Chapter 6 Passband Data Transmission

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Laurence Milton Marshall
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1 Chapter 6 Passband Data ransmission Different methods of digital modulation Outline PSK(Phase-shift keying), QAM(Quad. amp. mod), FSK(Phase-shift keying) Coherent detection of modulated signals in AWGN Carrier phase; Bit timing Noncoherent detection of modulated signals in AWGN: phase information is disregarded Modems for transmission and reception of digital data over PSN (public switched telephone network) Sophisticated modulation techniques over a wideband channel with medium to severe ISI Carrierless amplitude/phase modulation Discrete multitone echniques for synchronizing receiver to transmitter 6 -

2 Introduction Baseband Pulse transmission: a discrete pulse-amplitude modulation (PAM) is transmitted over a low-pass channel Digital passband transmission Data stream is modulated into a carrier with fixed frequency limits imposed by a band-pass channel of interest. Major issue is the optimum design of the receiver in the presence of noise. Communication channel: microwave radio, satellite Modulation processswitching (keying) amplitude, phase, frequency hree basic signaling schemes: ASK(Amp.-shift keying), PSK(Phase-shift keying), FSK(Phase-shift keying) > Unlike continuous-wave modulation, it is not difficult to distinguish between PSK and FSK signals. > Unlike ASK signals, PSK and FSK signals have a constant envelopeimperious to amplitude nonlinearities (microwave radio, satellite channels)preferred to ASK for passband data transmission over nonlinear channels. 6 - Modulation schemes ASK(Amp.-shift keying) PSK(Phase-shift keying) FSK(Phase-shift keying) 6-3

3 Hierarchy of digital modulation M-ary schemes: conserve bandwidth at expense of increased power M-ary modulation techniques: M-ary PSK, FSK, ASK Hybrid form: M-ary amplitude-phase keying (APK) special case: M-ary quadrature-amplitude modulation (QAM)special case: M-ary ASK Linear modulation: M-ary PSK, M-ary QAM > M-ary PSK: constant envelopenonlinear band-pass channel > M-ary PSK: changes in carrier amplitudenonlinear band-pass channel Classification of digital modulation techniques: the receiver is equipped with a phase-recovery circuit or not Coherent modulation techniques > M-ary PSK, QAM, FSK Noncoherent modulation tech. (no carrier phase information is needed) > ASK, FSK: impractical to maintain carrier phase synchronization > Differential phase-shift keying (DPSK): noncoherent form of PSK 6-4 Power Spectra o study power spectra of resulting modulated signals is particularly important in two contexts - Occupancy of channel bandwidth - Cochannel interference in multiplexed systems Bandpass signal (modulated signal) complex envelope s t s I t cosf c t s Q t sinf c t Res t expjf c t s t s I t + js Q t S B f: power spectral density of complex envelope s t Baseband power spectral density Power spectral density of s t S s f S B f + S B f + Bandwidth efficiency: ration of data rate in bits per second to effectively utilized channel bandwidth R b ---- bits/s/hz - multilevel encoding; spectral shaping B f c amp. f c power / /4 6-5

Passband ransmission Model Bandpass communication channel - Channel is linear: bandwidth is wide enoughs i (t) no distortion - Channel noise w(t): white Gaussian with zero mean and power spectral

4 Passband ransmission Model Bandpass communication channel - Channel is linear: bandwidth is wide enoughs i (t) no distortion - Channel noise w(t): white Gaussian with zero mean and power spectral density / reverses operations performed in transmitter Minimize effect of channel noise Linear AWGN 6-6 Binary Phase-Shift keying (rror rate) Binary Phase-shift keying (BPSK) s t b cosf c t s t b cosf c t t ---- cosf c t f c n c t f Xj x j m i x j s ij N i, exp o j f X x b exp ---- x j s N z x + b o p f X x x d exp ---- x + b dx N o exp z d z ----erfc b b p p P e P e p e p p + p p -----erfc ---- b n c : fixed integer s s s t t s t t b A b s A ---- b P e s b b signal-space diagram for coherent BPSK;n c transmitted signal energy per bit 6-7

