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 (iii) a signal, si(t) , sent during each symbol period, Ts = n.Tb (ii) each symbol mapped to signal si(t), M possible signals: s1(t),…,sM(t)

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

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) = 0  t Ts, i = 1,2,…M Es = log2MEb energy per symbol Ts = log2MTb symbol period written in quadrature form as: si(t) = for i = 1,2,…M Basis Signal ? 2 Ts

Mary Phase Shift Keying defined over 0  t  Ts 2(t) = Orthogonal basis signals sMPSK(t) = i = 1,2,…M MPSK signal can be expressed as

Mary Phase Shift Keying MPSK basis has 2 signals  2 dimensional constellation M-ary message points equally spaced on circle with radius MPSK is constant envelope when no pulse shaping is used 2(t) 1(t) MPSK signal can be coherently detected  = Arctan(Y/X) Minimum | I -  | non-coherent detected with differential encoding

Mary Phase Shift Keying Probability of symbol error in AWGN channel – using distance between adjacent symbols as Pe = average symbol error probability in AWGN channel Pe  When differentially encoded & non-coherently detected, Pe estimated for M  4 as: 2 Q 4 Es No sin  2M Pe 

Power Spectrum of MPSK Ps(f) = ¼ { Pg(f-fc) + Pg( -f-fc) } Ts = Tblog2M - Ts = symbol duration - Tb = bit duration Ps(f) = ¼ { Pg(f-fc) + Pg( -f-fc) } PMPSK(f) = PMPSK(f) =

PSD for M = 8 & M = 16 rect pulses RCF Increase in M with normalized PSD (dB) fc-½Rb fc-¼Rb fc fc+¼Rb fc+½Rb -10 -20 -30 -40 -50 -60 fc-⅔Rb fc-⅓Rb fc+⅓Rb fc+⅔Rb PSD for M = 8 & M = 16 rect pulses RCF Increase in M with Rb held constant Bnull decreases  B increases denser constellation  higher BER

MPSK Bandwidth Efficiency vs Power Efficiency 2 4 8 16 32 64 B = Rb/Bnull 0.5 1.0 1.5 2.5 3 Eb/N0 (dB) 10.5 14.0 18.5 23.4 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

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

Mary- Quadrature Amplitude Modulation allows amplitude & phase to vary general form of M-ary QAM signal given by 0  t  Ts i = 1,2,…M si(t) = 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

Mary- Quadrature Amplitude Modulation Assuming rectangular pulses - basis functions given by 1(t) = 0  t  Ts 2(t) = QAM signal given by: ai1(t) + bi2(t) si(t) = 0  t  Ts i = 1,2,…M coordinates of ith message point = and (ai, bi) = element in L2 matrix, where L =

Mary- Quadrature Amplitude Modulation e.g. let M = 16, then {ai,bi} given based on ai1(t) + bi2(t) {ai,bi} = 1(t) + 2(t) s11(t) = -3 3 0  t  Ts 1(t) + 2(t) s21(t) = -3 0  t  Ts

16 ary- Quadrature Amplitude Modulation -1.5 -0.5 0.5 1.5 2(t) 1(t) 1.5 0.5 -0.5 -1.5 1011 1010 0001 0011 1001 1000 0000 0010 1110 1100 0100 0101 1111 1101 0110 0111 QAM: modulated signal is hybrid of phase & amplitude modulation each message point corresponds to a quadbit Es is not constant – requires linear channel

Mary- Quadrature Amplitude Modulation In general, for any M = L2 ai1(t) + bi2(t) {ai,bi} =

Mary- Quadrature Amplitude Modulation The average error probability, Pe for M-ary QAM is approximated by assuming coherent detection AWGN channel no fading, timing jitter Pe  In terms of average energy, Eav Pe  Power Spectrum & Bandwidth Efficiency of QAM = MPSK Power Efficiency of QAM is better than MPSK

Mary- Quadrature Amplitude Modulation M-ary QAM - Bandwidth Efficiency & Power Efficiency Assume Optimum RC filters in AWGN Does not consider fading, jitter, - overly optimistic 28 5 1024 33.5 24 18.5 15 10.5 Eb/N0 (BER = 10-6) 6 4 3 2 1 B = Rb/Bnull 4096 256 64 16 M

Mary Frequency Shift Keying MFSK - transmitted signals defined as si(t) = 0  t Ts, i = 1,2,…M fc = nc/2Ts nc = fixed integer si(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

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 

Mary Frequency Shift Keying MFSK non-coherent detection using matched filters followed by envelope detectors average probability of error, Pe Pe = bound Pe  use leading terms of binomial expansion Pe 

MFSK Channel Bandwidth Coherent detection B = Non-coherent detection B = 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

M-ary QAM - Bandwidth Efficiency & Power Efficiency 28 5 1024 33.5 24 18.5 15 10.5 Eb/N0 (BER = 10-6) 6 4 3 2 1 B = Rb/Bnull 4096 256 64 16 M Coherent M-ary FSK - Bandwidth Efficiency & Power Efficiency 7.5 0.29 32 6.9 8.2 9.3 10.8 13.5 Eb/N0 (BER = 10-6) 0.18 0.42 0.55 0.57 0.4 B = Rb/Bnull 64 16 8 4 2 M

Summary of M-ary modulation in AWGN Channel B 3 2.5 2 1.5 1 0.5 MPSK/QAM Coherent MFSK 0 4 8 16 32 64 M 30 25 20 15 10 5 MPSK QAM Coherent MFSK (BER = 10-6) EB/N0

log10 Eb/ N0 16 PSK 15 12 16 QAM BFSK 9 6 4 FSK 4 PSK/QAM 3 BPSK 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 -3 -2 -1 0 1 2 3 log2 C/B 15 12 9 6 3 Error Free Region 16 PSK 16 QAM 4 PSK/QAM BPSK 4 FSK 16 FSK BFSK -1.6dB

Power & BW Efficiency Bandwidth Efficiency B = Power Efficiency Eb/N0 = energy used by a bit for detection B = Bandwidth Efficiency

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 = EbC is the average signal power (measured @ receiver) N = BN0 is the average noise power Eb = STb is the average received bit energy at receiver N0 = kT (Watts Hz–1) is the noise power density (Watts/Hz), - thermal noise in 1Hz bandwidth in any transmission line