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Recovery of antenna pattern loss in a search radar
Signal Processing 65 (1998) 329—336
Recovery of antenna pattern loss in a search radar A. Farina*, M. Valeri Systems Analysis Group, Alenia Difesa, Via Tiburtina Km. 12.400, 00131 Rome, Italy Received 9 July 1997; revised 27 November 1997
Abstract An algorithm is presented for the maximisation of the signal-to-noise power ratio (SNR) at the output of a radar receiver when the target direction of arrival (TDOA) is unknown. The target detection is affected by receiver noise and clutter. The radar has a beam which is scanning with a priori known pattern. For the maximisation of SNR, the knowledge of TDOA would be needed; in practice, the TDOA is unknown and the detection is also affected by the so-called antenna pattern loss. In fact when the antenna pattern sweeps across the target, the received signal experiences an amplitude modulation in lieu of a constant amplitude. It is the purpose of this paper to exploit this modulation to estimate the TDOA and conceive a detection processor which recovers the antenna pattern loss. ( 1998 Elsevier Science B.V. All rights reserved. Zusammenfassung Es wird ein Algorithmus zur Maximierung des Signal-zu-Rausch-Verha¨ltnisses am Ausgang eines Radarempfa¨ngers pra¨sentiert, wenn die Zieleinfallsrichtung (TDOA) unbekannt ist. Die Zieldetektion wird durch das Empfa¨ngerrauschen und den Clutter beeinflu{t. Das Radar hat eine Antennenkeule die nach einem a-priori bekannten Muster scannt. Fu¨r die Maximierung des S/N-Verha¨ltnisses wu¨rde die Kenntnis der Zieleinfallsrichtung (TDOA) beno¨tigt; in der Praxis ist die Zieleinfallsrichtung (TDOA) unbekannt und die Detektion wird zusa¨tzlich durch den sogenannten Antennenkeulenverlu{t beeinflu{t. In der Tat erfa¨hrt das Echosignal eine Amplitudenmodulation wenn die Antennenkeule u¨ber das Ziel schwenkt. Es ist die Absicht dieses Beitrages diese Modulation fu¨r eine Scha¨tzung der Zieleinfallsrichtung (TDOA) auszuwerten und einen Detektionsalgorithmus vorzustellen, der den Antenendiagrammverlust zuru¨ckgewinnt. Dieser Beitrag ist folgenderma{en gegliedert. Zuerst wird das problem beschrieben (Kapitel 1) und der Alogrithmus zur Maximierung des S/N-Verha¨lthnisses vorgestellt (Kapitel 2); anschlie{end wird die Leistungsfa¨higkeit des Algorithmusses mit Hilfe einer Monte Carlo Simulation u¨berpru¨ft. ( 1998 Elsevier Science B.V. All rights reserved. Re´sume´ Nous pre´sentons un algorithme pour la maximisation du rapport signal sur bruit (SNR) a` la sortie d’un re´cepteur radar lorsque la direction d’arrive´e de la cible (TDOA) est inconnue. La de´tection de la cible est affecte´e par du bruit et du fouillis au niveau du re´cepteur. Le mode de balayage du faisceau radar est suppose´ connu a priori. Pour maximiser le SNR, il est ne´cessaire de connaıˆ tre la TDOA; en pratique, celle-ci est inconnue et la de´tection est e´galement affecte´e par ce
* Corresponding author. Tel.: #39-6-41502279; fax: #39-6-41502665; e.mail: [email protected]. 0165-1684/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 5 - 1 6 8 4 ( 9 7 ) 0 0 2 2 9 - 6
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A. Farina, M. Valeri / Signal Processing 65 (1998) 329–336
qu’on appelle la perte de mode de balayage de l’antenne. En fait lorsque le mode de l’ antenne balaye la cible, le signal rec7 u encoure une modulation d’amplitude au lieu de rester a` une amplitude constante. Le but de cet article est d’exploiter cette modulation pour estimer la TDOA et concevoir un processeur de de´tection qui e´limine la perte de mode de balayage d’antenne. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Radar detection; Adaptive signal processing
1. Problem formulation The paper is organised as follows. The problem is stated in this section. The algorithm to maximise the SNR is presented in Section 2; mathematical details are given in Appendix A. Subsequently, the performance of the algorithm is examined by resorting to Monte Carlo simulation (Section 3); a perspective for future research (Section 4) concludes the paper. The definitions of frequently used functions are as follows: a, a N-dimensional steering vector and its h derivative with respect to the angle h b, bK Complex valued amplitude of target and its estimate d, s, z N-dimensional vectors of disturbance, target echoes and received radar signal samples F derivative of scalar function F with respect h to the angle h G(h) two-way antenna gain M N]N-dimensional covariance matrix of interference P,P probability of detection and probability $ &! of false alarm PRF pulse repetition frequency of radar SNR signal-to-noise power ratio h target direction of arrival TG u angular rotation speed of antenna R w N-dimensional vector of weights Consider a radar antenna, with an azimuth beam width h , that rotates at the angular velocity u . B R Suppose that a target is present at the TDOA h . TG The received baseband signal at time t is z(t)"s(t)#d(t)"bG(h )e+2pf$t#d(t), (1) TG where b is the unknown target signal complex valued amplitude, G(h ) is the two-way antenna TG
gain at the TDOA, f is the target Doppler frequency $ (which is assumed known throughout the paper) and d(t) is the global interference (clutter, electromagnetic interference and receiver noise). d has a Gaussian probability density function (pdf) with zero mean and known auto-correlation function. In the time on target (ToT), the TDOA varies between h !h /2 and h #h /2. The N echoes scattered TG B TG B back by the target during the ToT are collected in a column vector z; the nth component is
A
B
h u z(n)"bG h ! B#(n!1) R e(+2pf$ n~1)@PRF TG 2 PRF #d(n¹), n"1,2, N,
(2)
where ¹"1/PRF, N"PRFh /u and PRF is B R the radar pulse repetition frequency. With vector notation the previous equation is rewritten as follows: z"s#d"ba(h )#d; TG
(3)
the nth component of the N-dimensional vector a is
A
B
h u a (h )"G h ! B#(n!1) R e(+2pf$ n~1)@PRF, TG n TG 2 PRF n"1,2,N.
