Performance Upper Bounds of High Altitude Platform Systems over a Two-State Switched Channel

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1 Performance Uer Bounds of Hgh Alttude Platform Systems over a Two-State Swtched Channel Hen Th Thu Nguyen, Hung Vet Nguyen, Thang Nhat e Faculty of Telecommuncatons, Posts and Telecommuncatons Insttute of Technology, Km, Nguyen Tra Street, Ha Dong Dstrct, Hano, Vetnam. Postgraduate Studes Faculty, Posts and Telecommuncatons Insttute of Technology, Km, Nguyen Tra Street, Ha Dong Dstrct, Hano, Vetnam. Abstract Ths aer resents erformance uer bounds of Hgh Alttude Platform (HAP) systems over two-state swtched channel model wth QPSK modulaton. e roose the modfed trells dagram for unctured convolutonal code to derve erformance uer bounds of HAP systems usng IrCC-URC-QPSK codng scheme. Ths codng scheme acheves better erformance than codng scheme n CAPANINA roject []. Furthermore, the results of uer bounds are also the bass for further verfcaton of the effectveness of the IrCC-URC-QPSK codng scheme. Keywords: Hgh Alttude Platform, Uer bounds, Seral Concatenated Convolutonal Code, Two-State swtched channel model, Bt error robablty. INTRODUCTION The demand for wreless communcaton s ncreasng day by day. Peole want a hgh seed of communcaton n less tme. Everyone wants a fast communcaton from anywhere to anyone. Even rural area also requres nternet faclty. It s too hard to establsh a Base staton for artcular small vllage for broadband communcaton or any wreless communcaton. Even t s too costly to launch a satellte for artcular rural area. So Hgh Alttude Platform (HAP) broadband communcaton s exected to become a oular soluton for the wreless communcatons nfrastructure []. HAP networks have been ncreasngly layng an mortant role n suortng broadband wreless communcaton systems, namely fourth generaton ong Term Evoluton (4G-TE) and ffth generaton (5G) networks []. HAPs are communcaton facltes stuated at an alttude of 7 to km and at a artcular ont relatve to the Earth. They have the caablty of carryng dverse relay ayload suortng multurose communcatons. HAPs n a fully deloyed confguraton are caable of rovdng servces and alcatons rangng from broadband wreless access, navgaton and ostonng systems, remote sensng and weather observaton/montorng systems, moble telehony as well as dgtal TV [4]. Nowadays, there are no recommendatons on the secfc codng and modulaton used n the HAP system. Though, there have been several studes evaluatng the erformance of HAP usng a arallel concatenated convolutonal codes (PCCC), seral concatenated convolutonal codes (SCCC) codng scheme and DPC [],[5]. However, these studes only evaluate erformance by smulaton wthout theoretcal erformance analyss. Based on the codng scheme roosed n [], we roose a codng scheme that uses an rregular convoluton code (IrCC) [6] n an outer encoder to ncrease the channel caacty-near effcency. To acheve ths, we use the EXIT chart tool [7],[8],[9] to select subcodes wth dfferent code rates and determne the ercentage of nformaton bts encoded by these subcodes. These subcodes are formed by addng more generators (, ) and uncturng for lower rates or only uncturng hgher rates whle maxmzng the free dstance from memory- mother code defned by generators (7, 5). th the smulaton results of the roosed codng scheme, we wll derve analytcal uer bounds to the erformance of t. For smlcty, the conventonal unon bound technque s used n ths aer. Aganst ths background, the novel contrbuton of ths aer s that we make use of the unon bound technque [,] to derve erformance uer bounds of HAP systems over a twostate swtched channel model. Frst, we roose the modfed trells dagram for unctured convolutonal code to fnd the transfer functon and weght enumenator. Then, we derve the erformance uer bounds for IrCC-URC-QPSK codng scheme wth subcodes havng varous code rates. The derved erformance bounds are comared wth