a digital demodulator for psm signals

8/14/2019 A Digital Demodulator for PSM Signals

1/9

1352 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-21, NO. 12, DECEMBER 973

A Digital Demodulator for PSM SignalsVILJO 0 .HENTINEN, PASI P. LAIHO, A N D RISTO M A R K I L A H T I

Abstract-This paper describes a digital demodu lator for phase-shift-keyed (PSK) signals in which the phase difference betweenhe receivedsignal and a carrier reference is found by measuring digitally the timeinterval between the zero crossings of the signal and the reference., Inthe case of coherent detection the reference is locked to one of t he Mpossible phases of the signal.The advantage of this demodulator is that bulky low-pass filters anddelay lines as well as critical threshold devices can be dispensed with.On the other hand, digital measurement of the phase and the use of afinite-width sampling window lead to degradation of the error perfor-mance. This degradation is theoretically analyzed for both coherent andphase-comparison detection. The quantizing error proves to be smallenough when the phase difference is coded into a 6-b binary number,and he heoretical results are in good agreement with measurementstaken from an experimental phase-comparison demodulator for four-and eight-level PSK signals. The experimental demodulato r is also de-scribed in the pzper.

D I. INTRODUCTIONATA MODEM S can be ma de considerably smaller as wellas less expensive if integration echn iques are utilized.Although the integrat ion of bulky LC filters, delay lines, etc.,is hardly feasible, the prob lem an be overcome by using digitalprocessing of signals and/o r active filters. Van Gerven and vander Wurf, for example, have studied data modem s equippedwith digital filters and modulators [11 . In this paper we intro-duce a digital demodulator for phase-shift-keyed (PSK) signalstha t requires neitherlow-pass filters nor delay lines.In the digital PSK dem odulator describe d, the phase differ-ence be twe en the received signal and a local reference of ap-proxim ately the same frequency is derived from the digitallymeasured interval betw een their zero crossings. Depending onwhe ther or not the reference is phase locked to o ne o f t h e Mpossible phases of the signal, we have co her ent or phase-com-parisondifferentially coherent)detection , respectively. Theprinciple of oper ation is explained in mor e detail in Sec tion 11.

The digital detec tor considered ha s two inhe rent deficiencies.First, he digital mea surem ent of he phase difference intro -duces quantizing errors. Secondly, the zero crossing, which isobserved during a finite time interval T 1 , oes n ot necessarilyoccur at he mom ent when he signal-to-noise ratio is maxi-mum and/or theintersymbol interference s minimum. Rather,the zero crossings are rando mly located in the ampling windowof width T 1 . These effects lead to a degrad ation of error per-for m anc e as discussed in Sect ion 111. Add itive Gaussian noiseis assumed thro ugh out he analysis. For he numerical corn -.

the IEEE Communications Society for publication without oral presen-Paper approved by the Associate Editor for Data Communications oftation. Manuscript received March 28, 1 9 7 3 ; revised June 1 4 , 1 9 7 3 .The authors are with Oy okia Ab Electronics, Helsinki, Finland.

putat ions it is assumed tha t he signal has raised-cosinespectr um after passing throug h the receiving filter.A 2400/3600 b/s experim ental phase-comparison dem odula-

tor is described and hemeasurem ents aken are comparedwith the theore tical calculations in Section IV.

11 DESCRIPTIONF THE DEMODULATORA implified block diagram of the digital dem odulato r is

shown n Fig. 1. The received M-ary PSK signal is passedthroug h a bandpass filter and imiter to a m odulo-two addertogetherwith a ocal eference. The referencehas the sameor about the same freque ncy as the carrier wave and, in thecase of coherent detection, ts phase is locked to one of Mpossible phases of th e received signal. The wid th o f the outp utpulses equals the interval betwe en he zer o crossings of hesignal and reference, and is thus proportional to the absolutevalue of their phase difference.