(4)
The N-dimensional interference vector d has a Gaussian pdf with zero mean and known N]N-dimensional covariance matrix M"EMddHN, where H denotes transpose-conjugate. In the current radar technology, the vector z of received echoes is linearly combined with a set w of N weights, the resulting signal is u"wHz"wH(s#d)"wHs#wHd.
(5)
A. Farina, M. Valeri / Signal Processing 65 (1998) 329—336
331
The corresponding signal-to-noise power ratio (SNR) at the output of the weighting filter is EMDwHsD2N SNR "10 log 6,$B 10EMDwHdDN2 wHEMssHNu "10 log 10 wHMw EMDbD2NwHa(h )aH(h )w TG TG , "10 log 10 wHMw
(6)
where EM ) N denotes statistical expectation of ( ) ). The signal is envelope detected and compared to a suitable detection threshold to ascertain whether a target is indeed present in the direction of the beam. The detection probability P for a Swerling $ 0 target model is P "Q(J2SNR , J!2lnP ), (7) $ 6 &! where Q( ) , ) ) is the Marcum Q function and P is &! the prescribed level of false alarm. Because the target echoes are amplitude modulated by the antenna pattern, the weight vector w should take into account the modulation. This is not done in the current technology; in fact, the amplitude of each element of the weight vector w is either constant or has a tapering selected to give low side lobes in the Doppler frequency domain to suppress the clutter returns. As a consequence, up to 2 dB pattern losses are experienced [1]; this means a 11% reduction of the radar’s maximum range. It is the purpose of this paper to present an algorithm that recovers the antenna pattern losses. From a practical point of view this means that it is not needed to increase the radar transmitted power of up to 2 dB (i.e.: 1.6 times the nominal power) to achieve the nominal detection performance.
2. TDOA estimation and maximisation of SNR 6 We look for an optimum weight vector w that, 015 when replacing w, can maximise the output signalto-noise power ratio, in the following indicated with SNR . If we would know the TDOA, the 6 optimum filter weight that maximises the SNR 6 value would be readily available from the technical
Fig. 1. Estimation of TDOA to maximise the SNR . 6
literature; see, for instance, Eq. (6) of [4] and the following Eq. (9) in this paper. In practice, however, the TDOA is not known and should be estimated by exploiting the knowledge of the antenna beam pattern and the measured amplitude modulation impressed on the signal scattered by the target. The proposed processing scheme is depicted in Fig. 1. The algorithm to estimate the TDOA is formally equivalent to that derived by Davies, Brennan and Reed (DBR) in [2]. The problem studied by DBR refers to the estimation of the TDOA on the basis of echoes captured by a linear array of antennas; the exploited information is the progression of the phase in the signals received by the array. In the problem considered here, as shown in Fig. 1, a tapped-delay line replaces the linear array of antennas; the exploited information is the amplitude modulation of the target echoes. The application of the DBR method to the problem described in this paper is explained in detail in Appendix A; here, only the key results are quoted. The pdf of z conditioned to the unknowns b and h is TG 1 pz s(z; b, h )" e~*z~a(hTG)b+HM~1*z~a(hTG)b+. (8) @ TG nNdet(M) The estimation of the TDOA is done via the maximum likelihood principle: thus the estimated TDOA will maximise the above pdf. Maximisation of Eq. (8) is equivalent to minimising its exponent.
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A. Farina, M. Valeri / Signal Processing 65 (1998) 329–336
Because the exponent is a function both of b and h , at first the value of b which minimises the TG exponent is found. The minimum over b is given by Eq. (6) of [4]: aH(h )M~1z TG bK " , (9) aH(h )M~1a(h ) TG TG Inserting Eq. (9) in Eq. (8), we obtain a function of only the variable h ; the estimation algorithm TG requires the evaluation of a non-linear function F(h) (defined in Eq. (A.2)) for all the look directions h to identify the direction along which the function has a minimum. This is impracticable for a real time solution, therefore a sub-optimum solution is sought. In Appendix A the following expression is obtained (see Eq. (A.8)): hK "k#h , (10) 1 where hK is the estimated TDOA and k (which is the right-hand side term of Eq. (A.8)) is a correction factor depending on the quantities R"aHM~1z,
d D" (R)"aHM~1z, h dh
(11)
h denotes the look direction of the antenna. In the 1 theory of linear adaptive arrays, these quantities are the output of the adapted sum and difference beams; thus, the angle hK is obtained via a generalisation of the monopulse processing. In the problem studied here, the quantities (11) are referred to as pseudo-sum and pseudo-difference patterns in the time domain rather than in the spatial domain; the pseudo patterns are adapted to reject the interference. The estimated TDOA can be inserted in Eq. (9) which gives the estimated value of the target complex amplitude. Eq. (9) can be interpreted as the output of a filter having the following weights: M~1a(hK ) w " . 015 aH(hK )M~1a(hK )
(12)
This weight vector inserted in Eq. (6) gives the sub-optimum value for the SNR ; inserting SNR 6 6 in Eq. (7) gives the achievable value of P . $ In summary, the proposed detection algorithm proceeds according to the following steps: f estimation of TDOA (Eq. (10)); f calculation of the steering vector a(hK ) (Eq. (4)); f calculation of the filter weights (Eq. (12));
f
calculation of the envelope of the filter output (Eq. (5)), and f evaluation of the detection performance (Eqs. (6) and (7)). In the work above it was tacitly assumed that the clutter covariance matrix M, used in Eq. (12), was a priori known; in practice, the matrix has to be estimated on line from the data. The proposed detection algorithm can be made adaptive by substituting the estimated covariance matrix MK in lieu of the a priori known M; it can be shown that the corresponding processing filter (5) and (12) enjoys the CFAR (constant false alarm rate) property [4]. Of course, the on-line estimation of the matrix M is computationally demanding. Thus, it is worth to replace the adaptive filter (12) with either a nonadaptive filter which rejects the clutter via very low sidelobes or a non-linear filter with very limited adaptation needs as explained in [3]; this is a theme for future research.