smulaton results to further demonstrate the erformance of the desgned code. The aer s structured as follows. Secton resents twostate swtched channel model for HAP system. Secton derve analytcal uer bounds to the bt-error robablty of HAP systems usng a SCCC codng scheme over the two-state swtched channel. The concluson s gven n Secton 4. Two-state swtched channel model for HAP et us now consder a sngle transmsson lnk between an HAP and a hgh-seed movng tran n CAPANINA roject [] that assocated wth the transmtted and receved sgnals of x and y, resectvely. The receved sgnal can be reresented as y hx n () where h=h s.h f s the comlex-valued fadng coeffcent that 9

2 fluctuates on a symbol-by-symbol bass, and n s the AGN rocess havng a varance of N / er dmenson. In order to account for dfferent roagaton condtons n ths scene, a two-state swtched model was consdered n []. It s a Markov model characterzed by two states, good (G) and bad (B). Good condtons corresond to a Rce fadng model, whle bad channel condton corresonds to a Raylegh dstrbuted fadng. As far as the Rce fadng model s concerned, the Rcan factor K s determned from measurements []. Suffce t to say that for rural envronments t takes on values on the order of db, whle for suburban envronments t s K = db. Indeed, the latter value of K has been used n ths aer for system erformance verfcaton. Fgure : Two-state swtched channel model for HAP n CAPANINA roject []. e assume that states of channel are selected at the begnnng of the transmsson. The frst state (G) wth statonary robablty π G=.565 consders frequency-flat uncorrelated real Rcan fadng channel wth Rce factor K = db, whle the second state (B) consder frequency-flat uncorrelated real Raylegh fadng condtons wth statonary robablty π B =- π G =.45. But once a channel state s selected, t wll not change for the entre transmsson erod. As seen n Fg., at the transmtter sde, an nformaton frame of N bts are encoded by the CC encoder, n order to roduce an outut frame havng a frame length of N/R c bts. The frame s then nterleaved before beng encoded agan by the URC encoder. The frame from outut of the URC encoder s modulated by a modulaton scheme emloyng η bts for resentng a symbol, before transmttng to a receve sde. At the recever sde, the sgnals receved durng a sngle frame are de-maed then decoded by the URC decoder before enterng the teratvely decodng rocess of J teratons occurrng between the URC decoder and the convolutonal decoder. For 8-subcode IrCC, we denote these subcodes wth the tules r, w, w,..., l,,,... where,,...8 or 7, w, j,,, denotes the j frequency of occurrence of g j n the generator matrx, l s the uncturng erod and s the uncturng attern assocated to g j (n octal) as follow []:.,,,,,,,,,,.4,,,,,,,,.5,,,,,,.6,,,, 7,,.7,,,7, 77,5,.8,,,4, 7,,.9,,,9, 777,..,,,,,, 7,7,7,, Performance uer bounds of HAP systems over a twostate swtched channel model In ths secton, a channel codng schemes for HAP are roosed and evaluated. Our channel codng scheme (IrCC- URC-MOD) s SCCC codng scheme conssts of a rregular convolutonal code (IrCC) and a rate- convolutonal code (URC) that joned by an nterleaver of length N bts, the outer code (IrCC) wth rate R c s a rregular convolutonal code contaned eght subcodes wth code rates from. to.9. These subcodes are formed from mother code that s eghtstate (7,5) recursve, systematc convolutonal encode. URC s a one-state (,) recursve, systematc convolutonal encoder (Fg.). Fgure : EXIT chart of 8 subcodes from mother code CC(,5/7). HAP Two-state swtched channel Channel d m x m t Source Modulator Encoder IrCC Encoder URC Modulator ht nt rt d m Demodulator Demodulator URC Channel Decoder User termnal Snk IrCC Decoder Fgure : IrCC-URC-MOD Codng Scheme for HAP system. Fgure 4: EXIT chart for 8-subcode IrCC(7,5)-URC-QPSK at R c=.67. 9

