Cross-correlation tracking for Maximum Length Sequence based acoustic localisation

Transcription

1 Cross-correlation tracking for Maxiu Length Sequence based acoustic localisation Navinda Kottege Research School of Inforation Sciences and Engineering The Australian National University, ACT, Australia Uwe R. Zier Research School of Inforation Sciences and Engineering The Australian National University, ACT, Australia Abstract While Maxiu Length Sequence based cross-correlation and localisation ethods are specifically robust with respect to ultiple fors of disturbances, the ethod is still subject of becoing 'side-tracked' by echoes, reverberations and other resonance effects. This article details a ethod to track the correct reading throughout a series of easureents which is specifically designed for the cross-correlation localisation technique as deployed in underwater environents by the Serafina project. The ethod is based on the selection of the ost likely candidate rather than Kalan filters or other low-pass filters and extrapolation ethods. Therefore every individual reading is an actual easureent, rather than an extrapolation. As transducers with non-linear frequency responses are coon in energy-efficient underwater acoustic setups, the resulting signal deforations are explicitly addressed and corrected by eans of inverse frequency transforations. The article discusses in detail a series of experients perfored in a highly reverberant underwater environent. The achieved perforances include standard deviations of a less then 3 degrees in aziuth estiation, and 11 c in range estiations throughout the presented sequence of experients. 1. Introduction In attepting to design and ipleent localisation systes for autonoous vehicles, the transducer characteristics as well as the operating environent contributes to the perforance of the syste. This is especially true in underwater environents where localisation is perfored using acoustic signals. The underwater acoustic localisation syste proposed in [Kottege and Zier, 7; Kottege and Zier, 8a] and later developed in [Kottege, 8; Kottege and Zier, 8b] uses low cost transducers as projectors and hydrophones while using acoustically transitted Maxiu Length Sequence (MLS) signals [Peterson et al., 1972]. The accuracy and precision of the estiates produced by this syste is affected by the non-linear frequency response of the low cost transducers as well as the cluttered underwater environents which produce ultiple reflections of the acoustic signal. The two-fold outlier rejection schee described in the following sections anage to copensate the effects of the transducer frequency response while recovering estiates which otherwise would be lost due to reflected signals. 2. Cross-correlation of MLS signals The ain tool used in [Kottege, 8; Kottege and Zier, 8b] for estiating the aziuth, range and heading of a signal source is tie-doain cross-correlation of MLS signals. The theory associated with the statistical properties of MLS signals [Borish and Angell, 1983; Borish, 198a; Borish, 198b; Bradley, 1996] recoends the use of a longer sequence to iprove the definition of the cross-correlogra peak. However, using signal chirps with a longer duration attracts undesirable reverberation effects in cluttered and enclosed acoustical environents. As a coproise, a 127 point MLS signal with a duration of 1.3 s when sapled at 96 khz is used. The peak position of the cross-correlogra reains unaffected by the introduction of uncorrelated noise on to the source signal channels while the peak height is reduced as a result. Figure 1 shows the cross-correlogra produced by cross-correlating two MLS signals with a relative shift of 8 saples which were ixed with white Gaussian noise with a signal to noise ratio of db.

2 Saples db re 1V μpa AQ- Sensitivity Response , 1, 1, Frequency (Hz) Figure 2: The frequency response curve of the hydrophone reproduced fro the AQ- data sheet [] Saples 3.1. Inverse frequency response filtering To test the effect of this potential frequency filtering effect introduced by the transducers, the frequency response shown in Figure 2 was epirically odelled 1 and ipleented as an FFT filter. The filter was then applied to white noise 1. a) MLS Generator Transducer response filter Shift / Delay. White noise addition b) MLS Generator Transducer response filter Shift / Delay Inverse transducer response filter Figure 1: Plot shown in c) results fro the cross-correlation of the shifted noisy MLS signals shown in a) and b). In each plot, the y-axis reprinter noralised aplitude. White noise addition Cross-correlation 3. Effect of transducer frequency response As it was shown by Figure 1.c, containation by additive white noise does not contribute to a noticeable deterioration of the cross-correlation perforance of MLS signals, apart fro a slightly lower height for the peak. In theory, addition of two flat frequency spectra should again result in a flat frequency spectru, hence the spectral properties of the MLS signal which provides the narrowness of the peak are preserved. However, this is not necessarily the case when these signals are transitted and received via transducers with a non-linear frequency response within the bandwidth of the signal. Due to the sapling rate used, the signals are band liited by 48 khz. The frequency response of the Benthos AQ- transducers which are used as transitting projectors as well as receiving hydrophones by the localisation syste is shown in Figure 2. As seen fro this logarithic plot, the transducers have a resonance near khz and an anti-resonance near 2 khz which results in a highly non-linear response within the signal bandwidth. c) d) MLS Generator Signal preparation & aplification Projector MLS Generator Signal preparation & aplification Projector Pre-aplification & analogue to digital conversion Hydrophones Pre-aplification & analogue to digital conversion Hydrophones Cross-correlation Inverse transducer response filter Cross-correlation Figure 3: The four different setups producing the four different cross-correlogras shown in Figure The shape of the frequency response curve was replicated as the shape of a frequency response curve of an FFT filter.

