GENERALIZED WAVELET-BASED SYMBOL RATE ESTIMATION FOR LINEAR SINGLE- CARRIER MODULATION IN BLIND ENVIRONMENT

GENERALIZED WAVELET-BASED SYMBOL RATE ESTIMATION FOR LINEAR SINGLE- CARRIER MODULATION IN BLIND ENVIRONMENT Rima Hatoum LIP6/UPMC Uiversity of Paris VI; 4 Place Jussieu, Paris, Frace Alaa Ghaith EDST/Lebaese Uiversity; Beirut, Lebao Guy Pujolle, PhD LIP6/UPMC Uiversity of Paris VI; 4 Place Jussieu, Paris, Frace Abstract I o-cooperative commuicatio, sigal parameters are ukow at the receiver frot-ed. This eviromet is cosidered as blid. Hece, estimatio algorithms become very importat for efficiet ad automatic real-time services. Ideed, kowig the fudametal sigal parameters is ecessary for data recoverig correctess ad demodulatio reliability. I this paper, we address a symbol period estimatio approach geeralized for all liear modulatio schemes. This approach is based o the wavelet trasform due to its high capacity to detect discotiuity structures ad zoom o the sigal abrupt chages. A pre-estimatio of the carrier frequecy is itroduced, allowig the operatig i the totally blid eviromet. From simulatios we coclude that this approach presets high estimatio accuracy ad high efficiecy for very low SNR levels ad it is applied for all types of liear modulatios i practical Rayleigh chael case. Keywords: Symbol period estimatio, liear modulatio, wavelet trasform, discotiuities detectio 1. Itroductio Recet wireless commuicatio systems provide real-time applicatios requirig a high flexibility ad accuracy techiques. Itelliget receiver is the basic compoet achievig these issues. The modulatio type is the sigature that characterizes received sigals (Trees, 1. It is classified to aalog ad digital modulatios. Modulatio order (biary or M-ary order is a importat factor that 49

proportioally affects trasmissio data rate. Moder commuicatio systems widely itroduce digital modulatio schemes that are divided to liear modulatio schemes, Amplitude Shift Keyig (ASK, Phase Shift Keyig (PSK ad Quadrature Amplitude Modulatio (QAM; ad to oliear modulatio schemes, Frequecy Shift Keyig (FSK. I blid eviromet, where o iformatio exchage exists betwee the trasmitter ad the receiver, itelliget receiver is based o the sigal processig algorithms i order to estimate ukow sigal parameters such as carrier frequecy, symbol period, ad sigal badwidth ad so o (Dayoub, Okassa-Mfoubat, Mvoe & Rouvae, 7; Yu & Bai, 1. I order to correctly recover trasmitted data ad demodulate received sigal, receiver requires a accurate ad efficiet estimatio of the symbol period. Relevat researches have paid their attetio to the symbol period estimatio issue. Symbol period ca be estimated by filterig process whe the received sigal pass through a filter bak ad the through a oliear uit (Yu, Shi & Su, 5. This approach treats a basebad sigal assumig kowig carrier frequecy i priori ad suffers from high complexity. Origially, a mathematical tool is widely used to image processig ad compressio, it is the Wavelet Trasform (WT. With the itroductio of this trasformatio i the telecommuicatio fields, some authors beefit from this tool for the estimatio objectives. I (Cha, Piews & Ho, 1997, authors use the Haar wavelet trasform to estimate the symbol rate of the M- ary PSK sigal. I (Deg & Liu, 8, they estimate the symbol rate accordig to the relatio betwee the sigal badwidth ad the symbol rate. The method is based o the Haar wavelet whe the scale value is optimized by a Mote Carlo simulatio. Therefore, i (Xu, Wag & Wag, 5 authors proposed a improvemet of the symbol rate estimatio based o the wavelet trasform of the basebad sigal. This proposal is very sesitive to frequecy errors ad so, it is ot stable. Authors i (Gao, Li, Huag & LU, 9 overcome the stability problem of such algorithms ad estimate the symbol rate of the M-ary PSK sigal takig ito accout Doppler frequecy effect. The choice of wavelet scale is critical ad it depeds o the carrier frequecy. Thus, the eviromet is cosidered as pseudo-blid sice the sigal carrier frequecy that is ecessary to choose the most appropriate scale value, is assumed kow. Therefore, a pre-estimatio of the carrier frequecy becomes a importat task i totally blid eviromet (Xu, Yu, Liu, Zhag & Guobig, 9, Yu, Shi & Su, 4. However, the estimatio fails if the frequecy estimated from the iterval search is ot icluded i the sigal s badwidth. This idicates that the search takig place i a wrog medium (Hatoum, Ghaith & Pujolle, 1. Thus, we must accurately determie the spectrum occupacy boudaries of the sigal. To achieve this, we are based o the 5

