Adaptive Radar Signal Detection with Integrated Learning and Knowledge Exploitation

Transcription

1 Integrated Learning and Knowledge Exploitation Hongbin Li Departent of Electrical and Coputer Engineering Steven Intitute of Technology, Hoboken, NJ 73 USA Muralidhar Rangaway AFRL/RYAP Bldg 6, 4 Avionic Circle, WPAFB, OH USA uralidhar.rangaway@u.af.il ABSTRACT We conider the proble of weak ignal detection in trong diturbance with a ubpace tructure. Unlike conventional ubpace detection technique relying on the availability of a large aount of training data, we conider a knowledge-aided (KA) ubpace detection approach for training liited cenario by incorporating partial prior knowledge of the ubpace. A unique feature of the propoed approach i that it can identify the iing ubpace bae and recover the full ubpace tructure by uing only the tet ignal, thu bypaing the need for training data. The propoed approach utilize a Bayeian hierarchical odel for knowledge repreentation. The odel i integrated within a pare Bayeian fraework, which proote parioniou ubpace repreentation of the oberved data. A variational Bayeian inference algorith i developed baed on the propoed odel to recover paraeter and ubpace tructure aociated with the diturbance, which are then brought into a generalized likelihood ratio tet (GLRT) to perfor ignal detection. Nuerical reult are preented to illutrate the perforance of the propoed ubpace detector in coparion with everal notable exiting ethod.. INTRODUCTION Detecting a weak ignal in trong diturbance (noie, interference, clutter, jaing, etc.) i a fundaental proble in radar, onar, and any other application. A popular approach i baed on uing an etiated covariance atrix of the diturbance obtained fro training data for diturbance itigation. Thi ha led to a faily of covariance atrix baed (a.k.a. fully adaptive) detector (ee [] and reference therein). One liitation of thee detector i that they require heavy training to enure the accuracy of the covariance atrix etiate. Training requireent can be reduced by exploiting tructure of the diturbance. A frequently conidered one i when the diturbance (approxiately) ha a low-rank ubpace tructure. When the ubpace i fully known, ignal detection ay proceed by projecting the obervation into the orthogonal copleent of the ubpace, followed by cro-correlating with the target ignal and energy noralization. Thi lead to a beta tet tatitic [], which i optiu in the ene that it i uniforly ot powerful (UMP) invariant [3]. When the ubpace i unknown, it can be etiated by uing, e.g., the principal eigenvector of the aple covariance atrix STO-MP-SET

