ANALYTICAL APPROACH IN ESTIMATING ERROR PERFORMANCE OF PARTIALLY COHERENT PSK RECEIVER OVER KAPPA-MU FADING UDC ( :519.

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1 FACTA UNIVERSITATIS Series: Automatic Cotrol ad Robotics Vol. 15, N o 1, 016, pp ANALYTICAL APPROACH IN ESTIMATING ERROR PERFORMANCE OF PARTIALLY COHERENT PSK RECEIVER OVER KAPPA-MU FADING UDC ( :519.74) Gora Đorđević 1, Milica Petković 1, Časlav Stefaović 1 Faculty of Electroic Egieerig, Uiversity of Niš, Niš, Republic of Serbia Faculty of Natural Scieces ad Mathematics, Uiversity of Prištia, Kosovska Mitrovica, Republic of Serbia Abstract. I this paper, a ovel aalytical expressio for the symbol error probability (SEP) of M-ary Phase-Shift Keyig (MPSK) sigal trasmissio is derived, by usig the Fourier series method (FSM). The sigal is trasmitted over - fadig chael. The hardware imperfectios i PSK demodulator are take ito cosideratio, ad are preseted through the phase oise described by Tikhoov distributio. Based o the derived SEP expressio, umerical results are preseted ad discussed. It is illustrated that the existece of phase oise leads to the irreducible SEP floor which degrades the system performace to a large extet. Key words: - distributio, M-ary Phase-Shift Keyig (MPSK), phase oise, symbol error probability (SEP), Tikhoov distributio. 1. INTRODUCTION The pheomeo of fadig, which represets the variatio of istataeous value of the received sigal, is oe of the mai reasos for degradatio of the system performace. I order to properly describe the differet propagatio eviromets imposed by fadig pheomea, may statistical models have bee proposed i techical literature [1]. The - distributio has recetly bee proposed ad adopted as a coveiet model for describig liear, lie-of-sight (LOS) eviromets with several clusters [] [5]. It is also proved to be a geeral model, sice may other fadig distributios ca be obtaied for differet distributio parameters. For example, whe =0 - distributio is reduced to Received December 31, 015 Correspodig author: Gora T. Đorđević Faculty of Electroic Egieerig, Aleksadra Medvedeva 14, Niš, Republic of Serbia gora@elfak.i.ac.rs

2 34 G. ĐORĐEVIĆ, M. PETKOVIĆ, Ĉ. STEFANOVIĆ the Nakagami-m distributio, for =1 it approximates Rice distributio, ad for =0 ad =1, it is reduced to the Rayleigh distributio. I order to provide high spectral efficiecy, the M-ary phase-shift keyig (MPSK) modulatio is commoly used i practical commuicatio systems. The mai flaw of the systems based o PSK modulatio is the imperfect extractio of the carrier referece phase at the receiver durig coheret detectio [1], [6]. Although the mai cause of the performace degradatio is the existece of additive white Gaussia oise (AWGN), as well as the ufavorable fadig chael coditios, the hardware imperfectios of PSK modulator ad demodulator are also of great importace. The PSK modulator imperfectios are usually weaker tha the PSK demodulator imperfectios. The mai reaso for the PSK demodulator imperfectios is the imperfect referece sigal recovery whe coheret detectio is performed, which is preseted through the existece of phase oise [7] [9]. Cosiderig MPSK sigal trasmissio iflueced by - fadig chael, the ovel symbol error probability (SEP) expressio is derived. The hardware imperfectios i PSK demodulator represeted through the phase oise are take ito accout. The Tikhoov probability desity fuctio (PDF) is used for describig the impact of the phase oise. The SEP expressio is derived by usig the Fourier series method (FSM) preseted i [7] [9]. Furthermore, the SEP expressio for the system uder ivestigatio, whe the phase oise is small ad eglected, is also derived. O the basis of the derived expressios, umerical results are preseted. The rest of the paper is orgaized as follows. Sectio describes the system ad chael model. The average SEP aalysis is preseted i Sectio 3. Numerical results with discussios are offered i Sectio 4. Sectio 5 gives some cocludig remarks.. SYSTEM AND CHANNEL MODEL The aalysis of the system uder ivestigatio cosiders the PSK sigal trasmissio over the - fadig chael. Sice - distributio is geeral ad ca be reduced to simpler cases, such as Nakagami-m, Rayleigh, ad Rice fadig distributios, the followig aalysis is coveiet for describig PSK sigal trasmissio iflueced by differet fadig coditios. Beside the impact of - fadig, the PSK system performace aalysis takes ito accout hardware imperfectios represeted through phase oise. It is assumed that x deotes the fadig evelope of the desired sigal, which experieces -distributio, so the PDF of x is give by [] 1 ( 1) ( 1) ( 1) fx ( x) x x I x 1 1 exp( ) exp 1, (1) where ad are Rice factor ad fadig severity factor of the sigal evelope respectively, is the local mea power of x, ad I v () deotes the modified Bessel fuctio of the first kid ad order [10, eq. (8.406)].

