This paper concerns with an optimal pulse shaping filter in time-discrete multipath Rayleigh fading environment. The shift orthogonal pulse shaping filter is proposed and proved to be the optimal envelope in the sense of least average bit error probability. It is suited to the Rayleigh fading channel with an arbitrary strength/delay profile of multiple time-discrete paths for BPSK and QPSK signaling etc. Numerical results show that shift orthogonal waveform based pulse shaping filter has lower bit error rate than general shaping waveforms such as square root raised cosine (SRRC) waveform

井, 研治

徐, 平平

楊, 慰民

樊, 祥寧

畢, 光国

AIT Associated Repository of Academic Resources (Aichi Institute of Technology) / 愛知工業大学学術情報リポジトリ

愛知工業大学研究報告第33号B 平成10年An Analysis of Opti盟 U盟 PulseShaping Filte:r in TiIne-Discrete M ultipath Rayleigh Fadin富Channels. マルチパスをともなう通信路における最適なパルス整形フィノレタ明1軍iminYANG+ Xiangning FAN+ Pingping XU+ Guangguo BI + 楊愚民+、 繋祥寧+、 徐平平+、 畢光国+、K邑njiINOMOTO件井研治++Abstract: This paper concems wi出回 optimalpulse shaping filter in time-discrete multipath Rayleigh fading environment百leshift orthogonal pulse shaping filter is proposed and proved to be the optimal envelope in the sense of least average bit error probability. It is suited to也eRayleigh fading channel WIth an arbiむarys位ength/delayprofile of multiple time圃discretepaths for BPSK加 dQPSK signaling etc. Numerical results show that shift orthogonal waveform based pulse shaping filter has lower bit 間 orrate th田 generalshaping waveforms such as square root raised cosine (SRRC) waveform. Key words: matched白lterbound， equally spaced Rayleigh fading channel model， pulse shaping filter， shift or仕lOgonality1. Introduction In m畠nydata communication systems such as digital mobile radio and terestrial broadcast systems， data signals are always a:fected by multipath fading組 drequire some form of channel compensation for the resulting signal distortionτ'he severe performance degradation e:fects associated with multipath おdingin radio channels are well known. However， the e:fects offading c朋 besubstantially mitigated through similarly旦sm釦 ywell known techniques such :戸P町E吋叩伽吋伊巴∞ωc中国東南大学通イ信言研究室(中国南京市)十+愛知工業大学情報通信工学科(豊田市〉出 diversity，coded modulation， pilot， and equalization， which have been widely studied and employed. Different methods are adopted to compensate 也e fading effects in distinct channels， while now， we deal with the issues 丘omanother viewpoint， in which the pulse shaping filter design2) is concemed once more. Indeed， signal design has an increasing important role to play in wireless communication systems for a host of applications. In也is paper， the time shift orthogonal pulse shaping filter is deduced and proved to be an optimal waveform in the sense 229 230 愛知工業大学研究報告、第33号B、平成10年、 Vol.33-B、日ar. 1998 ofleast av号ragebit e汀orrate in the time噌discreteRayleigh fading ch怨mels目百貨 optimalbaseband wav思formis independent of the path paramet怠rsexcept白紙也君。rthogonalsh出 distanceis decided by the path spacing.