A Study on Performance Analysis for Error Probability in SWSK Systems

Transcription

1 556 Tae-Il Jeog, et al.: A STUDY ON PERFORMANCE ANALYSIS FOR ERROR PROBABILITY IN SWSK SYSTEMS A Study o Performace Aalysis for Error Probability i SWSK Systems Tae-Il Jeog, wag-seo Moo, ad Jog-Nam Kim, Member, KIMICS Abstract This paper presets a ew method for shift eyig usig the combiatio of scalig fuctio ad wavelet amed scalig wavelet shift eyig (SWSK). A algorithm for SWSK modulatio is carried out where the scalig fuctio ad the wavelet are ecoded to ad i accordace with the biary iput, respectively. Sigal eergy, correlatio coefficiet ad error probability of SWSK are derived from error probability of frequecy shift eyig(fsk). The performace is aalyzed i terms of error probability ad it is simulated i accordace with the id of the wavelet. Based o the results, we ca coclude that the proposed scheme is superior to the performace of the covetioal schemes. Idex Terms Correlatio Coefficiets, Frequecy Shift Keyig(FSK), Error Probability, Scalig Wavelet Shift Keyig(SWSK), Wavelet Trasforms. I. INTRODUCTION THE wavelet trasform is maily used i the field of audio ad image-sigal processig. It is recetly used i digital commuicatios. A promisig applicatio of wavelet trasforms is i the field of digital wireless commuicatios where they ca be used to geerate waveforms that are suitable for trasmissio over wireless chaels. This type of modulatio is ow as wavelet trasform. The advatage of this scheme emerges from its diversity strategy; wavelet modulatio allows trasmissio of the data sigal at multiple rates simultaeously []. Worell ad Oppeheim outlied the desig of the trasmitter ad receiver for wavelet modulatio []. The performace of wavelet modulatio i a additive white Gaussia oise (AWGN) chael was also evaluated i Worell s wor. Ptasisi ad Fellma simulated wavelet modulatio usig the Daubechies wavelet ad measured the performace of the bit error rate (BER) usig a AWGN chael. Ptasisi s wor showed the performace of wavelet modulatio to be equivalet to that calculated by Worell[3]. Mauscript received July, ; revised August, ; accepted August,. Tae-Il Jeog ad Kwag-Seo Moo are with the Departmet of Electroics Egieerig, Puyog Natioal Uiversity, Pusa, , Korea ( tijeog@daum. et ad smoo@pu.ac.r ) Jog-Nam Kim(Correspodig Author) is with the Departmet of IT Covergece & Applicatio Egieerig, Puyog Natioal Uiversity, Pusa, , Korea ( jogam@pu.ac.r) The simplest wavelet is Haar [4][5] ad the commuicatio sigals cosidered are amplitude shift eyig (ASK), frequecy shift eyig(fsk) ad phases shift eyig(psk), which are trasmitted over a AWGN chael. Amplitude shift eyig, frequecy shift eyig ad phase shift eyig are the covetioal digital commuicatio scheme. I frequecy shift eyig, the frequecy of the carrier varies i accordace with the biary iput. For biary trasmissio the carrier assumes oe frequecy for ad aother frequecy for. This type of o-off modulatio is called frequecy shift eyig. Recetly some attempts have bee made to apply the wavelet trasform i digital commuicatio [6][7]. Referece [8] provides the modulatio scheme which uses the mother wavelet istead of the carrier frequecy such as phase shift eyig. That is, the phase of the mother wavelet is varied accordig to the source sigal. The modulatio schemes which use the scalig fuctio ad wavelet, ad the demodulatio schemes which use the biary matched filter are itroduced i [9][]. I this paper, we propose a ew shift eyig usig the scalig fuctio ad the wavelet. The three parameters: sigal eergy, correlatio coefficiet ad error probability are also itroduced, ad are derived from the formulas of the frequecy shift eyig. The performace of SWSK is aalyzed i terms of the error probability, which is obtaied from the average eergy ad correlatio coefficiet. This paper is orgaized as follows: Sectio Ⅱ reviews the mathematical prelimiaries. A algorithm for modulator ad demodulator for SWSK is preseted i Sectio Ⅲ. Simulatio results ad performace aalysis are preseted i Sectio Ⅳ. Fially, coclusios are draw i Sectio Ⅴ. II. MATHEMATICAL PRELIMINARIES A. Error probability of frequecy shift eyig I frequecy modulatio, the frequecy of the carrier varies i accordace with the biary iput. For biary trasmissio the carrier assumes oe frequecy to be ad aother frequecy to be. This type of o-off modulatio is called frequecy shift eyig. The frequecy shift eyig sigals are defied by [][]

