PULSE-TRAIN BASED TIME-DELAY ESTIMATION IMPROVES RESILIENCY TO NOISE

Transcription

1 PULSE-TRAIN BASED TIME-DELAY ESTIMATION IMPROVES RESILIENCY TO NOISE 1 Nicola Neretti, 1 Nathan Intrator and 1,2 Leon N Cooper 1 Institute for Brain and Neural Systes, Brown University, Providence RI Physics Departent, Brown University, Providence RI {Nicola_Neretti, Nathan_Intrator, Leon_Cooper}@brown.edu Abstract. Tie-delay estiation accuracy in echolocating systes decays for increasing levels of noise until a breakpoint is reached, after which accuracy deteriorates by several orders of agnitude. In this paper we present a robust fusion of tie-delay estiates fro ultiple pings that significantly reduces the signal-to-noise ratio corresponding to the accuracy breakpoint. We further show that a siple average of the tie-delay estiates does not shift the breakpoint to a lower signal-to-noise ratio. The proposed fusion of ultiple pings has the potential of iproving the resilience to noise of echolocating systes such as sonar, edical ultrasound and geo-seisic surveys, hence increasing their potential operating range and reducing health/environental hazards. INTRODUCTION The theory of optial receivers shows that the atched filter receiver axiizes the output peak-signal-to-ean-noise (power) ratio [1], and is the optiu ethod for the detection of signals in noise. Inforation about the distance of the target is extracted by coputing the tie at which the cross- correlation between the echo and a replica of the pulse is a axiu. This delay is converted into a distance by eans of the sound velocity in the particular ediu in consideration (e.g. water or air). This type of receiver is generally referred to as a coherent receiver. The classical theory of optial receivers describes the accuracy of tiedelay estiation by atch filtering via the well-known Woodward equation, which can be derived by using a variety of ethods [1]. For sall SNR's, one of the paraeters in the classical equation i.e. the bandwidth has to be odified, and the receiver is then called seicoherent. In [2] it was shown that the transition between the two types of behaviors occurs at different SNR's depending on characteristics of the pulses such as bandwidth and center frequency. With this observation, a novel syste based on an adaptive choice of the pulse was proposed [2] that can iprove accuracy in the case of relatively low SNR, when abiguity in the choice of the correct peak of the cross-correlation function cannot be avoided.

2 Due to the nonlinear nature of the tie-delay estiation proble, when the SNR drops below certain critical values threshold effects take place. Threshold effects can be characterized by a sharp deterioration of the tie-delay estiator variance. Several statistical bounds have been used in the past to describe the accuracy of a atched filter receiver perforance in between the above-entioned SNR critical values. These include the Craer-Rao lower bound [3], the Barankin bound [4], and the Ziv-Zakai bound [5-8]. Such bounds have been applied both to the proble of tie-delay estiation [9-16] and of frequency estiation [17-19]. In particular, the Barankin bound has been used to define the SNR breakpoints corresponding to the change in behavior of the optial receiver as the SNR decreases for the case of a single ping and single echo [10, 12, 13], a single ping and ultiple echoes [15, 16], and ultiple pings and a single echo [11, 14]. In this paper we analyze the SNR breakpoint, studying the probability of choosing the correct peak fro the noisy cross-correlation function with a ethod siilar to the one in [18], where the threshold effect was related to the existence of highly probable outliers far fro the true tie-delay value. This will enable us to extend the result to the case of ultiple pings without a priori knowledge on the tiedelay itself. This approach is different fro that used in previous work on the ultiple pings and single echo case [11, 14], where the ultiple echoes for a single object are obtained artificially via ultiple receivers and a unique ping. In fact, the bounds found in [11, 14] are valid only if the noise at the different receivers is totally uncorrelated or if the distance between transducer and receiver is constant, both conditions difficult to realize in practice. SINGLE PING BREAKPOINT Model for the autocorrelation function In order to copute the probability of outliers, we introduce a odel for the noiseless cross-correlation function envelope in the context of a seicoherent receiver. In this odel the autocorrelation function is approxiated by a piecewise constant function, with aplitude equal to A within the central interval I of length, and zero elsewhere. When white Gaussian noise is added to the echo the cross correlation envelope vector has a ultidiensional Gaussian distribution with centers at zero for all values that are outside of the central interval, and equal to A inside that interval. We also need to consider the width of the a priori window of the cross-correlation. The width of the window corresponds to the echolocating range. While the potential error in delay estiation is reduced when the width is reduced, so is the range. If the a priori window has a length of 2L and the sapling frequency is f s, then there will be NN A +N 0 2L f s points, N A f s of which will be within the central bin ( correct bin ), and N 0 outside the central bin but within the a priori window. Hence, the probability of selecting a given tie location in the cross-correlation function is given by α within the central interval I and (1-α)/N 0 β/n 0 elsewhere (Figure 1, top).

