Cramér‐Rao lower bound with P d < 1 for target localisation accuracy in multistatic passive radar
2014; Institution of Engineering and Technology; Volume: 8; Issue: 7 Linguagem: Inglês
10.1049/iet-rsn.2013.0213
ISSN1751-8792
AutoresValeria Anastasio, A. Farina, Fabiola Colone, P. Lombardo,
Tópico(s)Target Tracking and Data Fusion in Sensor Networks
ResumoIET Radar, Sonar & NavigationVolume 8, Issue 7 p. 767-775 ArticleFree Access Cramér-Rao lower bound with Pd < 1 for target localisation accuracy in multistatic passive radar Valeria Anastasio, Corresponding Author Valeria Anastasio [email protected] Selex ES S.p.A., Via Tiburtina KM. 12,400, 00131 Rome, ItalySearch for more papers by this authorAlfonso Farina, Alfonso Farina Selex ES S.p.A., Via Tiburtina KM. 12,400, 00131 Rome, ItalySearch for more papers by this authorFabiola Colone, Fabiola Colone DIET Department, University of Rome 'La Sapienza', Via Eudossiana 18, 00184 Rome, ItalySearch for more papers by this authorPierfrancesco Lombardo, Pierfrancesco Lombardo DIET Department, University of Rome 'La Sapienza', Via Eudossiana 18, 00184 Rome, ItalySearch for more papers by this author Valeria Anastasio, Corresponding Author Valeria Anastasio [email protected] Selex ES S.p.A., Via Tiburtina KM. 12,400, 00131 Rome, ItalySearch for more papers by this authorAlfonso Farina, Alfonso Farina Selex ES S.p.A., Via Tiburtina KM. 12,400, 00131 Rome, ItalySearch for more papers by this authorFabiola Colone, Fabiola Colone DIET Department, University of Rome 'La Sapienza', Via Eudossiana 18, 00184 Rome, ItalySearch for more papers by this authorPierfrancesco Lombardo, Pierfrancesco Lombardo DIET Department, University of Rome 'La Sapienza', Via Eudossiana 18, 00184 Rome, ItalySearch for more papers by this author First published: 01 August 2014 https://doi.org/10.1049/iet-rsn.2013.0213Citations: 18AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract By exploiting the sources of opportunity not designed for radar applications, the passive radar system often operates with a Pd in the range from 60 to 85% for acceptable false alarm rates. Multistatic radar networks can be used to both increase detection capability and localisation accuracy. The standard Cramér-Rao lower bound (CRLB) cannot be used for a realistic assessment of the target localisation accuracy in a surveillance area since it provides optimistic predictions. In this study, the authors derive the CRLB with missing observations, namely for Pd < 1, for the multisensory case. The results are illustrated for the case study of a multistatic passive radar exploiting FM radio transmissions where each bistatic pair within the network is assigned with a detection probability for each target position in the surveillance area. The obtained CRLB with Pd < 1 is then employed to compare the localisation performance achievable by a multistatic passive radar when using different sets of measurements: (A) only range measurements, (B) range and Doppler frequency measurements and (C) Doppler frequency measurements only. The reported results show that the proposed CRLB with Pd < 1 yields a more reliable prediction of the achievable performance thus allowing a fair comparison of the above different strategies for target localisation. 1 Introduction Bistatic passive radar systems based on the transmitters of opportunity have received an increased interest in the radar community because of their characteristics of covert operation, low cost, absence of contribution to e.m. pollution etc. [1, 2]. However, non-negligible drawbacks are present in such systems, among which are the use of non-optimised waveforms, the use of broad antenna beams, the low radiated power of the transmitters of opportunity, the high cochannel interference and the dependence on the specific bistatic geometry and radar setup, [3, 4]. Usually, these drawbacks do not prevent an effective target detection by the passive radar, but they imply a detection probability (Pd) typically in the range from 60 to 85% for the acceptable false alarm rates. The low cost of the passive radar suggests the use of multistatic systems, obtained either by employing a network of receivers or by exploiting the signals emitted by different transmitters of opportunity. By fusing the detections obtained by the multiple bistatic sensors, the overall multistatic system can become more reliable and provide increased detection probability [5]. Moreover, the multistatic passive radar network can be used to obtain an accurate, fairly homogeneous, two-dimensional (2D) target localisation, by exploiting the measurements collected by all the bistatic pairs. As is well known, the accuracy of the passive radar measurements can show significant variations since it depends both on the measurements resolution and on the signal-to-noise