spectrum sesig techiques to search the occupied spaces (ulike the CR cotext, the to detect their boudaries; givig ease to the search of sigificat sigal parameters. I this paper, we preset a geeralized symbol period estimatio for all types of digital ad liear modulatio (ASK, PSK ad QAM ad all modulatio order (biary ad M-ary. I additio, we pre-estimate the sigal carrier frequecy, ad the the estimatio of the symbol rate of the liear modulatios becomes totally blid ulike i the case of (Gao, Li, Huag & LU, 9 where they operate i pseudo blid eviromet ad restrict the estimatio for M-PSK schemes oly. The sequel of the paper is orgaized as follows: i sectio, we preset ad describe the wavelet trasform tool. I sectio 3 we preset the system model. The, we explai the pre-estimatio task of the carrier frequecy i sectio 4. I sectio 5 we preset ad describe the blid symbol period estimatio for the liear modulatios. Simulatio results are show ad aalyzed i sectio 6. Fially we coclude i sectio 7.. Wavelet Trasform Overview Wavelet Trasform is a mathematical tool that has bee domiatig i the field of sigal processig ad aalyzig. It has the excellet capability idetifyig, i terms of time ad frequecy, the local characteristics of a sigal (Malat, 9. Ulike Fourier Trasform, that globally characterizes a sigal, wavelet Trasform is adopted to aalyze o-statioary sigals. It reflects the istataeous behavior of each frequecy compoet with locally characterizatio (Torrece & Compo, 1998. The wavelet trasform is characterized by a variable resolutio suitable to the sigal variatios. Based o the widow fuctio, called wavelet mother fuctio, we cosider differet types of the Wavelet Trasform fuctio, e.g. Gaussia, Haar, Morlet, Hammig, Haig, Mexica Hat, etc. The mother wavelet must be scaled ad traslated to sca the etire sigal i multi-resolutio way, we obtai a family of wavelet fuctios represeted by Eq. 1 where a ad b are the scale ad traslatio factors respectively. Scalig factor must be a iteger as a power of : 1 t b ψ (1 a, b ( t = ψ ( a a This trasform is a kid of filterig process represeted by the covolutio: + * ( S = X ( t. ( t dt a, b ψ a, b Filters coefficiets are based ad determied by the wavelet family fuctios. The applicatios field is very large, for image processig ad compressio; also it applied for discotiuities detectio, de-oisig, smoothig fuctio, features extractio, data aalysis, distributio estimatio, etc. 51