3 For code rate R c=.67, we use 4 subcodes wth code rates.,.6,.7 and.9. Furthermore, we also determne weghtng coeffcents of 8-subcode IrCC(7,5),,..., 8 =[ ]. It means that each subcode encodes a fracton of rn nut nformaton frame to N code bts, where, r are weght and code rate of th subcode resectvely, N s number of nformaton bts,,,..., wth =8. As SCCC are lnear, so unon bound can be used to obtan an analytc exresson for the robablty of error. Assume that nut of SCCC s a sequence of N bts conssted of nformaton bts and K- tal bts. A weght enumerator BD, s defned for the termnated SCCC as, B m, D c. B D () thout loss of generalty, we assume that the all-zero codeword was sent, and we wrte the uer bound on bt error robablty [4] as m m P. c. P X, X () b, m m,two-stateswtched N m where cm, s the number of error events comosed of a sngle error event wth outut Hammng dstance m and nut Hammng dstance and s gven by N o cb, l. cl, d cm, l N () l o where c bl, s the number of error events comosed of a sngle error event wth outut Hammng dstance l and nut Hammng dstance b of outer coder; c ld, s the number of error events comosed of a sngle error event wth outut Hammng dstance d and nut Hammng dstance l of nner coder and Pm,Two-stateswtched, X X s the arwse error robablty of the transmtted sequence X and the estmated sequence X over two-state swtched channel. Note that equaton () stems from the unon bound that s based on the fact that the robablty of the unon of a number of ndvdual events s less than or equal to the sum of robabltes of the ndvdual events. The sums of the ndvdual robabltes of the equatons are not robabltes themselves and can assume values greater than one. The bounds are also based on maxmum-lkelhood decodng, whereas the SCCCs are decoded usng a dfferent, subotmum algorthm. Ths aarent nconsstency can be resolved through heurstc valdaton from a large number of smulatons, whch show the convergence of the smulated erformance toward the analytcal bounds for large random nterleavers [4]. Consder the QPSK sgnal constellaton shown n Fg. 5. Assume that the all zero message s transmtted ( x j ). et X dffer from X by exactly m symbols, whch consst of m symbols of m symbols of S j A weght enumerator. S j and B, D,D s defned for the termnated SCCC usng QPSK modulaton scheme as m m B D c B D D (4),D,. m m, m, m Fgure 5: QPSK sgnal constellaton Then the uer bound on bt error robablty s gven by P. c. P X, X (5) where b, m, m mm,qpsk,two-stateswtched N m m c, m, m s the number of error events comosed of a sngle error event wth outut Hammng dstances m, m and nut Hammng dstance and s gven by c, m, m N l c. c o b, l l, d, d N l o where c bl, s the number of error events comosed of a sngle error event wth outut Hammng dstance l and nut Hammng dstance b of outer coder; c s the number of l, d, d error events comosed of a sngle error event wth outut Hammng dstances d, d and nut Hammng dstance l of nner coder and Pm,QPSK, Two-stateswtched, (6) X X s the arwse error robablty of the transmtted sequence X and the estmated sequence X over two-state swtched channel. eght enumerator of subcodes formed from mother code CC(,5/7) Inut D D D Outut Puncturng Pattern Fgure 6: Dagram of encoder CC(,5/7) wth code rate R c=

4 Consder a convolutonal encoder (,5/7) wth bt memory elements, code rate R c=.67 (Fg.6). In ths case, assume that the rate ½ convolutonal code CC(7,5) s unctured wth a uncturng attern = [ ; ] and uncturng erod l= to obtan rate.67 convolutonal code CC(,5/7). It means that the thrd coded symbols are unctured. To obtan the weght enumerator for comutng uer bounds of bt error robablty, t s necessary to do as follows: () e roduce a modfed trells dagram of convolutonal code wth a certan code rate (e.g. CC(,5/7), R c=.67 n Fg.7b) from trells dagram of mother convolutonal code (e.g CC(,5/7), R c=.5 n Fg.7a) combned wth gven uncturng attern and uncturng erod. Secally, we label the branches between two states wth the owers of B and. The exonent of B denotes the Hammng dstance between the