ratio (SNR). The range resolution of these sensors is directly dependent on the frequency bandwidth of the exploited transmission of opportunity. It ranges from about 2 km, typically provided by the FM-based passive radar systems [6], to about 40 m, provided by the digital TV-based sensors [7, 8]. The Doppler frequency resolution, which depends on the coherent integration time, ranges typically from 0.5 to 1 Hz of the FM-based sensors, to 1–4 Hz of the digital TV-based sensors. It is interesting to note that, even if the measurements accuracy is accurately modelled, the low Pd values, typical of the passive radar operation, make the localisation accuracy predicted by the standard Cramér-Rao lower bound (CRLB) too optimistic, since it neglects the (highly probable) possibility of missing detections. Therefore an appropriate modification is required for the CRLB to provide a realistic prediction of the localisation performance. For this purpose, we exploit the enumeration approach of [9-11] that is here extended and applied to the case of a multistatic passive radar. This is appropriately considered to derive a CRLB with Pd < 1, able to yield a realistic evaluation of the achievable positioning accuracy that fully includes the effects both of the system geometry and of the SNR. For this purpose, an independent detection test is applied for each bistatic TX–RX pair forming the considered passive radar network, so that the contribution provided by the corresponding measurements is averaged over the detection probability evaluated for that bistatic pair. Note that a first attempt towards this direction was presented by Anastasio et al. [12] where the CRLB with Pd < 1 was introduced for a passive radar network exploiting the range measurements only to perform target localisation over a given target trajectory. In this paper, the theoretical derivation is extended to include also other sets of measurements. Specifically, we consider the exploitation of both the range and the Doppler frequency measurements as well as the case of the Doppler frequency measurements only. The reported results clearly sustain the message of this paper, which is, 2-fold: (a) The CRLB with Pd < 1 should be considered as the appropriate measure to assess the localisation performance of a multistatic passive radar network; in fact, by taking into proper account the missing detections yields a more reliable performance prediction. (b) Using the CRLB with Pd < 1 allows a fair comparison of the localisation performance of the considered passive multistatic radar when exploiting different sets of measurements; in particular it yields a more accurate prediction of the performance improvement resulting from the exploitation of the increased sets of measurements. In particular, the latter point is useful for selecting the most appropriate set of measurements to be used for target localisation. Both the range and the Doppler frequency measurements are to be considered only where their joint use provides a better accuracy than using only either the range or the Doppler frequency. The Doppler frequency-only sensors can be useful for operating with very narrow band signals of opportunity, such as the VOR (very high frequency omni-directional range) radio-aids signals. The paper is organised as follows. After introducing the standard CRLB in Section 2, we derive the CRLB with Pd < 1 for the case of a multistatic passive radar. In Section 3, we introduce the considered case study that is representative for an FM-based multistatic passive radar including three transmitters and three receivers. Then, in Section 4, we report the accuracy predictions obtained for the considered case study under the three operational conditions: (A) using only the range measurement to estimate the target position, (B) using the range and the Doppler frequency measurements to estimate the target position and velocity and (C) using only the Doppler measurements to estimate the target position and the velocity over an area of interest (AoI). Finally, we draw our conclusions in Section 5. 2 CRLB with Pd < 1 for a multistatic passive radar The prediction of the ideal positioning accuracy for a localisation system is typically obtained by resorting to the standard CRLB [13]. For a multistatic passive radar system exploiting N bistatic transmitter–receiver pairs, we assume that N bistatic range measurements, , and/or Doppler frequency measurements, , are available (1) with the expected values given by the ideal measurements. In particular, the ideal measurements provided by the nth bistatic couple, that is, the bistatic range Rn(x, y) and the Doppler frequency fn(x, y, vx, vy), have well known expressions as a function of the target position (x, y) and the velocity components (vx, vy) (2) where and are the coordinates