An Improved Algorithm and It s Application to Sinusoid Wave Frequency Estimation

Size: px

Start display at page:

Download "An Improved Algorithm and It s Application to Sinusoid Wave Frequency Estimation"

Warren Butler
5 years ago
Views:

1 Iteratioal Coerece o Iormatio ad Itelliget Computig IPCSI vol.8 () () IACSI Press, Sigapore A Improved Algorithm ad It s Applicatio to Siusoid Wave Frequecy Estimatio iaohog Huag ad Li Zhag College o Iormatio Egieerig, Hebei Uited Uiversity, agsha,hebei,chia College o Electrical Egieerig, Hebei Uited Uiversity, agsha,hebei,chia tshxh@63.com ad Zhagli @6.com Abstract. A improved algorithm based o Kay s estimator ad its applicatio to Siusoid Wave requecy estimatio are ivestigated. Firstly a ew method o spectrum aalysis is itroduced, which has excellet perormace i suppressig spectral leaage ad the property o phase ivariat, the a hybrid All Phase Kay(ApIay) algorithm is proposed,which merges Kay s estimator ad phase uwrappig. he improved algorithm is applied to the requecy estimatio o a siusoid, the requecy perormace is better tha Kay ad Iay. Whe SR(<7 db), he simulatio results show that the mea square error o the ew requecy estimator is improved 4dB tha Kay, db tha Iay. Whe SR(>7 db), the perormace o ApIay obtais CRLB quicly, ad the perormace is stable i the whole requecy rage. Keywords: requecy estimatio, Kay, phase uwrappig, siusoid, CRLB. Itroductio Estimatig the requecy o a sigle siusoid corrupted by additive, white, Gaussia oise (AWG) is a importat ad classical problem i commuicatios, radar ad soar sigal Processig. Maximum lielihood (ML) requecy estimators i requecydomai were studied by Rie ad Boorsty i [], which has large complexity. he timedomai estimators i [8] are derived rom the ML priciple. retter proposed uwrappig the sigal phase ad perormig liear regressio to obtai a requecy estimate [], but ca oly wor well at high sigaltooise ratio (SR). Kay addressed the phase uwrappig problem by oly cosiderig the phase diereces ad preseted a simple requecy estimatio algorithm, amely, Kay s estimator [3], which ca approach the CramerRao lower boud (CRLB) at high SR, but this estimatio method has obvious threshold i reality applicatio ad has relatio with the requecy. he perormace becomes bad whe requecy is close to, hal o sample requecy, sample requecy. Iay i [9] improved Kay s estimatio perormace. I this paper, we improve the Iay urther, ApIay is proposed based o Kay s estimator ad a ew method o spectrum aalysis ad phase uwrappig, which has better perormace that the MSE o requecy estimatio improves about 4dB tha Kay s ad db tha Iay i the low SR(<7 db), ad close to CRLB whe (>7 db) quicly.. A ew FF Spectrum Aalysis A ovel algorithm o spectrum estimatio is put orward i the literature [], amed ApFF, which improves the data trucatig way o traditioal DF spectrum aalysis ad reduces the leaage greatly. he bloc diagram is show i the bottom o the igure.he order widow uctio is the covolutio o two same symmetric order widow. 97

2 Figure. Figure the diagram o ApFF spectrum aalysis First, we deduce the amplitude o a sigal cosistig o a sigle requecy.i the sigal with sigle requecy is x j π s = e, where is the sigal requecy, s is the sample requecy. o oe sample poit x( ) i the time sequece, there are vectors o dimesio icludig this sample poit: = [ x ( ) = [ x ( = [ x () ) x ( x () + ) x ( ) x ( x ( x ( )] Cycle shit every vector, shit the sample x( ) to the irst positio o the sequece ad get the other vectors o dimesio: = [ x ( ) = [ x ( ) = [ x ( ) x ( x ( + ) + ) x ( ) x ( x ( )] x ( ) )] )] ] ) We ca get all phase data vector by addig vectors aimig at x() AP = [ x( ) ( ) x( + ) + x() x( ) + ( ) x( )] Accordig to the shit property o discrete Fourier trasorm, there has clear relatioship betwee the i ()ad i (), where i()is the discrete Fourier trasorm o i(i=,, ) ad i ()is the discrete Fourier trasorm o i (i=,, ). ( ) i πi j ( ) e ApFF is made up o the sum o i(),so: i ] = () ( ) ( ) ( ) e AP = i = i i= i= ( ) i j π + i π π j j s = e e e i= = = e j s Si π ( ) s Si π ( ) s π i j Accordig to (), the amplitude o all phase spectrums is as ollows ( ) ( j π j π i j π ) s s s e e e i= = = π () si π ( si π ( traditioal DF requecy spectrum amplitude, which is beeit to reducig the spectrum leaage. s s ) ),it is the square o Aother importat character o ApFF spectrum aalysis is that its phase is costat ad is t ilueced by the requecy shit, so the phase eed t to be corrected. hat meas the real phase o sigal ca be obtaied by ApFF spectrum aalysis whe the sigal is oiteger trucated. he measured phase value ad the real phase value have less error. ae the sigal cos(. π/6t+π/8) as a example to search the reaso o so little phase error about ApFF spectrum aalysis. We ca get samples:

3 he iput sigal o all phase is made up o 6 groups o =6 samples. he irst group cosists o the last 6 samples amog all samples, the secod group cosists o aother 6 samples which let shit value, but.736 should right shit to the irst positio, other groups could be get as the same way he phases o the samples o the 6 group o =6 sigals are : Because the requecy is., we should observe the secod phase i every group. hree are bigger ad three are smaller tha the real phase () durig 6 phases. Iput data o ApFF is the average o the above 6 groups sigals, phases are couteract each other, which mae phase dierece zero. So the phase got by ApFF is the sigal real phase. he result o experimet shows that whe the sigal is iterperiod sampled, the phase got by ApFF is perect. I this case, the phase o sigal got by ApFF with widow (aiser(,9.5) covolute Kaiser(,9.5) ) is as ollows: I this case, the real phase is, the measured phase is.69, so the error is oly.69%, ad we ca thi them very similar. Accordig to the above example, we ca get the coclusio that all phase has perect phase aalysis property, especially whe the sigal is oiterperiod sampled, the phase aalyzed by this method is almost the real value; while the phase aalyzed by traditioal method is delect rom the real value. So a method o ApFF phase dierece is proposed i [], which ca estimate sigal s parameters with less error, but has ot good result i the low SR. 3. Improved Algorithm based ApFF ad Kay A moocompoet siusoid cotamiated by AWG ca be modeled as: r ( = s( + w( = A exp{ jφ( } + w(, =,... φ ( = π c + θ c Where A is the amplitude, c θ ad c are requecy ad iitial phase, is sample cycle. We ca get phase rom (3) rom the received sigal: Im[ r ( )] φ( = arcta Re[ r ( )] (5) ( ) But this is ot the true phase, the true phase φ is obtaied through phase uwrappig o the measured phase φ (. I the coditio o oise, the relatio betwee true phase ad measured phase is as (6),where ( Z ) is the umber o period o the th sample. φ( φ ( π he phase dierece o adjacet samples Δφ is:6 = (6) Δ φ =Δ φ ( ) π (7) ( ( 99

4 Where Δ φ( = φ( φ( ), Δ φ() = φ() Δ φ () (), = φ, =, =.Whe φ ( ) ad φ ( ) are i the same period Δ φ( >,otherwise, Δ φ <. hus the true phase ca be recovered accordig to the phase dierece betwee adjacet samples. he true phase o the th sample is: i Δφ(, = φ ( = φ( + π, (8) i Δφ( <, = + he better the perormace o phase uwrappig; the closer to π the phase dierece o adjacet samples. By usig this perormace, a improved algorithm based o ApFF ad Kay is proposed, the step o this algorithm is as ollow: a) Obtai R( ) by Perormig Fourier trasorm o r ( ), the estimate the ceter requecy c o r,compute ( ) the shit value = s c,where s is the samplig requecy. (i) ae samples ad shit to ceter requecy, aother sigal ca be obtaied: ~ z ( ) = r ( ).*exp( j π c ),where is the samples legth o r ( ). (ii) Perorm ApFF o z( ( ) i order to get phases Z,the legth o which is. (iii) ( ) Uwrap Z φ ( ) by usig ormula (8) to get the true phase ZI. (iv) Get Phase dierece Δφ( through two slices o phase data with legth, which is shit oly oe poit. (v) Get the ubiased estimator o requecy = h( Δφ (,where = (vi) Get the true requecy o r: ( ) 4. Prepare Your Paper Beore Stylig ^ ^ =. 3 ( ) h ( = [ ]. Do some simulatios By Matlab to test the improved algorithm. ae two groups o siusoid wave as example, requecies are Mhz ad 4Mhz separately, samplig requecy is s=mhz ad the legth o samples is *=63, estimate requecies o this two siusoid waves by Kay estimator ad Iay ad ApIKay(i order to simple, we amed the method o this paper ApIKay ), uder the low SR coditio(sr<7db),do MoteCarlo simulatios to get mea square error. he umber o simulatio rus is set to or each case; the result is show i table ad. ABLE I. FREQUECY ESIMAIO COMPARISO ( c = MHz,UI:KHZ) db db db 3 db 4 db 5 db 6 db Kay IKay ApIay CRLB ABLE II. FREQUECY ESIMAIO COMPARISO ( c = 4MHz,UI:KHZ) db db db 3 db 4 db 5 db 6 db Kay IKay ApIay CRLB From able ad able, we ca coclude that the mea square error(mse) o IKay ad ApIay improve 3dB ad 4 db or so tha that o Kay separately. From able, we ca see that the MSE o Kay is very large whe the sigal requecy is close to / samplig requecy, while Iay ad ApIay are ar better tha Kay. his is because Kay s estimator is sesitive to requecy. ow we compare the perormace relatioship betwee MSE ad requecy by usig three methods. Simulatio coditios: SR= db, requecy step is 5 Mhz, the rage o requecy is rom Mhz to Mhz, there are requecy poits. Compute MSE o this requecy poits, the result is show i the igure. From Figure (), we ca see that