The pulse stream from the adder is sampled once for eachsymbol by observing those pulses whose width is least affectedby noise. Fo r practical signals and filte rs the sampling wind owis usually located in themiddleof hesymbol .Theactualsampling instants, determined by the zero crossings, fluctu atearound he best ampling mom ent. This is com parable osampling with timing jitter in conventional demodulators. Thewidth of th e sampling window T 1 must be chosen so that oneand only one zero crossing of the signal and reference appearsin th e sampling window. Thu s we have

where f, = 0,/2n is the carrier frequency.The width of thesampled pulses, i.e., the absolu te value of

the phase difference betw een the signal and re feren ce, is me a-sured digitally b y using equally spaced c ountin g pulses (Fig. l).The sign of this phase difference is dete rm ined by mea suringthe phase difference ,betw een he signal and the 90 shiftedreference.

Counters give the phase difference as a binary numb er. Forcohere nt detection the decision abo ut signal phase is ma de onthe basis of this binary num ber, whereas for phase-comparisondetection it is based on the difference between two adjacentnumbers .

The digital measurem ent of the phase difference introducesquantizing error whose value depen ds on the num ber of bitsin the binary counte rs. If n-b binary coun ters are used, it is

in the sampling window, k = 1,2, .. K , where K is the largest integerIn general, T 1 can be chosen so that k and only k zero crossings areT I T 1 . However, k = 1 gives the lowest performance degradation.


2/9


3/9

1354 . . IEEE TRANSACTIONS ON COMMUNICATIONS, DECEMBER 1973

M - p h a s es i g n a ld a t a___

Carr ierr e f e r e n c e

(a)r - _ _ _ _ _

o n n ' e c i r c u i t

L - - - - - - - - - - -(b)

Fig. 2 . (a) Digital demodulator with coherent detection. @) Digital demodulator with phase-compaison detection.t l ( t z ) as uniform distribution and is independen t of the quan- and, in the best case. istizing error qi ( 4 ; ) .k T + t is given by (A 8)Th e kth received noisy signal elemen t at the mom ent t ' =Z k ( t ) = r1 k [ t , k , n k ( t ) ] ' cos {act + U c k T k + As seen in Fig. 4, the zero crossing occurs n he shadedarea

N - (incorrect decision) when the phase difference a t the mome nt+ @ k ( t , k ) + q k [ t , k , n k ( t ) ] I 9) t l exceeds n/M, or at the ' moment t 2 is less than - n / M . In

Nwhere k = ( k - N , , k-N,+1, . . . , k + N , ) is the sequence of other words,-Nthe ransm itted phase angles. & is due to intersymbol nter-

ference and q k primarily due to noise, as . indicated in Appen -dix A. Com paring Z k ( t ) with Sk( t ) in (4), we see that the phasedifference is q k + . However, because the quantizing errorqis inclu ded , th e measured phase difference becomes

q k [ t , k , n k ( t ) l + @ k ( t , k )+ 4 -N yAssuming that he cou ntin g pulses are phase locked neitherwith the reference nor w ith the received signal, then the qua n-tizing errors of bo th th e signal and reference have un iform dis-tribution in the interval (-A/2 , A/2) (Fig. 5 ) . Because the ref-erence is phase loc ked o he signal, the prob ability densityfunction of the tot al qua ntizing error, n the worst case, is .

or7?k' [ t2 k , nk( t2)I + @ k ( f 2 k ) + 4


4/9

HENTINEN e t a l . : DEMODULATOR FO R PSK SIGNALS 355, ~ g n a l n o l s e

Timing of t h e IC- I / T k T f k + l / Tb i sampling wrndows

f k + 2 / T

ci 0 , g r t a l r z e d r g n a l

O ~ g , t a l , r e dr e f e r e n c e 9 0 1

g / s l g n a i ond 90-M o d - 2 su m at t h er e f e r e n c e n n t u u u u u u U U L P(e )=

J p h a s e s u l s e sThe s , g n a l of i h ek i p h a s e

T h e s , g n a l 01 t h e P Ul w UUM+ I

Fig. 3 . Waveforms in the digital demodulator.

Fig. 5. Quantizing error in coh erent detection .The rand om variables k , t l , t 2 , and q in (12) are inde pen -dent, and t l and t2 hTve the same probability density func-tions as to in (6). The limits for the density function of q areshown in ( loa ) and ( lob) .

