Improving Broadband Acoustic Doppler Current Profiler with Orthogonal Coprime Pulse Pairs and Robust Chinese Remainder Theorem

Transcription

1 Improvng Broadband Aoust Doppler Current Profler wth Orthogonal Coprme Pulse Pars and Robust Chnese Remander Theorem Cheng Ch, Har Vshnu and Koay Teong Beng Aoust Researh aboratory, Tropal Marne Sene Insttute, Sngapore Abstrat Broadband aoust Doppler urrent proflers (ADCPs) are nstruments that are wdely used n underwater observaton. However, onventonal broadband ADCPs fae the problem of beng lmted n auray due to veloty ambguty. Ths paper proposes a method based on usng orthogonal oprme pulse pars and the robust Chnese remander theorem to mtgate ths lmtaton. The proposed method breaks through the lmt of veloty ambguty of onventonal broadband ADCP and mproves the performane sgnfantly. The smulatons show that our proposed method an derease the standard devaton of urrent veloty measurement by nearly three tme, when ompared to the onventonal method. Keywords- Broadband aoust Dopper urrent profler, urrent measurement, robust Chnese remander theorem, orthogonal oprme pulses, veloty ambguty I. INTRODUCTION Aoust Doppler urrent proflers (ADCPs) are wdely used n measurng oean urrents []-[5]. Modern ADCPs usually operate n broadband due to ther lower measurement varane as ompared to narrowband ADCPs [], []. The transmtted sgnal of broadband ADCPs usually onssts of a seres of oded pulses represented n Fg. (a), whh orrelate well around a hosen measurement lag, namely the length of the oded pulse, and have lttle orrelaton at other lags []. For broadband ADCPs, the pulse-par (or ovarane ) method [], [6], [7] s generally used to estmate Doppler shft by extratng the orrespondng phase at the hosen measurement lag. Unfortunately, for onventonal broadband ADCPs, usng the pulse-par method entals a trade-off between auray and maxmum measurable veloty (ambguty veloty) desrbed below, whh lmts the performane. For the pulse-par method, the extrated phase may only fall wthn -π to π range. Any other observed value (whh an arse due to a hgh true veloty of urrent beng observed) wraps around and falls wthn the (-π, π) lmt, leads to a phase ambguty. Ths phase ambguty n turn leads to an ambguty n the estmaton of Doppler shft. Thus, the maxmum measurable veloty of onventonal broadband ADCPs s the maxmum veloty that an be determned unambguously, alled ambguty veloty. As analyzed n Seton II, the ambguty veloty s determned by the length of the oded pulse, namely the hosen measurement lag. However, havng a hgher ambguty veloty requres the oded pulse to be shorter, whh results n worsenng auray []. We also hghlght ths problem n our smulatons n Seton V. In summary, the ambguty veloty lmts the performane of onventonal broadband ADCPs. Conventonal broadband ADCPs requres the users to onfgure the ambguty veloty before system deployment. Beause the ambguty veloty must aommodate all possble velotes that we wsh to measure [], ths results n the seleton of oded pulse lengths that unfavorably lmts the ahevable measurement auray. In ths paper, we fous on fndng a soluton to mprove broadband ADCPs by breakng the onventonal lmt of the ambguty veloty for a gven sgnal length. To mprove the performane of broadband ADCPs, we propose to transmt two orthogonal oprme pulse pars smultaneously as shown n Fg. (b) and employ the robust Chnese remander theorem (CRT) [8]-[] to resolve veloty ambguty. The smulatons show that our proposed method an derease the measurement standard devaton by nearly three tmes. Ths paper s organzed as follows. In Seton II, we brefly desrbe the bas theory of onventonal broadband ADCPs. Seton III wll ntrodue the robust CRT. Our proposed method s shown n Seton IV. Seton V presents the smulaton study omparng the proposed method to the onventonal method. Fnally, onlusons are gven n Seton VI.