3. Performance evaluation Two detection algorithms are compared: the conventional that does not estimate the TDOA, and the proposed algorithm which estimates and exploits the TDOA. The conventional detection algorithm differs from the proposed algorithm because the weight vector (12) does not depend on the estimated TDOA hK ; i.e.: the nth component of the N-dimensional vector a is a "e(+2pf$ n~1)@PRF, n"1,2,2, N, (13) n which differs from Eq. (4) that includes the antenna pattern shape and the TDOA. By resorting to a Monte Carlo simulation the following performances of the algorithms are evaluated: f mean and standard deviation values of the estimated TDOA, f detection probability for a prescribed probability of false alarm, and f signal-to-noise power ratio SNR upstream from 6 the detection threshold. A number of parameters have been selected and kept fixed for all the simulations: f antenna beam width: $1°, f N"17 pulses during the ToT,
A. Farina, M. Valeri / Signal Processing 65 (1998) 329—336
333
f f f
target type: Swerling 0, Doppler frequency of target: 0.5PRF, time auto-correlation function of the clutter: Gaussian, f Doppler frequency of clutter: 0, f clutter-to-noise power ratio: CNR"50 dB, f one lag clutter auto-correlation coefficient: o"0.99, and f probability of false alarm: P "10~6. &! Figs. 2 and 3 show the mean and root mean square (rms) of the TDOA estimate for different values of the TDOA h and of the SNR at the input of the TG antenna for a single pulse. This is defined as EMDsD2N SNRI "10 log , $B 10EMDdD2N
(14)
where s and d represent signal and interference for a single echo received by an isotropic antenna (i.e,.: without the radar antenna pattern filtering). Specifically, SNRI"4 dB refers to Fig. 2 and SNRI"8 dB is for Fig. 3. These results have been obtained by averaging 100 independent Monte Carlo trials. It is noted that the estimate is unbiased and the rms value is a fraction of the beam width when the TDOA is within the antenna beam; on the skirt of the antenna the estimation accuracy deteriorates because of reduction of the SNR and the limited accuracy of the Taylor series expansion. Of course, the estimate improves with higher SNRI.
Fig. 3. Mean error and rms of the TDOA estimate for SNRI"8 dB.
Fig. 4(a) and Fig. 5(a) display the detection probability P versus several values of TDOA and for $ two values of SNRI; namely, Fig. 4 refers to 4 dB, while SNRI"8 dB pertains to Fig. 5. Two curves are shown in each figure; one refers to the proposed algorithm, the other is related to the conventional algorithm (i.e., the one that does not estimate the TDOA. From the figures it is noted that for each target position there is an advantage in using the optimum set of weights (12). Note also that when the TDOA is on the peak of the beam, the amplitude modulation is reduced and there is practically no advantage in using the optimum set of weights. Fig. 4(b) and Fig. 5(b) compare the signal-to-interference plus noise power ratio SNR at the filter 6 output; an advantage up to 2.3 dB is noted.
4. Conclusions
Fig. 2. Mean error and rms of the TDOA estimate for SNRI"4 dB.
A processing scheme has been presented that exploits the amplitude modulation of the scattered echoes from the target to estimate the TDOA and to determine a set of weights that maximises the signal-to-interference plus noise power ratio at the output. An advantage of up to 2.3 dB in SNR is gained with respect to the conventional detector that does not estimate and exploit the TDOA.
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A. Farina, M. Valeri / Signal Processing 65 (1998) 329–336
Fig. 4. (a) Detection probability for the conventional and proposed detection algorithms; SNRI"4 dB. (b) SNR for the 6 conventional and proposed detection algorithms; SNRI"4 dB.
Fig. 5. (a) Detection probability for the conventional and proposed detection algorithms; SNRI"8 dB. (b) SNR for the 6 conventional and proposed detection algorithms; SNRI"8 dB.
A by-product of this paper is the algorithm for estimating the TDOA without the need of the hardware required by the classical monopulse; this algorithm can be named pseudo-monopulse because it uses the pseudo-sum and pseudo-difference patterns. The pseudo-monopulse has the following advantages with respect to the classical monopulse: (i) it does not need a monopulse feed (for reflector antennas) or a beam forming network (for phased arrays) to generate the difference pattern orthogonal to the sum pattern, (ii) it does not need an additional (tightly matched) receiving channel for capturing the radar echoes from the difference channel, (iii) it
can reasonably operate also when the TDOA is outside the !3 dB beamwidth but on the skirt of the antenna main beam. The main disadvantages of the pseudo-monopulse are (a) it needs that the antenna rotates, (b) it does not work when the signal received by the sum channel is so strong to deny the amplitude modulation induced by the antenna scan, while in this case the classical monopulse has more chance to properly work, (c) it may have degraded performance when the target amplitude scintillates from pulse to pulse, (d) for the previous reason the radar cannot operate in frequency agility mode.