3. System Model Our aim is to estimate the symbol period Ts i a blid eviromet we cosider the wireless Rayleigh chael. A pre-estimatio of the carrier frequecy is ecessary. The processig tool is the wavelet trasform; particularly the mother wavelet fuctio used is the Morlet oe due to its high flexibility ad high precisio. At the trasmitter, digitally ad liearly modulated data symbols s pass through a pulse shapig filter g (t. The passbad trasmitted jω ct+θ sigal s (t is give i Eq. 3: s ( t = s (3 g( t Ts e I this case we cosider the root raised cosie (RRC filter as pulse shapig filter. f c is the carrier frequecy ad θ is the carrier phase. To simplify, θ = rd. This filterig gives the rectagular shape for the sigal i the frequecy domai. Thus, the spectrum is clearly characterizes by its discotiuity structure ad abrupt chages. Coveyed sigal is itercepted by the receiver with o prior iformatio about trasmitted sigal characteristics ad it give by the followig Eq. 4: y( t = s( t w( t (4 α τ k k k + where α k ad τ k are the gai ad delay k th path characteristics respectively due to the Rayleigh chael effect. w(t is the Additive White Gaussia Noise (AWGN. The liear modulatio type is ukow by the receiver. It may be ASK, PSK or QAM modulatio, biary or M-ary order. For ASK sigal, we cosider the case of the cotiuous phase sigal; as costellatio, the ASK symbols are real values. Therefore, i the liear modulated sigal, chages ca appear at each symbol period (Ts ad its multiple values. Thus, this time domai sigal is characterized by a abrupt variatios ad discotiuity structure. Maily, our objective is, to pick clearly the positios of these discotiuities (sigularities i frequecy domai sigal for the carrier frequecy estimatio ad time domai sigal for the symbol period estimatio. Cotiuous Wavelet Trasform (CWT is cosidered i order to reflect this discotiuity characteristic ad the chose Morlet wavelet fuctio has the followig form: (5 ϕ( t = e jωϕt where ω ϕ is the cetral agular frequecy that characterizes the Morlet fuctio. This parameter is adjustable which offers high processig flexibility. Choice of this frequecy value is much critical sice Smooth t e 5

sigal features produce relatively large wavelet coefficiets at scales where the oscillatio i the wavelet correlates best with the sigal feature. I time domai, the Morlet fuctio scas the etire sigal i order to idetify the abrupt chages, if exist, at each symbol period. As first step, we estimate the sigal carrier frequecy that is the basic to the symbol period estimatio. I particular, these holes may be licesed spectrum bads, ot curretly i a exploited state. I additio, these holes may be guard bads to avoid iterferece betwee user trasmissios. Badwidth edges are accurately detected; the, we determie by filterig a part of the received large bad where we decide to paid a attetio ad fid the carrier frequecy. Oce this carrier frequecy is estimated, we begi searchig of the symbol period. 4. Carrier Frequecy Estimatio Carrier frequecy kowledge has a essetial role i sigal processig, dow-coversio ad demodulatio of the received wireless commuicatio sigal. The carrier frequecy estimatio is the basic task allowig estimatig other sigal parameters. We have itroduced a ew method to estimate the carrier frequecy i totally blid eviromet based o the high wavelet trasform capability to detect discotiuities edges. Morlet wavelet is used ad o restrait o the choice of the scale value but the scalig factor must be a iteger as a power of. For our purpose, basig o the spectrum sesig approach, we examie the presece of a itercepted sigal through a particular badwidth which may cotai several sigals of differet systems. The processed sigal to be cocered is the Power Spectral Desity (PSD of the received sigal. Ideed, this spectral structure supports discotiuities represeted by edges of each sub-bad, we mea the frequecies f ad f 1 where is the occupied badwidth umber i the cosidered spectrum. The estimatio steps are as follow: - First, we sese the presece of sigal eergy iside a determied frequecy rage with respect to a determied eergy threshold. - The, the detected sigal is auto-correlated i order to reduce the white oise effect. - The basic domai is the frequecy oe. Thus, we obtai the autocorrelated sigal spectrum through the Fourier Trasform: S corr ( f. - For spectrum edges detectio purpose, we itroduce the Wavelet Trasform (WT of the autocorrelatio spectrum: we apply the covolutio product betwee the autocorrelatio S corr ( f ad the 53