5 Binary Phase-Shift keying (Power spectra) in-phase component +g(t), -g(t) g t b, t b, otherwise Power spectral density ---- G f s G f b sinc f G f b sin c f S B f b sin c f b sin f f falls off ~ inverse square of frequency BW(PSK) < BW (MSK or FSK) Sidelobe(PSK) > Sidelobe(FSK) Binary PSK transmitter Coherent BPSK receiver 6-8 Quadrature Phase-Shift keying (QPSK) Goal in the design of digital communication: Low probability of error efficient utilization of channel bandwidth. BPSK QPSK: N, M4 s i t cos f c t + i -----, t 4, elsewhere cos i cosf c t sin i sinf c t 4 orthonormal basis functions t --- cos f c t t --- sin f c t t i 3 4 signal vector cos i s i sin i A A ---- : symbol duration; : nergy/symbol f c n c / for some fixed integer n c Gray-encoded set of debits:,,, Gray-code 6-9

6 QPSK received signal Observation vector X in-phase x Quadrature x x t t + w x t t + w rror Probability of QPSK x t s i t + w t bit error in in-phase and Quadrature channels are statistically independent P c P ' average prob. of a correct decision Symbol error probability if» P e P c erfc erfc erfc BR ----erfc BR (BPSK) BR (QPSK) with the same b /, but BW(QPSK) / BW(BPSK) white Gaussian noise with power spectral density / wo BPSK with b / and power spectral density / b P b -----erfc ---- P ' ----erfc BPSK QPSK s s bit error rate for each channel 6 - Binary sequence is divided into two other sequences - wo waveforms may individually be viewed as a BPSK signal Generation and Detection of QPSK QPSK ransmitter QPSK Receiver 6 -

7 Symbol shaping function g t in-phase Quadrature power spectra, t, otherwise G f sincf S BI S BQ Power Spectra of QPSK G f sin cf S B f S BI + S BQ sin cf 4 b sin cf in-phase and Quadrature components have a common power spectra in-phase and Quadrature components are statistically independent BW(PSK) < BW (MSK or FSK) Sidelobe(PSK) > sidelobe (MSK or FSK) no spike in spectra 6 - M-ary Phase-Shift keying (M-ary PSK) cos f s i t c t + i --, t M, elsewhere i i i M A A ---- M : symbol duration; : nergy/symbol f c n c / for some fixed integer n c M message points are equally spaced on a circle of radius for the case of M 8 d d 8 sin -- M Average symbol error P e f X x j x d erfc sin -- M P e d erfc ---- sin -- M (M-ary PSK) M4 8-PSK he approximation becomes extremely tight, for fixed M, as / is increased. - For M 4, the same form for QPSK 6-3

8 rror Probability of M-ary PSK r t Acosf c t + k + n x t cosf c t n y t sinf c t Received M-ary PSK P e 8 Pn y Asin --+ P ny Asin 8 -- P Common Area 8 P n y A sin-- + P ny A sin erfc Asin n y --erfc Asin 8 n y erfc A sin -- erfc sin erfc sin-- P M e M M ---P e 8 erfcsin -- M n 8-ary PSK signal-to-noise radio S N S A N n B quivalent noise bandwidth as B for integrate-and-dump circuit, matched filter and correlator S S S N B ---- s Q 4 n y n y n A x A sin 8 8 I nx ny n B n t n x t + jn y t Common Area B 6-4 Power Spectra of M-ary PSK S B f sin cf b log Msin c flog M normalized power spectral density S B S B f b log M : symbol duration Channel bandwidth required to pass M-ary PSK signal (main lobe, null-to-null) / Bandwidth efficiency (spectral efficiency): the ratio of data rate to channel bandwidth fficient modulation maximize bandwidth efficiency - achieve this bandwidth at a minimum of average signal power or average SNR Bandwidth efficiency R b ---- B B log M Data rate R b (bits/s/hz) Channel bandwidth R b log M f b vs normalized frequency f R b ---- B log M M is increased, bandwidth efficiency is improved at the expense of error performanceincrease b / f 6-5