A. Farina, M. Valeri / Signal Processing 65 (1998) 329—336
Acknowledgements Dr. R. Sanzullo (Alenia) has experimentally validated the algorithms described in this paper by processing live data recorded by an operating phased-array radar.
Appendix A. This appendix is an application of Eq. (42) of [2]; here, a quick recall of the analytical derivation is provided. The maximum-likelihood principle gives the estimates b and h which maximises Eq. (8) in TG the test. The maximisation of the pdf is equivalent to the minimisation of its exponent:
A
zHM~1z!
B
K
K
DzHM~1aD2 zHM~1a 2 #aHM~1a b! . aHM~1a aHM~1a (A.1)
In this expression the only term dependent on the complex amplitude b is the last; thus, the estimate bK is given by Eq. (9) in the text. Inserting Eq. (9) in Eq. (A.1), the following function of h is obtained: DzHM~1a(h)D2 F(h)"zHM~1z! . a(h)HM~1a(h)
(A.2)
Note that the second term on the rhs is the correlation between the set z of received echoes and the expected set a(h) of samples from a target having a direction of arrival equal to h; the presence of
G
335
indicate with F(h) the second term on the rhs of Eq. (A.2). To find the maximum of the correlation, which indicates the TDOA, a search over a proper angular interval encompassing the main beam would be needed; this method is not practical for an operational system. A sub-optimum solution is obtained by resorting to the second-order Taylor series expansion of F(h) around a known angle, e.g., the angle h denoting the look direction of the 1 antenna:
K
K
dF 1 d2F F(h)"F(h )# (h!h )# (h!h )2. 1 1 1 dh 2 dh2 h1 h1 (A.3) This quantity is maximised when dF(h)/dh"0, or
K
F (A.4) (hK !h )"! h , 1 F hh h1 where F and F are the first and second derivatives h hh of F evaluated in h ; hK is the angle at which the 1 function F(h) takes the maximum value, i.e., it is the estimate of h . Because F(h) depends on the TG measurement z, the process of differentiation often results in a noisier estimate of the quantity; consequently, the denominator of Eq. (A.4) is approximated by the mean value of the second derivative of F in lieu of the second derivative itself: F h . (hK !h )"! 1 EMF N hh The first derivative of F(h) is
(A.5)
H
dF(h) d (zHM~1a(h))(a(h)HM~1z) " ! dh dh a(h)HM~1a(h)
(zHM~1a )(a(h)HM~1z)#(zHM~1a(h))(aHM~1z) (zHM~1a(h)) (a(h)HM~1z) d h h "! # (a(h)HM~1a(h)) (a(h)HM~1a(h) dh (a(h)HM~1a(h))2
K
K
zHM~1a(h) 2 d a(h)HM~1z zHM~1a(h) "! (zHM~1a )! (aHM~1z)# (a(h)HM~1a(h)) h a(h)HM~1a(h) dh a(h)HM~1a(h) a(h)HM~1a(h) h "!bK (zHM~1a )!bKM (aHM~1z)#DbK D2[aHM~1a(h)#a(h)HM~1a ], h h h h M~1 ensures the cancellation of the interference. The estimate of the TDOA is obtained by maximising the second term which is the only one depending on h; for simplicity we continue to
(A.6)
where the over-bar denotes complex conjugate and a indicates the gradient of the vector a with h respect to the angle h. With similar calculations the second derivative of F can also be calculated;
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A. Farina, M. Valeri / Signal Processing 65 (1998) 329–336
the mathematical expression of its average value is not reported here for the sake of brevity. We introduce the following notations: d R"aHM~1z, D" (R)"aHM~1z, h dh
AB
k"
R D
, (A.7)
A A
B
aHM~1a aHM~1a h aHM~1a aHM~1a h h h b b " 11 12 . b b 21 22 Remember that all a-vectors and derivatives appearing in Eqs. (A.6) and (A.7) have to be evaluated at h . After some algebra, it can be shown that the 1 estimation of the TDOA is
B"EMkkHN"
B
[b2 (RM D#DM R)!RM R(b #b )b ] 12 21 11 , hK !h " 11 1 [2RM R(b b !b b )] 11 22 12 21 (A.8) which is the expression (10) of the text with i equal to the rhs term of Eq. (A.8). Note that when M"p2I, i.e., the interference is just the system noise, Eq. (A.8) reduces to an equation which is similar to the classical expression of the monopulse. In fact, we find the following equalities: aHz aHz R" , D" h , p2 p2
aHa b " , 11 p2
aHa aHa b " h , b " h h. 22 21 p2 p2
aHa b " h, 12 p2 (A.9)
Substituting Eq. (A.9) in Eq. (A.8) we obtain
C A
D
zHa aHa aHa h! h Re zHa DaHzD2 hK !h " . 1 aHa DaHa D2 h h! h aHa (aHa)2
B
(A.10)
If h "0, the PRT is constant and the antenna 1 pattern is sampled symmetrically around the peak, it follows that aHa "0. (A.11) h Thus Eq. (A.10) reduces to the standard monopulse equation operating on the set of echo samples which are amplitude modulated by the rotating antenna pattern: ReMaHz/aHzN h hK !h " . 1 aHa h h aHa
C D
(A.12)
References [1] L.V. Blake, Prediction of radar range, in: M.I. Skolnik (Ed.), Radar Handbook, 2nd Ed., McGraw-Hill, New York, 1990, Chapter 2, pp. 2.46—2.47. [2] R.C. Davis, L.E. Brennan, I.S. Reed, Angle estimation with adaptive arrays in external noise fields, IEEE Trans. Aerospace Electron. Systems AES-12 (2) (March 1976) 179—186. [3] A. Farina, Linear and non-linear filters for clutter cancellation in radar systems, Signal Processing 59 (1) (1997) 101—112. [4] F.C. Robey, D.R. Fuhrmann, E.J. Kelly, R. Nitzberg, A CFAR adaptive matched filter detector, IEEE Trans. Aerospace Electron. Systems AES-28 (1) (January 1992) 208—216.