2 INTERNATIONAL JOURNAL OF KIMICS, VOL. 9, NO. 5, OCTOBER 557 S FSK s( = Acosω(, t Tb, for ( = s( = Acosω(, t Tb, for where, s ( ad s ( correspod to the biary symbols ad, respectively. The average eergy(e) ad the correlatio coefficiet( ρ ) per bit for coheret matched filter is give as, respectively E = + [ s ( s ( t ] ) [ s ( s( ] dt dt () () ρ = (3) E The probability of error yields E( ρ) = A T Pe = erfc erfc b (4) η 4η where, erfc is complemetary error fuctio, ad deotes the oise power i watt/hz. A is the amplitude of carrier sigals. It is assumed that the oise is additive white Gaussia oise with a two-sided power spectral desity of η /, mea zero ad variace σ = η /. For the coheret detectio, matched filter detectio is optimum as illustrated i [][]. B. Discrete Wavelet Trasforms The discrete wavelet trasform algorithms mae use of ba of quadrature filters. The filter coefficiets are based o the mother wavelet, ψ(, ad the scalig fuctio, φ( derived from the mother wavelet. The mother wavelet ad scalig fuctio are used to implemet high pass filters ad low pass filters respectively. The procedure for decompositio of a sigal usig discrete wavelet trasform at first, ivolves the covolutio of the sigal with a pair of quadrature filters to obtai the approximatio coefficiet from the low pass filter ad to obtai the detailed coefficiets from the the high pass filter. The outputed data sequeces are the decimated by a factor of two. To compute the ext level of resolutio, the decimated sequece of detailed coefficiets are iputted to the ext set of quadrature filters. I this fashio the wavelet coefficiets at the th level of decomposio ca be obtaied[4][5]. Let the sigal processig fuctios of the high pass filters ad the low pass filters be deoted by the operators H ad G. Now let {h()} be a square-summable sequece of coefficiets which defies the liear operator H, ad similarly {g()} for G. The relatioship betwee the mother wavelet, the scalig fuctio ad the filter coefficiets are defied as [7] Ψ( = g() ϕ(t ) ϕ( = h() ϕ(t ) This methodology for implemetig the discrete wavelet trasform is ow as multiresolutio aalysis. The filter coefficiets have to satisfy the coditios as follows[4][5]: h [ ] = (5a) h [ ] = (5b) g [ ] = (6a) g [ ] = (6b) where h [] is the scalig fuctio ad correspods to the coefficiet of the low pass filter, g [] is the wavelet ad correspods to the coefficiet of the high pass filter. We will ow discuss a mothod for costructio of scalig fuctio usig successive approximatio. i-th iteratio h [ ] h [ ] h [ ] h ( i) [ ] Fig.. Discrete impulse respose for scalig fuctios. The discrete impulse respose h ( i) i [ ] = * h [ ] (7) = ca be obtaied i the i-th iteratio step by i-fold covolutio of the dyadic upsampled ipulse respose h[ m] if = m h [ ] = otherwise ( ) The impulse respose h i i+ [ ] has ( N )( ) + coefficiets, where N is the umber of the coefficiet of h []. If the umber of iteratio is icresed i = the the discrete impulse respose is cosiderd to be a cotiuous sigal. Hece equ.(7) ca be writte as[4][5] Similarly h g ( ) ( ) [ ] = = [ ] = = * h [ ] ϕ( * g [ ] Ψ( where, * meas a covolutio. I practical applicatios, few iteratio steps are eough to obtai the scalig fuctio i Fig.. (8) (9)