9 M-ary Quadrature Amplitude Modulation (QAM) M-ary QAM is a two-dimensional generalization of M-ary PAM a k b k 3 5 s i t a k cosf c t b k sinf c t t k orthonormal basis functions : nergy of signal with the lowest amplitude ( t --- or ) channel cos f c t t d min A wo distinct QAM constellations - Square constellations: number of bits per symbol is even - Cross constellations: number of bits per symbol is odd minimum distance between any two message points in constellation Amplitude for I channel cosf c t or Q Channel sinf c t d min t --- sin f c t M-QAM for M 6 with Gray-encoded t corresponding 4-PAM 6-6 rror probability of Square QAM With an even number of bits per symbol, L M (positive integer) - M-ary QAM square constellation Cartesian product of a one-dimensional L-ary PAM probability of symbol error probability of correct detection for M-ary QAM P c P e ' P e ' : probability of symbol error for corresponding L-ary PAM P e ' M erfc ---- P e P c P e ' P e ' probability of symbol error for M-ary QAM erfc ---- M average value of transmitted energy symmetric L av i L f i, f M Average Symbol nergy P e erfc av M M M-QAM for M 6 with Gray-encoded corresponding 4-PAM 6-7

rror probability of Cross QAM Cross constellations: number of bits per symbol is odd d Construct a signal constellation with n bits per symbol - A square constellation with n - bits per symbol -

10 rror probability of Cross QAM Cross constellations: number of bits per symbol is odd d Construct a signal constellation with n bits per symbol - A square constellation with n - bits per symbol - xtend each side of square constellation by adding n-3 symbols - Ignore corners in the extension -3d -d -d d 3d x: 4x n-3 n- n- + n- n, n bits per symbol It is not possible, it is not possible to express a QAM cross constellation as product of a PAM constellation - It is not possible to perfectly Gray code a QAM cross constellation - Complicates determination of symbol error probability. P e M erfc ---- o N 6-8 rror Probability of M-ary QAM M-ary PAM r t a k cosf c t+ n x t cosf c t n y t sinf c t -3d -d d M P e P n y d M -- M P n M + y d P n y d M M erfc d M erfc S M erfc M M M M --- S N M a M k ---- M m d M d P AV S s t m 3 -- M d S S d P e ---- erfc S -PAM 4-PAM P N N e --erfc N M-ary QAM r t a k cosf c t b k sinf c t+ n x t cosf c t n y t sinf c t P e d M erfc P AV S s t d S M erfc N M d 3 M 4-QAM P e erfc S N --- S N d -3d -d -d d --- S N 3d d 3d 6-9

11 Coherent Binary Frequency-Shift keying (BFSK) Symbols and are distinguished from each other by transmitting one of two sinusoidal waves that differ in frequency by a fixed amount Sunde s FSK: continuous-phase signal phase continuity is always maintained including inter-bit switching timescontinuous-phase frequency-shift keying (CPSK) i b cosf s i t i t, t b, elsewhere b : transmitted signal energy per bit n f + c i i n c : integer i i t cosf i t s b, i j ij, ij M, N i j b A b A rror Probability of BFSK s b s b d Y X + X b Y X + X b Var Y Var X + Var X f y y exp y + b p Py symbol f y y y d b x x symbol : symbol : y b exp dy N o z exp d z ----erfc b b P e -----erfc b P e BFSK P e BPSK d b b N d x t t x t t x x y x x symbol : x x y symbol : x x y N --- o N o 4 4 b x x BPSK b 6 -

12 Generation and Detection of Coherent BFSK On-off level encoder: - volts; volts b Inverter: - f on, f off; f off, f on f i and f are chosen to equal different integer multiples of bit rate / - f i and f are synchronized - A single keyed (OSC) oscillator - modulated wave is shifted he detector consists two correlators - correlator outputs are subtracted - y > symbol y < symbol BFSK transmitter BFSK Receiver 6 - s t b cos f c t----- t indep. of input binary wave Power Spectra of BFSK b t cos cosf c t b t sin sin f c t for all f b f f t g f 8 bcos f b f t f f S B f b f f b cos f f f b f f 4 FSK (continuous phase) falls off ~ f -4 does not produce as much outside signal of interest FSK (Discontinuous phase) falls off ~ f - - f and f operate independently FSK has a smoother pulse shape and lower sidelobes than PSK Falls off ~ f -4 Smoother Lower sidelobes 6-3

13 Coherent Minimum-Shift-Keying (MSK) Coherent detection of BFSK - phase information is not fully exploited - other than to provide for synchronization of receiver and transmitter - proper use of phaseimprove noise performance of receiver - his improvement is achieved at expense of increased receiver complexity Sunde s FSK - Deviation ratio is exactly unity (f -f / ) - Phase change over one bit interval is radians - here is no memorychange occurred in previous bit interval provides no help in current bit interval waveform of MSK signal 6-4 Coherent MSK vs CFSK Continuous-phase frequency-shift keying (CFSK) b cosf t + symbol s t t b cosf t + symbol Another useful way of representing CFSK s t b cosf c t + t continuous function of time including switching time t -----t h f c f c h f h f b + symbol ;symbol f c -- f + f h f f Deviation ratio h symbol h symbol t phase Continuity Sunde s FSK h h / Phase rellis even odd 6-5