of the receiver and the transmitter. The above measurements are assumed to be affected by the zero-mean random Gaussian errors and , respectively, with standard deviations (3) (4) depending on the frequency bandwidth, B, of the signal of opportunity, on the coherent integration time, Tint, and on the SNR corresponding to the transmitter–target–receiver path for the specific nth bistatic pair. To obtain a compact formulation of the CRLB, we define the set of the estimation parameters, Θ = [x, y, vx, vy], and arrange the bistatic range and the Doppler frequency ideal measurements into the N × 1 vectors and . Therefore the element (j, h) of the Fisher information matrix (FIM) can be written as (see the Appendix) (5) where γR = 1 if the bistatic range measurements are used and γD = 1 if the bistatic Doppler frequency measurements are considered The CRLB is given by the inverse of the FIM in (5). In particular, the variances of the estimation errors of the parameters of interest are the elements on the main diagonal of this matrix, namely. For the multistatic passive radar case, this standard CRLB does not provide a realistic prediction of the localisation accuracy, since the following considerations are in order: (1) The single passive radar sensor has often a low detection probability because of many reasons, for example, the low transmitted power levels not intended for a two-way operation, the waveform not under control of the radar designer, the wide antenna beams and the potentially high number of cochannel interferences and so on. (2) Despite the low Pd of the single sensor, a network of the passive radar can be very effective to cover the low height flight regions (especially as a gap-filler in specific areas), because the missed detections are compensated for by the multiple observations by different sensors. Moreover, because of the low cost of a single sensor, the use of multiple sensors in a network is feasible. (3) The operational condition of the multistatic passive radar is therefore especially subject to the detection losses, thus, for properly evaluating the localisation performance of the passive radar network, it is important to take into account the missing observations. (4) For this motivation it is appropriate to derive the modified version of the CRLB for the multistatic passive radar that accounts for Pd < 1. To take into account the Pd < 1, an extension of the enumeration approach described in [9-11] to the multisensory case is derived as follows. We assume that, for each bistatic pair, the radar receiver takes an independent decision on the target presence. Hence, each target can be detected at most by the N bistatic pairs. When it is detected by a bistatic pair, the corresponding measurements are available, that is, range measurement and/or Doppler frequency measurement (depending on the specific type of the sensor considered in the analysis). To represent the decision of the nth bistatic sensor, we introduce a binary variable dn that encodes the event where the target is detected (6) Therefore the set of the N binary variables {dn, n = 1,…, N} can assume one of the L = 2N possible determinations. To account for the missing observations, for the lth determination S(l), the measurements of the nth bistatic pair are considered only if . Consequently, the value of the element j, h of the FIM, conditional to S(l) is given by (7) By defining as the detection probability of the nth bistatic pair, we have dn = 1 with the probability and dn = 0 with the probability. In particular, the lth determination S(l) has the following probability of occurrence (8) Finally, to obtain an unconditional expression for the FIM, we average (7) over the probability of the determination S(l), which yields the desired expression (9) that can be used for a realistic evaluation of the target localisation performance. It is worth mentioning that (9) provides a realistic evaluation of the localisation performance, under the initial assumption that the measurement error follows the Gaussian distribution. This is generally an appropriate model when the sidelobes of the waveform ambiguity function are kept to a low level, no grating lobes are apparent and the very low Pd cases are not included. This is the case, that is considered in detail in the following sections of this paper. When this assumption is not guaranteed, which might well be the case with the passive radar in the low-Pd case, depending on the specific waveform of opportunity and the signal processing chain, the presence of outliers might appear. This implies that the Gaussian hypothesis is not valid when the Pd is low and the RMS error might sensibly increase with respect to the CRLB in (9), which is obtained under a Gaussian assumption, as illustrated for the maximum likelihood [14]. In these cases, the estimation performance of the 2D target location cannot be predicted by the presented CRLB, and a prediction of the RMS estimation error should be provided with an appropriate statistical technique (see, e.g. the approaches in [15, 16]). 