2 contructed fro training data. The reulting ubpace detector i called the eigencanceler [4], which belong to a highly ucceful cla of reduced-rank (a.k.a. partially adaptive) detector [5]-[8]. Exploiting prior knowledge i another way to reduce training requireent. Technique following thi direction i uually dubbed knowledge-aided (KA) proceing [9], where the prior knowledge often refer to a prior etiate of the diturbance covariance atrix. Mot thee technique often eploy a KA etiate of the diturbance covariance atrix via colored loading, which involve linearly cobining the prior etiate with the aple covariance atrix. The weighting coefficient can be deterined via a Bayeian approach by treating the covariance atrix a a rando atrix aigned with a conjugate prior, e.g., the invere Wihart ditribution [], []. Then, the poterior etiate take a linear cobining for, with the cobining coefficient deterined by a paraeter that repreent the reliability of the prior knowledge. Mot above KA baed technique can be thought of a extenion of the fully adaptive detection, ince they rely on an iproved etiate of the fulldienional covariance atrix. Becaue of their fully adaptive nature, thee KA detector till require coniderable training unle the data dienion i fairly all or the prior knowledge i ufficiently accurate. In typical et-up, the training ize of a KA detector can be reduced roughly by a factor of two copared with it non-ka counterpart (e.g., []). While ipreive, the reduction ay be inufficient for radar detection in nonhoogeneou environent, where training data i carce ince the clutter i location dependent and ay vary ignificantly around the tet cell. In thi paper, we conider ubpace detection with partial prior knowledge of the diturbance ubpace, and develop new partially adaptive KA ethod for detection with extreely liited data (i.e., no training). Our tudy i otivated by the fact that in practice, we often have oe prior knowledge of the diturbance ubpace, either fro prior obervation or etablihed databae of the environent being oberved, e.g., patial location of doinant clutter catterer (ajor natural or an-ade tructure) in the urveillance area, and the angle- Doppler trace of the clutter pectru (a.k.a. clutter ridge) oberved by an airborne phaed-array, which can be deterined by otion paraeter of the oving ening platfor [, Section.6..]. Such inforation tranlate to knowledge of oe of the ubpace bai vector. To incorporate uch partial prior knowledge for detection, we introduce a hierarchical odel for knowledge repreentation, which i integrated within a Bayeian fraework for inference, leading to parioniou ubpace repreentation of the oberved data. Our Bayeian fraework for KA proceing i baed on pare Bayeian learning [3] and i ditinctively different fro the Bayeian fraework ued for covariance atrix baed KA proceing [], []. We develop a variational Bayeian inference algorith to recover paraeter and tructure aociated with the diturbance, which are then brought into a generalized likelihood ratio tet (GLRT) to perfor detection. Notation: Vector (atrice) are denoted by boldface lower (upper) cae letter. All vector are colun * T H vector. Supercript (), () and () denote coplex conjugate, tranpoe and coplex conjugate tranpoe, repectively. I denote an identity atrix. denote the cardinality of a et. Gaa( x; ab, ) denote the Gaa ditribution of rando variable x with cale and rate paraeter a and b, repectively: Gaa( xab ;, ) =Γ ( abx ) a a e bx () where Γ ( ) a t a = t e dt denote the Gaa function. Finally, N( x; μ, Φ ) denote the circularly yetric coplex Gauian probability denity function (PDF) of rando vector x with ean μ and covariance atrix Φ : H N( x; μ, Φ) =π Φ exp ( x μ) Φ ( x μ). () { } STO-MP-SET-4

3 . PROBLEM FORMULATION Conider the hypothei teting proble of detecting a known ultichannel ignal in diturbance: H : y = d (3) H : y = κ+ d N where y denote the obervation (a.k.a. tet data), the target ignal which i aued known but with an unknown coplex-valued aplitude κ, and d the diturbance ignal. The ultichannel obervation y ay conit of aple taken in pace (with ultiple antenna), tie, or jointly in both doain a in STAP [5]. In phaed-array and MIMO yte, i often referred to a the teering vector paraeterized by the radar look angle and/or target Doppler frequency, while d ay include clutter, jaing and noie. The teering vector i known ince for typical radar operation, the above hypothee are teted for pecific value of angle and Doppler frequency [4]. In any cae of practical interet, the diturbance d ay have a low-rank ubpace repreentation [3]: d= Hβ + n (4) N L where H L conit of L< N linearly independent bai vector of the ubpace, β contain the ubpace coefficient, and n i coplex white Gauian noie vector with zero ean and covariance σ I. For the hypothei teting (3) to be eaningful, it i aued that / pan( H ). Clearly, Hβ i a low-rank coponent in d. A noted before, jaing and clutter in radar often have a low-rank ubpace tructure. While the detection proble (3) ha been well tudied under the condition H i exactly known (e.g., [3]), we conider a practically otivated cae where only partial prior knowledge of H i available. To odel uch N M prior knowledge, we eploy an overcoplete dictionary atrix, M N, uch that Hβ = x (5) where x i an M pare vector with parity L. The dictionary atrix can be fored on a fine grid covering the entire paraeter pace that paraeterize, e.g., -dienional (D) direction-or-arrival (DOA) in beaforing or -dienional (D) angle-and-doppler plane in STAP. To focu on the ain proble (i.e., ubpace detection with inaccurate prior knowledge) without cauing exceive raification, we aue M i ufficiently large and will not conider the grid-iatch proble due to finite dicretization on the paraeter pace, which can be addreed by a nuber of recent technique (e.g., [5]-[7]). The prior knowledge can be repreented a a group of colun of. The knowledge i incoplete in that the ubet ay i oe colun that are neceary to repreent. More preciely, let {,,, M}, (6) denote the index et that indexe the colun of. If denote the true index et for, the knowledge can be denoted by a ubet. We do not aue knowledge of, the cardinality of and, equivalently, the rank of H. The proble of interet i to identify iing bae in via Bayeian learning, which are ued jointly with the prior knowledge to olve the hypothei teting (3). STO-MP-SET