3 558 Tae-Il Jeog, et al.: A STUDY ON PERFORMANCE ANALYSIS FOR ERROR PROBABILITY IN SWSK SYSTEMS C. Cross-correlatio Receiver The optimum receiver for a ow sigal i a addititive white gaussia oise is the correlator or matched filter. The correlator performs a cross-correlatio of the received sigal r(, with each of the prototype of the trasmitted sigal s m (, o the iterval t, producig m outputs which are the compared by the detector. The decisio circuit determies the largest magitude output from the output of the sampler ad declares it as the trasmitted symbol [4]. Ⅲ. SCALING WAVELET SHIFT KEYING A. SWSK modulatio The covetioal modulatio schemes for frequecy shift eyig require two carrier frequecies. The high frequecy is ecoded to ad the low frequecy is ecoded to for a iput biary data. SWSK system requires a scalig fuctio ad wavelet istead of usig two carrier frequecies i frequecy shift eyig. The scalig fuctio is ecoded to ad the wavelet is ecoded to for iput biary data. It is importat that scalig fuctio ad wavelet satisfy equ.(8) ad (9), if the umber of iteratio of scalig fuctio ad wavelet are more tha that obtaied from the smoothig waveforms. The shift eyig system usig the scalig fuctio ad the wavelet is called scalig wavelet shift eyig(swsk). We are defied as S SWSK s( = ϕ(, ( = s ( = ψ (, for biary for biary () For biary trasmissio the carrier assumes scalig fuctio for ad wavelet for as represeted i equ.(). Trasmitter ad receiver of SWSK systems are preseted i Fig.. If the biary data is applied to SWSK modulator, it is ecoded to the scalig fuctio ad wavelet by SWSK modulator. The modulated sigal is added to the additive white Gaussia oise before trasmissio by the trasmitter. represeted i Fig. 3 which is similar to frequecy shift eyig. If the biary iput is the the modulator is ecoded to scalig fuctio, ad if the biary iput is the ecoded to the wavelet. ψ ( φ( Fig. 3. SWSK system modulator. S SWSK ( B. SWSK demodulatio I SWSK demodulatio, we fid the sum of the correlatio for the iterval t, later we compare it with the give threshold. If the sum is larger tha the give threshold, the result would be by the comparator, ad if it is smaller, it would be [9]. The algorithm for SWSK demodulatio is described as follows: ) Receive the SWSK sigal which is icluded i additive white Gaussia oise. ) Compute simultaeously the correlatio by each correlator. 3) Obtai the matched filter outputs. 4) Recostruct the biary output by the comparator (decisio circui. 5) Iterate step ) 4) util the last sigal is achieved. C. Calculatio of the sigal eergy, correlatio coefficiet ad probability of error I this sectio, we derive the sigal eergy, the correlatio coefficiets ad the probability of error for SWSK. To determie the probability of error for the matched filter, we eed the sigal eergy ad the correlatio coefficiets of the sigal to be calculated i SWSK schemes. The average eergy(e) ad the correlatio coefficiets( ρ ) per bit for SWSK system is derived as[] E = ψ + dt () [ ( ϕ ( ] [ ϕ( ψ ( t ] ) dt ρ = E () From equ.(5a) ~ (6b), (8), (9) ad Fig., we ca defie the scalig fuctio ad wavelet as follows: Fig.. Bloc diagram of SWSK systems. The SWSK receiver cosists of a matched filter ad a comparator (decisio circui. Coheret detectio of SWSK is accomplished by comparig the outputs of two matched filters. The modulatio algorithm for SWSK is Tb Tb ϕ ( = (3a) ϕ ( = (3b) Tb Ψ( t ) = (4a)