3 containated MLS signals earlier shown in plots a) and b) of Figure 1 and the two filtered signals were cross-correlated. This setup is illustrated in Figure 3.a. The effect it has on the cross-correlogra shown in Figure 4.a can be copared to the cross-correlogra shown in Figure 1.c, which was produced by the sae source signals albeit the frequency filtering. Resonance of the transducer near khz appears as the doinant frequency in the resulting cross-correlogra. For coparison, Figure 4.c shows the cross-correlogra of two signal channels received via two AQ- transducers a) Adjacent side lobe peaks Main peak Adjacent side lobe peaks a) 1.. b) 1. Main peak b) c) Figure : Cross-correlogra plots resulting fro a) unfiltered signal channels and b) filtered signal channels, showing the ain peak and adjacent side lobe peaks caused by the resonance frequency of the transducers. The y-axes on these plots represents the noralised aplitude. d) Figure 4: Plots in a) and b) represents the cross-correlogras of the shifted MLS signals containated with additive white noise, first filtered with the transducer frequency response, then filtered with the inverse of that filter. Plots c) shows the crosscorrelogra resulting fro two actual signal channels (with a shift of +8 saples) which were transitted and received using the transducers. Plot d) shows the resulting cross-correlogra when the inverse transducer filter was applied to the signal channels prior to cross-correlation. In each plot, the y-axis represents the noralised aplitude of the signals. (separated by.3 ) when an MLS (length-127, duration 1.3s ) signal was transitted via another AQ- transducer.the signal travelled a distance of 2. underwater and the cross-correlation reveals a delay of +8 saples (83.33 μs) between the channels. This setup is scheatically illustrated in Figure 3.c. Both the cross-correlogras, one fro cross-correlating the real and the other fro cross-correlating the siulated signals, shows the doinance of the khz resonance of the transducer throughout the plots. Even though, the position and height of the peak is not affected, the uniqueness of the peak had been lost by being surrounded by an envelope of decaying side-lobes (Figure ). This decreases the accuracy of the localisation syste when used in enclosed and cluttered environents due to peaks caused by reflected signals. In order to address this issue, another filter was epirically odelled which had the inverse frequency response of the one odelled earlier to represent the response of the transducer. The results of applying the new FFT filter 1 to the siulated and the real signals used earlier and cross-correlating the channels are shown in plots b) and d) of Figure 4 respec- 1. The filter was applied to the two signal channels prior to being cross-correlated.

4 a) b) c) d) e) f) Nuber of estiates Nuber of estiates Figure 6: Contour plots shows series of cross-correlogras resulting fro three different experients shown on three rows. tively. The setups used in these instances are scheatically depicted in Figures 3.b and 3.d. As depicted by the plots the uniqueness and narrowness of the peak are restored aking it possible to unabiguously locate it. It ust be ephasised that the filter process only accounts for the transducer characteristics and not those of the propagation ediu. Since channel characteristics of the underwater ediu greatly varies with depth, teperature and salinity as well as environental features such as the coposition and texture of the botto (sedient / sand / vegetation), odelling the transducer characteristics are ore practical. In the context of an autonoous underwater vehicle, it is far ore convenient to account for the characteristics of onboard sensors than to have access to a odel of the channel characteristics of the operating ediu. However, the underwater channel does indeed have an effect on the trans-