2 Fg.. Spetrum of the orthogonal oprme FM pulses. Fg.. Shemat of onventonal transmtted sgnal (a) and our proposed two orthogonal oprme pulse pars (b). II. CONSTRAINTS OF CONVENTIONA BROADBAND ADCP Fg. (a) llustrates a shemat of the transmtted sgnal of onventonal broadband ADCPs, whh onssts of repeated oded pulses. The length of eah oded pulse T 0 determnes the ambguty veloty of the onventonal broadband ADCP. Generally, the total length of the transmtted sgnal, whh onssts of multple oded pulses, determnes the depth ell sze of broadband ADCPs []. For onventonal broadband ADCPs, the measurement veloty obtaned by usng the pulse-par method [5], [6], [9] s expressed as v = Δ, 4π ft φ () 0 0 where s the sound speed n water, f 0 s the entral frequeny of the oded pulse, T 0 s the pulse length, Δ φ s the phase hange at the measurement lag T 0. For v to be meanngful, the range of the phase hange should be - π Δφ π. () If Δ φ s larger than π or lower than -π, t leads to a phase ambguty, whh translates to veloty ambguty. Therefore, the maxmum measurable veloty (referred to as ambguty veloty) of onventonal broadband ADCPs an be expressed as vmax 0 (3) 0 0 For a hosen f 0, to nrease v max 0, T 0 needs to be dereased. However, the derease of T 0 wll nrease the measurement devaton. Ths s re-demonstrated n Seton V. III. ROBUST CRT The tradtonal CRT demonstrates that gven ntegers, where M, M,, M are oprme, and a postve nteger N < lm{m, M,, M } where lm represents the least ommon multple, N an be unquely reonstruted from ts remanders modulo the postve ntegers [8], [9]. However, the tradtonal CRT s not robust, whh means a small error from any remanders may result n a large reonstruton error [8]- []. To mtgate the effet of remander errors, several robust methods have been proposed reently, whh are referred to as robust CRT [8]-[]. The robust CRT an be generalzed to be appled to real numbers. Our paper proposes to use the robust CRT to reonstrut the real velotes from the measured, and possbly erroneous velotes. The losed-form robust CRT [9] for real numbers s ntrodued brefly n the followng. et R be a postve real number, whh an be wrtten as R = nmγ + r, (4) where n s an unknown foldng nteger wth, and all Γ are known postve ntegers and oprme, M s a known real-valued normalzaton fator deded by the system desgn and r s the real-value remander wth 0 r < MΓ. The goal of robust CRT s to robustly reonstrut R from gven nosy real-valued remanders r ˆ, whh are the erroneous verson of r. The losed-form robust CRT algorthm s explaned n the followng Step alulate the dfferenes q ˆ,

3 Fg.3. Shemat desrbng our approah for smulatng the reeved ehoes and subsequent proessng. rˆ ˆ r qˆ, =, M where [] stands for the losest nteger rounded off to. Step alulate the remander of qˆ, Γ, modulo Γ,,, (5) ˆ ξ = q ˆ Γ mod Γ, (6) where Γ, s the modular multplatve nverse of Γ modulo Γ, and an be alulated n advane. Step 3 alulate ˆn = ˆ ξ,, γ = Γ nˆ b γ mod, (7) where b, s the modular multplatve nverse of γ / Γ modular Γ, γ s γ Γ Γ / Γ =Γ Γ. (8) Step 4 Calulate n ˆ for ; nˆ Γ qˆ, nˆ Γ (9). IV. PROPOSED METHOD For onventonal broadband ADCPs, operatng wth a hgh ambguty veloty results n hgh unertanty n measurements. Choosng probe sgnal wth a low ambguty veloty an help nrease the measurement auray, but when the settng s used to measure a large veloty wthout resolvng the ambguty robustly, t an lead to serous errors. We propose to transmt two orthogonal oprme pulse pars smultaneously. The am of hoosng orthogonal sgnals s to guarantee two ndependent veloty values an be measured at the same tme. The ambguty velotes of the two orthogonal pulse pars are desgned to be oprme so that any phase ambguty an be resolved by usng the robust CRT. When measurng a urrent veloty that s larger than the two ambguty velotes of the two pulse pars respetvely, the true urrent veloty an be reonstruted from the two velotes (erroneous remanders) measured by usng the two orthogonal oprme pulse pars, respetvely. The proposed orthogonal oprme pulse pars are llustrated n Fg. (b). The desgn of orthogonal sgnals has been studed by the researh ommunty. Many exstng orthogonal sgnals n the open lterature an be good anddate for ths work. In ths paper, we hoose the orthogonal lnear frequeny modulated sgnals, as shown n Fg.. The oprme property of the pulse pars wll be explaned n the followng. et the pulse lengths of the two pars shown n Fg. (b) be T and T. Denote the entral frequenes of the two pars as f and f. Aordng to () and (), the ambguty veloty of the pulse par an be wrtten as v max (0) The ambguty veloty of the pulse par an be wrtten as