scaled Morlet fuctio ϕ a ( f : WT( f = Scorr ( f ϕa ( f (6 The covolutio i frequecy is equivalet to the product i the time domai. The Wavelet Trasform ca be obtaied i a alterative way by the Fourier Trasform of the multiplicatio of: the sigal autocorrelatio (i time domai ad iverse Fourier Trasform (FT of the wavelet fuctio. Ideed, it is cosiderably faster to do the wavelet trasform calculatio based o the multiplicatio way. Therefore, the wavelet trasform ca be obtaied by the Eq. 7: WT( f = F. T{ Scorr ( τ. φ( aτ } (7 - The importat trasitios i the sigal are reflected by a high modulus level of the wavelet trasform, Fig. 1. Hece, we detect the badwidths boudaries by pickig the local maxima of the wavelet trasform PSD modulus. I real-time applicatios, maually idetificatio of the local maxima is ot satisfied; the results are ot expected to be accurate, especially i oisily coditios. Thus, we itroduce a automatic method to accurately detect frequecy edges from the wavelet trasform local maxima. The basic idea is that a maximum is searched withi a widow with a fixe size L. This widow scas the etire cosidered spectrum. Each modulus value iside the widow is compared to the average of all modulus values i the widow. If the compariso ratio is greater tha a pre-defied decisio threshold g, the correspodig modulus is cosidered as a local maximum ad the correspodig frequecy is idetified as a badwidth edge. The followig equatios resume the local maxima selectio coditios: (L. WT( fi > g L WT( f + WT( f ( i i k= WT ( f i 1 < WT ( f i ad WT ( f i+ 1 < WT ( f i L ad g are the robustess parameters, their choice is very critical: - Icreasig L gives more method accuracy - High value of level g may ot show the existece of a sigificat peak. Low value of g, may lead to a false decisio whe cosiderig low peak levels, which are a oise peaks, as local maxima (bads boudaries. 54

Fig. 1 Received sigal PSD (top ad wavelet trasform modulus (bottom. Due to this search, frequecy bad edges are automatically idetified with very high precisio. These edges allow us to clearly locate spectrum occupacy of a desired sigal i a totally blid eviromet. We remid that the commuicatio sigals are symmetric sigals where the carrier frequecy is the ceter of the badwidth occupatio. Thus, oce the badwidth edges are accurately determied, the carrier frequecy is automatically foud by the ceter of the determied ad surely bad (Hatoum, Ghaith & Pujolle, 1. 5. Symbol Period Estimatio We develop a method to estimate the symbol period of a liear modulated sigal. Such sigal is characterized by a discotiuous structure. Ideed, the iformatio data is carried by the amplitude ad/or the phase of a siusoidal oscillatio at each symbol period. The Ts estimatio is based o the wavelet trasform, sice this aalysis tool has excellet capability to reflect the discotiuities i the sigal oscillatio. These discotiuities are traslated to the abrupt chages of the wavelet trasform. This wavelet behavior depeds o the relatio betwee the wavelet fuctio ad the studied sigal: at scales where the wavelet oscillatio correlates best with the sigal feature, largest are the wavelet coefficiets. For siusoidal oscillatios, the CWT coefficiets display a oscillatory patter at scales where the oscillatio i the wavelet approximates the period of the sie wave (Torrece & Compo, 1998. Thus, the choice of scale is very importat for the symbol period estimatio. For Morlet fuctio, at scale equal to: ω ϕ a = (8 ω c 55

