Modulations in Digital Communications PDF

Modulations in digital communications Binary Phase Shift Keying (BPSK) Quadrature Phase Shift Keying (QPSK) Multiple Phase Shift Keying (M-PSK) Minimum Shift Keying (MSK) Gaussian filtered MSK (GMSK) Quadrature Amplitude Modulation (QAM) Orthogonal Frequency Division Mul...

Modulations in digital communications Binary Phase Shift Keying (BPSK) Quadrature Phase Shift Keying (QPSK) Multiple Phase Shift Keying (M-PSK) Minimum Shift Keying (MSK) Gaussian filtered MSK (GMSK) Quadrature Amplitude Modulation (QAM) Orthogonal Frequency Division Multiplexing (OFDM) Amplitude shift keying (ASK) Simplest case of two level amplitude shift keying: 2 Ei S DAM (t ) = cos(2pf c t ), i = 1,2; 0 < t £ T T Amplitude shift keying (ASK) Error probability T 1 E + E1 - 2 r E1 E 0 E 0 = ò [ s 0 (t )] 2 dt = 0 p(e) min = erfc 0 2 4N 0 0 T E1 = ò [ s1 (t )] 2 dt = E1 0 T 1 r= E1 E0 òs 0 0 (t ) s1 (t )dt = 0 On-off ASK signal error probability is expressed as: 1 E p(e) = erfc 2 2 N0 Here E = E1 / 2 average bit energy, N0- noise spectral density Binary phase shift keying (BPSK) Signal shape and spectrum In case of BPSK, information is stored in signal phase: e.g. RF carrier phase is 0, when transmitting logical 1, and π -- when transmitting logical 0. BPSK signal is expressed: ì 2E ü ï cos(2pf ct ), ï ï T ï S DFM (t ) = í ý0 £ t £ T ï- 2 E cos(2pf t ).ï ïî T c ïþ For coherent BPSK radio pulse of duration T contains a whole number of carrier periods, i.e.: n T= fc Binary phase shift keying (BPSK) m(t) 1 0 1 0 1 t SDFM(t) t 2E S DFM (t ) = m1 (t ) cos(2pf c t ) T It is similar to two-sided amplitude modulation  signal spectrum sDFM corresponds to spectrum of square pulse series, which is symmetricaly distributed with respect to carrier frequency éæ sin p ( f - f )T ö 2 æ sin p ( f + f )T ö 2 ù f >0 sDFM ( f ) = const êçç c ÷÷ + çç c ÷÷ ú êëè p ( f c - f )T ø è p ( f c + f )T ø úû s, dB 20 0 -20 -40 fc-6R fc-4R fc-2R fc fc+2R fc+4R fc+6R BPSK transmitter and receiver Binary BPSK sequence Modulator signal Bipolar signal (Multiplier) Carrier A cos( 2pf c t ) b Received BPSK Correlator T signal x(t) x Decision X ò0 device 1, if x > 0 Local carrier 0, if x < 0 A cos( 2pf c t ) x2(t) Bandpass filter Frequency divider /2 Carrier recovery circuit BPSK bit error rate (BER) Bit error rate for noisy BPSK signal: 1 E 0 + E1 - 2 r E1 E 0 p(e) min = erfc 2 4N 0 T E 0 = ò [ s 0 (t )] 2 dt = E 0 T For antipodal signals: E1 = ò [ s1 (t )] 2 dt = E E1=E, E0=E and ρ=-1, 0 s0 (t ) = - s1 (t ) 1 T E r= E1 E0 ò0 s0 (t )s1 (t )dt = - E1E0 1 E p (e) = erfc 2 N0 With the same BER level BPSK signal can be decoded from 3 dB higher noise (lower SNR) compared to ASK signal BPSK bit error rate (BER) Exercise 1: Find bit error rate for BPSK modulation at Eb/No = 8 dB. Exercise 2: Find bit error rate for BPSK modulation at SNR = 8 dB for 54 Mbps rate service over 22 MHz channel bandwidth. BER of BPSK with noisy