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Communications______________________________________________________________________ An Asymptotically Unbiased E-Pulse-Based Scheme for Radar Target Discrimination David Blanco, Diego P. Ruiz, Enrique Alameda, and María C. Carrión Abstract—This communication proposes an E-pulse-based scheme for radar target discrimination that provides asymptotically correct results for any level of additive white noise contaminating the radar signal. After multiple sampling of the signal dispersed by the target, it is analytically shown that the cross correlation between the output signals of the E-pulse designed for the standard target, corresponding to two different sampling periods, is asymptotically null, regardless of the amount of contaminating noise. The results obtained by simulation have allowed us to propose a discrimination criterion that produces better results than the original E-pulse technique at very low signal-to-noise ratio (SNR) levels.
Fig. 1.
Original E-pulse scheme.
Index Terms—Extinction pulse, natural resonances, radar target discrimination, radar target identification.
I. INTRODUCTION In recent decades, various techniques for the discrimination of radar targets based on the concept of natural target resonances have been proposed in the literature. Particularly noteworthy is the extinctionpulse (E-pulse) technique [1]–[9]. The basis of this scheme consists in synthesising an E-pulse Ep(t) such the convolution of the transient, late-time response of a specific or standard target and Ep(t) is null [1]. This technique has been analyzed in several subsequent works [2]–[4], and several authors have proposed changes and studied its performance when the target response is contaminated by noise; see for example [5], [6], and references therein. Moreover, the applicability and feasibility of this technique has been studied in several published articles [7]–[9]. The present communication first carries out a study of the asymptotic performance of the E-pulse technique in the presence of contaminating white noise, analysing the causes of resulting deterioration at low SNR. We then propose a modification of the E-pulse discrimination scheme that provides asymptotically correct results for any amount of signal-tonoise ratio (SNR). II. PERFORMANCE ANALYSIS OF THE E-PULSE SCHEME Let r(t) be the complex late-time [1] radar return scattered by one of the N conducting targets of the library. In the original E-pulse scheme, an E-pulse filter is constructed for each of the N targets such that the output of r(t) is null when it passes through the filter designed for the target from which the signal derives (Fig. 1). After passing r(t) through the N E-pulse filters fEpi (n)gi=1;...;N ; N output signals fyi (n)gi=1;...;N are obtained and their energy fZi (n)gi=1;...;N is calculated. Finally, the target with the lowest energy after normalizing to the total energy in the scattered pulse, Zi is chosen as correct [2].
Manuscript received October 1, 2002; revised March 6, 2003. This work was supported in part by the Ministry of Science and Technology of Spain under Project TIC 2001-2902. The authors are with the Department of Applied Physics, Faculty of Sciences, University of Granada, 18071 Granada, Spain (e-mail:
[email protected]). Digital Object Identifier 10.1109/TAP.2004.827251
Fig. 2. Proposed E-pulse scheme.
The radar signal is often contaminated by noise, so that x(n) = r(n) + v (n), where v (n) is an additive, complex white gaussian noise. In this case, the output signal becomes
yi (n) = Epi (n) 3 r(n) + Epi (n) 3 v (n):
(1)
Calculating the energy of yi (n) in a time interval of span W = M T; M being the number of data points and T the sampling period, and applying Parseval’s theorem [10], we obtain in the frequency domain
Zi
=
1
W
M
k)X (k)j2 + v2 jEpi (k)j2
jEpi ( k=0
+ 2 v R(X (k ))jEpi (k )j
2
(2)
where v is the noise standard deviation, Ep(k) and X (k) are the discrete Fourier transform of the signals Ep(n) and x(n), respectively, and ( 1 ) denotes the real part. If the correct target is p, for i = p the first and third terms of expression (2) are null. Therefore, Zp depends on the noise standard deviation v and on the filter energy Epp (n). When synthesising the E-pulse filters, they must all have the same energy so that the comparison between Zi provides the desired information. Therefore, the second term will be identical for the different terms. However, the value of the third term for i 6= p depends on channel i, on the dispersed signal x(n) and on the variance of the noise. Specifically, it can give a negative contribution to the energy, which could produce a lower energy value for a false channel than that obtained for the real channel, as long as the contribution of the first term is small enough.
R
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Fig. 3. Percentage of misclassification for a thin wire scattered signal using subsectional pulse basis functions. discrimination scheme.