3 Case study and its geometry In this section, we introduce an appropriate case study that will be then exploited in Section 4 to compare the performance predictions obtained by using the different versions of the CRLB. For this purpose, we consider the passive radar network represented in Fig. 1, composed of three omni-directional broadcast transmitters and three receivers operating with a wide antenna beam (possibly approaching 180°) that covers the circular shaped area under surveillance with a radius of 60 km (the white region in Fig. 1b). We assume that each receiver exploits only two transmitters far away about 100 km from it. In particular, the following combinations of system elements are considered: RX1 with TX1 and TX3, RX2 with TX1 and TX2 and RX3 with TX2 and TX3, respectively, thus obtaining a multistatic radar network composed of six bistatic pairs. Fig. 1Open in figure viewerPowerPoint Sketch of the case study The geometry of the selected bistatic pairs is compliant with the following constraints for the design of a passive multistatic radar [17]: (i) the whole AoI shall be illuminated by the frontlobe of the receiving antenna; and (ii) the direct signal from the exploited transmitter shall arrive in the low level backlobe region of the receiving antenna (so that the direct signal is attenuated by the antenna pattern). The SNR observed at the nth bistatic pair is evaluated as follows (10) For the considered case study of a passive radar based on FM radio broadcast transmissions, the parameters in (10) are selected as follows. Assuming to exploit the transmitters characterised by a medium power level, their equivalent isotropically radiated power PtGt has been selected to be equal to 10 dBW; the wavelength is λ = 3 m; and the target radar cross section is σ = 7 m2, respectively. The parameters adopted for the receivers have been derived from the experimental prototypes developed by Lombardo et al. [18]. Specifically, we assume that the receivers are characterised by an antenna gain Gr = 8 dB, a bandwidth B = 150 kHz and an equivalent noise figure NF = 35 dB; moreover, they operate with a false alarm probability Pfa = 10−4, by using an integration time Tint = 2 s. The SNR affects both the measurements accuracy, as expressed in (3) and (4), and the detection capability. In particular, by assuming a Gaussian disturbance with a known power level and a Swerling I target model, the target detection probability for the nth bistatic pair can be evaluated as [19, 20] (11) As an example, Fig. 2a reports the detection probability obtained in the surveillance area with the bistatic pair TX1–RX1 using the above parameters. As is apparent, only a limited sector of the considered area is characterised by the Pd values higher than 0.8. As previously mentioned, exploiting a passive radar network can increase the reliability of the resulting system by providing an improved detection probability; this is shown, for example, in Fig. 2b which reports the probability that a target in the surveillance area is detected by at least two out of the six bistatic pairs. Obviously, when the requirement on the minimum number of common detections increases, the resulting probability rapidly degrades as is shown in Fig. 2c which reports the probability that a target is detected by at least four bistatic pairs. Fig. 2Open in figure viewerPowerPoint Detection capability of the multistatic passive radar network in the surveillance area for the considered case study a Detection probability obtained with the single bistatic pair TX1–RX1 b Probability that the target is detected by at least two out of the six bistatic pairs c Probability that the target is detected by at least four out of the six bistatic pairs 4 Example of the localisation performance analysis using the CRLB with Pd < 1 The derivation of the realistic expression of the localisation accuracy, obtained in Section 2, allows us to obtain reliable performance assessments, especially for the passive radar where the detection probability is typically far from unity. Therefore a first objective of this section is to assess whether the new realistic bound gives a performance prediction sensibly different from the standard CRLB that does not account for the Pd < 1. A second objective is to assess the realistic performance improvement achievable by exploiting the Doppler frequency measurements both in conjunction with the range measures and assuming that they are the only measurements available. This is also especially relevant for the passive radar systems, since they are