4 INTERNATIONAL JOURNAL OF KIMICS, VOL. 9, NO. 5, OCTOBER 559 Tb Ψ ( t ) = (4b) By substitutig equ.(3a)~(4b) i equ.() ad () respectively, we get the average eergy ad the correlatio coefficiets per bit for SWSK system. Therefore, the probability of error (Pe) for SWSK ca be derived as[] = E( ρ) T Pe erfc = erfc b (5) η η (c) Ⅳ. SIMULATION AND PERFORMANCE ANALYSIS Computer simulatios were performed to examie the probability of error for the SWSK system i the additive white Gaussia oise chael. The simulatio was maily performed by Daubechies5 wavelet. The results were obtaied usig the MATLAB software. Equatios of average eergy ad correlatio coefficiet are used i equ.() ad () i order to obtai the probability of error, which is used i equ.(5). The SNR is betwee to db, ad we set the give threshold to. We cosidered the followig three modulatio schemes: ) ASK, A=, ρ = ; ) FSK, A=, ρ = ; 3) PSK, A=, ρ = -; where, A is the amplitude of sigal Fig. 4 shows the differet ids of wavelet used i this paper. The umber of tap is. If the umber of iteratio is icresed the each figures are cosidered to be a cotiuous sigal which satisfied equ.(3a)-(4b), (d) Fig. 4. Examples of differet wavelet ad scalig fuctio: (a)biorthogoal, (b) Symlet, (c) Coiflet3, (d) Daubechies. The experimetal results are give i TablesⅠ,Ⅱ ad Fig. 5~8. Fig. 5 shows the output waveform for the bit stream at SNR=6[dB] by SWSK scheme. The output waveforms are related as show i Fig.. The waveform of the origial data is observed i Fig. 5 (upper). It is the passed through the matched filter ad received by the SWSK s receiver as show i Fig. 5 (middle). Fially, the waveform data is recostructed as depicted i Fig. 5 (bottom). We cofirm that the iputted trasmitted sigal is perfectly recostructed by the SWSK receiver. (a) (b) Fig. 5. The output waveform for bit stream.

5 56 Tae-Il Jeog, et al.: A STUDY ON PERFORMANCE ANALYSIS FOR ERROR PROBABILITY IN SWSK SYSTEMS Table Ι presets the performace of error probability for SWSK agaist the covetioal commuicatio schemes with Tb=.sec. It is observed that the performace of SWSK scheme is superior to the covetioal schemes as bit error rate (BER) is.6 at SNR of 6dB. The error probability of phase shift eyig ad amplitude shift eyig used equ.(6) ad (7), respectively [][]. covetioal schemes, if the duratio of the period is loger, the performace of SWSK is improved. For example, at Tb=sec, the BER of SWSK usig coheret matched filter is.4. The BER of ASK, FSK, PSK 3 are also.9,.4 ad.5, respectively. A T Pe = erfc b η (6) A T Pe = erfc b 8η (7) TABLE Ι THE COMPARISON OF BER PERFORMANCE AGAINST SNR SNR (db) SWSK (ρ=) ASK (ρ=) FSK (ρ=) PSK (ρ=-) Fig. 6 shows the BER agaist SNR for SWSK scheme at Tb=.sec. The performaces of covetioal schemes are also icluded i the figure for compariso. It is observed from the figure that the performace of SWSK is better tha the covetioal schemes. For example, at SNR of 6dB, the BER of SWSK usig coheret matched filter is.3, where the BER of ASK, FSK, PSK are.7,.7 ad.9, respectively. Fig. 6. BER versus SNR for differet modulatio schemes. Fig. 7 shows the performace compariso for error probability at SNR of 6dB. Comparig with the Fig. 7. BER versus period for differet modulatio schemes. If amplitude (A) i equ.(6) is the equ.(5) ad (6) are equal. But the simulatio experimet is differet. Although the formula of error probability for SWSK is similar to phase shift eyig, the error probabilities are differet, ad it is maily due to the oise power. Table Ⅱ shows the oise power agaist SNR for SWSK ad phase shift eyig schemes. It is oticed that the oise power of SWSK is about times smaller whe compared with the phase shift eyig. Note that because the scalig fuctio ad wavelet are satisfied from (3a) to (4b), the oise power for SWSK sigal is robust compared with phase shift eyig scheme. TABLE Ⅱ THE COMPARISON OF NOISE POWER SNR(dB) SWSK PSK The compariso of BER performace for differet wavelets for =.sec is show i Fig. 8. I this paper, the Daubechies, Coiflet3, Symlet5, ad Biorthogoal wavelets are used. The results of Fig. 8 show experimetally that the Coiflet3 wavelet was superior to the other wavelets. The umber of filter taps for the Coiflet3 wavelet is 8, Daubechies ad Biorthogoal3_ wavelets have, ad the Daubechies5, Biorthogoal_5 ad Symlet5 wavelets have taps. Icreasig the umber of filter taps causes the differeces i error probability, which also depeds o the wavelet characteristics.