local carrier phase If carrier recovery circuit introduces carrier phase error, signal amplitude at the output of modulator (and at the same time at the output of correlator) is reduced by cos( Dj ). This is equivalent to bit energy reduction by cos 2 ( Dj ) 1 æ E ö p (e) = erfcçç cos(Dj ) ÷÷ 2 è N0 ø Suppose Dj statistics is defined by Gaussian distribution with dispersion s j. Then error probability 2 1 ¥ æ E ö pj (e) = ò erfcçç cos(Dj ) ÷÷ f (Dj )dDj 2 -¥ è N0 ø 1 é (Dj )2 ù f ( Dj ) = exp ê- 2 ú. sj 2p êë 2s j úû BER of BPSK with noisy local carrier phase BER 10-2 s j = 30 0 10-3 10-4 s j = 20 0 10-5 s j = 00 2 4 6 8 10 12 14 16 S/N, dB 10-6 Bit error rate for noisy BPSK signal and random local carrier phase Quadrature phase shift keying (QPSK) Two types exist: ì 2E ï cos[ 2pf c t + ( 2i + 1) p ], 0 £ t < T s i (t ) = í T s 4 Standard ï0, kitur î otherwise ì 2E ï cos(2pf c t + (i + 1) p ), 0 £ t < T Rotated s i (t ) = í Ts 2 ï otherwise î0, kitur Ts = 2Tb Ts = n , i = 0,1,2,3 fc The phase of the signals is referenced from the beginning of the symbol; thus, the two forms differ by the location of the jumps within cosine function. In the case of QPSK, each symbol represents a so-called dibit - a combination of two adjacent bits. Dibit sets for QPSK Q 1,1 0,1 modulation -1,1 1,1 I Modulated Modulated carrier phase carrier phase Data dibit (standard (rotated 0,0 QPSK) QPSK) -1,-1 1,0 1,-1 11 π/4 π Q 0,1 01 3π/4 π/2 0, 2 0,0 - 2, 0 1,1 00 5π/4 0 2 ,0 10 7π/4 3π/2 I 1,0 0,- 2  QPSK modulation ì 2E ï cos( 2pf c t + i p + j 0 ), 0 £ t < T General case s i (t ) = í Ts 2 ï î0, kitur otherwise where i = 0,1,2,3; j 0 = p for rotated and j 0 = p 4 for standard QPSK. 2 Using sum angle equation: 2E 2E s i (t ) = cos( 2pf c t ) cos( i p + j 0 ) - sin( 2pf c t ) sin( i p + j 0 ) Ts 2 Ts 2 I In-phase signal Q Quadrature signal ì 2E ü s i (t ) = Reí exp j[(i p + j 0 )] exp( j 2pf c t )ý 2 î Ts (complex amplitude) þ  ì 2E ü s i (t ) = Reí exp j[(i p + j 0 )] exp( j 2pf c t )ý 2 î Ts þ 2E Complex amplitude A= exp j[(i p + j 0 )] Ts 2 Modulation Q Q constellations 1,1 0,1 0,1 1,1 -1,1 0, 2 0,0 - 2, 0 1,1 2 ,0 I I 0,0 -1,-1 1,0 Standard 1,-1 1,0 Rotated 0,- 2 QPSK modulation a b Standard Max jump: 2 Rotated Max jump: √2 Instantaneous signal voltage jumps resulting in out-of-band emissions QPSK transmitter and receiver Formation of dibit sequences Bits of a0,a1, a2,a3,a4,…aN T duration I-bits Q-bits Dibits of a0,a2,a4,…aN-1 a1,a3,a5,…aN 2T duration QPSK Binary sequence Bipolar signal Demultiplexer I Q sequence sequence Carrier A cos(2πf c t ) X Phase shifter A sin(2πf c t ) π2 X QPSK Σ signal QPSK transmitter π [T. S. Rappaport, Wireless communications: Principles and practice. Prentice Hall, 2002] QPSK receiver π [T. S. Rappaport, Wireless communications: Principles and practice. Prentice Hall, 2002] Offset QPSK Binary Dvejetainė Bipoliarinis sequence seka BipolarNRZ Demultiplexer koderis Demultiplekseris NZR