4 v max () et us denote v max / v max as Γ / Γ.The two pulse par sgnals are oprme f Γ and Γ are oprme. Gven that they are oprme, the true urrent veloty an be reonstruted from the velotes measured by usng the robust CRT. The reonstrutng proess s desrbed brefly n the followng. Wthout onsderng the errors of the measured velotes, the real estmated veloty an be formulated as vˆ= vˆ + n( vmax), vˆ vˆ n( vmax) = + () where ˆv s the real veloty, ˆv and ˆv are the velotes measured by the two pars, n and n are foldng ntegers orrespondng to eah pulse par. The normalzaton fator M an be wrtten as Fg. 4. Standard devatons of urrent veloty estmates usng sgnals wth ambguty velotes (V max) under the same depth ell (transmtted sgnal length). v v M = = Γ Γ max max. (3) When onsderng the errors of the measured velotes, assume that e and e are the errors of ˆv and ˆv. The robust CRT [8]-[] an guarantee that f M e < 4, (4) M e < 4 n and n an be unquely determned. In our smulaton, we fnd that the error bound of robust Chnese remander theorem an be realstally fulflled. Fg. 5. Standard devatons of urrent veloty estmates usng the onventonal method and the proposed method. to 0.50 and 0.45 m/s, whh are an order smaller than the ambguty veloty settng n the onventonal method. V. SIMUATIONS AND RESUTS We undertake a smulaton study to ompare our method wth the onventonal method of estmatng urrent velotes. Smulaton ondtons are summarzed as follows. Depth ell sze s set to 4 m. The transmtted sgnals are set to a tme length, orrespondng to the depth ell sze. The entral frequeny s set to 500 khz. The system bandwdth s 00 khz. For the onventonal pulse par, the ambguty veloty s set to 5 m/s. We hoose the lnear frequeny modulated (FM) sgnals as the transmtted pulses. In ths paper, orthogonalty of the two FM pulses s realzed by makng sure they le n two dfferent sub-bands, as shown n Fg.. The ambguty velotes of the pulse par and shown n Fg. (b) are set We smulate the aoust data reeved by the broadband ADCP, as onsstng of ehoes from the urrent and ambent nose. The shemat desrbng our methodology of smulatng the ehoes, whh has also been used n [4] s llustrated n Fg. 3. The ehoes are taken as the superposton of Doppler-resampled baksattered sgnals of a mass of satterers. We frst verfy that for the onventonal broadband ADCP, the ambguty veloty lmts the auray. Fg. 4 shows that the numeral smulaton aptures the effet of ambguty velotes (V max) that lmts the auraes of the urrent estmaton n the onventonal method. We then replae the onventonal method wth our proposed method n the smulaton but keep the remanng setup

5 unhanged. Fg. 5 shows the performane enhanement of the proposed approah ompared to onventonal approah. It an be found that the proposed method outperforms the onventonal method, redung the standard devaton by nearly 3 tme at a SNR of 0 db. Also, the SNR lmt at whh performane breaks down, s redued n the proposed method. In summary, the smulaton results demonstrate the sgnfant mprovement of estmaton auray of our proposed method. VI. CONCUSION The veloty ambguty n onventonal DVs lmts the mprovement of measurement auray. To redue the measurement devaton, we propose a method based on desgnng orthogonal oprme pulse pars, and use the robust CRT to resolve veloty ambguty. Smulatons verfy that, the measurement standard devaton of the proposed method s dereased by nearly 3 tmes, ompared wth the onventonal method. REFERENCES [] B. H. Brumley, R. G. Cabrera, K.. Denes, and E. A. Terray, Performane of a broad-band aoust Doppler urrent profler, IEEE J. Oean. Eng., vol. 6, no. 4, pp , Ot. 99. [] P. Wans, B. Brumley, J. Gast, and D. Symonds, Soures of measurement varane n broadband aoust Doppler urrent proflers, n Pro. MTS/IEEE OCEANS Conf., Seattle, WA, USA, Sep. 00, DOI 0.09/OCEANS [3] R. Pnkel and J. A. Smth, Repeat sequene odng for mproved preson of Doppler sonar and sodar, J. Atm. and Oean Teh, vol. 9, no., pp , Apr. 99. [4] C. Ch, Z., and Q., Desgn of optmal multple phase-oded sgnals for broadband aoust Doppler urrent profler IEEE J. Oean. Eng., vol. 4, no., pp , Apr. 06. [5] K. B. Therault, Inoherent multbeam Doppler urrent profler performane I-Estmaton varane, IEEE J. Oean. Eng., vol. OI-, no., pp. 7-5, Jan [6] S. S. Abeysekera, Performane of pulse-par method of Doppler estmaton, IEEE Trans. Aerosp. Eletron. Syst., vol. 34, no., pp.50 53, Apr [7] S. M. Kay, A fast and aurate sngle frequeny estmaton by lnear predton, IEEE Trans. Aousts, Speeh and Sgnal Proess., vol. 37, no., pp , 989. [8] X. W., H. ang, and X.-G. Xa, A robust Chnese remander theorem wth ts applatons n frequeny estmaton from undersampled waveforms, IEEE Trans. Sgnal Proess., vol. 57, no., pp , Nov [9] W. J. Wang and X.-G. Xa, A losed-form robust Chnese remander theorem and ts performane analyss, IEEE Trans. Sgnal Proess., vol. 58, no., pp , Nov. 00. [0]. Xao, X.-G. Xa, and W. J. Wang, Mult-stage robust Chnese remander theorem, IEEE Trans. Sgnal Proess., vol. 6, no. 8, pp , Sep. 04. [] W. J. Wang, X. P., W. Wang, and X.-G. Xa, Maxmum lkelhood estmaton based robust Chnese remander theorem for real numbers and ts fast algorthm, IEEE Trans. Sgnal Proess., vol. 63, no. 3, pp , Jul. 05. []. Xao, X.-G. Xa, and H. Huo, Towards robustness n resdue number system, IEEE Trans. Sgnal Proess., vol. 6, no. 8, pp , Mar. 07.