5 itted MLS signal as seen in Figure 4.d where the filtering process does not copletely reconstruct the original crosscorrelogra (Figure 1.c). Furtherore, the power of the received signals are greatly reduced by the inverse filtration process since ost of the transitted signal power is around the resonance frequency of the transducer. This attenuation affects the distance which the MLS signals can be effectively transitted at a given transission power Effectiveness of filtering Figure 6 shows contour plots of noralised cross-correlogras resulting fro cross-correlating the two experientally recorded hydrophone channels corresponding to the MLS chirps eitted by a projector. The receiving hydrophone pair was rotated about the id point of the line connecting the during the experient 1. Depending on the direction of rotation, the tie difference of arrival (TDOA) between the two channels either increase or decrease during the experient resulting in either an increasing or decreasing acoustic pathlength difference. The three rows shows three different experiental runs while the first colun is without any filtering and the second colun is with the inverse frequency response filter applied. In each of the contour plots, each vertical slice is a noralised cross-correlogra such as the ones shown in Figure 4, with the peaks oving diagonally fro top to botto (first row) or vice versa (second and third row) as the experient proceeds with increasing estiates. For clarity, closely cropped versions of the cross-correlogra plots c) and d) in Figure 4 is presented in Figure. The three different experients whose results are depicted in Figure 6 were selected to show varying levels of effectiveness and necessity of the filter being applied. The instance shown in plot a) has at least three adjacent peaks copeting for proinence characterised by red parallel ridges continuing diagonally across the plot. This is due to the doinant frequency coponent introduced by the resonance of the transducers. This behaviour of the cross-correlogra peaks causes the precision of the relative localisation syste estiates to decrease (the standard deviation of errors increase). As a result of the filter, the side lobe peaks in the cross-correlogras subsides leaving only the ain ridge of peaks as shown in plot b). Plot c) depicts an experient where the peaks of the cross-correlogras were affected by actual reflected signals (off the test tank walls in which the experient was conducted in) apart fro the ultiple adjacent peaks due to resonance as in the previous case. In the first instance of reflection between estiates 8 to 1, the peaks are lost while in the 1. Experients conducted in a cylindrical test tank (Diaeter 4.2, depth 1. ) with corrugated etal walls and filled with freshwater. See [Kottege and Zier, 8b] and [Kottege, 8] for details of experiental setup. second instance between estiates 14 to 16, a series of outlier peaks appear further away fro the continuing ridge of peaks. As a result of the filter, the adjacent peaks subside leaving the ain ridge as before but the outlier peaks still reain as seen fro plot d). In plot e), though there are ultiple ridges flanking the ain ridge caused by adjacent side lobe peaks in the crosscorrelogras, the ain ridge is continuously higher than its flanks. As expected, the filtered signals causes the side lobe peaks to subside leaving only the proinent ain ridge shown in plot f). The experient represented in the first row clearly benefits fro the filtering schee as it helps to subdue unwanted side lobe peaks. In the experient shown in the third row, due to the height of the ain ridge copared to its flanks, filtering does not necessarily introduce any iproveent to the estiates even though the side lobe peaks are suppressed in the process. In the experient depicted in the second row, while filtering contributes to reducing the standard deviation of estiation errors, it does not iprove perforance in areas affected by reflections (i.e. peak drop-offs and outliers). Such situations necessitate an additional schee for outlier rejection. 4. Outlier rejection via peak tracking The localisation syste suggested in [Kottege and Zier, 7] and [Kottege and Zier, 8a] uses a siple axiu search routine coupled with sub-saple spline interpolation for finding the peak of a cross-correlogra and the corresponding position of the peak. The following peak tracking schee builds on this ethodology and contributes to effectively reject outliers arising due to interference. The ain feature of this schee is that it enforces continuity assuptions of the estiated quantities by considering the physically possible variation of each raw estiate (in saple space) within one estiation step. Here R s1 s 2 refers to the full range cross-correlation of two length N signal channels s 1 n and s 2 n which includes both positive and negative discrete lags in saple space spanning N N. If x n denotes an eleent of the resulting cross-correlation with a lag of n saples and can be expressed as: xn x N x N = R s1 s 2 (1) Xn is defined as a set containing all ordered pairs of lags and corresponding value of the cross-correlogra as follows: Xn = nxn n N N (2) Another set X Sorted is fored by sorting X n in descending order by the value of x n as: X Sorted = y N N (3) Therefore, the following conditions are satisfied by the eleents of X Sorted and Xn :