14 Signal-Space Diagram of MSK h /, frequency deviation, difference between two signaling frequencies f and f, equals half bit rate. f f. - Minimum frequency spacing allows two FSK signals representing symbols and ; coherently orthogonal - CPFSK with a deviation ratio of one half minimum shift keying (MSK) s t b cost cos f c t b sint sin f c tt, t, t b or h / s I t b cost b cos cos t b cos t t half-cycle cosine pulse b s Q t b sint b sin b sin t b t cos t half-cycle sine pulse b, b symbol, b symbol, b symbol, b symbol 6-6 s I t b cos t t s Q Signal-Space Diagram of MSK t b cos t t s t b cost cos f c t b sint sin f c t t ---- cos t ---- sin t t cosf c t sinf c t t s t s t + s t t integral for time interval s s t t b cos t s s t t b cos t 6-7

rror Probability of MSK, b symbol, b symbol, b symbol, b symbol Both integrals for a interval Lower and upper bound for s and s - s shifted by to s t is common to both integrals - and b are defined

15 rror Probability of MSK, b symbol, b symbol, b symbol, b symbol Both integrals for a interval Lower and upper bound for s and s - s shifted by to s t is common to both integrals - and b are defined Average Probability of errors x x t t s + w, N t x x t t s + w, t b P e -----erfc he same as PSK, QPSK, Detection ~ observation over 6-8 Generation and Detection of MSK s t s t + s t, t t ---- cos t cosf c t b t t sin sinf c t b wo phase-coherent sinusoidal waves at f and f and for h / wo narrow-band filters orthonormal basis functions t and t. a and a : bit rate / h f c f h f c f t integration interval t 6-9

16 Power Spectra of MSK s t b t cos cos f c t b t sin sin f c t g I t g Q t g I t, gi f b cos f g Q t b f gq f g f S B f g f b b cos f b f MSK produces less interference than PSK - MSK falls off ~ f -4 PSK ~ f - - he desired characteristics of MSK especially when operates with a bandwidth limitation GMSK: satisfy the stringent requirements of certain applications such as wireless communication normalized to 4 b t t Falls off ~ f -4 Smoother Lower sidelobes 6-3 Gaussian-Filtered MSK (GMSK) Desirable properties of MSK - Constant envelope - Relatively narrow bandwidth - Coherent detection performance equivalent to that of QPSK Power spectrum a compact form: premodulation low-pass filter (Pulse-shaping filter) - narrow bandwidth; sharp cutoff - impulse resp. with relatively low overshoot - volution of a phase trellis MSK Pulse-shaping filter Gaussian function Adjacent channel interference (ADJ) of wireless communication system using MSK is not low enough H f exp log -- f W frequency-shaping pulse g t log dB h t W exp W t log log g t W log exp W log t shifted in time by.5 6-3

17 Power spectra of GMSK Frequency-shaping pulse g t ---- t h t ht d Wexp W log log t ime-bandwidth product (W ) - play role of a design parameter - W is reduced, time spread of frequency-shaping pulse is increased - W MSK - W more of transmit power is concentrated inside passband of GMSK Undesirable feature of GMSK - signal is no longer confined to a single bit intervalpulse spreadisi Power spectra of MSK and GMSK for varying time-bandwidth product (W ) 6-3 rror Probability of GMSK ISI increases with decreasing W a trade-off between spectral compactness and performance loss is a constant whose value depends on time-bandwidth product W - log (/) in db: a measure of performance degradation compared to MSK - W MSK W.3,performance degradation ~.46 db - a small price to pay for highly desirable spectral compactness of GMSK signal - An important applicationgsm rror probability of GMSK in the presence of AWGN W.46.3 P e ----erfc b a small price to pay for highly desirable spectral compactness 6-33