nut sequence, that generated the codeword sequence n queston, and the all-zero nut sequence. The exonent of D s the Hammng dstance between the coded branch outut and an all zero branch outut. Furthermore, for each trells deth t = to (N + K - ) comute B B B F, F,.G, St S S t St-, S t St t where F, St denotes the coeffcents of the olynomal of state S t at deth t; G B S,S, t t s the label of the branch between the state S t at deth t- and the state S t at deth t; and S t s the set of states at K deth t of the trells dagram, St,..., B B B 4 (a) B B (b) Fgure 7: Trells Dagram of CC(,5/7) wth code rate R c=.5 (a) and R c=.67 (b). () e slt the zero state nto two searate states, namely the start state S and the end state S, as dected n Fg.8. e then exress each state of the dagram as a functon of the other states, so as to obtan the state equatons: S S S S S S S S S S S + S S S S S S S S S S S S S 4 S S + S S S S B S + B S B S B S 6 S B S + B S B S B S S 7 S6 S S 4 S S B B S 5 S B B B B B B Fgure 8: The augment state dagram for CC(,5/7) wth R c=.67. () Uon solvng these equatons for the rato S S, we obtan the transfer functon for convolutonal code wth a certan code rate. From ths transfer functon, we determne a weght enumerator of the CC(,5/7) wth code rate R c=.67 as (B,)= B + + +B +B + + B +5B B +B +B + B + B + B + B + B The same rocedure ales to convolutonal codes wth a code rate of R c =.,.6,.7,.9 from mother convolutonal code CC(,5/7) that have uncturng attern and uncturng erod, we also determne the weght enumerator of these codes as B B B B B B B... S 95

5 B B... B, B B B B B B B B B B 7 5 B B B... B, B B B B B B 7B B 6B 6B B B B 4B B 7B 6B B B, B 6B 6B 4B 65B 5B B 85B B 77 B... B 6B B B B B B B B ( B B B B B B 8 6 B B B B )... eght Enumerator for URC usng QPSK modulaton Fg. 9a shows the state dagram for URC wth constrant length K= and the generator olynomals G=[/]. In ths case we consder two dummy varables D and D. The resultng branch metrcs for each branch of URC are shown n Fg. c, where the exonent of B reresents the Hammng dstance of the nut sequence. As shown n Fg.9c, for QPSK modulaton scheme (M=4), the merged nner trells has two states and M/ arallel edges between every dstnct startng and endng state. Accordngly, the weght enumeratng functon of the nner code (URC) can be gven by l m m,d,d = b D D l,m,m l m m where b s the number of error events conssts of m l,m,m symbols of m Hammng dstance and m symbols of m Hammng dstance when the nut Hammng dstance s. Then, the weght enumeratng functon of the nner code URC(/) wll be (7), D, D D ( D D ) D D D D D D D D D D D D D D D 5D 4D D D D D 6D 5D 4D D D D D 5D D 6D D D 5D D D D D D 56D 5D D D 4D D 5D D D 4 D D... D (87D 495D 65D 45D 9D ) D 495D 65D 45D 9D D D (D 75D D D D ) D 75D D 55D D D 5 4 D (D D 86D 66D D ) 5 4 D D 86D 66D D D 5 4 D (468D 65D 64D 78D D ) D 65D 64D 78D D D D (688D 8D 455D 9D D ) D 8D 455D 9D D D D (8568D 8D 56D D 4D ) D 8D 56D D 4 D D / / D (a) / / / / / / (b) D D D D D D D D Fgure 9: Trells mergng for the nner code. The weght enumerator for a SCCC wth code rates R c=.67,.,.6,.7 and.9 emloyng QPSK modulaton s (c) 96

6 B, D, D BD 5D 4D D D D N BD 6D 5D 4D D D N D B B 5B B D 5D D 6D D N D B B 5B B 5D D 6D D D N D B B 56D 5D D D 4D N D B B 5D D D 4 D D... N B, D, D B D (87D 495D 65D 45D 9D ) N B D 495D 65D 45D 9D D N B D (D 75D D 55D D ) N B D 75D D 55D D D N B D (D D 86D 66D D ) N B D D 86D 66D D D N B D (468D 65D 64D 78D D ) N B D 65D 64D 78D D D N B, D, D D B B B 5D 4D D D D N D B B B 6D 5D 4D D D N B, D, D D D D D D D B B B N D D D D D D B B B N D 5D 4D D D D B B B N D D D D D D B B B N B B B B.9 B, D, D D 6D 5D 4D D D 6 N 8 B... 5B 6B B B B B D 5D 4D D D D 6 N 8 B... 5B 6B D D D D D D ( B B B B B ) N D D D D D D ( B B B B B ) N Parwse Error Probablty over a two-state swtched channel A two-state swtched model dscussed n Secton wth the same arameters as noted n [] n relaton to the transton