Recovery of antenna pattern loss in a search radar A. Farina*, M. Valeri Systems Analysis Group, Alenia Difesa, Via Tiburtina Km. 12.400, 00131 Rome, Italy Received 9 July 1997; revised 27 November 1997
Abstract An algorithm is presented for the maximisation of the signal-to-noise power ratio (SNR) at the output of a radar receiver when the target direction of arrival (TDOA) is unknown. The target detection is affected by receiver noise and clutter. The radar has a beam which is scanning with a priori known pattern. For the maximisation of SNR, the knowledge of TDOA would be needed; in practice, the TDOA is unknown and the detection is also affected by the so-called antenna pattern loss. In fact when the antenna pattern sweeps across the target, the received signal experiences an amplitude modulation in lieu of a constant amplitude. It is the purpose of this paper to exploit this modulation to estimate the TDOA and conceive a detection processor which recovers the antenna pattern loss. ( 1998 Elsevier Science B.V. All rights reserved. Zusammenfassung Es wird ein Algorithmus zur Maximierung des Signal-zu-Rausch-Verha¨ltnisses am Ausgang eines Radarempfa¨ngers pra¨sentiert, wenn die Zieleinfallsrichtung (TDOA) unbekannt ist. Die Zieldetektion wird durch das Empfa¨ngerrauschen und den Clutter beeinflu{t. Das Radar hat eine Antennenkeule die nach einem a-priori bekannten Muster scannt. Fu¨r die Maximierung des S/N-Verha¨ltnisses wu¨rde die Kenntnis der Zieleinfallsrichtung (TDOA) beno¨tigt; in der Praxis ist die Zieleinfallsrichtung (TDOA) unbekannt und die Detektion wird zusa¨tzlich durch den sogenannten Antennenkeulenverlu{t beeinflu{t. In der Tat erfa¨hrt das Echosignal eine Amplitudenmodulation wenn die Antennenkeule u¨ber das Ziel schwenkt. Es ist die Absicht dieses Beitrages diese Modulation fu¨r eine Scha¨tzung der Zieleinfallsrichtung (TDOA) auszuwerten und einen Detektionsalgorithmus vorzustellen, der den Antenendiagrammverlust zuru¨ckgewinnt. Dieser Beitrag ist folgenderma{en gegliedert. Zuerst wird das problem beschrieben (Kapitel 1) und der Alogrithmus zur Maximierung des S/N-Verha¨lthnisses vorgestellt (Kapitel 2); anschlie{end wird die Leistungsfa¨higkeit des Algorithmusses mit Hilfe einer Monte Carlo Simulation u¨berpru¨ft. ( 1998 Elsevier Science B.V. All rights reserved. Re´sume´ Nous pre´sentons un algorithme pour la maximisation du rapport signal sur bruit (SNR) a` la sortie d’un re´cepteur radar lorsque la direction d’arrive´e de la cible (TDOA) est inconnue. La de´tection de la cible est affecte´e par du bruit et du fouillis au niveau du re´cepteur. Le mode de balayage du faisceau radar est suppose´ connu a priori. Pour maximiser le SNR, il est ne´cessaire de connaıˆ tre la TDOA; en pratique, celle-ci est inconnue et la de´tection est e´galement affecte´e par ce
* Corresponding author. Tel.: #39-6-41502279; fax: #39-6-41502665; e.mail: [email protected]. 0165-1684/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 5 - 1 6 8 4 ( 9 7 ) 0 0 2 2 9 - 6
330
A. Farina, M. Valeri / Signal Processing 65 (1998) 329–336
qu’on appelle la perte de mode de balayage de l’antenne. En fait lorsque le mode de l’ antenne balaye la cible, le signal rec7 u encoure une modulation d’amplitude au lieu de rester a` une amplitude constante. Le but de cet article est d’exploiter cette modulation pour estimer la TDOA et concevoir un processeur de de´tection qui e´limine la perte de mode de balayage d’antenne. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Radar detection; Adaptive signal processing
1. Problem formulation The paper is organised as follows. The problem is stated in this section. The algorithm to maximise the SNR is presented in Section 2; mathematical details are given in Appendix A. Subsequently, the performance of the algorithm is examined by resorting to Monte Carlo simulation (Section 3); a perspective for future research (Section 4) concludes the paper. The definitions of frequently used functions are as follows: a, a N-dimensional steering vector and its h derivative with respect to the angle h b, bK Complex valued amplitude of target and its estimate d, s, z N-dimensional vectors of disturbance, target echoes and received radar signal samples F derivative of scalar function F with respect h to the angle h G(h) two-way antenna gain M N]N-dimensional covariance matrix of interference P,P probability of detection and probability $ &! of false alarm PRF pulse repetition frequency of radar SNR signal-to-noise power ratio h target direction of arrival TG u angular rotation speed of antenna R w N-dimensional vector of weights Consider a radar antenna, with an azimuth beam width h , that rotates at the angular velocity u . B R Suppose that a target is present at the TDOA h . TG The received baseband signal at time t is z(t)"s(t)#d(t)"bG(h )e+2pf$t#d(t), (1) TG where b is the unknown target signal complex valued amplitude, G(h ) is the two-way antenna TG
gain at the TDOA, f is the target Doppler frequency $ (which is assumed known throughout the paper) and d(t) is the global interference (clutter, electromagnetic interference and receiver noise). d has a Gaussian probability density function (pdf) with zero mean and known auto-correlation function. In the time on target (ToT), the TDOA varies between h !h /2 and h #h /2. The N echoes scattered TG B TG B back by the target during the ToT are collected in a column vector z; the nth component is
A
B
h u z(n)"bG h ! B#(n!1) R e(+2pf$ n~1)@PRF TG 2 PRF #d(n¹), n"1,2, N,
(2)
where ¹"1/PRF, N"PRFh /u and PRF is B R the radar pulse repetition frequency. With vector notation the previous equation is rewritten as follows: z"s#d"ba(h )#d; TG
(3)
the nth component of the N-dimensional vector a is
A
B
h u a (h )"G h ! B#(n!1) R e(+2pf$ n~1)@PRF, TG n TG 2 PRF n"1,2,N.