usually characterised by highly accurate Doppler frequency measurements, because of the long coherent integration times used in their processing. Moreover, the use of the Doppler measurements could only be of interest for the sensors that are not able to provide the range measurements, as for example, the continuous waves radar which uses no (or very limited) frequency modulation. Therefore the following three strategies are considered and compared in the following for target localisation: Case A: exploitation of the range measurements only. Case B: joint exploitation of the range and the Doppler frequency measurements. Case C: exploitation of the Doppler frequency measurements only. To show in an effective way the localisation accuracy in the whole surveillance area, we combine the positioning error along the two axes, x and y, into a single horizontal accuracy parameter σH(12) The results obtained for the case study described in Section 3 in Case A are reported in Fig. 3. Fig. 3Open in figure viewerPowerPoint Case A – horizontal accuracy obtained by varying the target position on the AoI considering a multistatic passive radar collecting only range measurements with a Pd = 1, σɛR = 500 m b Pd = 1, σɛR = f(SNR) and c Pd = f(SNR), σɛR = f(SNR), respectively Specifically, Fig. 3a shows the performance prediction obtained by the standard CRLB using the FIM elements in (5) (γR = 1, γD = 0), assuming a constant range measurements accuracy equal to one-fourth of the range resolution cell. This is tantamount to assuming a constant SNR, equal to 9 dB, over the surveillance area for each bistatic pair in the network. In the considered case of an FM-based passive radar, the range measurements accuracy is assumed to be equal to . Fig. 3b shows the standard CRLB using the FIM elements in (5), assuming that the range measurement accuracy at the nth bistatic pair varies with the target position inside the surveillance area as a function of the local SNRn, as described in (3). Therefore, in this case, the SNR is taken into account to determine the range measurements accuracy, but it is assumed that detection is always guaranteed. Finally, Fig. 3c shows the CRLB with Pd < 1, obtained by using the FIM elements in (9) (γR = 1, γD = 0). Therefore, in this case, the varying SNR is assumed to affect both the range measurements accuracy and the detection probability. The following considerations are in order: When the SNR is not taken into account (see Fig. 3a), the predicted 2D localisation accuracy is fairly constant over the AoI. This is due to both the symmetry of the considered geometry and to the joint exploitation of the measurements provided by the six bistatic couples. When the SNR is only assumed to affect the range accuracy but not the detection probability (see Fig. 3b), the accuracy of the range measurement provided by each bistatic couple depends on the specific target position inside the AoI. In particular, it tends to be much better than the range resolution [see (3)] in the proximity of the passive radar baseline because of the highly expected SNR, which potentially yields a high detection probability (see Fig. 2a). Therefore the global localisation accuracy also changes with the specific target position, going approximately from a few meters to 270 m in the centre of the AoI. This appears to be generally a very optimistic performance prediction. Fig. 3c, obtained by considering the uncertainty of the target detection through (9), shows that Pd < 1 causes a significant accuracy degradation. More specifically, it appears that the accuracy increases with respect to Fig. 3a when the global integrated SNR is very high, whereas it sensibly degrades in the areas where the SNR is lower, up to about 440 m at the centre of the AoI. This can be explained by noting that in such a region the number of the measurements jointly exploited can be much smaller than six. On comparing Fig. 3c to Figs. 3a and b, it is apparent that neglecting the missing observations leads to an unrealistic and far more optimistic system performance assessment. The analysis above shows the importance of using our modified CRLB with Pd < 1 in place of the standard CRLB. This appears to be quite important in the passive radar case, where Pd is typically far from unity. To investigate the accuracy improvement potentially achievable by exploiting the Doppler frequency measurements, cases B and C are considered in the following. The corresponding results are reported in Figs. 4 and 5, respectively. In both the cases, we use the measurements potentially provided by the six bistatic couples to estimate the four target motion parameters (x, y, vx, vy). As in Fig. 3, sub-figures Fig. 3a are obtained by assuming Pd = 1 and the constant measurements accuracies equal one fourth of the resolution