matrx of the Markov rocess wth statonary robabltes π = ( ). The frst state wth statonary robablty.565 consders frequency-flat uncorrelated real Rcan fadng channel wth Rce factor K = db, whle the second state consder frequency-flat uncorrelated real Raylegh fadng condtons. So Parwse Error Probablty (PEP) usng QPSK modulaton over ths two-state swtched channel P, mm,qpsk,two-stateswtched P mm,qpsk,two-stateswtched m m X, X X X can be exressed as,565. Pm m, QPSK, Rcan X, X,45. Pm m, QPSK, Ray X, X For Raylegh Fadng Channel, PEP s exressed as P X, X mm,qpsk, Ray sn sn Es N m sn sn Es N m d For Rcan Fadng Channel, the condtonal arwse error robablty can be wrtten as P X, X a Q mm,qpsk, Rcan Es N m m a l l xl xl a k k xk x k For usng QPSK modulaton, m symbols of S and l l (8) (9) () x x for xk xk 4 for msymbols of S. Substtutng these values n (7) and usng the exact Q functon ntegral, can be wrtten as 97

7 ,, P X X E P X X a mm,qpsk, Rcan mm,qpsk, Rcan m Es N a l l ex... sn a am a am m Es N a k...ex k d sn... d... d... d... d a am a am a a a a m m () substtutng the normalzed Rcan robablty densty functon, we have I a K K K Pm,, a e... m,qpsk Rcan X X K da Es N exa K sn I a K K K...a K e da d Es N exa K sn whch after ntegraton reduces to sn mm,qpsk, Rcan X, X K ex sn sn P where K ; K E N E N s sn K ex sn sn s m m m m d () () So bt error robablty of IrCC-URC-QPSK codng scheme n HAP system over two-state swtched channel model usng 8-subcode IrCC (,5/7) as Pe _ IrCC URC QPSK Pe _. 5 Pe _.6 6 Pe _.7 8 P (4) e_.9 where e _. P, P e _.6, P e _.7, Pe _.9 s calculated n (5) and weghtng coeffcents s determned n Fg.4. Results of erformance analyss over the uer bound wth smulaton results for the HAP system usng QPSK modulaton over a two-state swtched channel model are shown n Fg.. It s shown that the bt error robablty s closer to the smulated BER erformance as SNR ncreases. Furthermore, bt error robablty decreases consderably as frame length ncreases and bt error robablty of system IrCC-URC-QPSK sgnfcantly mroves comared to system CC-URC-QPSK. However, the derved analytc exresson s a weak uer bound on the bt error robablty. Because the decodng algorthm and tye of nterleaver are used n smulaton and analyss s dfferent. Fgure : Uer bound on bt error robablty for code rate-.67 CC(,5/7) and IrCC(,5/7) codes wth QPSK modulaton scheme n two-state swtched channel model of HAP systems. CONCUSIONS In ths aer, we have roosed a modfed trells dagram for unctured convolutonal code to determne the transfer functon. From ths we derved uer bounds for IrCC-URC- QPSK codng scheme for HAP system over two-state swtched channel model n terms of maxmum-lkelhood bt error robablty. The obtaned results, comared to the smulaton results. Ths comarson demonstrates the accuracy of the resented analyss and vce versa. Although, the derved analytc exresson s a weak uer bound on the bt error robablty. But, t stll shows the effects on the code by frame length, and when usng rregular convolutonal code. Therefore, ths method of analyss s stll consdered as a tool to desgn the channel code. ACKNOEDGEMENT The authors would lke to thank to Motorola Solutons Foundaton for suortng ths research. REFERENCES []. A. Boch, M. addomada, M. Mondn, and F. Daneshgaran, Advanced channel codng for ha-based broadband servces [nternetworkng and resource management n satellte systems seres], IEEE Aerosace and Electronc Systems Magazne, vol., no. 9,. C 7, 7. []. F. Dong, Y. He, X. Zhou, Q. Yao, and. u, Otmzaton and desgn of HAPs broadband communcaton networks, n 5 5th Internatonal Conference on Informaton Scence and Technology (ICIST), , IEEE, 5. []. D. Grace and M. Mohorcc, Broadband Communcatons va Hgh Alttude Platforms. John ley & Sons,. [4]. F. A. Olvera, F. C.. d. Melo, and T. C. Devezas, Hgh alttude latforms resent stuaton and technology trends, Journal of Aerosace Technology 98

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