3 Aalytical Approach i Estimatig Err. Perf. of Partially Coheret PSK Receiver over Kappa-Mu Fadig 35 After applyig the stadard techique of trasformig radom variables, the PDF of the istataeous sigal-to-oise-ratio (SNR) is derived by usig fx ( x) f ( ) as x x 1 1 ( 1) ( 1) ( 1) f ( ) exp I 1 1 1, exp( ) () where deotes the istataeous SNR per symbol of - distributio, ad represets the average SNR per symbol. Sice the cosidered system employs M-ary PSK (M is modulatio order), ote that b log M ad b log M, where b ad b are the istataeous ad the average SNR per bit, respectively. The hardware imperfectios are represeted through the phase oise, which is modelled by the Tikhoov PDF give by [7] [9] exp(cos ) I0(1 ) f ( ),, (3) where is the variace of the phase error. Sice the followig aalysis is performed by usig FSM method, the Tikhoov PDF is preseted i the Fourier series form as [7] [9] 1 f ( ) c cos( ),, (4) 1 where the Fourier coefficiet is defied as [7] [9] c I (5) (1/ ). I0(1/ ) 3. THE SEP ANALYSIS I order to determie the aalytical expressio for the average SEP of the system uder cosideratio, the FSM is applied, based o the PDF of the phase of the composite received sigal give i the Fourier series form as [7] [9] 1 f ( ) b cos( ),, (6) 1 where the Fourier coefficiet b is depedet o the fadig chael coditios. For the cosidered system iflueced by - fadig, the PDF of the phase ca be obtaied as f ( ) f ( / ) f ( )d, (7) 0

4 36 G. ĐORĐEVIĆ, M. PETKOVIĆ, Ĉ. STEFANOVIĆ where f () represets the PDF of the istataeous SNR give by eq. (), ad f ( /) is a Fourier series form of the coditioal PDF of the received sigal phase due to AWGN, defied as 1 f ( / ) a ( )cos( ),. (8) The Fourier coefficiet for AWGN 1 a is preseted as [9] 1 a F! ( ) 1 exp( ) 1 1 1; 1;, where () is Gamma fuctio defied as [10, eq. ( )] ad hypergeometric fuctio defied as [10, eq. (9.1)]. 1 1 ;; (9) F is the cofluet After substitutig eqs. (), (8) ad (9) ito (7), the PDF of the phase is obtaied as 1 1 ( 1) 1 f ( ) 1 cos( ) 1 1 1! exp( ) (10) 1 ( 1) ( 1) exp exp( ) 1F1 1; 1; I 1 d. 0 I order to solve the previous itegral, after utilizatio [11, eq. ( )], the modified Bessel fuctio of the first kid i eq. (10) is trasformed as I 1 d 0 d! ( d ) d 1 ( 1) 1 ( 1), (11) so the PDF of the phase i eq.(10) is ow give i the form 0 d d d 1 ( 1) f ( ) 1 cos( ) d d 0 1! d! ( d )exp( ) d 1 ( 1) exp exp( ) 1F1 1; 1; d. The expoetial fuctio i eq. (1) is represeted i terms of Meijer s G fuctio [10, eq. (9.301)] by usig [11, eq. ( )] as ( exp 1) G ( 1), 0 (1) 1,0 0,1 (13) while the product of the expoetial fuctio ad the cofluet hypergeometric fuctio with the same argumet is trasformed i terms of Meijer s G fuctio by [11, eq. ( )] as