百lerefore，也soptimum is a universal principle in time-discrete Rayleigh fading chan問 Is目百leorg加 izationof the paper is as follows. In section 2， we describe time・discreternultipath Rayleigh fading channels and the lower matched fi1ter bound of the average 出回orrate. Section 3 develops the optimal shaping filter and points out that sm食orthogonalwaveform is叩 optimalO問.In Section 4， the nurnerical results of the optimurn envelope conducted over P組副European Digital Cellular (GSM) channels such as吐letypical urban σU) and hilly teπain (HT) models are presented， in cornparison with general square root raised cosine (SRRC) waveforms which is employed in the Nor註1America Digital Cellular 8t旦ndardIS-54. Finally， we present the conc1uding remarks in section 5 2. Channel model and the lower matched filter bOl.md We begin with a time-discrete Rayleigh おdingchannel at b品sebandwhich consists of p equally spaced paths. With a baseband model， it C阻 beexpressed by an impulse response as 1)ユ)h(t) =三α;(t)z; (t)b"(t-'!";) (1) where勾(t)is a slow tim刊訂ymg，zero mean， unit varianceラ complex Gaussian random proce民 τz組 dαi(t)抗出edelay and the root mean square (r.m.s目) value of i-th path a枕enuationcoe笠icientat time t. The delays '!"i ar岳部sum信dto be constants and equally spaced by M for a particular channel， i.e. τi+l -Tj = /)，τ， i=l，・ーラP園LSupposing that BPSK is preferred and the transmit土日dinformation is denoted出勾=:tlラ thebaseband signaling waveform is skg(t)， g(t) is a shaping pulse. In view of the above description， ifthe transmitted pulse passed也roughthe channel (1) corrupted by additive white Gaussian noise， the r巴ceivedsignal at baseband is r(←三α;z;g(t-τ;)Sk +n(t) (2) where n(t) is the zero rnean， complex wh出Gaussian random process with power spectral density No. We also国S山間也atthe slowly varying randorn pro関 s z;(t) does not change Wl"由m吐leduration of the pulse. As we know， the matched filter bound (MFB) is a bound on 也eoptimal perform組 ceov阜ra communication channel that assumes白紙 thetransmitted pulses are su伍cientlyseparated 80 that no intersymbol interference 0∞ursl).τllUs， the MFB is a lower bound， which may not be achiev旦blefor specific implernentations. However， it h且stheoretical and practical importance for measuring the lirniting performance of receivers under considerations n isshown in 1) that the lower MFB of average bit error rate can be obtained as follows: 九四ω=î;~k{1-~Å;4(1十川] (3) where， another variable M has been introduced， M is non-negative definite p * p Hermitial1 rnatrix with elements 内 =αJ4R(τ;-'!"j)，んj=l;..，pラR時刷(←付@サ)ト=F-1戸一→T唱宇ヤ[和ド炉h阿阿|伊阿附附附州G何叩附州州(←付州キ→4イ川t灯12門つ]i臼s 批 a制uω仰tω…Oa白加m削lC叩tio叩工noぱfg訂(t)ト， G何(伊例ω吋)i白s出eFouri問巴町r位加sお如rm 1ぱ g(蛇州か付t).We i叩紅m坤lp刷判凶lici此均t開 V刊阜l叩e g(t) is real valued ， hence R(一τ)= R(サ阻d師同副lyM is a symmetric Hermitian mairix with al elements not less than 田町 i.e叫ミ0λ;(λI判訂E也eeigenvalu巴Sof M and they are assumed to contain only 1 P singular ones (no multiple).ρ=-;:-;:-;-Lλ; c組2No ;:ibe proved to be just the averaged received signal ωnOlS巴r抗io.it is assumed to be constant for a glV叩modelfor the convenience of ev昌l四 tion丸一品 λI ‘ -a7A4λ-λ )=1，)芋 jλI訂enormalized eigenvalues， i.e 一 I P A ム=λJすλ=一二Lで'/ )::1 j TrlMj In the following， we will rninimize the lower MFB (3) versus eigenvalues of matrix M. Since αi depends on the channel， and we want to derive組 optimumwaveform for the R且yleighfading channel which consists of multiple time嗣discrete paths with an arbitrary strengthJd巴laypro自lefor BPSK signaling， we have to optimize Optimum pulse shaping filter in time-discrete multipath Rayleigh fading channels 231 th潟己 ob対j邑cti討vefi.