3 α βn 0-1 L B 2 B 3 B 4 B 5 B 1 Figure 1 - The noiseless cross-correlation function is approxiated by a piecewise constant function such that the probability of selecting a given tie location in the cross-correlation function is given by α within the central interval I of length and (1-α)N 0-1 elsewhere (top figure). We then divide the a priori window into intervals of length equal to the size of the correct bin, to obtain intervals B 1, B 2,, B, with B 1 I representing the correct bin (botto figure). Without loss of generality, we consider the case where N A and N 0 are integers. We define a rando vector such that the first N A rando variables correspond to the aplitudes of the points within the correct bin, while the last N 0 correspond to the aplitudes of the points outside. For a white Gaussian noise, the joint probability density function for the vector of n rando variables is then given by: 2 x A 1 2 p 2 ( x1, x2,, xn ) N e σ, v A, A,, A,0,0,,0 ( σ 2π ) (1) NA N0 The desired probability of tie-delay estiation within the correct bin of the center of the cross-correlation function is given by T a α P arg ax { X j} I P( Xi > Xk, k i) Na + 1 j N i 1 xi xi xi i k1 k2 kn 1 i 1 ( x A 2σ ) 2 N 2 x v 1 2 2σ N dx dx dx dx e + ( σ 2π ) Na 1 A ( 2 ) σ [ ( )] N A N0 N 1 e 1+ erf x 1+ erf x dx 2 π (2)

4 Accuracy breakpoint Let T be the rando variable (RV) whose probability distribution is given by equation (2). Let σ be the standard deviation (STD) of the distribution of the echo location in the central (correct) bin, and let σ 0 be the STD of the distribution which is outside the central bin. Now, suppose we saple fro the original distribution whose cuulative function is given by equation (2). For n observations, a fraction of α of the falls in the correct bin on average, while βn fall outside. The standard deviation of the distribution will then be given by: T α T β T ασ std ( ) std ( ) + std ( ) + βσ (3) We define the breakpoint (BP) as the level of noise for which the contribution of T 0 to the total error becoes doinant. Thus, the root-ean-square error (RMSE) will be significantly larger than the one given by the unifor distribution on I alone when α<σ /(σ + σ 2 0 ). We then define the probability breakpoint to be: 2 ( ) 1 ( ) α0 σ0 σ + σ0 σ It is possible to find the SNR breakpoint as the SNR value for which equation (2) equals the value in (4) (4) WHY DOES THE MEAN FAIL Since the easureents fro different pings are independent and identically distributed, the central liit theore (CLT) iplies that the standard deviation (error) of the averaged RV should be n 1/2 ties saller than the error ade by each of the n easureents separately. This is indeed the case before the breakpoint. However, this process does not iprove the situation after the breakpoint and, in particular, does not shift the breakpoint to lower SNR s. Thus, while averaging iproves accuracy, it does not increase noise tolerance. Below we provide a atheatical analysis which explains why the breakpoint does not change. The easureent process described above is equivalent to sapling for a unifor distribution F on the interval I with probability α and fro a unifor distribution F 0 on the interval I 0, with a central gap corresponding to I, with probability β. Suppose we saple n ties to obtain T 1, T 2,, T n and use the saple ean as our estiate for the delay. On average, αn values will be in the correct bin, T 1, T 2,, T αn, while βn will be sapled fro the unifor distribution, T 1 0, T 2 0,, T βn 0. Then the saple ean can be decoposed into two parts: 0 T n 1 n T 1 n i 1 i n α T βn i 1 i T + i 1 i (5)