The wavelet coefficiets chage abruptly ad preset high level modulus at the amplitude ad/or phase chage positio of the sigal, where ωc = πf c represets the carrier frequecy. This motivates us to preestimate the carrier frequecy as preseted i sectio above. After obtaiig the carrier frequecy, the scale value is calculated i order to havig better estimatio performace. Locally characterizatio aalysis of the wavelet fuctio allows a clear detectio of the istataeous sigal feature. Morlet Wavelet fuctio makes zoomig o the abrupt sigal chages. ASK sigal is give by the equatio below: jωct s ( t = A g( t Ts e (9 ASK +,1,..., M { 1} A step A A. (1 where A is the amplitude value durig the period. Ts, A is the referece amplitude value ad A step is the amplitude separatio i the ASK costellatio ad M is the modulatio order. At each symbol period, oly the amplitude may chage; let cosider Α = A+1 A the differece amplitude betwee two cosecutive symbols; thus, the wavelet trasform ca be expressed as follow: b t τ t τ ( jω ( jω t ϕ c a a Ae e e dt 1 WTASK τ = + (11 a + t τ t τ ( jω ( jωct ϕ a a ( A + Α e e e dt b b is the time differece that determies the relative distace betwee the Morlet fuctio ad the trasitio positio i the modulated sigal at the multiple of symbol period. Now, let itroduce the scale value calculated from the Eq. 8. We cosider two cases: 1 if Α =, o amplitude chage betwee two jωcτ cosecutive symbols: WTASK τ = A πae (1 ad the wavelet modulus is cosidered as costat with respect to the amplitude ad it equal to: WT ASK τ = A πa (13 If Α, we obtai the equatio below: 56

b t τ ( a e dt jω cτ Ae WTASK τ = + a + t τ Α ( a (1 +. e dt A b (14 For expressio simplicity, let cosider the variable Α chage: Β = 1+. With itegral calculatio ad with referred to the error A fuctio complemet below erfc( x = π x e t dt Let us cosider ρ ( b.5erfc( b / a, the we obtai the followig a = WT τ = A πa 1 + ρ ( b( B 1 e (15 jωcτ expressio: ( ASK Now, we search the poits that maximize the modulus fuctio of the wavelet trasform i order to pick the amplitude trasitios positio i the WT τ ASK modulated sigal with respect to the follow derivatio. = ρ a ( b ω ϕ Therefore, for a =, the modulus of this wavelet trasform is as follow: ωc A WTASK TS = A πa (1 + Α (16 For the PSK sigal, the expressio is: s t A g t Ts e j e jωct PSK ( = (. ϕ. (17 1 M 1 ϕ, π, π,... π M M M (18 Similarly, we cosider ϕ = ϕ +1 ϕ the differece phase betwee two cosecutive symbols. Thus, the wavelet trasform is give by: b t τ ( a e dt jϕ Ae WT PSK τ = +. e a + t τ ( j ϕ a e e dt b a jω τ c (19 57

After calculatio, we obtai the followig expressio: j ϕ j( ωct+ ϕ WT τ = A aπ 1+ ( e 1 ρ ( b e ( PSK [ ] a Hece, the maximized modulus is obtaied whe ρ a ( b =. 5 for all values of ϕ. By aalyzig this value, we coclude that it is achieved whe b=, thus, whe the cetral of the Morlet wavelet coicides with the abrupt chage i the sigal. We obtai the Eq.1: A aπ j ϕ WTPSK Ts = 1+ e (1 For QAM sigal where amplitude ad/or phase ca chage: s t A g t Ts e j e jωct QAM ( = ϕ (.. ( A ad ϕ are as show i Eqs. 1 ad 18. Wavelet trasform becomes as: b t τ ( a e dt jϕ Ae jωcτ WT QAM τ = +. e (3 a + t τ ( j ϕ a B. e e dt b Ad the modulus is maximized ad preseted below: j ϕ WT T = A πa 1 + ρ. B. e 1 (4 QAM S Where ρ is the maximizatio factor: 1- ( B.cos( ϕ ρ = (5 B + 1-. B.cos( ϕ For Α ad ϕ = we retur to the ASK case Eq. 16; ad if Α = ad ϕ we retur to the PSK case Eq. 1. After calculatig ad maximizig the modulus of the Morlet wavelet trasform for the modulated sigal i time domai, we detect the local maxima poits. They reflect variatios of the modulated sigal, thus the miimum value of the differece betwee the peaks is cosidered as the symbol period. Due to the oise, peaks searchig is ot efficiet sice small chages are itroduced. This effect ca be overcome by computig the cyclo-autocorrelatio fuctio of the WT τ. This fuctio reduces effectively the oise level ad clearly shows maximum chages; i additio, it presets a periodicity that makes easier the calculatio of the differece betwee peaks positios Fig.. 58