If this negative value of the third term occurs, as it is multiplied by the noise variance, the only requirement is to increase v , i.e., decreasing the SNR, so that this term annuls the contribution of the first term and a misclassification will occur. III. PROPOSED E PULSE-BASED RADAR TARGET DISCRIMINATION SCHEME The proposed discrimination scheme takes advantage of the statistical properties of white noise with the aim of obtaining a correct E-pulse output that is asymptotically null, even in the presence of noise, that is, regardless of v (Fig. 2). The signal scattered in the late-time x(t) = r(t) + v (t) is sampled in two different time intervals T1 and T2 obtaining two, statistically independent signals x1 (n) = r1 (t) + v 1 (t) and x2 (n) = r2 (t) + v 2 (t). These two signals pass through the E-pulses corresponding to the N library targets samples at periods T1 and T2 as well, producing fEpi1 (n)gi1;...;N and fEpi2(n)gi1;...;N , respectively. 2N signals are thus obtained
zis (n) = Epis (n) 3 xs (n) = dis (n) + vis (n)
(3)
with
dis (n) = Epis (n) 3 rs (n) vis (n) = Epis (n) 3 v s (n)
i = 1; . . . ; N
and
s = 1; 2:
If we calculate the cross correlations [10] corresponding to the same target, but with different sampling periods, and consider p to be the index of the correct target, we obtain
ri ( ) =
n di (n)di (n + ) + 1 2 n vp (n)vp (n + ) 1
2
n vi (n)vi (n + ) 1
2
for
i 6= p (4)
where it can be observed that the cross correlations have no random component and are therefore independent of additive noise [which is different to that occurs in (3)]. Based on these cross correlations, we can define as an identification criterion the filter that provides the least energy. Thus, given an interval of maximum time delay 0 , the correct target will be that for which Ei is minimal, defining Ei as
Ei
=
jr i ( ) j : 2
0
=
(5)
Original E-pulse scheme.
Proposed
A selection criterion for the correct target is thereby obtained that is valid and asymptotically independent of the amount of additive noise. In order to improve the discrimination accuracy and to reduce the misclassification probability, the contribution of the deterministic component in (4) should be always positive. In practice, this is achievable if the minimum cross correlation without lag ( = 0) is taken as the discrimination criterion instead of the output energy given in (5), T1 and T2 are close enough and their corresponding sampling frequencies are large compared to the natural frequency corresponding to the maximum target resonance. Fig. 2 shows the proposed scheme, using the zero lag = 0 in the cross correlation (4) as the discrimination criterion.
IV. SIMULATION RESULTS To carry out a performance study of the proposed scheme, we have chosen for the discrimination test three conducting wires of lengths L; 0:9L and 1:1L, and a radius a given by L=a = 200. The wire impulse response has been synthesized using the first four modes of the first branch [11], and the sampling periods T1 = 0:02 L/c and T2 = 0:019 L/c have been chosen. In the original E-pulse scheme, T1 was used as the sampling period. The election of the sampling periods T1 and T2 should be as close as possible, whenever the noise is actually white. Closer the two periods, greater the contribution of the signal in the wrong channel to the crosscorrelation in (4) and lesser the probability of the noise contribution to be bigger than the signal contribution in (3). Then, natural E-pulses corresponding to the subsectional pulse base functions [1] have been synthesized for the three conducting targets. In the simulations, we have calculated the number of times that the original scheme and the proposed one mistakenly identify target L as correct in 100 Monte Carlo runs, as a function of the SNR. The results obtained for 60 pulse incidence angle are shown in Fig. 3. In all the simulation the correct target is the target of length L. The choice of the correct target affects only in the absolute values of the results but not in their relative values. The length of the target response is taken in all cases as 20 L/c. As can be observed in the figure, the proposed scheme is an important improvement over the original E-pulse scheme, allowing a completely correct identification up to SNR as low as 05 dB and always showing fewer errors than the original scheme. The same conclusions can be obtained if other types of E-pulses are used for discrimination. For example, if quadratic basis functions are used to expand the polynomial E-pulses [4], the percentage of misclassification are shown in Fig. 4. As in the preceding case, the results show
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Fig. 4. Percentage of misclassification for a thin wire scattered signal using polynomial (quadratic) basis functions. discrimination scheme.
the capability of the proposed method to extend the discrimination to a broader SNR range. V. CONCLUSION Based on a study of the performance of the E-pulse technique, a new discrimination scheme for radar targets has been proposed based on the use of cross correlation as a discrimination criterion in the E-pulse method. It is shown analytically that the proposed scheme provides asymptotically correct results regardless of the noise level contaminating the target response. Simulation results corroborate this enhancement and indicate that the proposed scheme produces fewer discrimination errors than the original E-pulse in all cases.
