BSTJ 60: 10. December 1981: On the Performance of Phase-Shift-Keying Systems. (Prabhu, V.K.; Salz, J.)

11.6 MB·English

On The Performance of Phase-Shift-Keying
Systems

By V. K PRABHU and J. SALZ

(Moruscritrecolved June 18, 1951)

Coherent phase-sift Keying (cpsx) and differential phase shit
eying (ovsk) are widely used modulation methods in digital com

‘munications. Bandwidth effcieney, good noise immunity, constant
‘envelope, and simplicity of implementation make these schemes par-
ticularly attractive for use over the satellite, terrestrial radia and
‘oiceband telephone channels. While avetetancyses abouid in the
literature, treatment io usually restricted to the additive Causian
channel Importand issues determining ultimate performance, such
{8 the joint eect of intersymbol interference and the acquisition of
‘carrier phase have not been adequately addressed. The mai purpose
ofthis paper i to develop analytical tools that can he used 10 asses
‘atom performance under practi operating eanditions, Pure cr

herent demodulation schemes auch a8 chk are ideals which are
rarely achieved in practice, and earrier phase must be estimated
prior to andjar during data transmission. This requiree start-up
ie, a well a cule wqnipment, cid the fidelity of the phase
featimate ulimately determines performance. In contrast, Deh is
independent of carrier phase, since decisions are made on phase
differences. Hexcever, this comes ata price, and itis moun Dat ideal
multiphase vrsk suffers an asymptote performance penalty of 8d
fn signal-to-noise ratio (a/n) over ideal ersn. We develop a new
rigoraus method for calculating the error rates of hoth crax and
sk, under a variety of operating conditions. In particular, we find
that the intersymbol interference penalty for quaternary Drsx tz about
1B worse in s/n than for open. We demanstrae thatthe detection
tifcieney of cPeR approaches the ideal, provided that the s/n ofthe
[phase recovery eieut i about 10 UR more Hane that at the recetoer
Input. Alteratively, forthe same a/n. a 1O-houdl plese lucked Lop
integration time is required te achieve near ideal performance

ner
1. TRoDUCTION

Coherent yhase-shift keying (ersm} nd diferential phase shift key
ing tops} are tro techniques oftun ved in digital communications
‘over channels such as satelite, lerresiril radio, and voicoband vele-
‘phone, The iterate ahownds in anaes of their performance under
[ variey of condition, A soenple collection of some of this literature
‘nay be found in Ref. 1, The chief vessons fr the widespread use of
thete techniques are simplicity of implementation, suporior perform.
tee over the adie Gaussian noise chennel, minimal bandwidth
fccupaney, and minimal envelope variation

"The roletive performance of casi and OFSK sysioms ie well under
stood only in the presence of additive Gaussian noise. Tn thie case, the
election ellienty af Desi ia known to be about 1 4B (in 3/n) below
that of cos for binery madlnion and this degradation approaches 3
AR for mulilevel eyatems. Ia uolicatione where 3-48 lose in s/n is
imporeant, auch atin down-link satelite, pace communications, and
terrestrial radio under deep fuling conditions, cPsk is the preterred
‘method, In crsi, however Use generation and extraction of a Tocal
‘caror-phaso reference at the receiver is required. A coherent phase
fetimave i wally obtained by using pheeelockedl loop (Pt) tech-
niques, and becaure of frequency instabilities and phase iter inherent
in transmtcer and reosiver systems, enrier recovery Joop bandwidehs
taannot ie made arbitrarily small. Conserendy, in practico a noisy
have esleite ip obtained and only pari coherent reception can be
‘aimed, The reason for suing DPS is immonicy from slow carier-
phase Muceuations therefore, the phase recovery problem inherent in
Cran is avoided, However, the delection ficiency of DPS may ap-
proach that of eFSr under noisy phase estimation cunditions and
Intereymbol interference (9, The noel (understand tia phenome
fon ont fusdaentel level isthe principal ebjectve of cis papor.

"hs bandoidth occupancy is alwave Important, the effects of 1%
generated by the use of band iting Mitere must-be takon into account
in any analyee of these systems. Reeause of the linear nature of the
‘demodulation process in CPSK, the effec of 151 bas been treated in
treat detail, Since nes demodulation inherently nonlinear, the
tnalyais of performance is very Aiffcule and no adequace analticl
‘hethods are currenily waable. Als, the combined effects of imper.
fect phase estimation nnd 1st on CPs must be determined ao thatthe
relative detection efficiencies of band-limited DnsK and crsK ean be
Tiny sessed.