The total probabilityof error can be calculated from the con-ditional error probability [2]

r

I l l

'T1/2 T 1 / 2 - T I C 2 0 T { / 21 I

* *k T t ' k T f

Fig. 4. Decision boundaries in coherent detection.form because q k has a symm etric density functio n. Thu s (BS)P 1 and P2 an be written as follows:

where erfc ( u ) = 1 - erf ( u ) = ( 2 / G ) J;P e- dt and K [ I isgiven in (B4). Assuming the sym bol s to be indepen dent withequal probability, he probability of a particular symbol se-quence ki is

c o u n t i n gu l s e swc t '-

m e a s u r e dh o s e l f f e r e n c eI 1a c t u a l h a s e r f f e r e n c e-

r rmP 1

1.-- [Pl (e ki , t l 4 ) P2(e I ki , t l , 411 dt ldq. (14)Tl - NEquation (14) represents the case of maximum error proba-bility; the m inimu m er ror pro bability is obtained by replacingA with A/2. The integrationof (14) is perfo rmed with a digitalcom put er. Th e signal is assumed t o have a raised cosine spec-trum afte r he receiving filter having a cosine characteristic.Bo th he signal and the receiving filter are optim um or anideal de mod ulator [4], but not necessarily op timu m for hedigital demod ulator. he signal spectrum etermineshewaveform g ( t ) , which in this case is supposed to cause signifi-cant ntersymbo l nterference only o he preceding and fol-lowin g pulse, i.e., N , = N 2 = 1. The noise power is reducedto the bandwidthnumerically equal to the bit ate.

The degradation of the signal-to-noise ratio, compared w ithideal detection, is shown in Fig. 6as a function of the numberof bits in binarycounters.The calculationswere made fora 24 00 b/s 4-phase modem using the A-code and the carrierfrequency1800 Hz, as recommended by he nternationalTelephoneand TelegraphConsultative Com mitt ee (CCITT).The width of the samplingwindow for hisdemodulator isT/3,where Tis the signaling interval.

As illustrated in Fig. 6, there is nothing to be gained by in-creasing th e num ber of bits in the binary counter beyond 6.


5/9

356 IEEE TRANSACTIONS O N COMMUNICATIONS,DECEMBER 1973

1 2 . 4 5 6 7 8 9 7 0 1 1 1 2' N u m b e r o f b l t s rn c o u n t e r s

Fig. 6 . Degradation of thesignal-to-noise ratio as a unction of thenumber of bits in binary counters for coherent detection, T1 = T / 3 .

The degradation is about 0.6 dB for an error probability ofThe performan ce of the igital demodu lator can be improved

by modulating the received signal to a higher frequency, whichmakes thesampling window narrower, (1): The degradation ofthe signal-to-noise ratio as a functio n of the sampling windowT 1 s shown in Fig. 7.B. Phase-ComparisonDetection

In phase comp arison the decision is based on the measuredphase .difference betw een wo adjac ent symbols. Erro r ariseswhen the phase difference deviates more than +n/M from theunpe rturbed value.In phase-comparison dete ction , the decision boundaries forthe kth symb ol can also b e found, but now they depen d onthe deviation of he zer o crossing of the previous (k - 1)thsymbol (Fig. 8). .The zero crossing of the (k - 1)th sym bol isdenoted by t l owithout noise and by t l with noise. The corre-spondingquantities of thek thsymbol are t 2 i and t 2 . Thezero crossings tha t precede and follow th e above men tionedzero crossing have bee n denoted by indices - and +. The zerocrossings t l o and t zo depen d on each other and fulfill, accord-ing to (A6), the equations

10-4 .

u,t10 + w,(k - 1) T + J / k - 1 + 0 = n /2 +in (15a)uCt2o+ w,kT + k + 0 = n/2 + ir (15b)

where i and j are chosen so that I t l o , I< T1 /2 . rom (15 )follows tha tt 2 o = t 1 o - TI J/k/n+ Tt J / k - l / n + iT1 (1 6

where i is chosen so that Itzo < T1 2. We denote by A t the,deviation of the zero crossing of the (k - 1)th symbol from itsnoiseless value. Assuming P( ]At1> T1/2) 0 , he deviationof the zero crossing is

At = t l - t l o+ iT1 (17)where is chosen so that At


6/9

HENTINEN e t d l . : DEMODULATOR FOR PSK SIGNALS 1357

where and j are chosen so that I t2 I, I t 2 2 I G T I 2.The effect of quant izing on the decision boundaries can bedealt w ith in the same way as in the coherent detection . How-ever, since its effect was fo un d to be negligible for coher entdetection with 6-b counters, the quantizin g error is neglectedin the following analysis.