血mcω:tiぬonwith r跨号s叩pe即ctto noth悩ingb叫 g(的t).By s叩omem悶a創組捌n悶11やp抑u叫da鉱抑t討iぬon眠 we know 也拭p L，ki = l( by integrating equ (11) in 1) ) and p_ Zλi = 1 . Hence Pave in (3) can be simplified部以外=~-îiÉki[ ~-liP/(lゐjJ(4) 3. Optimum signaling enveiope in multipath Rayleigh f:畠dingch.畠阻neisMinimizing }い(月 in(4) is equiv且l側 tomaximizing the second item 『? ? ?」? ?????????Sinc昌江canbe deduced that p ，-I 11 一一、 l P ，_ 11 一一寸:Lkil，{λip/!l+λip!1 and :Lkdλ，.p/!I+λ，.pll !. A; ( whぽe k; = 百_'"( _ ) are of the same 1λ ーλJj=l，j*; "'; -"'j gradie事1tvariation trend to individual λ; in our situations ( i.e.，社leyincrease or decrease at the same time to each pointλ; ). The goal now can be changed into maximizing the function ル(刈=iK11zp/(I+む)]， while ル (M)=ikt[丈作?、??????? ? ? ?? ??????? ? ??????????????? ??????? ?????????」??????????????wh阜reQ(M) =白(Aj+〆1)_ p-P =λ1λ2Aλp +〆1(A)2A"Ip-l +AJI324A 2p+A+2223A 2p) 十戸(A)2A2p-2 +A12425A 2p叫 +2324A2p) +A+ρ (ーP-I)(AI + 2十八+2p)Note that 0::;λ; :;1，γi ， which implies that 一一一一 一一一一一- p一λIλJミんλj九品注目λ1，1=1 we can obtain that Q(M) ミ(1+ρ勺 +ρ吋小ρ(P-2)C;-2) (6) 自工+p-(p-J) ;=1 80 (6) can be expressed in term of matrix determinant.τnat is Q(M)2g-. '竺-'+ρヤ 1)(7) [Tr(MW where， g=1+p-1cb +ρ-2cj+A+ρ一(p-2)cF-2〉 OAs a result we c加 obtaIn也拭 Q(M)increa悶w油 theincreme凶 of!M!3)From the matrix theory 3)， we know that the determinant of a non-negative definite matrix with order p arrives maximum when the matrix is diagonal ( IMI::; lII1lj ，叩th叩 aliti=1 if M is diagonal， i.e. IMI =かl;i<=>叫=0， t7附， 1n仙 rw叫 R(ri -r j ) = 0 whenever的It means that signals企omdifferent paths are uncorrelative， and the epoch difference to distinct path equals integer multiple of the path spacing M.官1Isis why we call them shi武orthogonality. It is obvious that， without loss of general辻y，the diagonal matrix ( IMI=品目i=自λ;) 232 愛知工業大学研究報告、第33号B、平成10年、 Vol.33-B、Mar. 1998 satisfies inequality (7) as weH. Fu巾 ermore，the diagonal matrix corr自spondsto the maximized lower bo凹 d.祖国， the maximization of lower bound of (6)叩 d(5) has been obtained. Under these conditions， the maximizing bound maximizes the objective itself. The fおollowingis tωop戸r悶剛ov刊巴 t也h拭由em鵬a凱悶X幻i加町m悶 dv四alu田le0ぱfQ到(M)in (7) Is be ofuniqueness 3). [Gershgo出1Theorem] Let丹="[，j，j#1匂，then its eigenv畠luelies within a circle of radius ηcentered at 11; . If some circles overlap，社lenthe overlapping circles contain just as many eigenvalues， counted according to 也eumultiplicity τne circles condense to points if M is diagonal. So the maximum lower bound corresponds only one set of eigenvalues (There does not exit四yother. eigenvalues which c拍make (7) maximum).