(4)
The N-dimensional interference vector d has a Gaussian pdf with zero mean and known N]N-dimensional covariance matrix M"EMddHN, where H denotes transpose-conjugate. In the current radar technology, the vector z of received echoes is linearly combined with a set w of N weights, the resulting signal is u"wHz"wH(s#d)"wHs#wHd.
(5)
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331
The corresponding signal-to-noise power ratio (SNR) at the output of the weighting filter is EMDwHsD2N SNR "10 log 6,$B 10EMDwHdDN2 wHEMssHNu "10 log 10 wHMw EMDbD2NwHa(h )aH(h )w TG TG , "10 log 10 wHMw
(6)
where EM ) N denotes statistical expectation of ( ) ). The signal is envelope detected and compared to a suitable detection threshold to ascertain whether a target is indeed present in the direction of the beam. The detection probability P for a Swerling $ 0 target model is P "Q(J2SNR , J!2lnP ), (7) $ 6 &! where Q( ) , ) ) is the Marcum Q function and P is &! the prescribed level of false alarm. Because the target echoes are amplitude modulated by the antenna pattern, the weight vector w should take into account the modulation. This is not done in the current technology; in fact, the amplitude of each element of the weight vector w is either constant or has a tapering selected to give low side lobes in the Doppler frequency domain to suppress the clutter returns. As a consequence, up to 2 dB pattern losses are experienced [1]; this means a 11% reduction of the radar’s maximum range. It is the purpose of this paper to present an algorithm that recovers the antenna pattern losses. From a practical point of view this means that it is not needed to increase the radar transmitted power of up to 2 dB (i.e.: 1.6 times the nominal power) to achieve the nominal detection performance.
2. TDOA estimation and maximisation of SNR 6 We look for an optimum weight vector w that, 015 when replacing w, can maximise the output signalto-noise power ratio, in the following indicated with SNR . If we would know the TDOA, the 6 optimum filter weight that maximises the SNR 6 value would be readily available from the technical
Fig. 1. Estimation of TDOA to maximise the SNR . 6
literature; see, for instance, Eq. (6) of [4] and the following Eq. (9) in this paper. In practice, however, the TDOA is not known and should be estimated by exploiting the knowledge of the antenna beam pattern and the measured amplitude modulation impressed on the signal scattered by the target. The proposed processing scheme is depicted in Fig. 1. The algorithm to estimate the TDOA is formally equivalent to that derived by Davies, Brennan and Reed (DBR) in [2]. The problem studied by DBR refers to the estimation of the TDOA on the basis of echoes captured by a linear array of antennas; the exploited information is the progression of the phase in the signals received by the array. In the problem considered here, as shown in Fig. 1, a tapped-delay line replaces the linear array of antennas; the exploited information is the amplitude modulation of the target echoes. The application of the DBR method to the problem described in this paper is explained in detail in Appendix A; here, only the key results are quoted. The pdf of z conditioned to the unknowns b and h is TG 1 pz s(z; b, h )" e~*z~a(hTG)b+HM~1*z~a(hTG)b+. (8) @ TG nNdet(M) The estimation of the TDOA is done via the maximum likelihood principle: thus the estimated TDOA will maximise the above pdf. Maximisation of Eq. (8) is equivalent to minimising its exponent.
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Because the exponent is a function both of b and h , at first the value of b which minimises the TG exponent is found. The minimum over b is given by Eq. (6) of [4]: aH(h )M~1z TG bK " , (9) aH(h )M~1a(h ) TG TG Inserting Eq. (9) in Eq. (8), we obtain a function of only the variable h ; the estimation algorithm TG requires the evaluation of a non-linear function F(h) (defined in Eq. (A.2)) for all the look directions h to identify the direction along which the function has a minimum. This is impracticable for a real time solution, therefore a sub-optimum solution is sought. In Appendix A the following expression is obtained (see Eq. (A.8)): hK "k#h , (10) 1 where hK is the estimated TDOA and k (which is the right-hand side term of Eq. (A.8)) is a correction factor depending on the quantities R"aHM~1z,
d D" (R)"aHM~1z, h dh
(11)
h denotes the look direction of the antenna. In the 1 theory of linear adaptive arrays, these quantities are the output of the adapted sum and difference beams; thus, the angle hK is obtained via a generalisation of the monopulse processing. In the problem studied here, the quantities (11) are referred to as pseudo-sum and pseudo-difference patterns in the time domain rather than in the spatial domain; the pseudo patterns are adapted to reject the interference. The estimated TDOA can be inserted in Eq. (9) which gives the estimated value of the target complex amplitude. Eq. (9) can be interpreted as the output of a filter having the following weights: M~1a(hK ) w " . 015 aH(hK )M~1a(hK )
(12)
This weight vector inserted in Eq. (6) gives the sub-optimum value for the SNR ; inserting SNR 6 6 in Eq. (7) gives the achievable value of P . $ In summary, the proposed detection algorithm proceeds according to the following steps: f estimation of TDOA (Eq. (10)); f calculation of the steering vector a(hK ) (Eq. (4)); f calculation of the filter weights (Eq. (12));
f
calculation of the envelope of the filter output (Eq. (5)), and f evaluation of the detection performance (Eqs. (6) and (7)). In the work above it was tacitly assumed that the clutter covariance matrix M, used in Eq. (12), was a priori known; in practice, the matrix has to be estimated on line from the data. The proposed detection algorithm can be made adaptive by substituting the estimated covariance matrix MK in lieu of the a priori known M; it can be shown that the corresponding processing filter (5) and (12) enjoys the CFAR (constant false alarm rate) property [4]. Of course, the on-line estimation of the matrix M is computationally demanding. Thus, it is worth to replace the adaptive filter (12) with either a nonadaptive filter which rejects the clutter via very low sidelobes or a non-linear filter with very limited adaptation needs as explained in [3]; this is a theme for future research.