4 3. KA SUBSPACE SIGNAL DETECTION We conider a generalized likelihood ratio tet (GLRT) approach for the detection proble by incorporating knowledge of the ubpace. The likelihood function under the H and H hypothee given obervation y are p (,, ; ) ( ;, ), β H σ y = N c y Hβ σ I (7) p (,,, ; ) ( ;, ). κ β H σ y = N c y κ+ Hβ σ I (8) The tet variable of the GLRT, given by ax p {,,, } ( κ, β, H, σ ; y) κ β H σ, (9) ax p ( β, H, σ ; y) { β, H, σ } require finding etiate of the unknown paraeter under both hypothee, which are dicued next. Under H, it i eay to ee the axiu likelihood etiate (MLE) of the aplitude κ conditioned on H and β i H ( ) ˆκ = y H β () H Plugging ˆκ in (8) and axiizing the reulting likelihood with repect to (w.r.t.) σ give the MLE of the noie variance a ˆ σ = Py PH β, () N H H where the ubcript indicate the etiate i obtained under the hypothei H and P I ( ) denote the projection atrix that project to the orthogonal copleent of. Subtituting () and () back into (8), we can obtain the MLE of H and β by { H, } argin H, P β = β y P Hβ. () The above leat-quare (LS) fitting iplie the following interpretation for the etiation. Specifically, after concentrating out κ and σ fro the likelihood function, the paraeter etiation proble under H reduce to an equivalent and iplified one that involve etiating only H and β by uing the tranfored data P y : P y PHβ e, where the N noie vector e conit of independent and identically = + ditributed (i.i.d.) zero-ean coplex Gauian entrie. Note that in the original etiation proble, the real fitting error P y PHβ coputed at the true value of H and β are lightly correlated. Thi interpretation will be eployed in Section IV. The etiation under H proceed in a iilar anner by uing (7). Specifically, the MLE of the noie variance conditioned on H and β i ˆ σ =, N y Hβ (3) STO-MP-SET-4

5 where the ubcript ignifie the etiate i obtained under hypothei H. In turn, the MLE of H and β are given by { H, β } = argin H, β y Hβ. (4) Clearly, () and (4) are iilar, but neither can be uniquely olved without additional inforation of the unknown, which are too any relative to the data ize. One approach i to exploit a paraetric odel for the L ubpace atrix H, e.g., the DOA φ of the interference ource in a beaforing etup, in which the proble becoe to jointly etiate φ and β. Equivalently, we can ue the pare repreentation (5) and write the cot function in () and (4) in the following unified for: in z Ax, (5) x where z P y and A P under H, while under H, z y and A. The iniization in (5) ha to be perfored with a parity contraint on x. The parity recovery proble can be olved by uing a wealth of technique fro greedy ethod to -nor baed procedure (e.g., [8]). However, thee technique are difficult to incorporate prior and potentially containated knowledge for ubpace recovery. The proble i deferred to Section IV, where we develop new technique to incorporate uncertain prior knowledge of the ubpace tructure to etiate H and β. Once { H, β } and { H, β } have been obtained, they can be ubtituted in (), () and (3) to copute the etiate of the ignal aplitude κ and noie variance σ. Uing thee paraeter etiate in (8) and (7), it i eay to how that the tet variable (9) can be iplified to a ratio of the noie variance etiate, and the GLRT i given by H ˆ σ TGLRT τ (6) ˆ σ H where τ denote a tet threhold. 4. KNOWLEDGE-AIDED SUBSPACE RECOVERY 4. A Bayeian Knowledge Model A hown in Section III, the ubpace etiation proble () and (4) under both hypothee can be cat in one fraework baed on the following eaureent odel: z = Ax+ e, (7) N N M where z denote the obervation, A M a known dictionary atrix, x an unknown pare vector with unknown parity L, and e the eaureent noie with ditribution N (, ) c γ I, where $\gaa$ denote the invere variance, which i alo unknown. Spare Bayeian learning (SBL) [3] i a popular approach that can be ued to recover the pare vector x fro (7). However, SBL doe not ipoe any prior knowledge on the parity pattern of x. We need oe extenion to incorporate prior knowledge to recover x. To facilitate dicuion, a brief review of SBL (ee [3] for ore detail) i ueful. The approach ue a Gauian invere Gaa hierarchical odel. Specifically, the pare vector x i odeled a conditional Gauian with PDF given by M p( x α ) = Nc( x;, α ), (8) = STO-MP-SET