百lesolution brings about maximized P~珂 (M) 1 and minimIzation of ι.e(P) as well. It Is evident 白紙theevaluation of optimal baseband waveform corresponding to least BER does not depend on the chaImel coefficientsαi (t) but determined only by shaping pulse g(t). It is suitabJe for any eq叫 Iyspac巴dtime-discrete Rayleigh fading channel with arbitrary strength /delay profile. As a result， the 1壬ffiOREMon shift orthogonality can be stated asヲ[ THEOREM ] In the sense of 1開 staverage bit error probabilityョtheshi食orthogon晶lshaping :filter is阻 optimalbaseband waveform over any time-discrete equally spaced multipath Rayleigh fading channel with arbitrary strength/delay pro:file. Viz， the baseband waveform is required to be orthogonal to its own shifts by integer multiple of也epath sp品cmgIlr. 圏4. Numerical results ov母rGSM ch晶rmelsWe consider the GSM TU and HT channel models for both IS・54signaling and GSM signaling‘ Although TU叩dHT models have several variations in the practice， we choose at will a simpler 6・pathform model as our examples 1) ，6).官leamplitude and delay profiles ofthe channel models are shown in table 1. We 出 sumethat也etransmitter shaping filter has a square root raised cosine (SRRC)企equencyresponse with excess b叩 dwid出 α=0.35 in the calculation of the average bit e汀orprobability in GSM signaling拍 dIS・54signaling. This kind of SRRC waveform is specified by 1S-54.百leperformance of也eshi食 orthogonalenvelope transmitted at the s創nerat岳部 IS偏54and GSM are also computed over也etwo GSM channels， i.n comparison with the aforementioned SRRC shaped IS-54田 dGSM signaling. For IS-54 sign昌ling，although n-/4QPSK is adopted， we assume that the symbol is transrnitted by BPSK for the sake of comparison with other situations. So the bit duration of IS-54 is about 41.152μs. By Jacobi methods1 we can get也at也enormalized eigenvalues for the TU 拍車nneland the HT channel shown as Table2 Table 1. Delay and Amplitude Profile ofGSM Channels (profile sampling duration is O. 813us) 1Umodel HTmodel Path詳 Delay(us) Power Delay (us) Power 0.000 1.000 0.000 1.000 2 0.813 0.669 0.813 0.251 3 1.626 0.448 1.626 0，060 4 2.439 0.300 15.447 0.258 5 3.252 0.200 16.260 0.177 6 4.056 0.134 17.073 0.122 Table 2.官lenormalized eigenvalues in the TU channel and the HT channel for IS圃54signaling TU channel for IS・54 HT channel for IS国540.99839544 0.96204072 0.00000076 0.00000807 0.00000020 0.00000164 0.00160289 0.03794055 0.00000029 0.00000317 0.00000039 0.00000581 The symbol interval for GSM signaling is 3.692μS.百lenormalized eig巴nvaluesfor GSM TU channel and for也eHT shown as Table 3. We c皿 alsoattaIn the results for shi武orthogonal waveform. are eigenvalues for TU channel阻 d也eHT chmmel shown as Table 4. Table 3.百lenormalized eigenvalues in the TU chmmel皿 dthe HT channel for GSM signaling Optimum pulse shaping filter in tim告-mscretemultipath Rayleigh famng channels 233 TU channel for GS恥f HT channel for GSM 0.86080128 0.894089228 0.00094519 0.01018010 0.12472734 0.00010650 0.00028550 0.08765873 0.01306165 0.00019492 0.00017903 0.00777048 By substituting above eigenvalues into (3) respectively， two curves of average bit error probability versus signal to noise ratio (SNR) over different channels are given in Figure 1 and Figure 2. Fi忠ue 1 shows the performance of waveform shaped by Shift orthogonal and SRRC envelopes over GSM TU channels without any forward correction coding. The comparison of the也reecurves clearly depicts也at也eaverage bit eロorrate of shi食 orthogonalwaveform is les也阻thegeneral SRRC ones. The SNR gain is evident especially when SNR is greater than 5dB.