Tn Section T of chia paper, we dosribe a technique for determining
the degradation in -ary trai operating inthe presence of is, additive
‘Gaussian nose, and imperfect carrer phase. In Section IT, we consider

2908 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1863
the performance af Meaty nPSK subjest lo 11 and additive Gaussian
1. COHERENT DETECTION
2.1 System description of CPSK

Figure 1 shows the M-ary crs aystem that we conser. The signa,
(2) before the transmit ier ean be represented as

510) = Rol axtthexpitaete + wih, d= S=F, 0
where the baseband modulation signa

xiy— 3 exptiagrecd ¢— 47/7, ei

and the constants A, fv pare the erie amplitude, frequency (in
Hid), und phuse, respectively. Also, rect) is the rectangular window

fac] eurrespondts tothe dats soquonce to be transmitted. Without las
of generality, we asuame thatthe Mf phise values of ay are uniformly
Sierbuted with equal probability between (=n, 7]. Soy a takes on
value in the set aye,

PIASE-SHIFTKEYING 2200
We also nesume tha the eats phasas in different true slots are
eatetcallyindependenc.

Tn our model, the ranamit ker, transmission chan, and receive
fitter ae linear and time invariant, Thorefore, the complex envelope,
(th at the output ofthe receive iver may he wrican as

le) = x10) OME) + mle + HALE)

Ale) = br(0) hell) @ halt,
where c(t, helt and het) are, respectively the impulae response of
Zhe transmit itr, the channel, and che roceive filer, ‘The symbol
{S dennten eonelution, Alo, n(t) +24 (2 is the complex envelope of
the Gaussian noieo passed ehrough the receive Mer. For symmetrical
‘cers, n() and (t) new independently and identically dribuced (nd)
Gaussian random variables with mean zero, and variance

2 Fina

‘where Ne ia the double-sided spectral density of the original white
ine and H(i the baseband equivelont transfer function ofthe
‘ecsive filter

2241 Detection in CPSK

Assuring thatthe recovered carrier i expli(2e/.t +f) wheres
sv estimate of yin oq, (1), the detector operats on the ignal, (2()
represented a3

wt) = 5 alt — AP expfitan + a) + E+ c

hor and ae i ams oid vviable with mea sr, and
seni
is ne pase ero a
6
st = fri @ ett attr re)
“To esate che tate phat on = Be a= Oyen el me

stetectoe meseures the phase #f wid} ut (= fs and a comet decision
resulte when

2510 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1051
2.1.2 ror rte for ary CPSK
Here, we briefly review some known results for crSK and then
develop new results applicable to our more general model
‘Error-race calculations for ide csi in added Gnussian noise can
bbe found in Refs. 207. References and 9 provide numerical methods
for calculating the probability of error in he presen of tt Reference

10 takes into account ist ad demoulaun phase eror, ut the results
‘aro restricted to only binary and quaternary aystema, We naw gener=
‘lias theve results.

Using the union bound and the representation ofthe received signal,
6g (2), it fllows trom en, (4) Uaat the probability of error, Pet),
ven thatthe phase © is transmitted,

maxi P,P) < Pell) =P. + Ps

where

nan i(o-weg) 0]

-nfoeon (6-3)
nenlecona)s

a smut oe Dl @

ean ngr el abaya ei
Penge Y Peed)
‘But, since tho signal constellation is award to bo circularly aymnet

te, Pet)0) is independent of
‘For convenience, we shall now assume that © = 2/1, Hence,

rinPy(Infonsl(e+Z9)]
+2nenf (tem) [}+o<a),

raexplif) = alts ATI

and 3 denocea che exclusion of the tur k= 0, 8 similar expression
can be writen for Ps,

where

PHASE-SHIFT KEYING 2311
Accurate estimations of Py and Pe are easy co obtain in the presence
of only Gavesian nae, bur are ie diffiule when Ist is ade and
{re ever more tedious when the csteibucion of earricr-phase e170,
‘ust be taken into account.

"In the next section we derive an exponentially tight upper bound on
‘these quantities for a fied cnrrier-phase error and thea perform
ltsymptotc [large signal-to-noise ratio (e/n}] anlyses on these upper
bounds fr a given distribution nf earier phase error.