6 where x denote the -th eleent of x and α T eployed for the invere variance vector α [ α,, α M ] : M = it invere variance. Meanwhile, a Gaa prior i p( α ) = Gaa( α; ab, ), (9) 6 where uggeted choice for hyperparaeter a and b are very all value, e.g.,, uch that the prior i unifor (over a logarithic cale) [3]. Such a broad prior over the hyperparaeter allow the poterior probability a to concentrate at very large value of oe of α, which effectively drive the correponding x (deeed irrelevant to data) to zero, thu leading to a pare olution. With prior knowledge on the upport of x, it i no longer eaningful to et the prior p( α ) to be identically non-inforative acro different. For ubpace coefficient x belonging to the knowledge et, i.e.,, where i defined in Section II, we hould avoid uing broad and parifying prior p α ), which caue the poterior ean to becoe unbounded. The pread of the Gaa ditribution can be reduced by chooing a larger value for the rate paraeter b. Therefore, we propoe to replace (9) with a fixed b by the following prior odel to incorporate prior knowledge: M p( α ) = Gaa( α; ab, ), () = where b i allowed to vary with. Specifically, we chooe a relatively larger value for b, e.g., b [.,], if, o that the prior i non-parifying over, while the other b reain all. The reulting odel i referred to a the ubpace knowledge (SK) odel: b,, SK: b = () 6 c,, c where b [.,] and denote the copleent of. Finally, the invere variance γ of the noie in (7) can be jointly etiated in the Bayeian approach by eploying a prior for γ. We ue a non-inforative Gaa prior for γ a in [3]: p( γ) = Gaa( γ; cd, ), () 6 where c= d =. 4. A Bayeian Knowledge Model Let θ {, x α, γ} denote a vector containing all paraeter to be etiated. With the probabilitic odeling dicued in Section IV-A, thee paraeter are treated a latent variable. A tandard Bayeian inference procedure would proceed to copute the poterior p( z θ) p( θ) p( θ z) =, (3) p( z) ( STO-MP-SET-4