Original E-pulse scheme.
Proposed
[9] G. S. Wallinga, E. J. Rothwell, K.-M. Chen, and D. P. Nyquist, “Enhanced detection of a target in a sea clutter environment using a stepped, ultra-wideband signal and E-pulse cancellation,” IEEE Trans. Antennas Propagat., vol. 49, pp. 1166–1173, Aug. 2001. [10] A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1999. [11] K. M. Chen, D. P. Nyquist, E. J. Rothwell, L. L. Webb, and B. Drachman, “Radar target discrimination by convolution of radar returns with extinction-pulses and single-mode extraction signals,” IEEE Trans. Antennas Propagat., vol. 34, pp. 896–904, July 1986.
ACKNOWLEDGMENT
Calculation of Higher Order Diffracted Fields for Multiple-Edge Transition Zone Diffraction
The authors wish to thank the anonymous referees for carefully reading this communication and for their valuable comments.
Peter D. Holm
REFERENCES [1] C. E. Baum, E. J. Rothwell, K.-M. Chen, and D. P. Nyquist, “The singularity expansion method and its application to target identification,” Proc. IEEE, vol. 79, pp. 1481–1492, Oct. 1991. [2] D. P. Ruiz, A. Gallego, and M. C. Carrion, “Extinction pulse and resonance annihilation filter: Two methods for radar target discrimination,” Radio Sci., vol. 34, no. 1, pp. 93–102, Jan. 1999. [3] S. L Primak, J. LoVetri, Z. Damjanschitz, and S. Kashyap, “Auto-regressive filter-based E-pulse discriminating scheme,” IEEE Trans. Antennas Propagat., vol. 47, pp. 216–218, Jan. 1999. [4] M. C. Carrion, A. Gallego, J. Porti, and D. P. Ruiz, “Subsectional-polynomial E-pulse synthesis and application to radar target discrimination,” IEEE Trans. Antennas Propagat., vol. 41, pp. 1204–1211, Sept. 1993. [5] E. J. Rothwell, K.-M. Chen, D. P. Nyquist, P. Ilavarasan, J. E. Ross, R. Bebermeyer, and Q. Li, “A general E-pulse scheme arising from the dual early-time/late-time behavior of radar scatterers,” IEEE Trans. Antennas Propagat., vol. 42, pp. 1336–1341, Sept. 1994. [6] Q. Li, P. Ilavarasan, J. E. Ross, E. J. Rothwell, K.-M. Chen, and D. P. Nyquist, “Radar target identification using a combined early-time/late-time E-pulse technique,” IEEE Trans. Antennas Propagat., vol. 46, pp. 1272–1278, Sept. 1998. [7] J. E. Mooney, Z. Ding, and L. S. Riggs, “Performance analysis of an automated E-pulse target discrimination scheme,” IEEE Trans. Antennas Propagat., vol. 48, pp. 616–628, Mar. 2000. [8] P. Ilavarasan, J. E. Ross, E. J. Rothwell, K.-M. Chen, and D. P. Nyquist, “Performance of an automated radar target discrimination scheme using E pulses and S pulses,” IEEE Trans. Antennas Propagat., vol. 41, pp. 582–588, May 1993.
Abstract—A well-known problem in conventional uniform theory of diffraction (UTD) is multiple-edge transition zone diffraction. Here, higher order diffracted fields have to be used in order to improve the result, and these fields are added by means of a series. Unfortunately, this series will converge slowly for many edges, which means that a good enough result might require a large number of terms. However, by summing up the series properly, the computation time can be kept reasonable, which will be shown for knife edges. The summation technique is though valid for wedges as well. Index Terms—Geometrical theory of diffraction, multiple-edge diffraction, transition zone diffraction.
I. INTRODUCTION Wave propagation over hills in rural areas, or over buildings in urban areas, will very often mean diffraction by multiple obstacles. It very often also means multiple-edge transition region diffraction. For the uniform theory of diffraction (UTD) method considered herein, this is a case of diffraction that needs some special care, as conventional UTD is not accurate enough for multiple transition region diffraction [1]. Manuscript received March 13, 2000; revised July 2, 2003. The author is with the Swedish Defence Research Agency, Division of Command and Control Systems, Linköping SE-58111, Sweden. Digital Object Identifier 10.1109/TAP.2004.827489
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