Passand Daa ransmission I References Phase-shif keying Chaper 4.-4.3, S. Haykin, Communicaion Sysems, Wiley. G.
Inroducion Inroducion In aseand pulse ransmission, a daa sream represened in he form of a discree pulse-ampliude modulaed PAM signal is ransmied over a lowpass channel. In digial passand ransmission, he incoming daa sream is modulaed ono a carrier wih fixed frequency and hen ransmied over a and-pass channel. G.
Inroducion Inroducion he modulaion process making he ransmission possile involves swiching keying he ampliude, frequency, or phase of a sinusoidal carrier in accordance wih he incoming daa. here are hree asic signaling schemes: Ampliude-shif keying ASK Frequency-shif keying FSK Phase-shif keying PSK G.3
Inroducion ASK PSK FSK G.4
Inroducion Unlike ASK signals, oh PSK and FSK signals have a consan envelope. PSK and FSK are preferred o ASK signals for passand daa ransmission over nonlinear channel ampliude nonlineariies such as micorwave link and saellie channels. G.5
Classificaion of digial modulaion echniques Coheren and Noncoheren Digial modulaion echniques are classified ino coheren and noncoheren echniques, depending on wheher he receiver is equipped wih a phaserecovery circui or no. he phase-recovery circui ensures ha he local oscillaor in he receiver is synchronized o he incoming carrier wave in oh frequency and phase. G.6
Classificaion of digial modulaion echniques M-ary signaling In an M-ary signaling scheme, here are M possile signals during each signaling inerval of duraion. Usually, n M and n where is he i duraion. In passand ransmission, we have M-ary ASK, M-ary PSK, and M-ary FSK digial modulaion schemes. We can also comine differen mehods: M-ary ampliude-phase keying APK M-ary quadraure-ampliude modulaion QAM In aseand ransmission, we have M-ary PAM G.7
Classificaion of digial modulaion echniques M-ary signaling M-ary signaling schemes are preferred over inary signaling schemes for ransmiing digial informaion over and-pass channels when he requiremen is o conserve andwidh a he expense of increased power. he use of M-ary signaling enales a reducion in ransmission andwidh y he facor n log M over inary signaling. G.8
Coheren PSK he funcional model of passand daa ransmission sysem is m i Signal ransmission encoder si s i x x Signal Modulaor Channel Deecor ransmission decoder mˆ Carrier signal m i is a sequence of symol emied from a message source. he channel is linear, wih a andwidh ha is wide enough o ransmi he modulaed signal and he channel noise is Gaussian disriued wih zero mean and power specral densiy N /. o G.9
Coheren PSK he following parameers are considered for a signaling scheme: Proailiy of error A major goal of passand daa ransmission sysems is he opimum design of he receiver so as o minimize he average proailiy of symol error in he presence of addiive whie Gaussian noise AWGN G.
Power specra Coheren PSK Use o deermine he signal andwidh and co-channel inerference in muliplexed sysems. In pracice, he signalings are linear operaion, herefore, i is sufficien o evaluae he aseand power specral densiy. Bandwidh fficiency R Bandwidh efficiency ρ is/s/hz B where R is he daa rae and B is he used channel andwidh. G.
Coheren Binary Phase-shif Keying BPSK In a coheren inary PSK sysem, he pair of signals s and s used o represen inary symols and, respecively, is defined y s cosπf c s cosπf c + π cosπf c where, and is he ransmied signal energy per i. G.
Coheren Binary Phase-shif Keying BPSK For example, [ ] s d cos πf c d o ensure ha each ransmied i conains an inegral numer of cycles of he carrier wave, he carrier frequency f is chosen equal o n / for some fixed ineger n. c G.3
Coheren Binary Phase-shif Keying BPSK he ransmied signal can e wrien as s φ and s φ where φ π cos fc < G.4
Generaion of coheren inary PSK signals o generae a inary PSK signal, we have o represen he inpu inary sequence in polar form wih symols and represened y consan ampliude levels of + and, respecively. Signal ransmission si encoder Produc Modulaor s i φ cosπf c G.5
G.6 Generaion of coheren inary PSK signals his signal ransmission encoder is performed y a polar nonreurn-o-zero NRZ encoder. + inpu symolis inpu symol is i s he carrier frequency c n f / where n is a fixed ineger. i c i c i s f s s f s s if cos if cos π π
Deecion of coheren inary PSK signals o deec he original inary sequence of s and s, we apply he noisy PSK signal o a correlaor. he correlaor oupu is compared wih a hreshold of zero vols. x X x Decision device if if x x > < φ Correlaor G.7
G.8 Deecion of coheren inary PSK signals xample: If he ransmied symol is, cos f x c π and he correlaor oupu is c c c d f d f f d x x cos cos cos π π π φ
Deecion of coheren inary PSK signals Similarly, If he ransmied symol is, x. G.9
rror proailiy of inary PSK We can represen a coheren inary sysem wih a signal consellaion consising of wo message poins. he coordinaes of he message poins are all he possile correlaor oupu under a noiseless condiion. he coordinaes for BPSK are Decision oundary and. φ G.
rror proailiy of inary PSK here are wo possile kinds of erroneous decision: Signal s is ransmied, u he noise is such ha he received signal poin inside region wih x > and so he receiver decides in favor of signal s. Signal s is ransmied, u he noise is such ha he received signal poin inside region wih x < and so he receiver decides in favor of signal s. G.
rror proailiy of inary PSK For he firs case, he oservale elemen x is relaed o he received signal x y x [ s + w ] i x φ d + φ d w φ d x is a Gaussian process wih mean x : x i [ x i [ ] + w φ d] G.