5 If σ is the standard deviation of the distribution F, and σ 0 is the standard deviation of the distribution F 0, then, applying the CLT to the two sus in equation (5) we obtain: 2 2 αn T 2 n 0 i 1 i n β T n 2 α σ i 1 i 0 β σ std, std (6) The root-ean-square error will be significantly larger than the one given by the distribution F alone when βnσ 2 0 >αnσ 2, i.e. when α<σ /(σ + σ 2 0 ). It is observed that this bound does not iprove with the nuber of pings and is equal to the bound found for a single ping. This explains why averaging the tie-delay estiates fro ultiple pings have not been found useful for very low levels of SNR. It should be noted that it is possible to shift the breakpoint by estiating the tie delay for the averaged cross-correlation functions of all the observations, a process that would require an extreely good alignent of the cross-correlation functions. This is basically equivalent to reducing the noise level by averaging the echoes [17], which is not realistic in a situation where the sonar is not copletely still with respect to the target, or where it is not practical to store the entire echo wavefors for off line processing. However, a successful ipleentation of this averaging can be achieved by using ultiple receivers [11, 14]. USING THE MODE Suppose we divide the a priori window into intervals of length equal to the size of the correct bin, to obtain 2L/ intervals B 1, B 2,, B, with B 1 I representing the correct bin (Fig. 1, botto). If p 1, p 2,, p are the probabilities for an estiate to fall in each of the intervals and Y 1, Y 2,, Y are rando variables representing the nuber of estiates falling in each interval, then Y 1 + Y 2 + +Y n, and p 1 + p 2 + +p 1. The joint probability distribution for the nuber of estiates in each bin is given by the ultinoial distribution n! PY k Y k Y k p p p (7) 1 2 (,,, ) k k k k1! k2! k! The probability of choosing the correct bin using the ode is the probability that the nuber of estiates falling in the correct bin k 1, is greater than the nuber of estiates falling in any other bin k i, i 1: correct 1 ( j, 1) P P Y > Y j k1, k2,, k 1 2 k1 > kj, j 1, k j 1 j n n! k! k! k! l 1 k l l p (8) The su in equation (8) can be decoposed into two parts: 1) the probability P >50% that ore that half of the n points fall into the correct bin; 2) the probability P <50% that even if less than half of the n points fall in the correct bin, the nuber of

6 points in it is greater than that of any other bin, such that P(correct bin)p >50% +P <50%. The probability P >50% can be written as: n k1 n! kl n k1 > 50% p 1 l ( ) 1 1! 2!! l k p p j k k k 1 k1, k2,, k (9) k1> n/2 j 2 k1 > n/2, k j 1 j n P The probability p 1 of an estiate to fall outside the correct bin is unifor over the a priori window with probabilityβ(1-α), so that the probability for it to fall in any interval of size is p j (1-α)/(-1), j 1. Substituting these values into equation (9) we obtain P > 50% n k ( ) ( 1 ) n k k n/2 k α α, (10) > where α is a function of the SNR. The coputation of P <50% is ore coplicated. However, it is possible to derive an upper bound (SNR >50% ) on the SNR breakpoint for the tie-delay accuracy coputed by using the ode of n estiates, as the SNR for which P >50% α 0, where α 0 is given in equation (4): n k ( ) ( SNR> ) 1 ( SNR> ) n k k α α 50% 50% α0 (11) k> n/ Ping (BLACK) 10 Pings (RED) 25 Pings (BLUE) 50 Pings (GREEN) PROB. CORRECT SNR (db) Figure 2 Probability of aking the correct choice as a function of SNR for different nubers of pings. The arrows indicate upper bounds on the SNR breakpoints. The tighter bound corresponding to the total probability of choosing the correct bin will always be lower than the one derived fro equation (11), i.e. SNR BP SNR >50%. The breakpoint which the ode can achieve will be for a lower SNR than the one calculated above, which is already significantly better (Figure 2) than the breakpoint achieved by either a single ping or by averaging of the echo delay estiates of ultiple pings.