6 INTERNATIONAL JOURNAL OF KIMICS, VOL. 9, NO. 5, OCTOBER 56 [9] H. J. Oh, T. I. Jeog ad T. O. Lee, Wavelet Shift Keyig System Usig a Biary Matchig Filter, KIMICS, vol., o., pp , Nov. 8. [] T. I. Jeog ad E.J. Kim, Performace Aalysis for Wavelet i the Wavelet Shift Keyig Systems, KIMICS, vol. 3, o. 8, pp , Aug. 9. [] D. R. Smith, Digital Trasmissio System, d ed. Va Nostrad Reihold: New Yor, 993. [] S. Hayi, Commuicatio System, 4th ed. Joh Wiley ad Sos: New Yor, NY,. Fig. 8. The compariso of BER performace for wavelets, for Tb=.sec. Tae-Il Jeog received the B.S., MS., ad Ph.D degrees i departmet of electroics egieerig from Puyog Natioal Uiversity, Busa, Korea, i 994, 997, ad, respectively. His research iterests iclude image processig, multimedia commuicatio ad wavelet trasform. V. CONCLUSIONS The covetioal commuicatio schemes are preseted as ASK, FSK ad PSK. I this paper, we have theoretically ad experimetally ivestigated the error probability for SWSK scheme which is a ew shift eyig usig scalig fuctio ad wavelet. The formula of error probability for SWSK was derived from the frequecy shift eyig. The SWSK performace was compared with the covetioal commuicatio scheme i terms of error probability. The simulatio results show that the performace of BER is give as.3 at =.sec for SWSK scheme. Based o the results achieved we coclude that the error probability of SWSK scheme is superior to the covetioal schemes. Regardig Fig. 7, more theoretical cofirmatio will be added i future papers. REFERENCES Kwag-Seo Moo received the B.S., MS., ad Ph.D degrees i departmet of electroics egieerig from Kyugpoo Natioal Uiversity, Daegu, Korea, i 979, 98 ad 989 respectively. He is a professor at the departmet of electroics egieerig, Puyog Natioal Uiversity. His research iterests iclude image/video processig, multimedia commuicatio, wavelet trasform ad digital commuicatio. Jog-Nam Kim received the MS ad a Ph.D degrees from Gwagju Istitute of Sciece ad Techology(GIST), Korea, i 997 ad, respectively. He wored at the Techical Research Istitute(TRI) of Korea Broadcastig System(KBS) from to 3. Sice 4, he has bee a associate professor i the departmet of IT covergece & applicatio egieerig at Puyog Natioal Uiversity. His research iterests iclude patter recogitio, image/video processig, image/video compressio, image/video watermarig ad VLSI desig for real-time video applicatio. [] M. J. Maglai, Wavelet Modulatio i Gaussia ad Rayleigh Fadig Chaels, Thesis, Virgiia Tech. July,. [] G. W. Worell ad A. V. Oppeheim, Wavelet-based represetatio for a class of self-similar sigal with applicatio to fractal modulatio, IEEE Tras. Iformatio Theory, vol. 38(), pp , 99. [3] H.S. Ptasisi ad R. D. Fellma, Implemetatio ad simulatio of a fractal modulatio commuicatio system, Proc. IEEE SUPERCOMM/Iteratioal Commuicatios Coferece, vol. 3, pp , 994. [4] M. Vetterli, Wavelet ad Subbad Codig, Pretice Hall PTR, 995. [5] N. J. Fliege, Multirate Digital Sigal Processig, JOHN WILEY & SONS, 994. [6] R. J. Barsati, T. Smith, R. Lee, Performace of a Wavelet-Based Receiver for BPSK ad QPSK Sigals i Additive White Gaussia Noise Chaels, Proc. 39 th Southeaster Symp. Syst.Theory, Maco, Ga March 4-6, pp. 7-73, 7. [7] K. M. Ho, C. Vaz, D. G. Daut, Improved Demodulatio of Phase Shift Keyed Sigals Usig Wavelet Threshold, Saroff Symp. 8 IEEE, pp. 8-3, Apr. 8. [8] J. Olive, S. S. Kumari, V. Sadasivam, Wavelet for Improvig Spectral Efficiecy i a Digital Commuicatio System, ICCIMA'5, IEEE, 5.