i = 1,2,…M coordinates of ith message point = ai Emin and bi Emin (ai, bi) = element in L2 matrix, where L = M 12 Mary- Quadrature Amplitude Modulation e.g. let M = 16, then {ai,bi} given based on Emin {ai,bi} = ai1(t) + Emin bi2(t) ( 1,3) (1,3) (3,3) ( 3,3) ( 3,1) ( 1,1) (1,1) (3,1) ( 3,1) ( 1,1) (1,1) (3,1) ( 3 , 3 ) ( 1 , 3 ) ( 1 , 3 ) ( 3 , 3 ) s11(t) = -3 Emin 1(t) + 3 Emin 2(t) 0 t Ts s21(t) = -3 Emin 1(t) + Emin 0 t Ts 2(t) 13 16 ary- Quadrature Amplitude Modulation QAM: modulated signal is hybrid of phase & amplitude modulation • each message point corresponds to a quadbit 2(t) 1011 1001 1110 1111 1010 1000 0.5 1100 1101 -1.5 0001 -0.5 0 0.5 0000 0100 -0.5 1.5 1(t) 0110 0011 0010 -1.5 0101 0111 1.5 •Es is not constant – requires linear channel 14 Mary- Quadrature Amplitude Modulation In general, for any M = L2 Emin ai1(t) + {ai,bi} = Eminbi2(t) ( L 1, L 1) ( L 3, L 1) ( L 1, L 3) ( L 3, L 3) ... ... ( L 1, L 1) ( L 3, L 1) ... ( L 1, L 1) ... ( L 1, L 3) ... ... ... ( L 1, L 1) 15 Mary- Quadrature Amplitude Modulation The average error probability, Pe for M-ary QAM is approximated by Pe 41 1 Q M 2 Emin N0 • assuming coherent detection • AWGN channel • no fading, timing jitter In terms of average energy, Eav Pe 3Eav 1 41 Q M ( M 1) N 0 Power Spectrum & Bandwidth Efficiency of QAM = MPSK Power Efficiency of QAM is better than MPSK 16 Mary- Quadrature Amplitude Modulation M-ary QAM - Bandwidth Efficiency & Power Efficiency • Assume Optimum RC filters in AWGN • Does not consider fading, jitter, - overly optimistic M 4 16 64 256 1024 4096 B = Rb/Bnull 1 2 3 4 5 6 15 18.5 24 28 33.5 Eb/N0 (BER = 10-6) 10.5 17 Mary Frequency Shift Keying MFSK - transmitted signals defined as si(t) = 2 Es cos (nc i)t Ts Ts 0 t Ts, i = 1,2,…M • fc = nc/2Ts • nc = fixed integer si(t) = 2Es i cos 2 f c Ts 2Ts t 0 t Ts, i = 1,2,…M Each of M signals have • equal energy • equal duration • adjacent sub carrier frequencies separated by 1/2Ts Hz • sub carriers are orthogonal to each other 18 Mary Frequency Shift Keying MFSK coherent detection - optimum receiver • receiver has bank of M correlators or matched filters • each correlator tuned to 1 of M distinct carrier frequencies • average probability of error, Pe (based on union bound) Pe M 1Q Eb log 2 M N0 19 Mary Frequency Shift Keying MFSK non-coherent detection • using matched filters followed by envelope detectors • average probability of error, Pe Pe = M 1 (1) k 1 M 1 k 1 k 1 k kEs exp (k 1) N 0 bound Pe use leading terms of binomial expansion Pe Es M 1 exp 2 2 N0 20 MFSK Channel Bandwidth Coherent detection B = Non-coherent detection Rb ( M 3) 2 log 2 M B= Rb M 2 log 2 M Impact of increasing M on MFSK performance bandwidth efficiency (B) of MFSK decreases • MFSK signals are bandwidth inefficient (unlike MPSK) power efficiency (P) increases • with M orthogonal signals signal space is not crowded • power efficient non-linear amplifiers can be used without performance degradation 21 M-ary QAM - Bandwidth Efficiency & Power Efficiency M 4 16 64 256 1024 4096 B = Rb/Bnull 1 2 3 4 5 6 15 18.5 24 28 33.5 Eb/N0 (BER = 10-6) 10.5 Coherent M-ary FSK - Bandwidth Efficiency & Power Efficiency M 2 0.4 B = Rb/Bnull Eb/N0 (BER = 10-6) 13.5 4 0.57 10.8 8 0.55 9.3 16 0.42 8.2 32 0.29 7.5 64 0.18 6.9 22 Summary of M-ary modulation in AWGN Channel B 3 2.5 EB/N0 (BER = 10-6) 2 1.5 1 0.5 0 MPSK/QAM Coherent MFSK 0 4 8 16 32 30 25 20 15 10 5 0 64 M MPSK QAM Coherent MFSK 0 4 8 16 32 64 M 23 Shannon Limit: • Most schemes are away from Eb/N0 of –1.6 dB by 4dB or more • FEC helps to get closer to Shannon limit • FSK allows exchange of BW efficiency for power efficiency log10 Eb/ N0 15 9 BPSK 16 FSK 4 PSK/QAM 6 Error Free Region -3 16 QAM BFSK 12 4 FSK -1.6dB 16 PSK -2 3 -1 0 1 2 3 log2 C/B 24 Power & BW Efficiency Bandwidth Efficiency EbC C log 2 1 B = B N0B Power Efficiency Eb/N0 = energy used by a bit for detection 25 S Eb B ( dB ) ( dB ) ( dB ) N N0 R Power & BW Efficiency if C ≤ B log2(1+ S/N) error free communication is possible if C > B log2(1+ S/N) some errors will occur • assumes only AWGN (ok if BW << channel center frequency) • in practice < 3dB (50%) is feasible S = EbC is the average signal power (measured @ receiver) N = BN0 is the average noise power Eb = STb is the average received bit energy at receiver N0 = kT (Watts Hz–1) is the noise power density (Watts/Hz), - thermal noise in 1Hz bandwidth in any transmission line 26