7 which i however infeaible for the conidered proble ince the arginal ditribution p( z) = p( z θ) p( θ)dθ cannot be coputed analytically. To circuvent the difficulty, we eploy variational Bayeian inference that utilize an approxiation of the poterior p( θ z ). Variational Bayeian ethod have been ued with great ucce in variou application { } (ee [9] and reference therein). Specifically, we write θ in a partitioned for: θ θ, θ,, θ K, where K = 3 for the SK odel with: θ x, θ α, and θ3 γ. A popular approxiation of the poterior p( θ z ) i baed on the ean field approxiation (e.g, [9]): PDF i given by ( ) ( ) exp ln p( z, θ) l k qk( θk) =, exp ln p( z, θ) dθ K p( θ z) qk( θ k), where the coponent where p(, z θ) = p( z θ) p() θ denote the joint ditribution of z and θ, while l=/ kdenote the tatitical expectation w.r.t. ditribution q ( θ ), l =/ k, that i, l l l k k k = (4) ln p( z, θ) = ln p( z, θ) q ( θ )d θ. (5) l=/ k l l l l k Note that (4) i not an explicit olution ince the factor poterior q ( θ ) depend on the other factor q ( θ ), l =/ k. However, it point naturally to an iterative procedure for finding the factor. Specifically, we l l can tart by initializing q ( θ ) = p( θ ), for t =, where t i the iteration index; that i, the factor k k k poterior are initialized by their correponding prior ditribution. Then, for the t -th iteration, we can update ( t qk ( θ k) by uing q ) l ( θ l), l =/ k, in the right-hand ide of (4). Each iteration cycle through all factor fro k = to k = K, and the iterative proce top till a practical convergence criterion ha been et. Next, we conider variational Bayeian inference baed ubpace etiation by integrating the knowledge odel introduced in Section IV-A. With the SK odel, we have p(, zxα,, γ) = p( z x, γ) p( x α) p() α p() γ (6) where p( z x, γ) = Nc( z; Ax, γ I ), p( x α) i given by (8), p( α) by (), and p( γ ) by (), repectively. It can be hown the factor poterior qx ( x ), q α ( α ), and q γ ( γ ) are repectively Gauian, product Gaa, M and Gaa ditribution: q x ( x) = c( x; μ, Φ), qα ( α) = Gaa( α; a+, b ), ( ) qγ ( γ) = Gaa γ; c+ N, d, where b = b + μ +Φ,, =,, M, k k = H, ( ) μ = γ ΦA H z, Φ = γ A A+ D H d = d + z A μ + tr{ AΦA }. A uch, the update of the factor poterior boil down to the update of thee paraeter. We uarize the reulting etiator in Algorith. STO-MP-SET

8 Algorith - SK-baed ubpace etiator Initialize Let α = a/ b, =,, M, () repeat ) Set t = t+. ) Update the covariance and ean of 3) Update the rate and ean of where q α () γ = c/ d, q ( x ): x ( ) ( ) H ( ) = γ t + t, { } A diag α,, α M, and t =. () () () Φ A A D (7) ( α) μ denote the -th eleent of 4) Update the rate and ean of Until convergence () ( ) () μ t = γ t Φ t A H z. (8) ( ) b = b + μ +Φ,, (9) α = ( a+ )/ b, =,, M, (3) q γ ( γ ) μ and Φ the -th diagonal eleent of, H d = d+ z A + tr AΦ A, Φ. μ { } (3) γ = ( c+ N)/ d. (3) Convergence i reached when the difference of oe paraeter etiate over two conecutive iteration i ufficiently all. We ue μ, the poterior ean of q ( x ) and alo an etiate of the pare vector x. An etiate of the ubpace atrix (i.e., H under H or correponding to the upport of μ. 5. NUMERICAL RESULTS x PH under H ) can be obtained a the colun of A We now preent iulation reult to illutrate the perforance of the propoed detector. The diturbance d ha a ubpace tructure a in (4), where H i fored by L Fourier vector with frequencie centered around the zero frequency, i.e., a lowpa narrowband interference. Specifically, let denote an N M dicrete Fourier tranfor (DFT) atrix, and the ubpace atrix H conit of the following colun of :{,,, L +,, L M, M L + +,, M }, where and denote the floor and, repectively, ceiling operator. We et N = 3, M = 64, and L = 7 in iulation. The target ignal i a Fourier vector with a noralized frequency.3 Hz. A tandard auption i that the interference bandwidth doe not overlap with the ainlobe of the target repone (otherwie, the interference will be detected a target). Therefore, 4 colun of, which cover the ainlobe of the target repone, are reoved to for the $N\tie M$ dictionary atrix [cf. Section II], where M = 6. The ignal-to-noie ratio (SNR) and interference-to-noie ratio (INR) are defined a: SNR = N κ / σ and INR = N β / σ STO-MP-SET-4