7 SIMULATION RESULTS To test the atheatical results presented in the previous sections, we developed a set of Monte Carlo siulations using a cosine packet. We first analyzed the histogras of the errors in the delay estiate of the ideal receiver for different SNR's (figure 3). For high SNR ( 20dB) all the errors are sall and follow the Woodward equation that corresponds to values within the central bin in figure 3a. As the level of the noise increases, large errors in the estiates appear. The errors are uniforly distributed over the entire a-priori window, and the relative ratio between the correct estiates (central bin) and the level of the unifor distribution decreases with SNR (figures 3b, 3c, and 3d). However, even for high levels of noise the central peak is significantly larger that the rest of the distribution. (a) SNR20 (b) SNR8 (c) SNR4 (d) SNR1 Figure 3 Histogras of the errors in the delay estiate in a Monte Carlo siulation for different SNR's. For high SNR ( 20dB) all the errors are sall and follow the Woodward equation that corresponds to values within the central bin in figure (a). As the level of the noise increases, large errors in the estiates appear. The errors are uniforly distributed over the entire a-priori window, and the relative ratio between the correct estiates (central bin) and the level of the unifor distribution decreases with SNR, see figures (b), (c), and (d). However, even for high levels of noise the central peak is significantly larger that the rest of the distribution. Figure 4a shows the perforance of ideal receiver for a single ping. For high SNR the accuracy follows the Woodward equation corresponding to a coherent ideal as expected fro the theory of optial receivers. The perforance breaks for low

8 SNR around 17 db. Figures 4b, 4c and 4d show the analysis of the accuracy breakpoint for different nuber of pings, 10, 50 and 100 respectively. 1 Ping 10 Pings 10-4 (a) 10-4 (b) Pings Pings 10-4 (c) 10-4 (d) RMSE (s) Ave. XC Mean Median Mode SNR(dB) Figure 4 RMSE as a function of SNR and nuber of pings 1, 10, 50 and 100 respectively for the Cosine Packet. Notice how the SNR breakpoint for the average of ultiple pings (red line) does not decrease with the nuber of pings. The blue line describes the optial accuracy that can be achieved using crosscorrelation fro ultiple pings. Its breaking point represents the optial breaking point that could have been achieved using stationary sonar and target, and that could be predicted by using the Barankin bound as in [11, 14]. This breaking point however is not attainable, as it relies on careful registration of returns fro different pings. Such careful registration can only be done if the distance between object to target is kept constant, or if it is known for each ping in advance. It can be seen that robust fusion of ultiple pings based on the ode (light blue, and agenta lines) iproves noise resiliency while retaining close to optial achievable accuracy under ultiple pings. In general there is no significant iproveent in the resiliency to noise when a siple ean of the observations is used due the strong containation of the distribution fro outliers (red lines). This confirs the atheatical result presented in the preceding sections. Figure 5, shows a suary of the results for the different ethods. The breakpoint for the averaged cross-correlation function (blue squares) follows the ideal curve obtained by reducing the level of the noise (solid blue line) as described above. The breakpoint of the estiate obtained fro the ean does not

9 substantially change as the nuber of pings in increased. A ore robust statistics such as the edian iproves the resiliency to noise as the nuber of pings increases (green triangles). The best results are obtained by using the ode of the estiates fro the ultiple pings (agenta diaonds) Mean Breakpoint (db) 10 5 Ave. XC Median Mode 0 Ideal Nuber of pings Figure 5 Breakpoint in db as a function of nuber of pings for different ethods. The solid line corresponds to the ideal case of noise reduction by averaging the echoes. CONCLUSIONS In suary, we have deonstrated that ultiple pings are useful for iproving the accuracy and in particular the resilience of tie-delay estiation to background noise. In particular, we have deonstrated, that a robust statistics such as the ode of the distribution of echo delays which is obtained fro ultiple pings, significantly decreases the signal-to-noise ratio breakpoint. We have further shown that the ean of this distribution has the sae breakpoint as a single ping, thus not contributing at all to the resilience to noise. Acknowledgeents This work was supported in part by ARO (DAAD ), and ONR (N C-02960).

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