Wedenote k = ( k - N l - l , k - N 1 , ' ' , k+N,) a n d A =,{ k,t 1 t l o } . n Fig. 9 are displayed the two possible locations inthe sampling window for the decision boundaries t z l and t z 2and the onditional density functionsof he zero-crossingpoints t i , 2 , nd t: . The.re gion of incorrec t decision has beenshade d. We can express the conditio nal probab ility of error as

--x/ \

where

The error probability in (20) is of the type (SI); therefore it f ( t 1 I t l o , t ; ( ) = d t , 7 Jl rfc { ~ ~ [ t l , ~ k - l , n k - l ( t l ) lcan be interpreted as -[:; 70 - t 2 i < T1 / 2erfc{dK [ t 2 i , k j n k ( t 2 i ) lP(r2 < 2i A) = ' sin [ac (TO - t 2 i> - @ k ( t Z i , k ) I 1,

i>

170. - t 2 i l TI / 2 ( 2 2 )

where ro is chosen as above.From the elements of the set A , only t l depends on the ran-dom variables and t l o theothers are indepe ndent. The

total error probab ility is [ 2 ]

' sin [ w c ( t l o - t l ) - @ k-1 ( t l 9 f k - l I 1. ( 2 6 )The error probability for phase-comparison detecti on is cal-culated from ( 2 3 ) as a fu nction of the signal-to-noise ratio ; the

parameter values are the same as those used in Sectio n 111-A forcoherentdetection.The noise comp onen ts in two adjacentsampling windows are assumed to be independe nt. However,when the sampling w indow is relatively wide, such an approxi-mation is poor. The calculated probability of error is shownin Fig. 10togeth er w ith th e measured results.

The frequencies of the received carrierand the referencemay differ because of the frequen cy instability of the referenceoscillator and the frequenc y errorsrising in a carrier fr equen cysystem , should such a system be used. The frequency error inthe last case may be 6 Hz [3] ,which causes a phase deviationof 1.62 between wo adjacentsymbols with he previouslymentioned parameters.This phase deviat ion,when assumed


7/9

1358 IEEE TRANSACTIONS ON COMMUNICATIONS, DECEMBER 1973

dBFig. 10. Calculated and measured error rate of the digital phasecom-parison demodulator as a function of the signal-to-noise ratio. ( A. 2400 b/s 4-phase modem with 1800-Hz carrier frequency,T I = T / 3 . )

uniformly distributed in the interval (- 1.62 , 1.62 ), leads toa degradation less .than th at caused by the quantizing error inthe case of a 6-b counter.1v. E V A L U A T IO NF T H E E X P E R I M E N T A L D E M O D U L A T O RThe experimen tal 2400/3600 b/s demodulator for 4/8 phasemodu lation uses digital phase-comparison detec tion with 7-b

counters. The demo dulator satisfies the CCITT recom menda -t ions for 2400 b/s modems. Thehange from code A to code Bas well as the change of the bit rate an be made with a remote-contr ol signal.

The actua l dem odula tor [see Fig. 2(b)] is built on a single100 X 20 0 mm' printed circuit board using TT L circuits.Analog c ircuits are used on ly for the receiving filter, the ampli-fier, and pa rt of he bit synch roniza tion device. Preliminaryestimates indicate that this demod ulator will be cheaper thanconventional demo dulators with corresponding characteristics.

The measured error rate as a functio n of the signal-to-noiseratio'is show n in Fig. 10. With an error rate of I O the cal-culated and measured degradations, as compared w ith an idealdem odulato r, are about 0.6 and 1.3 dB, respectively.

The effect of varying the frequency of the received carrierhas also been ested and the dem odulator found o o leratevery well frequency variations of k10 Hz.

V . CONCLUSIONIn the digital PSK demod ulator under consideration, bulky

low-pa ss ilters and delay lines as well as critical hresho lddevices aredispensed with . An analysis of theerrorperfor-

mance shows that, for the proposed 240 0 b/s digital dem odu-lator, he heore tical degradation of the signal-to-noise ratiowith heerrorprobabilityof is abou t0.6 dB. Theeffectof the quan tizing error is insignificant w hen coun ters with atleast 6 b are used for 4-phase mo dulatio n. When the receivedsignal is mo dulate d t o a higher fr eque ncy it is possible to ob-tain,with6-bcounters ,a degradation of the signal-to-noiseratio of less than 0.15 dB.