3. Performance evaluation Two detection algorithms are compared: the conventional that does not estimate the TDOA, and the proposed algorithm which estimates and exploits the TDOA. The conventional detection algorithm differs from the proposed algorithm because the weight vector (12) does not depend on the estimated TDOA hK ; i.e.: the nth component of the N-dimensional vector a is a "e(+2pf$ n~1)@PRF, n"1,2,2, N, (13) n which differs from Eq. (4) that includes the antenna pattern shape and the TDOA. By resorting to a Monte Carlo simulation the following performances of the algorithms are evaluated: f mean and standard deviation values of the estimated TDOA, f detection probability for a prescribed probability of false alarm, and f signal-to-noise power ratio SNR upstream from 6 the detection threshold. A number of parameters have been selected and kept fixed for all the simulations: f antenna beam width: $1°, f N"17 pulses during the ToT,
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f f f
target type: Swerling 0, Doppler frequency of target: 0.5PRF, time auto-correlation function of the clutter: Gaussian, f Doppler frequency of clutter: 0, f clutter-to-noise power ratio: CNR"50 dB, f one lag clutter auto-correlation coefficient: o"0.99, and f probability of false alarm: P "10~6. &! Figs. 2 and 3 show the mean and root mean square (rms) of the TDOA estimate for different values of the TDOA h and of the SNR at the input of the TG antenna for a single pulse. This is defined as EMDsD2N SNRI "10 log , $B 10EMDdD2N
(14)
where s and d represent signal and interference for a single echo received by an isotropic antenna (i.e,.: without the radar antenna pattern filtering). Specifically, SNRI"4 dB refers to Fig. 2 and SNRI"8 dB is for Fig. 3. These results have been obtained by averaging 100 independent Monte Carlo trials. It is noted that the estimate is unbiased and the rms value is a fraction of the beam width when the TDOA is within the antenna beam; on the skirt of the antenna the estimation accuracy deteriorates because of reduction of the SNR and the limited accuracy of the Taylor series expansion. Of course, the estimate improves with higher SNRI.
Fig. 3. Mean error and rms of the TDOA estimate for SNRI"8 dB.
Fig. 4(a) and Fig. 5(a) display the detection probability P versus several values of TDOA and for $ two values of SNRI; namely, Fig. 4 refers to 4 dB, while SNRI"8 dB pertains to Fig. 5. Two curves are shown in each figure; one refers to the proposed algorithm, the other is related to the conventional algorithm (i.e., the one that does not estimate the TDOA. From the figures it is noted that for each target position there is an advantage in using the optimum set of weights (12). Note also that when the TDOA is on the peak of the beam, the amplitude modulation is reduced and there is practically no advantage in using the optimum set of weights. Fig. 4(b) and Fig. 5(b) compare the signal-to-interference plus noise power ratio SNR at the filter 6 output; an advantage up to 2.3 dB is noted.
4. Conclusions
Fig. 2. Mean error and rms of the TDOA estimate for SNRI"4 dB.
A processing scheme has been presented that exploits the amplitude modulation of the scattered echoes from the target to estimate the TDOA and to determine a set of weights that maximises the signal-to-interference plus noise power ratio at the output. An advantage of up to 2.3 dB in SNR is gained with respect to the conventional detector that does not estimate and exploit the TDOA.
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Fig. 4. (a) Detection probability for the conventional and proposed detection algorithms; SNRI"4 dB. (b) SNR for the 6 conventional and proposed detection algorithms; SNRI"4 dB.
Fig. 5. (a) Detection probability for the conventional and proposed detection algorithms; SNRI"8 dB. (b) SNR for the 6 conventional and proposed detection algorithms; SNRI"8 dB.
A by-product of this paper is the algorithm for estimating the TDOA without the need of the hardware required by the classical monopulse; this algorithm can be named pseudo-monopulse because it uses the pseudo-sum and pseudo-difference patterns. The pseudo-monopulse has the following advantages with respect to the classical monopulse: (i) it does not need a monopulse feed (for reflector antennas) or a beam forming network (for phased arrays) to generate the difference pattern orthogonal to the sum pattern, (ii) it does not need an additional (tightly matched) receiving channel for capturing the radar echoes from the difference channel, (iii) it
can reasonably operate also when the TDOA is outside the !3 dB beamwidth but on the skirt of the antenna main beam. The main disadvantages of the pseudo-monopulse are (a) it needs that the antenna rotates, (b) it does not work when the signal received by the sum channel is so strong to deny the amplitude modulation induced by the antenna scan, while in this case the classical monopulse has more chance to properly work, (c) it may have degraded performance when the target amplitude scintillates from pulse to pulse, (d) for the previous reason the radar cannot operate in frequency agility mode.
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Acknowledgements Dr. R. Sanzullo (Alenia) has experimentally validated the algorithms described in this paper by processing live data recorded by an operating phased-array radar.