9 For brevity, the propoed detector baed on odel () i referred to a the SK detector. We copare with 4 other known detector a benchark, naely the clairvoyant ubpace detector of [], which aue perfect knowledge of the ubpace atrix H, the conventional KA detector, which take the ae for a the clairvoyant detector except that H i replaced by the prior knowledge of H, the adaptive ubpace detector (ASD), and a non-inforative SBL detector, which eploy a iilar Bayeian inference fraework a the propoed one but i not provided with any prior knowledge of the ubpace. The ASD i alo identical to the clairvoyant detector but replace H with an etiate coniting of the L principal eigenvector of the aple covariance atrix. The latter i contructed fro T = 8 target-free hoogeneou training ignal, which hare the ae ubpace atrix H. Note T = for the other detector, which require no training. For the KA detector, including the conventional KA and SK, we conider two cae for the prior knowledge including: Cae : =. The prior knowledge i perfect; Cae :. The prior knowledge for each iulation trial contain = 4 randoly elected indice of. 5. Cae : Full Knowledge Cae i of interet to how how the propoed SK behave in the preence of perfect prior knowledge. Figure (a) depict the probability of detection P d veru the SNR for the variou detector, where INR = 3 db and the probability of fale alar P f 3 =. The conventional KA in the current cae reduce to the clairvoyant detector. The perforance of the propoed SK i alo nearly identical to the clairvoyant, anifeting the benefit of the prior knowledge. The near optiality alo indicate that SK rarely reject c correct bai vector in or add erroneou bae in. Meanwhile, without uing the prior knowledge, SBL i the wort detector. The ASD i the only detector that require training. With T = 8 target-free i.i.d. training ignal, it perforance i till notably wore than the KA detector. Figure (b) how P d veru P f, i.e., the receiver operating characteritic (ROC) curve, where SNR = 5 db and INR = 3 db. The relation aong the variou detector are iilar to what were oberved before. (a) (b) Figure : Cae reult. (a) P d v. SNR with INR = 3 db and db and INR = 3 db. -3 P f =. (b)roc curve with SNR = 5 STO-MP-SET

10 5. Cae : Partial Knowledge We next conider a partial knowledge cae with and = 4, that i, the prior knowledge contain 4 randoly elected bae fro in each trial. The reult are hown in Figure. It i een that the SK detector i the cloet to the clairvoyant and ignificantly outperfor the conventional KA. Thi i becaue c SK i able to identify iing bae in and therefore can better reject the interference than the conventional KA detector. (a) (b) Figure : Cae (partial knowledge) reult. (a) P d v. SNR with INR = 3 db and P f = curve with SNR = 5 db and INR = 3 db. -3. (b)roc 6. CONCLUSIONS We preented a new knowledge-aided (KA) approach for ignal detection in trong diturbance by exploiting prior knowledge of the ubpace tructure of the diturbance. A unique feature i that the propoed approach account for the fact that the prior knowledge available in practice ay be incoplete. To addre uch uncertaintie, we introduced a Bayeian hierarchical odel for knowledge repreentation. The odel wa integrated in a Bayeian learning fraework, and a variational inference algorith baed on iple iteration tep wa developed to olve the aociated inference proble. Siulation reult, which cover two cae of full and, repectively, partial prior knowledge, deontrate that the propoed KA detector can benefit fro prior knowledge and ignificantly outperfor conventional detector when the available prior knowledge i incoplete. REFERENCE [] A. De Maio, Rao tet for adaptive detection in Gauian interference with unknown covariance atrix, IEEE Tranaction on Signal Proceing, vol. 55, no. 7, pp , 7. [] P. Wang, H. Li, and B. Hied, Moving target detection uing ditributed MIMO radar in clutter with nonhoogeneou power, IEEE Tranaction on Signal Proceing, vol. 59, no., pp , Oct STO-MP-SET-4

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