coder BitBito TT trukmė duration I sequence I seka. Bito Užlaikymo Bit duration Delay by per T trukmė 2T 2T T grandinė Nešlys Carrier Q seka. BitoQ sequence A cos(2pf c t ) X Bit duration 2T trukmė 2T Avoids phase Fazės Phasesukiklis shifter A sin( 2pf c t ) jumps by πtwo p 2 X times per symbol, e.g. 01→10 In O-QPSK Pastumtosios Offset QPSK S changes max 1 bit KFM signalas signal π/4-QPSK and π/4-DQPSK π/4-QPSK uses both constellations: when switching from symbol to symbol carrier phase is changed by ±π/4 Instead of phase changes π/2 and π in QPSK signal, here phase changes become ±π/4 and ±3π/4 [0,1] Q [1,1] I [0,0] [1,0] π/4-QPSK π/4-QPSK modulation avoids transitions over origin Differentialπ/4-QPSK: π/4-DQPSK Widely used is differential π/4 -QPSK form -- π/4 -DQPSK In the DQPSK signal, the i-th dibit value is associated not with the specific cosine phase in the symbol’s radio pulse, but with the phase change during the transition from the i-th radio pulse to the i + 1 radio pulse. The following table shows the correspondences between the modulated carrier phase change and the transmitted dibits. Dibit value Phase change 00 π/4 01 3π/4 10 -3π/4 11 -π/4 π/4-DQPSK example First radio pulse inπ/4 –DQPSK signal is transmitted with 0 rad phase. Determine pulse phases when transmitting logical series 0110100001. Solution: Transmitted dibits: 01, 10, 10, 00 ir 01. Corresponding phase changes from the table: 3π/4, -3π/4, -3π/4, π/4 and 3π/4. Therefore, cosine phase in 2nd radio pulse- (3π/4), 3rd- 0, 4th-(-3π/4), 5th- (-π/2), 6th– (π/4). Transmitted dibit value Phase change 00 π/4 01 3π/4 10 -3π/4 11 -π/4 π/4-DQPSK noncoherent decoder π/4 –DQPSK signal can be decoded using noncoherent decoder, which is simpler way compared to coherent decoding p –DQPSK π/4 signal -DKFM signalas 4 FazėsPhase matavimo įrenginys measurement device Užlaikymo Delay įrenginys device Dt=Ts=2T Keitiklis fazė- Phase-dibit dibito vertė converter I Q Decoded Atkurtojiinformation informacija Keitiklis Parallel-serial lygiagretus- converter nuoseklus Bit error rate (BER) When using coherent demodulators, the bit error rate under additive Gaussian noise for QPSK, O-QPSK and π/ 4-QPSK modulation forms is the same as for BPSK: 1 Eb p (e) = erfc [BER]. 2 N0 In case of M-ary modulations errors are defined usually by symbol error rate (SER). For four-level modulation, when symbol sorresponds to dibit, symbol error rate is expressed as: 2 p s (e)[ SER ] = 2 p (e) - p (e) Obtained from: 1 - p s (e) = [1 - p(e)][1 - p(e)]  Q Binary 1,1 series Bipolar 0,1 Demultiplexer 1,1 signal -1,1 I signal Q signal I Carrier X A cos(2 f t) c Asin(2 fct) 0,0 Phase shifter X π2 -1,-1 1,0 1,-1 To antenna Standard BPSK: 1 Eb QPSK: p (e) = erfc 2 N0 1 E p(e) = pI (e) + pQ (e) = erfc b 1 E 2 N0 pI ( e) = erfc b 4 N0 sS / N = 2 Eb 1 Eb N0 pQ ( e) = erfc 4 N0 QPSK signal spectrum fc-3R fc-2R fc-R fc fc+R/2 fc+R fc+2R fc+3R Exercise: QPSK modulation performance