5 Aalytical Approach i Estimatig Err. Perf. of Partially Coheret PSK Receiver over Kappa-Mu Fadig 37 ( 1) 1,1 1 exp( ) 1F1 1; 1; G 1,. 0, After replacig eqs. (13) ad (14) ito eq. (1), ad applyig [11, eq. ( )] as 1, the PDF of the phase is obtaied as d d d 1 ( 1) ( 1) f ( ) cos( ) d! d! ( d )exp( ) d 0 1 ( 1) 1 d. 0, d 1 1,0 1,1 G0,1 G 1, 0 0 The itegral i eq. (15) is solved by usig [11, eq. ( )], so the fial form of the PDF of the phase is derived as f 1 G 1,, d k d 1 e k1 k 1 d 0 d d!! 1, 1 d cos. k 1 0, Represetig the previous PDF i the Fourier series form as i eq. (6), the Fourier coefficiet b is derived as b (14) (15) (16) d k d ( 1) e k ( k 1) 1, 1, 1 d G,. d 0 ( k 1) (16) ( d ) d!! 0, Sice a perfect local referece phase is ot realizable i most practical scearios, the impact of the phase oise is cosidered. The coditioal SEP of MPSK is foud as [8] M 1 b PM ( ) 1 f ( )d 1 si cos( ). M M 1 M (17) After averagig the previous coditioal SEP over, the fial form of the average SEP of MPSK sigal trasmissio, i the presece of Tikhoov distributed phase error over - fadig chael ca be foud as 1 bc PM PM( ) f ( )d 1 si. M 1 M (18)

6 38 G. ĐORĐEVIĆ, M. PETKOVIĆ, Ĉ. STEFANOVIĆ Whe the phase oise is very small ad ca be eglected, the SEP of MPSK system ca be easily obtaied as [8] 1 b PM ( 0) 1 ( )d 1 si. / M f (19) / M M 1 M 4. NUMERICAL RESULTS I this sectio, umerical results obtaied by the derived expressio are preseted. The average SEP depedeces o modulatio order M, phase oise stadard deviatio ad -µ distributio parameters are observed. Fig. 1 shows QPSK SEP depedece o average SNR per bit for differet values of the phase error stadard deviatio. The system performace degradatio is oticed whe the value of is greater, meaig that the impact of the phase oise is stroger. Furthermore, the urecoverable error rate floor exists, which is a importat limitig factor that should be take ito cosideratio durig the PSK system desig. The error rate floor is evidet at lower values of SNR per bit whe the parameter has greater values SEP =5 0 =7.5 0 =10 0 =1.5 0 =15 0 = =0 0 k sr b [db] Fig. 1 QPSK SEP depedece o average SNR per bit for differet values of The SEP depedece o average SNR per bit for differet values of the phase error stadard deviatio, cosiderig BPSK, QPSK ad 8PSK modulatio, is preseted i Fig.. It ca be oticed that the ifluece of the phase oise is more expressed whe the modulatio order is higher. Observig the results obtaied for BPSK sigal trasmissio, it ca be cocluded that the SEP values are almost equal whe the phase error stadard deviatio is =5 0 ad =15 0, i the rage of low values of SNR b. Furthermore, the existece of error rate floor is oticed, appearig firstly whe the phase oise is stroger ad/or with greater modulatio order.