An experim ental 2400/3 600 b/s phase-comparison dem odu-lator is built on a single (100 X 200 mm') printedcircuit,boardwithTTL circuits.This demo dulator promises to becheaper thanconventionaldemodulators.The change fromcode A to code B as well as the change of the bit rate from2400 to 3600 b/s an be made by a remote-control s ignal .

The e rror rate for the experime ntal dem odulator s measuredat differ ent signal-to-noise ratios. Results sho w that the degra-dation is about 0.7 dB higher than the calculated one.

A PPE N D IX AAfte r passing throu gh the receiving f ilter , the M-ary phase-modulated signal can be written as [4]

ca~ ( t ' ) g ( t ' - n T COS (ac t Qn + e ( A I )n = m

where T is the signaling interval, g ( t ' ) is the pulse shape ofthe received signal, w, = 2nfc is the angular frequency, Q n sthe phase of the nth transm itted sym bol, and 6 is a randomangle, uniformly distributed over the interval -n,),.The kth received signal element at the mo men t t = k T + tIt I < T / 2 , s,


8/9

HENTINEN e t a l . : DEMODULATOR FOR PSK SIGNALS 1359

S k ( t ) = S ( t ' = kT + t )= g [ + ( k - n ) T ] COS (act w,kT + ',e . (A2)mn = m

We assume that g ( t ) differs from zero only for - N1 - l ) T


9/9

136 IEEE TRANSACTIONS ON COMM UNICATIONS,DECEMBER 1973

[31[4 1[ 51

Data transmission,White book, vol. VIII,Mar del Plata, 1968, IVthInternationalTelephone and Telegraph Consultative Committee,plenary assembly.W. R.Bennetand J . R . Davey, Data Transmission. New York:McCraw-Hill, 1965.C. R. Cahn, Combined digital phase and amplitude modulationcommunication systems, IRE Trans. Commun. Sys t . , vol. CS-8,pp. 150-155, Sept. 1960.

Viljo 0. Hentinen was born in Nilsia, Finland,onFebruary 1, 1934. He received the Dipl.Eng.degree nelectricalengineering,and theLic. Tech. and Dr. Tech. degrees from HelsinkiUniversity of Technology, Helsinki, Finland, in1960, 1968, and 1971,espectively.From 1961 he was a Research Engineer at theState Institute for Technical Research, Finland,working on the development of radio relay sys-tems. He then joined Oy Nokia Ab Electronics,Helsinki, Finland,wherehe wasHeadof theRadio Relay Laboratory. From 1966 to 1972 he worked with the De-partmen t of Elect rical Engineering, Helsinki University of Technology,wherehe was esponsible orundergraduateandgraduatecourses ncommunication theory and techniques. He also directed research con-cerned with digital communication systems. Since 1972 he as been theResearch Director of Oy Nokia Ab Electronics.

Dr. Hentinen is a member of the Finnish Society of Electronics Engi-neers and the Engineering Society in Finland (STS).

networks.

Pasi P. Laiho was born in Laitila, Finland, onOctober25,1947. He received the Dipl.Eng.degree nelectricalengineering romHelsinkiUniversity of Technology, Helsinki, Finland, in1971.From 1970 to 197 2 he was a Research Assis-tant atHelsinki University of Technology, ork-ing on digital communication systems. He thenjoined Oy Nokia Ab Electronics, Helsinki, Fin-land , where he is a Research Engineer workinginherea of dataransmissionnd ata

Risto M Siixkilahti was born in Helsinki, Fin-land,on December 13 , 1943. He received theDipl. Eng. degree in electrical engineering fromHelsinkiUniversity of Technology,Helsinki,Finland, in 1968.From 1968 he was an R & D Engineer at OyNokia Ab Electronics, Helsinki, Finland, work-ing in heareaofdata ransmissionanddatamodems.Now e is aDevelopmentProjectLeader in the Carrier Systems Department.

a digital demodulator for psm signals

Documents