百leperformance∞mparisons in GSM HT channel also illustrate也eadvantages of the proposed sh泊 orthogonalwaveform， just similar to Figure 1. Figure 1. Bit e汀orprobability curves for shi食 orthogonalenvelope， in comparison with two SRRC waveforms， corresponding to symbol duration of IS・54and GSM over GSM TU channel Table 4. The shi食 orthogonal wavefoml eigenvalues for TU channel and For the HT channel TU channel for shift HT channel for shi食orthogonal w品veform orthogonal waveform 0.55672437 0.84790456 0.24916813 0.05341883 0.11173681 0.00305246 0.05010520 0.02656400 0.02226898 0.02656400 0.02226898 0.01262021. _____j 1.0E+0 Numerieal Results (TU): 1.0E四1 : Shi武 Orthogonal:IS54 (SRRC) : GSM (SRRC) ????』?? ?1.0E-2 1.(TE-3 ?????』???1.0E-4 1.0E国51.0E-6 1.0E-7 5 10 15 Average SNR (dB) 。 20 234 愛知工業大学研究報告、第33号B、平成10年、 VoL33-B、Mar. 1998 Figure 1. Bit号XTorprobabiiity curves for shi食orthogonalenvelope， incomparison with two SRRC waveforms， corresponding to symbol duration of IS・54組 dGSMov釘 GSMTU channel. 1.0E-l ?? ? ?1.0E<Z 』。恒国.<;: 1.0E-3 悶ω on 国~ 1.0E-4 :> 〈1.0E-5 1.0E匂6Numerncal R記，suits(HT): : 1S54 (SRRC) : GSM (SRRC) :5 10 15 Average SNR (dB) Figure 2. Bit error probability cun明 forshift orthogonal envelope， incomparison with 1:¥市oSRRC waveforms， corresponding to symbol duration of IS置54and GSM over GSM HT channel 。5. Conduding R記marksThe optimal baseband signaling envelope based on shift orthogonal envelope for multipath Rayleigh fading channels has been presented in the paper.百1enumerical results show that the perform組問 isb甜 erthan general shaping waveforms such as SRRC w，且veforms，especially in the circumstance when the SNR islarger th叩5dB. Since we have the liberty to select a channel， and tms avoids也erestricting the generality目百1eresults cal1 also be generalized to the cases when the matrix M has multiple roots. Wavelet packet functions 4) are a kind of shift orthogonal functions. It does meet the optimal derτmnds. Furthermore，也eyare also a special type of Nyquist pulses， 80 they comply with Nyq四 tfirst criterion. However， the major disadvantages of the wavelet functions are their non-constant envelope wmch would impact the power consumptlOn 20 Refl芭re回ces1). Fuyun Ling， 'Matched filter-bound for time-discrete multipath Rayleigh fading chrumels，' IEEE Tr田 s. Commun.， vo1.43， No.2/3/4， pp.710-713， Feb.lMar.IApr.， 1995. 2). J. G. Proakis， Digital Communications， McGraw-Hill， New York， 1983. 3). K.E. Atkinson， An Introduction to Numerical Analysis， John Wiley & Sons， New York， 1989 4)目 1. Daubechiesヲ 'Orthonormal bases of compactly supported wavelets，' C.P.A.M， vol. XLI， pp.909時996，1988. 5). N町1. Fliege， Multirate Digital Signal Processing: Multirate Systems . Filter Banks . Wavelets， John Wiley & Sons， New York， NY，1994. 6). CEPT/CCHlGSM Recommendation 05.05， Transmission and Reception， Aprilラ 1988(受理平成10年3月20日〉

An Analysis of Optimum Pulse Shaping Filter in Time-Discrete Multipath Rayleigh Fading Channels

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An Analysis of Optimum Pulse Shaping Filter in Time-Discrete Multipath Rayleigh Fading Channels

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