Find receiver sensitivity in dBm required for QPSK modulation over AWGN channel to support bit rate of 54 Mbps with BER 0.001 at room temperature 290 K. Receiver noise figure 4 dB. Important property of quadrature modulator (frequency converter ) cos(a + b ) = cos a cos b - sin a sin b Gm (t) Hilbert(Gm) Carrier A cos(2 f c t ) X Phase shifter -A sin(2 f c t ) π2 X Only high-freq side lobe band Σ Constant envelope modulations Constant envelope signal  1010010 Dvejetainė Binary series seka Bipoliarinis Bipolar NRZ koderis NRZ Demultiplexer Minimum coder Demultiplekseris shift T II series seka 1 1 –1 -1 Delay Užlaikymas Q seka series -1 –1 1 keying 1 1 -1 by TT per 2pt A cos 4T -1 -1 1 X 2pt A sin Using half- 4T X cosine pulses Nešlys Carrier A cos(2pf c t ) X Fazės Phasesukiklis shifter A sin( 2pf c t ) p 2 X MM MSK signal Σ signalas  Minimum shift keying (MSK) æ 2pt ö æ 2pt ö S MM = bIUZ (t ) cosç ÷ cos(2pf c t ) + bQ (t ) sin ç ÷ sin (2pf c t ) è 4T ø è 4T ø Here fc - carrier frequency. Equation can be rewritten as: S MM = x1 (t )cos 2pf c t + x 2 (t )sin 2pf c t = x(t )cos[2pf c t - j (t )]. Here: x 2 (t ) x(t ) = x12 (t ) + x 22 (t ) j (t ) = arctg x1 (t ) æ 2pt ö æ 2pt ö x1 (t ) = bIUZ (t ) cosç ÷ x 2 (t ) = bQ (t ) sin ç ÷ è 4T ø è 4T ø æ 2pt ö 2 æ 2pt ö x(t ) = x12 (t ) + x 22 (t ) = bIUZ (t ) cos 2 ç 2 2 + ÷ Q b (t ) sin ç ÷ =1 è 4T ø è 4T ø. since bQ(t) and bIUŽ(t) may have only ±1 values  Minimum shift keying (MSK) What is MSK signal frequency? 1 æ 2pt ö æ 2pt ö S MM = bIUZ (t )[cosç 2pf c t + ÷ + cosç 2pf c t - ÷] + 2 è 4T ø è 4T ø 1 æ 2pt ö æ 2pt ö + bQ (t )[cosç 2pf c t - ÷ - cosç 2pf c t + ÷]. 2 è 4T ø è 4T ø From here it seems that instantenous frequency of MSK signal may have just two values: 1 , when bQ (t ) = -b I (t ) f+ = fc + 4T. 1 , when bQ (t ) =b I (t ) f- = fc - 4T It means that minimum shift keying at the same time is also a form of binary frequency shift keying (BFSK) MSK signal power spectrum 80 sMM, dB 23 dB 40 0 -40 fc-0.7 5 R fc+ 0 ,7 5 R f Binary frequency shift keying (BFSK) ì 2 Eb ï cos(2pfit ), when kai 0 £ t < T siDDM =í T ï0, otherwise kitur. î Here T and Eb - bit duration and energy, respectively, i=1,2. For Sunde’s coherent BFSK signal, frequency is expressed as: nc + i fi = T 0 T 2T 3T 4T t This is continuous-phase frequency shift keying h = T ( f1 - f 2 ) = Df / R = T / T1 - T / T2 (CPFSK) Deviation index BFSK signal spectrum 2 Eb æ pt ö s DDM (t ) = cosç 2pf c t ± ÷, 0 £ t < T T è Tø 2 Eb æ pt ö 2 Eb æ pt ö s DDM (t ) = cosç ± ÷ cos(2pf c t ) - sin ç ± ÷ sin( 2pf c t ) = T è Tø T è Tø 2 Eb æ pt ö 2 Eb æ pt ö = cosç ÷ cos(2pf c t ) ± sin ç ÷ sin( 2pf c t ). T èT ø T èT ø The first term represents two frequencies: 1 f1, 2 = f c ± 2T Second term -- g(t) multiplied by sin of carrier frequency ì 2 Eb æ pt ö ï sin ç ÷, 0 £ t < T g (t ) = í T èT ø ï0, otherwise kitur. î Sunde’s signal bandwidth: B = 2 f m + 2 f d = 2 / T + 2 / 2T = 3 / T = 3R BFSK signal