cells, that is, 0.125 Hz for the Doppler frequency measurements and 500 m for the range measurements where applicable; sub-figures Fig. 3b report the localisation accuracy predicted with the standard CRLB with Pd = 1 but the measurement accuracies are dependent on the local SNR as seen from (3) to (4); and finally sub-figures Fig. 3c show the achievable horizontal accuracy predicted by the CRLB with Pd < 1 [i.e. using (9) with γR = 1 and γD = 1]. It is also interesting to note that the figures are obtained by assuming a target velocity of 250 m/s with an angle of 100° from the x axis. This causes an apparent asymmetry in the figures that exploit the Doppler frequency information. Fig. 4Open in figure viewerPowerPoint Case B – horizontal accuracy obtained by varying the target position on the AoI considering a multistatic passive radar collecting range and Doppler measurements with a Pd = 1, σɛR = 500 m, σɛD = 0.125 Hz b Pd = 1, σɛR = f(SNR), σɛD = f(SNR) and c Pd = f(SNR), σɛR = f(SNR), σɛD = f(SNR), respectively Fig. 5Open in figure viewerPowerPoint Case C – horizontal accuracy obtained by varying the target position on the AoI considering a multistatic passive radar collecting only Doppler frequency measurements with a Pd = 1, σɛD = 0.125 Hz b Pd = 1, σɛD = f(SNR) and c Pd = f(SNR), σɛD = f(SNR), respectively By comparing the different sub-figures, as for case A, we can conclude that neglecting the missing observations provides unrealistic, far optimistic performance predictions also in cases B and C. In particular, in case C, the performance degradation is apparent when taking into proper account Pd < 1; in fact, in this case, there is a high probability for the target to be detected by an insufficient number of bistatic pairs (see Fig. 2c). Therefore the CRLB with Pd < 1 should be considered as the main reference for the following considerations. By comparing cases A to B, we observe that the use of the Doppler frequency measurements, in addition to the range measurements, provides a significant performance improvement. For example, by observing the central part of Figs. 3c and 4c, the accuracy parameter moves from about 440 to 215 m, which is more than a factor of two improvement. As expected, the high accurate Doppler frequency measurements of the passive radar can be used to improve the system performance. In contrast, using only the Doppler measurements provides a coarse localisation accuracy. Basically, better results are obtained by exploiting only the range measurements even if they are characterised by a poorer accuracy. This can be explained by observing that in case A (where the Doppler frequency measurements are not used) only the two position parameters (x, y) need to be estimated, instead of the four parameters (two positions and two velocity parameters). Moreover, the considered geometry does not show a good sensitivity to the Doppler frequency measurements. In contrast, at the centre of the AoI, all the Doppler frequency measurements give an important contribution to the global accuracy and their high accuracy provides a reasonable positioning accuracy (see Fig. 5c). Although the conclusions pertaining to the predicted values of the localisation accuracy are specifically for the considered case study, the reported example shows that only using the CRLB with Pd < 1 allows a fair comparison of the localisation performance of the considered passive multistatic radar when exploiting different sets of measurements. Specifically, it yields a more accurate prediction of the performance improvement resulting from the exploitation of increased sets of measurements (case B). Furthermore, we observe that using the standard CRLB would have led to different conclusions on the comparison between cases A and C; in fact, complementary performances are observed in the AoI when exploiting the range measurements only and the Doppler frequency measurements only (see Figs. 3b and 5b). To analyse in more detail the obtained performance prediction under the different cases considered above and to establish a comparative performance, we consider the localisation accuracy obtained along a selected target trajectory in the AoI, as depicted in Fig. 1b by using small dots. The results are reported in Fig. 6 for cases A–C using the CRLB with Pd < 1 and the measurements accuracy dependent on the SNR according to (3) and (4). Aiming at assessing the performance improvement resulting from the exploitation of the Doppler measurements as their accuracy decreases, in Figs. 6a–d we consider integration times equal to (a) 4, (b) 2, (c) 1 and (d) 0.5 s, respectively. Note that, when reducing the coherent integration time, both the SNR and the Doppler frequency resolution degrade accordingly; therefore the Doppler measurement accuracy decreases quadratically [see (4)]. In