G.3 and variance σ : ] [ ] [ o o o i i N d N ddu u u N ddu u w u w ddu u w u w d w x x φ φ φ δ φ φ φ φ φ σ
G.4 rror proailiy of inary PSK herefore, he condiional proailiy densiy funcion of x, given ha symol was ransmied is + o o N x N x x x f exp exp π σ πσ
rror proailiy of inary PSK and he proailiy of error is p f x dx x + exp πn o No Puing z x +, we have N p o π erfc / N o N o exp [ z ] dz dx u erfc u exp z dz π G.5
G.6 rror proailiy of inary PSK Similarly, he error of he second kind o N p p erfc and hence o e N p erfc
Quadriphase-shif keying QPSK QPSK has wice he andwih efficiency of BPSK, since is are ransmied in a single modulaion symol. he daa inpu d k is devided ino an inphase sream d I, and a quadraure sream d Q. G.7
d k Quadriphase-shif keying QPSK d I d Q G.8
Quadriphase-shif keying QPSK he phase of he carrier akes on one of four equally spaced values, such as π/4, 3π/4, 5π/4, and 7π/4. s i where i,,3,4. cos[πf c + i π / 4] elsewhere is he ransmied signal energy per symol; is he symol duraion; f c n / ; G.9
Quadriphase-shif keying QPSK ach possile value of he phase corresponds o a unique dii. For example, for i, for i, for i3 and for i4. only a single i is change from one dii o he nex G.3
Quadriphase-shif keying QPSK he ransmied signal can e wrien as s i where cos[πf cos[πf + i ]cos[i sin[πf si φ + si φ c c c π / 4] π / 4] ]sin[i π / 4] φ cos[πf c]; φ sin[πf c] G.3
Quadriphase-shif keying QPSK Inpu dii Phase of QPSK si s i π/4 / / 3π/4 / / 5π/4 / / 7π/4 / / G.3
G.33
Generaion of coheren QPSK signals he incoming inary daa sequence is firs ransformed ino polar form y a nonreurn-o-zero level encoder. he inary wave is nex divided y means of a demuliplexer ino wo separae inary sequences. he resul can e regarded as a pair of inary PSK signals, which may e deeced independenly due o he orhogonaliy of φ and φ. G.34
φ cosπf c s i X si Polar NRZ Demuliplexer + s s i X φ sinπf c G.35
Deecion of coheren QPSK signals x X x Decision device if if x x > < φ In-phase channel muliplexer Quadraure channel X x Decision device if if x x > < φ G.36
rror proailiy of QPSK he consellaion of QPSK is φ φ G.37
G.38 rror proailiy of QPSK he received signal is w s x i + and he oservaion elemens are + ± d w d x x φ φ + ± d w d x x φ φ
rror proailiy of QPSK As a coheren QPSK is equivalen o wo coheren inary PSK sysems working in parallel and using wo carriers ha are in phase quadraure. Hence, he average proailiy of i error in each channel of he coheren QPSK sysem is p erfc / N erfc o N o G.39
G.4 rror proailiy of QPSK As he i error in he in-phase and quadraure channels of he coheren QPSK sysem are saisically independen, he average proailiy of a correc decision resuling from he comined acion of he wo channels is + o o o c N N N p p erfc 4 erfc erfc
G.4 rror proailiy of QPSK he average proailiy of symol error for coheren QPSK is herefore / if erfc erfc 4 erfc >> o o o o c e N N N N p p
rror proailiy of QPSK In a QPSK sysem, since here are wo is per symol, he ransmied signal energy per symol is wice he signal energy per i, and p e erfc N o G.4
rror proailiy of QPSK he i error rae of QPSK is BR erfc N o herefore, a coheren QPSK sysem achieves he same average proailiy of i error as a coheren inary PSK sysem for he same i rae and he same / No u uses only half he channel andwidh. G.43
M-ary PSK During each signaling inerval of duraion, one of he M possile signals s i is sen. π cos πf c + i i,,..., M G.44
M-ary PSK he signal consellaion of M-ary PSK consiss of M message poins which are equally spaced on a circle of radius. For example, he consellaion of ocaphas-shif keying is P e π erfc sin M 4 No M G.45
Power specra of M-ary PSK signals he symol funcion is g oherwise where log M and is he i duraion. S f As he energy specral densiy is he magniude of he signal s Fourier ransform, he aseand power specral densiy is sin πf πf log M sinc f log M G.46
Power specra of M-ary PSK signals Normalized o f G.47
B Bandwidh efficiency he andwidh required o pass M-ary signal main loe is given y Qsinc log M R log M herefore, he andwidh efficiency is ρ R B log M G.48
Passand Digial ransmission Passand digial ransmission allows more efficien use of he allocaed RF andwidh, and flexiiliy in accommodaing differen aseand signal formas. xample Moile elephone Sysems GSM: GMSK modulaion is used a variaion of FSK IS-54: π/4-dqpsk modulaion is used a variaion of PSK G.49
Phase Recovery Carrier Synchronizaion wo ways in which a local oscillaor can e synchronized wih an incoming carrier wave ransmi a pilo carrier use a carrier-recovery circui such as a phase-locked loop PPL G.5