Fig. The wavelet trasform (a ad its cyclo-autocorrelatio (b However, we fid that the search i the frequecy domai is more robust. Therefore, we aalyze the autocorrelatio spectrum obtaied through the Fourier trasform calculatio. For PSK ad QAM modulated sigals, the first peak after the cetral frequecy (for basebad sigal, the cetral frequecy is equal to zero represets the symbol rate. As kow, this frequecy is the iverse value of the symbol period Rs = 1 Ts. I other had, For the ASK modulated sigals, first peak after the cetral frequecy is equal to the twice of the symbol rate. I time domai, differece betwee peaks i ASK autocorrelatio fuctio is equal to half of the symbol period. 6. Simulatio Results ad Aalysis The performace of the proposed scheme is studied ad evaluated by Matlab simulatio. Two performace criteria are cosidered: 1 The probability of the estimatio errors The Normalized Root Mea Square Error (NRMSE These values are calculated over 5 Mote Carlo iteratios ad preseted i fuctio of Sigal to Noise Ratio (SNR. At the trasmitter, the cosidered sigal parameters are as follows: the carrier frequecy is fc=1.5mhz ad the symbol rate is Rs=.5MHz (symbol period is 1/Rs. The samplig rate is fs=5mhz. The cetral agular frequecy of Morlet wavelet trasform is ω = 5rd with respect to efficiet rage cosidered ϕ as: ω ϕ 5. We assume 1 symbols as data legth ad SNR rage is - 1 14 db. The value of the symbol period to be estimated is obtaied by the miimum differece betwee the peaks of the time domai correspodig fuctio, it is the wavelet trasform cyclo-autocorrelatio of 59

the received sigal (Fig. b. This miimum is explaied by the fact that the data iformatio carried by the sigal features (amplitude ad/or phase ca chage at least oce at each symbol period. However, due to the oise effect, this calculatio for symbol period is uclear, thus we propose to itroduce the histogram of the peaks differece ad choose its maximum value correspodig to the most repeated differece value, as the estimated symbol period value. Nevertheless, i the frequecy domai the search is more robust. Thus, the estimated symbol period is calculated from the estimated symbol rate which is equivalet to the first peak uless the cetral frequecy i the PSD spectrum. I order to make more obvious the peak search, we itroduce a smoothig fuctio to avoid appearig udesired peaks. By a simple way, this peak value ca be foud, because it is the globally maximum. The performace of the symbol period estimatio is studied for both ideal chael (AWGN ad Rayleigh chael with Doppler frequecy fd=13 Hz. Fig. 3 shows the estimatio errors probability for the 16-PSK, 16- ASK ad 16-QAM, i the AWGN chael. By comparig, we observe that this estimatio algorithm is optimal after - db i a ideal chael (AWGN chael ad presets a excellet performace. Probability of errors AWGN 1.9.8.7.6.5.4.3. Ts Error Estimatio Probability: M=16, AWGN chael 16-PSK 16-QAM 16-ASK.1-1 -5 5 1 15 SNR (db Fig. 3 Symbol Period estimatio probability i AWGN chael, M=16. Thus, this algorithm is very suitable for very low SNR. As show, it is more efficiet for the PSK modulatio the for the QAM modulatio. For Rayleigh chael, Fig. 4 shows the probability of the estimatio error for all modulatio whe M=16. Similarly, this estimatio algorithm outperforms for the PSK modulatio the for QAM modulatio. Sice the ASK modulatio is more sesitive to oise that affects directly the amplitude value, it shows less performace by comparig with others liear modulatios. 6