contrast, the range resolution independent of the integration time and the expected degradation in terms of the measurement accuracy is only because of the SNR decrease. Fig. 6Open in figure viewerPowerPoint Horizontal accuracy obtained by using the CRLB with Pd < 1 when varying the target position along the considered trajectory, considering a multistatic passive radar in cases A–C, and integration time equal to a 4 s b 2 s c 1 s d 0.5 s, respectively Obviously, as the integration time decreases, the horizontal positioning accuracy degrades in all the considered cases. However, we are interested in understanding the comparative performance analysis when exploiting different sets of measurements. Note that, the results shown in Fig. 6b (Tint = 2 s) for the cases A–C, coincide with those reported in Figs. 3c, 4c and 5c on the considered target trajectory; therefore the same conclusions can be drawn: jointly using the range and the Doppler frequency measurements provides the best accuracy at any point of the trajectory, whereas using only the range or only the Doppler frequency provides the worst estimation accuracy. The use of the range measurements only gives an accuracy with limited variations along the trajectory, whereas the use of the Doppler frequency measurements only gives better results in the centre but largely worse results at the borders of the AoI. By using a coherent integration time of 4 s, Fig. 6a, the joint use of the range and the Doppler frequency measurements continues to show the best performance. Moreover, the very accurate Doppler frequency measurements make the Doppler-only case perform better than the range-only along the whole trajectory. In contrast, when the integration time decreases (see Figs. 6c and d), the degradation in the Doppler frequency measurements accuracy implies a degradation of the positioning accuracy obtained when using those measurements. Therefore the Doppler-only case yields a very limited localisation capability. Moreover, the case of the joint use of the range and the Doppler measurements is in the condition to attempt an estimate of the position and the velocity parameters, using also the very unreliable Doppler measurements (sometimes also missing since Pd < 1). It yields a performance comparable with the range-only case that just has to estimate the two position parameters, since in case B any missed detection removes two measurements (range and Doppler frequency) and the estimation of the larger vector of the parameters benefits from much less information. This shows that the target localisation based on both the range and the Doppler measurements is more affected by the uncertainty of the detection than the case of the range measurements only. Although the numerical values are specifically for the considered example the trend is clearly appropriate and the specific case study allows us to appreciate that, at least in some cases, the above effects can be significant. In this regard, we remark that the comparison performed by using the CRLB with Pd < 1 is fair and realistic in the passive radar network case, because the employed metric accounts for the missing measurements. In summary, we can conclude that the use of the Doppler frequency measurements is highly valuable only when a long coherent integration time can be used, so that a high accuracy is available. When this is not the case, the Doppler frequency information might not be helpful to significantly increase the positioning performance. Moreover, this improvement has to be traded for the required effort in terms of the system complexity and the computational load. Basically, we recall that the integration time can be kept at an upper limit by the expected target range migration and by the affordable computational burden. This especially applies to the passive radar systems based on the waveforms of opportunity with a wider frequency bandwidth (e.g. DAB, DVB-T, WiFi etc.). When the Doppler frequency resolution cell is good enough, an additional benefit of considering the Doppler frequency measurements in the localisation problem is the possibility to estimate the target speed with a good accuracy, as shown in Fig. 7 where on the left, the behaviour of the estimation accuracy of the target speed considering the range and the Doppler frequency measurements and, on the right, the results obtained by considering only the Doppler frequency measurements are reported. As for the target localisation problem, by using only the Doppler frequency measurements we obtain a degraded performance in terms of a target velocity components estimation, this is again because of the reduced set of the exploited measurements compared with the increased number of the parameters to be estimated. Fig. 7Open in figure viewerPowerPoint Target speed estimation error