Appendix A. This appendix is an application of Eq. (42) of [2]; here, a quick recall of the analytical derivation is provided. The maximum-likelihood principle gives the estimates b and h which maximises Eq. (8) in TG the test. The maximisation of the pdf is equivalent to the minimisation of its exponent:
A
zHM~1z!
B
K
K
DzHM~1aD2 zHM~1a 2 #aHM~1a b! . aHM~1a aHM~1a (A.1)
In this expression the only term dependent on the complex amplitude b is the last; thus, the estimate bK is given by Eq. (9) in the text. Inserting Eq. (9) in Eq. (A.1), the following function of h is obtained: DzHM~1a(h)D2 F(h)"zHM~1z! . a(h)HM~1a(h)
(A.2)
Note that the second term on the rhs is the correlation between the set z of received echoes and the expected set a(h) of samples from a target having a direction of arrival equal to h; the presence of
G
335
indicate with F(h) the second term on the rhs of Eq. (A.2). To find the maximum of the correlation, which indicates the TDOA, a search over a proper angular interval encompassing the main beam would be needed; this method is not practical for an operational system. A sub-optimum solution is obtained by resorting to the second-order Taylor series expansion of F(h) around a known angle, e.g., the angle h denoting the look direction of the 1 antenna:
K
K
dF 1 d2F F(h)"F(h )# (h!h )# (h!h )2. 1 1 1 dh 2 dh2 h1 h1 (A.3) This quantity is maximised when dF(h)/dh"0, or
K
F (A.4) (hK !h )"! h , 1 F hh h1 where F and F are the first and second derivatives h hh of F evaluated in h ; hK is the angle at which the 1 function F(h) takes the maximum value, i.e., it is the estimate of h . Because F(h) depends on the TG measurement z, the process of differentiation often results in a noisier estimate of the quantity; consequently, the denominator of Eq. (A.4) is approximated by the mean value of the second derivative of F in lieu of the second derivative itself: F h . (hK !h )"! 1 EMF N hh The first derivative of F(h) is
(A.5)
H
dF(h) d (zHM~1a(h))(a(h)HM~1z) " ! dh dh a(h)HM~1a(h)
(zHM~1a )(a(h)HM~1z)#(zHM~1a(h))(aHM~1z) (zHM~1a(h)) (a(h)HM~1z) d h h "! # (a(h)HM~1a(h)) (a(h)HM~1a(h) dh (a(h)HM~1a(h))2
K
K
zHM~1a(h) 2 d a(h)HM~1z zHM~1a(h) "! (zHM~1a )! (aHM~1z)# (a(h)HM~1a(h)) h a(h)HM~1a(h) dh a(h)HM~1a(h) a(h)HM~1a(h) h "!bK (zHM~1a )!bKM (aHM~1z)#DbK D2[aHM~1a(h)#a(h)HM~1a ], h h h h M~1 ensures the cancellation of the interference. The estimate of the TDOA is obtained by maximising the second term which is the only one depending on h; for simplicity we continue to
(A.6)
where the over-bar denotes complex conjugate and a indicates the gradient of the vector a with h respect to the angle h. With similar calculations the second derivative of F can also be calculated;
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the mathematical expression of its average value is not reported here for the sake of brevity. We introduce the following notations: d R"aHM~1z, D" (R)"aHM~1z, h dh
AB
k"
R D
, (A.7)
A A
B
aHM~1a aHM~1a h aHM~1a aHM~1a h h h b b " 11 12 . b b 21 22 Remember that all a-vectors and derivatives appearing in Eqs. (A.6) and (A.7) have to be evaluated at h . After some algebra, it can be shown that the 1 estimation of the TDOA is
B"EMkkHN"
B
[b2 (RM D#DM R)!RM R(b #b )b ] 12 21 11 , hK !h " 11 1 [2RM R(b b !b b )] 11 22 12 21 (A.8) which is the expression (10) of the text with i equal to the rhs term of Eq. (A.8). Note that when M"p2I, i.e., the interference is just the system noise, Eq. (A.8) reduces to an equation which is similar to the classical expression of the monopulse. In fact, we find the following equalities: aHz aHz R" , D" h , p2 p2
aHa b " , 11 p2
aHa aHa b " h , b " h h. 22 21 p2 p2
aHa b " h, 12 p2 (A.9)
Substituting Eq. (A.9) in Eq. (A.8) we obtain
C A
D
zHa aHa aHa h! h Re zHa DaHzD2 hK !h " . 1 aHa DaHa D2 h h! h aHa (aHa)2
B
(A.10)
If h "0, the PRT is constant and the antenna 1 pattern is sampled symmetrically around the peak, it follows that aHa "0. (A.11) h Thus Eq. (A.10) reduces to the standard monopulse equation operating on the set of echo samples which are amplitude modulated by the rotating antenna pattern: ReMaHz/aHzN h hK !h " . 1 aHa h h aHa
C D
(A.12)
References [1] L.V. Blake, Prediction of radar range, in: M.I. Skolnik (Ed.), Radar Handbook, 2nd Ed., McGraw-Hill, New York, 1990, Chapter 2, pp. 2.46—2.47. [2] R.C. Davis, L.E. Brennan, I.S. Reed, Angle estimation with adaptive arrays in external noise fields, IEEE Trans. Aerospace Electron. Systems AES-12 (2) (March 1976) 179—186. [3] A. Farina, Linear and non-linear filters for clutter cancellation in radar systems, Signal Processing 59 (1) (1997) 101—112. [4] F.C. Robey, D.R. Fuhrmann, E.J. Kelly, R. Nitzberg, A CFAR adaptive matched filter detector, IEEE Trans. Aerospace Electron. Systems AES-28 (1) (January 1992) 208—216.