5 the perormace o Kay is bad whe requecy is close to 5 MHz, while Iay ad ApIay are better i the whole rage o requecy. he mea MSE o Kay ad Iay ad ApIay durig the whole requecy are.mhz ad 358. KHz ad 76.5 KHz, ApIay s is most close to CRLB which is 7.7 KHz. Figure. Perormace relatioship betwee MSE ad requecy by usig three methods (the ollowig igure is zoom out igure) Whe SR>6dB, the estimate accuracy o ApIay closes to CRLB. ae the above sigal Mhz siusoid wave as example, compute the MSE o the sigal by usig these three algorithms at dieret SR. he result o simulatio is as the Figure (3). From Figure(3), We ca draw a coclusio that ApIay reduces the SR threshold o Kay ad Iay, that meas the MSE o ApIay obtais CRLB at SR=6dB,while Kay obtais CRLB at SR = db, Iay obtais CRLB at SR=8dB. Figure 3. Perormace o SR threshold ae aother sigal as example to test the requecy accurate by Kay,Iay,ApIay ad ApFF phase dierece [](amed ApPD). he sigal is complex expoet sigal, s is 3hz, we cosider three cases about sigal requecy, which is 6.736hz,.63hz, hz separately. he SR is db. Sigal legth is =56. he simulatio result is as able3. ABLE III. FREQUECY ESIMAIO COMPARISO

6 5. Coclusio db rue requecy Kay IKay ApIay ApPD his paper brigs orward ApFF, which has less leaage ad high precisio ad phase ivariat compared to the traditioal spectrum aalysis. A improved algorithm o siusoid wave uder the oise bacgroud is proposed, which has better requecy estimatio characteristic, improves the SR threshold o Kay, ad the perormace is stable i the whole requecy rage, which is a problem i Kay. So the research i this paper ca be used i parameter estimatio i the area o commuicatio ad Radar. 6. Acowledgmet his wor is supported by Sciece Foudatio o Hebei Provice (F85) ad also supported by Youth Foudatio o Educatio Departmet o Hebei Provice (58) 7. Reereces [] Rie DC, Boorsty RR. Sigletoe parameter estimatio rom discretetime observatio. IEEE ras I heory 974; I:5998. retter S. Estimatig the requecy o a oisy siusoid by liear regressio. IEEE ras I heory 985; I3: [] Kay S. A ast ad accurate sigle requecy estimator. IEEE ras Acoust Speech Sigal Process99; 39: 3 5. [3] Brow, Wag MM. A iterative algorithm or siglerequecy estimatio. IEEE ras Speech Sigal Process ; 5: 678. [4] Lovell BC, Williamso RC. he statistical perormace o some istataeous requecy estimators. IEEE ras Sigal Process 99; 4: 783. [5] Zhag Z, Jaobsso A, Macleod MD. Chambers JA. A hybrid phasebased sigle requecy estimator. IEEE Sigal Process Lett 5; : [6] Fowler ML, Johso JA. Phasebased requecy estimatio usig ilter bas. US Patet 64774,. [7] Fu H, Kam PY. MAP/ML Estimatio o the requecy ad phase o a sigle siusoid i oise. IEEE ras Sigal Process 7; 55 : [8] Deg Zhemiao,Huag iaohog. A Simple Phase Uwrappig Algorithm ad its Applicatio to PhaseBased Frequecy Estimatio.Ope Access, Recet Patets o Sigal Processig,,, 637 [9] Huag xiaohog,wagzhaohua, ew Method o Estimatio o Phase,Amplitude, ad Frequecy Based o All Phase FF Spectrum Aalysis, ISPAC7 ov 7

Complex Algorithms for Lattice Adaptive IIR Notch Filter

4th Iteratioal Coferece o Sigal Processig Systems (ICSPS ) IPCSIT vol. 58 () () IACSIT Press, Sigapore DOI:.7763/IPCSIT..V58. Complex Algorithms for Lattice Adaptive IIR Notch Filter Hog Liag +, Nig Jia