obtained by using the CRLB with Pd < 1 when varying the target position on the AoI in a Case B (exploiting the range and the Doppler measurements) b Case C (exploiting the Doppler frequency measurements only) 5 Conclusions To provide a realistic performance prediction of the localisation accuracy obtained by a multistatic passive radar system, the first result of this paper is the development of an appropriate expression of the CRLB with Pd < 1. Taking into account the missing observations was expected to be quite important for the case of the passive radar, since the individual sensor typically operates with a low Pd. This initial assumption has been made evident by showing that the prediction obtained by using the derived CRLB with Pd < 1 departs significantly from the standard CRLB, which provides far optimistic predictions. The derived expression of the CRLB has then been used to compare the localisation performance achieved by a multistatic FM-based passive radar over a specific observation area using, respectively, (A) range-only, (B) range and frequency and (C) Doppler frequency measurements only, collected by each sensor. The proposed CRLB with Pd < 1 allowed a fair comparison of the performance and a reliable assessment of the actual improvement to be expected when the Doppler frequency measurements are considered in addition to the range measurements. In particular, we showed that a significant benefit should be expected when a very high resolution is available for the Doppler frequency, namely when very long coherent integration times can be used. Under this condition, it is possible that using only the Doppler frequency measurements can provide a good performance. Moreover, the exploitation of the Doppler frequency measurements allows us to estimate the target motion components, which provides more information on the target. In contrast, as the integration time decreases, the Doppler frequency measurements do not provide a significant performance improvement, so that it can be convenient to consider only the range measurements. Although the main innovation in this paper is to take into account the missed detection in the evaluation of the localisation performance for a passive radar network, we must note that the results apply under the Gaussian hypothesis for the measurement errors. As mentioned in Section 2, there may be cases with a passive radar in the low-Pd case, – also depending on the specific waveform of the opportunity and the signal processing chain – where the Gaussian hypothesis is not valid and the RMS error sensibly increases with respect to the CRLB. The derivation of an appropriate performance bound for those cases, also including the missing information is the object of ongoing research activity and will be reported in future papers. 6 Acknowledgments The authors would like to thank the anonymous reviewer who addressed the importance of considering a possible non-Gaussian distribution of the passive radar error measurements to provide a prediction of the RMS error, and made it possible to clarify the range of application of the presented results. 8 Appendix Cramér-Rao lower bound Let us consider a set of N range measurements and Doppler frequency measurements , collected by N bistatic pairs. They are assumed to have expected values given by the ideal measurements, and , and to be affected by zero-mean random Gaussian errors and , respectively, with standard deviations and . Furthermore, the range and the Doppler measurements are assumed to be statistically independent, along with the measurements provided by different bistatic pairs. On defining the set of the estimation parameters, Θ, the derivative of the logarithm of the joint PDF of the available measurements with respect to the parameter Θj is given by (13) Consequently, the element (j, h) of the FIM can be written as follows (14) where the cross-terms have been neglected because of the statistical independence of the available measurements. Similarly, after straightforward calculations, we obtain By arranging the ideal bistatic range measurements and the Doppler frequency measurements in the N × 1 vectors R and f, the element (j, h) of the FIM can be written in a matrix notation as (15) where and . 7 References 1Willis, N.J.: ' Bistatic radar' (Technology Service Corp., Silver Spring, MD, 1995, 2nd edn.) 2Griffiths, H.D., Baker, C.J.: ' Passive coherent location radar systems. Part 1: performance prediction'. IEE Proc. Radar, Sonar and Navigation, June 2005, vol. 152, no. 3, pp. 153– 159 3'Special Issue on Passive Radar Systems'. IEE Proc. Radar, Sonar and Navigation, June 2005, vol. 152, no. 3, pp. 106– 223 4Griffiths, H.D., Baker, C.: ' The signal and interference environment in passive bistatic radar'. Proc. 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