6 6 Path length difference [] x δ 1 δ Error in aziuth [ ] Error in range [] x Nuber of estiates Error in heading [ ] 6 Path length difference [] x δ 1 δ Error in aziuth [ ] Error in range [] x Nuber of estiates Error in heading [ ] 6 Path length difference [] x δ 1 δ Error in aziuth [ ] Error in range [] x Nuber of estiates Error in heading [ ] Figure 7: The interediate values produced by the relative localisation syste which consists of the acoustic path length differences are plotted along with the errors of the final estiates which are calculated using these values. The first row is with no filtering, the second row is with inverse frequency response filtering applied and the third row is with filtering and peak tracking on the cross-correlogras enabled. N N n N N s.t. xn = y (4) n N N N N s.t. () y = xn N N 1 y y + 1 (6)

7 No filtering and no peak tracking Filtered with no peak tracking Filtered with peak tracking Std. dev. of aziuth error Mean of aziuth error Avg. dev. of aziuth error Std. dev. of range error r Mean of range error r Avg. dev. of aziuth error r Std. dev. of heading error Mean of heading error Avg. dev. of heading error Table 1 : Coparison of standard deviations, eans and average deviations of errors associated with aziuth, range and heading estiates corresponding to the interediate values plotted in Figure 7. where n N N x n y 1 1 and (4), () aintains bijectivity between Xn and X Sorted. A set M k corresponding to estiation step 1 k is defined as follows: Mk = N N s.t. (7) I k 1 Tolerance where is drawn fro the ordered pairs in the set X Sorted, Tolerance being a tolerance value based on the continuity assuptions of the quantity being estiated and I k 1 being the sub-saple interpolated lag at estiation step k 1. The lag of the new tracked peak of the cross-correlogra which aintains continuity with the previous estiates denoted by k is the iniu eleent of M k given as: k = in M k (8) This procedure essentially perfors a local axia search of the cross-correlogra within the neighbourhood of lags around the previously estiated lag. The size of the neighbourhood is decided by the value selected for Tolerance. The otivation behind the ipleentation described above which involves sorting the eleents of the cross-correlogra, was to provide a facility to dynaically liit the search doain for lags by restricting axiu to soe N restricted N in (7) (which in turn violates the bijectivity condition between X n and X Sorted ). This would specify a lower bound to the peak agnitude in the cross-correlogra 1. This also corresponds to the k th logical tie-step as well as the k th sending event. which can correspond to the lag returned by (8). However, this approach was not ipleented for the version of the localisation syste experientally evaluated in this text. Once the discrete lag is selected by (8), the sub-saple interpolated lag is returned by the cubic spline interpolation function I Spline which takes in n Int which is the nuber of sub-saple interpolation steps as a paraeter to produce the real lag at estiation step k : I k = I Spline k (9) where k N N I k N N. There are ultiple cross-correlations per estiation step which results in ultiple sets for M k used for estiating the aziuth, range and heading by the localisation syste. For each of these quantities, there is a corresponding tolerance value Tolerance calculated using the axiu possible variation (constrained by the physical capabilities of the experiental setup or autonoous vehicle used) of the lags in the saple doain corresponding to variation of angles and distances within the duration between two estiation steps. The underlying assuption is that these variations aintain continuity between estiation steps. As a boot-strapping technique, initially peak tracking is disabled and siple axiu searching is used to find the lags corresponding to the axiu agnitude peak in the cross-correlogra as no prior estiate is available at initialization. Once the position of the peak stabilises (e.g. detected by a result sequence which can be explained by the axial relative speeds of the vehicles, i.e. no discontinuities over a certain nuber of estiation steps), peak tracking is enabled.