18 GMSK for GSM Wireless Communications GSM: An important application of GMSK in a standardized wireless communication system; a time-division multiple-access system; W.3 - he best compromise between increased BW occupancy and resistance to CCI - 99% percent of power is confined to BW 5 khz, sidelobe~ outside this band CCI < 4dB Spectrum khz-wide subchannels - each subchannel at 7 kb/s - RF power spectrum (shaded subchannel) is down by an amount larger than 4dB at both adjacent subchannelseffect of CCI is practically negligible Power spectrum of GMSK signal for GSM wireless communications 6-34 Coherent M-ary FSK i M s i t cos --n c + it, t P e --- M erfc N good approximation for P e 3 Signal frequencies are separated by / M-ary FSK channel bandwidth B log M R b R B b M R b log M B log M M M M-ary PSK is spectrally efficient; M- ary PSK is spectrally inefficient - PSK: increase Mincrease - PSK: increase Mdecrease s i ts j tt d ij i t s i t n n c c : integer minimum distance d min f c Spectral analysis of M-ary FSK signals is much more complicated particular case frequency deviation k.5; M signal frequencies are separated by / Power spectra of M-ary FSK signals for M,4,8 6-35

19 Optimum Quadratic Receiver Coherent detectionassumptions - Perfectly synchronized to transmitter - only channel impairment is noise Uncertainty due to randomness - distortion in transmission medium - common parameter is carrier phase - especially true for narrow signals Synchronization with phase may be too costly - Disregard phase information at expense of degradation in performance noncoherent wo equivalent forms of Quadratic Receiver - quivalent matched filter - Noncoherent matched filter noncoherent Quadrature receiver using correlators noncoherent Quadrature receiver using matched filters Noncoherent matched filter 6-36 Noncoherent Orthogonal Modulation Noncoherent Orthogonal Modulation - noncoherent BFSK - Differential PSK (DPSK) ransmitted signal s i t cosf i t, Received signal f x t cosf i t + + w t Noncoherent Binary FSK and DPSK x t g t + w t, g t + w t, s tsent s tsent i t t Binary receiver for noncoherent orthogonal modulation i t andˆi t are orthogonal to each other i t m t cosf i t ˆi t m t sinf i t m(t) is a bandlimited message P e -- exp Quadrature receiver equivalent to either one of two matched filters 6-37

20 rror rate of Noncoherent BFSK and DPSK BFSK signal s i t cosf i t, upper matched to lower matched to cosf t cosf t f i n i t b P e -- exp DPSK signal s t symbol phase unchanged cosf c t cosf c t,, t t s t symbol phase unchanged cosf c t cosf c t +,, t t Noncoherent receiver for detection of BFSK DPSK is a special case of noncoherent orthogonal modulation with b b P e -- exp Generation and Detection of DPSK DPSK is noncoherent version of PSK - Incoming binary symbol b k is, leave symbol d k unchanged with respect to previous bit - Incoming binary symbol b k is, change symbol d k with respect to previous bit DPSK transmitter Signal-space diagram of received DPSK signal 6-39 DPSK receiver

Comparison of Digital Modulation schemes (Probability of error) BR decrease monotonically with increasing b /N - curves ~ shape in the form of a waterfall Coherent binary PSK, QPSK, MSK produce a

) At high b /N, coherent db less than noncoherent 3dB - BPSK vs DPSK - BFSK (coh) vs BFSK (incoh.

21 Comparison of Digital Modulation schemes (Probability of error) BR decrease monotonically with increasing b /N - curves ~ shape in the form of a waterfall Coherent binary PSK, QPSK, MSK produce a smaller BR than any of other schemes b /N (coherent) b /N (noncoherent)3db less - BPSK vs BFSK; DPSK vs BFSK (incoh.) At high b /N, coherent db less than noncoherent 3dB - BPSK vs DPSK - BFSK (coh) vs BFSK (incoh.) db 6-4 Comparison (Bandwidth fficiency) Power-bandwidth for coherent PSK - QPSK: best trade-off bet. power and bandwidth - M > 8: excessive power; complex equipment M-ary QAM is better than M-ary PSK for M >4 - QAM can be realized if channel is linear M-ary FSK: increasing M reduced power requirement increased channel bandwidth M6 signal constellations of M-ary PSK 6-4 M-ary QAM

MSK has three important properties. However, the PSD of the MSK only drops by 10log 10 9 = 9.54 db below its midband value at ft b = 0.

Gaussian MSK MSK has three important properties Constant envelope (why?) Relatively narrow bandwidth Coherent detection performance equivalent to that of QPSK However, the PSD of the MSK only drops by