Probability of errors 1.9.8.7.6.5.4.3 Ts Error Estimatio Probability:M=16, Rayleigh chael 16-PSK 16-QAM 16-ASK..1-1 -5 5 1 15 SNR (db Fig. 4 Symbol Period estimatio probability i Rayleigh chael, M=16. As it is clear, due to the fadig effect of the Rayleigh chael, the performace is ot optimal by comparig with the AWGN chael. But for practical Rayleigh chael, this method is very efficiet for low SNR values, above db. I other had, with respect to the NRMSE criterio, results are show i Figs. 5 ad 6 for AWGN ad Rayleigh chaels respectively. I additio, the NRMSE shows a high deviatio of the estimated symbol period value i a Rayleigh chael with respect to the AWGN chael, especially for SNR > db Fig. 6. We have compared with the results show i (Xu, Wag & Wag, 5 ad from the NRMSE values i the Fig. 7, we observe that our ew method outperforms the method based o the multi-scale Haar wavelet ad preseted i (Xu, Wag & Wag, 5. With respect to the modulatio order M, the performace of this estimatio algorithm is the same (show Fig. 8. This is due to the fact that the algorithm estimates the symbol period istead of the bit duratio ( T = T / log ( M b s. 7. Coclusio I this paper, a symbol period estimatio method is geeralized for all liear modulatio schemes. No-cooperative eviromet is cosidered where the sigal parameters are ukow by the receiver. The processig tool is the wavelet trasform that shows a excellet capability to detect discotiuities. 61

1.9.8.7 Ts Estimatio Normalized RMSE: M=16, AWGN chael 16-PSK 16-QAM 16-ASK NRMSE.6.5.4.3..1-1 -5 5 1 15 SNR (db Fig. 5 Symbol Period estimatio NRMSE i AWGN chael, M=16. 1.9.8.7 Ts Estimatio Normalized RMSE: M=16, Rayleigh chael 16-PSK 16-QAM 16-ASK NRMSE.6.5.4.3..1-1 -5 5 1 15 SNR (db Fig. 6 Symbol Period estimatio NRMSE i Rayleigh chael, M=16. These abrupt chages appear i the origial sigal at the symbol period ad/or at its multiple values. A pre-estimatio of the carrier frequecy is itroduced. I order to overcome the effect of the white oise, we itroduce the autocorrelatio fuctio. The local maximum positio of the autocorrelatio wavelet trasform i the frequecy domai correspods to the estimated symbol period value. This method shows a sigificat performace ad a very high accuracy at very low SNR values. It has a low computatio complexity ad it is suitable ad robust for a practical Rayleigh chael. 6

NRMSE 1.9.8.7.6.5.4 Compariso of the proposed method ad the Haar multi-scale method Our proposed method Haar multi-scale method.3..1-4 - 4 6 8 1 SNR (db Fig. 7 Compariso of our proposed method ad the HAAR multi-scale method i (Xu, Wag & Wag, 5:16-PSK, AWGN chael. Probability of errors 1.9.8.7.6.5.4.3 Ts Error Estimatio Probability: PSK modulatio, Rayleigh chael 8-PSK 16-PSK 3-PSK 64-PSK..1-1 -5 5 1 15 SNR (db Fig. 8 Probability of error estimatio i fuctio of the modulatio order M. Refereces: Va Trees, H.L.: Detectio, Estimatio, ad Modulatio Theory, Part I. Joh Wiley & Sos, 1. Dayoub, I., Okassa-Mfoubat, A., Mvoe, R. ad Rouvae, J.M.: A Blid Modulatio Type Detector for DPRS Stadard. Wireless Persoal Commuicatios 41, 7. Yu, Z. ad Bai, J.: Carrier Frequecy Estimatio i Automatic Modulatio Recogitio Eviromet. IEEE Iteratioal Coferece o Iformatio ad Automatio, 1. 63

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