7 Aalytical Approach i Estimatig Err. Perf. of Partially Coheret PSK Receiver over Kappa-Mu Fadig SEP =5 0 = sr b [db] Fig. MPSK SEP depedece o average SNR per bit for differet values of parameters ad various modulatio orders = SEP = = sr b [db] Fig. 3 QPSK SEP depedece o average SNR per bit for differet values of ad parameter Fig. 3 presets QPSK SEP depedece o average SNR per bit for differet values of the - distributio parameter. The ifluece of hardware imperfectios is determied by the phase oise stadard deviatio, which takes values =5 0, 10 0 ad Whe the parameter takes greater values, the fedig stregth is weaker ad system has better performace. I additio, it ca be observed that the impact of fedig stregth, determied

8 40 G. ĐORĐEVIĆ, M. PETKOVIĆ, Ĉ. STEFANOVIĆ by the parameter, is more domiat whe the phase oise stadard deviatio is lower ad hardware imperfectios are smaller. With greater values of, the phase oise is stroger, ad the parameter has a miimal impact o the average SEP performace. I additio, there exists the error floor, which is ot depedet o the value of the parameter. 10 -, sr b = 10 db SEP =5 0 sr b = 0 db = Fig. 4 QPSK SEP depedece o the fadig parameter for differet values of average SNR per bit ad the parameter Fig. 4 shows QPSK SEP depedece o the fadig parameter for differet values of average SNR per bit, whe the phase oise stadard deviatio is =5 0 ad =10 0. As it has already bee cocluded, greater values of the parameter correspod to the weaker fadig ad better coditios for trasmissio. Also, whe the average SNR per bit is greater, system has better performace. Furthermore, it ca be observed that the ifluece of the phase oise is more domiat whe the average SNR is greater. The SEP curve for =10 0 ad SNR b = 0 db has almost a costat value (there is a slight improvemet with icreasig the parameter ). Whe the phase oise is very strog, the value of the parameter has a mior impact o the system performace determiatio. This is i agreemet with the coclusios from Fig. 3. The MPSK SEP depedece o the phase error stadard deviatio for differet values of the average SNR per bit is observed i Fig. 5, assumig BPSK, QPSK ad 8PSK sigal trasmissio. Sice there are o chages i SEP values whe 0, the phase oise is weak ad ca be eglected. This SEP value is equal to the SEP value of the system without phase oise. The icreasig of results i the SEP performace worseig, so the hardware imperfectios i PSK demodulator are a importat factor durig the determiatio of the system performace. The greater values of the fadig parameters ad correspod to the weaker fadig coditios, which reflects i better system performace.

9 Aalytical Approach i Estimatig Err. Perf. of Partially Coheret PSK Receiver over Kappa-Mu Fadig SEP sr b =0 db [ 0 ] Fig. 5 MPSK SEP depedece o for differet values of the fadig parameters ad various modulatio orders 5. CONCLUSION I this paper, we derive a ovel aalytical SEP expressios of MPSK sigal detectio, which is iflueced by - fadig chael. The hardware imperfectios i PSK demodulator are cosidered, with the phase oise modeled by Tikhoov distributio. The preseted system performace aalysis is based o FSM. The derived SEP expressio is reduced to the case whe the phase oise is small ad eglected. O the basis of the derived results, umerical results are obtaied ad discussed. The ifluece of phase oise stadard deviatio, fadig parameters ad PSK modulatio order is observed. From the preseted results, it is cocluded that the existece of hardware imperfectios i PSK demodulator ca cause importat system performace deterioratio, especially whe a higher PSK order is cosidered. Furthermore, the urecoverable error rate floor appears as a cosequece of the phase oise existece, which ca seriously limit the SEP performace of the observed system. Ackowledgemet: This paper was supported by Miistry of Educatio, Sciece ad Techology Developmet of the Republic of Serbia uder Grats TR-308 ad III REFERENCES [1] M. K. Simo ad M. S. Alouii, Digital Commuicatio over Fadig Chaels: A Uified Approach to Performace Aalysis. New York, USA: Joh Wiley & Sos, Ic., 000. [] M. D. Yacoub, "The k-μ distributio ad the η-μ distributio," IEEE Ateas ad Propagatio Magazie, vol. 49, o. 1, pp , 007. [Olie]. Available:

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