2.1.3 Gounde on the eror rate
‘We bagin by writing 9. (8) 35

Pea wo +n -rain( +h +)

were (denotes expectation with reapect to ¢, and where,
He) = Pirvsin(Be + as +0 c)

Before we: can proceed with oa. (7, we need specific information on

the probability denaity function (pd) ofthe demodulating phase error

We shall assume thal the phase referenor ix derived from a pure

tone by a fit order prt» Ie well known’ hot the reulting paf for

the phase amor, eis

exp core)
ahi

-wborw isthe 6/n atthe inp to the ett multiplied by the reciprosal
ofthe banat,

pel = Ie 1

a

ve 9)

In eg, (1), isthe average power in the carrier, Ny isthe double
sided noise spectral density, and A, i che noise bandwidth of the
Tinoarie ri Alo, in eg, (9), (2) represents the modified Bessel
fnetion of the frst kind tal fender O. For 4 seconv-arder PLL, the
pul of eisalao approximately given by oq (8). We shall use this density
to abtain bounds on P

‘Since cis wayiuutetie random variable, eq (7) yields

rahe s veo.
were
My a SSH ee ala BO) gy

2912. THE BELL GYSTEM TECHNICAL JOURNAL, DECEMBER 1951
‘Using apne bouratng iwchninues al Laplace's method," we show
Appendie A ut

Pisa t ton
where
stax [once + Re ~ eo] + DA. cos cl)

BL (cones + BD) cos AIA + Be eI™

exp(—p' sink o/AN) + Bal) Ay
i coos

on

vao(1 +h
Note dtp? che 9 of the stem, Ako, D ean be regarded a he
fal of ithe pe recovery crcl to that inthe wa eye
the itoxration ime in bd,

Sinai, we a sow ha

Pradiat du

Tn muna, the seerage wynbo probability of eror, Pe, for Mary

eax nyiem ean be upper bonded by

2) OF enl-Drt co byte

pee OPTRA + fo) + DU coset
{os «5 GD) coe Mle) 4 Bed
WF TD coe eli B=

+2f)2a0= 8] see"

xa -rr|s =o
= pana( Ze afs
his upper Bound becomes

Pete] rant (5 +8) | aay

PHASESSHIFT KEYING 2819
smben phase estimation is perfect, Ds. Rquation (12) is the well-
Known Chemotf bound for Mary evs."
Tie obeervacion incervl of the wut is largo, D = 1, and if M2 1,

=dena(Gea[ Zonal)

2eostiniM) + Bl _ 2 coal
a 3

~»)))
+e -o'nie(5 +)
~( om)

og bfGea)D

aa
cng dw he aon 9
vine Sls kia aS
omelet

w]e

o-[{s

whens G+ Las D+ & ast shoul

° pie LM >2, D1 us)

22 Example of quaternary = 4) CPSK aystem

Lat us consider a quacerary (M = 4) Cesk syst and axume that
the chanel ie del

Tf pole Butcerworth transit and roceive filters are eed, the
reaulting average ayinbol prababilicy of error is plot in Fig. 2. Note
that the bound i furl tq and when the a/n of the phase recovery
‘inoue about 10 dB more than at the receiver input, the detection
ficiency of ens is sentially determined by Is alone, Alternatively,
fre can may Usa, forthe samme a/n, 0 10-baud rut integration time is
wired co achiove thie rstlimied performance. For this filter, tho 181
penalty is about 1.

TEM >, itis well mown chat the penalty in /n because of Gaussian
site alone ie aaymptticlly given By 1j[at’(=/)]

2914 THE BELL SYSTEM TECHNICAL JOURNAL, DEGEMBE;T 1881
Wi
\\ QUATERNARY crs
\
w\\ \ \\_epanerens
net
\e
Wy
\\ —

i

Fg 2 Prohaiy ferrgstrmary hve i Kyo nr lth tala
arbi Soe Gare omar ees sees
ecamr et ete haw eco) oo that i he pe
HEIN cata item rei ata eae

"The wpper bound in eq 413 indicates that i he definition of s/s
modified to take ito account the 1 power, Use addiional penal,
Deeause of imperfect phase estimation, ix

“This quantity ie plotted in Fig. 9. We observe that the /n penalty

PHAGESHIFT KEYING 2015
ee aan eae ate et a ea a

because of 351 is indapendent of M. In Fig. 8, also note that G)—»
fon i/M] a8 D>

I. DIFFERENTIAL DETECTION
5.1 System description of PSK

"The M-aty nn eputam is shown in Fig. 4 As before, the baseband
modulated bose signal cn be represented a

x(t) = 3 explingreotle ~ 471/7). @

Here, however, the sequence of phass [4] = [au.» ~ au] corresponds
to tho date sequence to be trensmited. Again, we assume that Mf

2816 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1081,