spectrum 40 S. dB f 1, 2 = f c ± 1 2T 20 0 -20 -40 f c -6R f c -4R f c -2R fc f c +2R f c +4R f c +6R BFSK modulator and demodulator a X Vienpolinio Unipolar signalo koderis cos(2pf1t ) BFSK signal coder DDM S signalas signal b Invertorius Inverter T X ò X 0 BFSK DDM + cos(2pf 2t) signalas signal cos(2pf1t ) To decision Į sprendimo S įrenginį device - T Bit error rate: X ò 0 1 Eb p (e) = erfc 2 2N 0 cos(2pf 2 t ) Again about minimum shift keying 1 f 1, 2 = f c ± h = T ( f1 - f 2 ) h = 0.5 4T 2 Eb s1 (t ) = cos(2pf1t ) T 2 Eb s 2 (t ) = - cos(2pf1t ) These signals are required to ensure T continuous signal/phase 2 Eb s3 (t ) = cos(2pf 2 t ) T 2 Eb s 4 (t ) = - cos(2pf 2 t ) T 5 B = 2 f m + 2 f d = 2 / T + 2 / 4T = = 2,5 R 2T The deviation index h = 0.5 is the smallest at which symbols with instantaneous frequencies f1 and f2 can be orthogonal (this is important as it enables the use of correlation-based receivers). This is where the name of minimum shift keying comes from. Fast frequency shift keying (FFSK) Binary unipolar pulse sequence Analog frequency MSK signal modulator with dev. index h=0.5 Gaussian minimum shift keying (GMSK) é f2ù H ( f ) = expê- 2 ú, ë B û Binary unipolar square pulse Gaussian low g(t) sequence Analog frequency pass filter GMSK modulator signal H(f) h = 1/2 h = Tb(f1-f2), if h = 1/2, then GMSK signal modulator using analog frequency modulator we will have Gaussian é æ ln 2 ö f 2 ù minimum shift H ( f ) = exp ê- ç ÷ 2 ú, keying (GMSK) ë è 2 øW û here W - bandwidth at -3 dB level  1é æ 2 æ t 1 öö æ 2 æ t 1 ö öù g (t ) = êerfc çç p WT ç - ÷ ÷÷ - erfc çç p WT ç + ÷ ÷÷ ú 2 êë è ln 2 è T 2 øø è ln 2 è T 2 ø ø úû Pulse at the output of Gaussian filter for different bandwidth - bit duration values MSK and GMSK signal power spectra Power density dependency on normalized frequency S, dB (f - fc)/R GMSK in GSM Gaussian filtered minimum shift keying, GMSK 270 kbps bitrate over 200 kHz bandwidth Spectral efficiency 1.36 bps/Hz [S.M. Redl et al., GSM and personal communications handbook, Artech House, 1998] GMSK signal modulator using quadrature modulation c(t ) Q(t ) sin[c (t )] ∫dt a (t ) Gaussian low g(t ) 900 pass filter H(f) Σ cos[c(t)] c(t ) I (t ) Carrier generator S MM = cos (ò g ( t ) dt )cos (2 p f c t ) + sin (ò g ( t ) dt )sin (2 p f c t) S MM = x1 (t )cos 2pf c t + x 2 (t )sin 2pf c t = = x(t )cos[2pf c t - j (t )]. = x(t ) cos[Q(t )].  x(t ) = x12 (t ) + x 22 (t ) = 1, x 2 (t ) j (t ) = arctg x1 (t ) ( ) = arctg [tg ò g (t )dt ] = ò g (t )dt. Instantaneous frequency: f = 1 æ dQ ö ç ÷ 2p è dt ø = f c - 1 d 2p dt (ò g (t )dt ) = f c - g (t ) 2p. When transmitting logical states signal g(t) acquires only two values: p g (t ) = ± 2Tb i. e. signal g(t) is symmetrical bipole signal with amplitude π/2Tb. Then: f 1, 2 = f c ± 1 4Tb This means that we have BFSK signal with deviation index h = 1/2. In order to obtain GMSK, such modulator needs to be complemented by Gaussian filter.