8 . Analysis As entioned earlier, unlike the siple search used to locate the peak which only considered the aplitude of the crosscorrelogra, with peak tracking, the history of the previous peak positions is incorporated in to the search paraeters and a higher proinence is given to the position rather than the aplitude of the peak. Peak positions estiated in this anner are refined as before using sub-saple interpolation. The cobination of inverse frequency response filtering and peak tracking significantly iproves the accuracy and precision of the estiates produced by the relative localisation syste. The effect of inverse frequency response filtering and peak tracking on the interediate values produced (path length differences) by the localisation syste and the corresponding errors associated with the final estiates are depicted in the plots shown in Figure 7. The first colun of Figure 7 contains plots of path length differences corresponding to TDOAs between the two hydrophone channels during an experient with two source signals transitted in sequence. In the experient depicted in Figure 7, the receiving hydrophone pair was rotated such that the aziuth of the source varied as : 9 9. Plot a.i) was obtained using unfiltered hydrophone channels as the inputs to the cross-correlation with no peak tracking on the cross-correlogras, plot b.i) was with the inverse frequency response filter applied but without any peak tracking while plot c.i) was with the filter applied and with peak tracking enabled. Plot a.i) shows the effects of the adjacent side lobe peaks which causes the outliers on either side of the ain path length differences. When the inverse frequency response filter is applied, a deterioration of the values around 1 and 3 estiates is clearly noticeable. These areas correspond to aziuths of 9 and 9 where the hydrophones on the observer are alost pointed in a direction perpendicular to projectors eitting the source signal. Due to the directivity pattern of the AQ- hydrophones which are non-oni-directional, the direct-path signals that are received in these regions would carry ost of their energy in frequencies within a narrow bandwidth centred at the resonance frequency of the transducers. The filtering process which ais to soothen out the received frequency spectru by attenuating spikes caused by resonance contributes to further reduce the energy of the direct-path signals. Under these circustances the signals reflected off the curved etal walls of the test tank, which ipinge on the hydrophones fro the front, tend to carry ore energy than the direct-path signals. Hence the cross-correlogra peaks due to reflections tends to have a higher aplitude than those due to direct-path signals in these regions causing the perforance of the estiation to deteriorate when the filter is applied. Due to causality, peaks caused by directpath signals occur before the peaks caused by reflected signals albeit with uch lower aplitude. The peak tracking algorith exploits this feature as explained in the previous section and anages to retrieve the peaks caused by direct path signals. The results can be seen by coparing plot b.i) and c.i) where applying filtering and peak tracking (with Tolerance = 1.2 ) copletely eliinates outliers in the path length difference easureents which are subsequently used for the angular estiations. The second colun in Figure 7 shows the errors in the final coponents of the pose vector, aziuth, range and heading calculated using the interediate values plotted in the first colun. As observed fro these plots, the standard deviation of the errors are greatly reduced when peak tracking is enabled while filtering does not lead to significant iproveents. The standard deviations and eans of these errors are copared in Table Conclusions As seen by the error plots given in Figure 7, the peak tracking schee and inverse frequency response filtering schee operating at an early stage in the estiation process is extreely effective in eliinating outliers and heavily contributes towards iniising estiation errors. As a future work, the peak-tracking algorith will be iproved to accoodate dynaically changing paraeters such as the search doain size which introduces a lower bound to the peak agnitude. Experients need to be conducted to further analyse the effect of the paraeter Tolerance on the estiates produced by the syste. References [Borish and Angell, 1983] Borish, J. & Angell, J. B., An efficient algorith for easuring the ipulse response using pseudorando noise; Journal of the Audio Engineering Society, 31(7), August 1983, pp [Borish, 198a] Borish, J., An efficient algorith for generating colored noise using a pseudorando sequence; Journal of the Audio Engineering Society, 33(3), March 198, pp [Borish, 198b] Borish, J., Self-contained crosscorrelation progra for axiu -length sequences; Journal of the Audio Engineering Society, 33(11), Noveber 198, pp [Bradley, 1996 ] Bradley, J. S., Optiizing the decay range in roo acoustics easureents using axiu-length-sequence techniques; Journal of the Audio Engineering Society, 44(4), 1996, pp [Kottege and Zier, 7] Kottege, N. & Zier, U. R.; Relative localisation for AUV swars, in proceedings of the IEEE International Syposiu on Underwater Technology 7 (SUT '7); Tokyo, Japan, April 7. [Kottege, 8] Kottege, N.; Underwater Acoustic Localisation in the context of Autonoous Subersibles; subitted at The Australian National University, Canberra, ACT, Australia, April 8.

9 [Kottege and Zier, 8a] Kottege, N. & Zier, U. R.; MLS Based Distributed, Bearing, Range and Posture Estiation for Schools of Subersibles, Experiental Robotics; Springer: Berlin; Heidelberg, March 8, pp [Kottege and Zier, 8b] Kottege, N & Zier, U. R.; Acoustical Methods for Aziuth, Range and Heading Estiation in Underwater Swars, In proceedings of Acoustics 8, Paris, France, July 8 [Peterson et al., 1972] Peterson, W. W. & Weldon, E. J. Jr., Error-correcting codes, MIT Press, Cabridge, Massachusetts, [Serafina, 8 ]