2014 Workshop on Mathematical Issues in Information Sciences. Cognitive Radar Signal Processing

Transcription

1 Cover 2014 Workshop on Mathematical Issues in Information Sciences Cognitive Radar Signal Processing Antonio De Maio, July 7, 2014 Professor, University of Naples, Federico II, Fellow IEEE A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

2 Introduction Introduction Cognition: Conscious mental activity that informs a person about his or her environment (US National Institute of Mental Health). It requires: perceiving, thinking, reasoning, judging, problem solving, and remembering; being smart and agile in the interaction with the environment. J. R. Guerci, Cognitive Radar:The Next Radar Wave?, Microwave Journal, January S. Haykin, Cognitive Radar: A Way to the Future, IEEE Signal Processing Magazine, January A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

3 Introduction Introduction It is necessary to develop a mapping among the mentioned biological cognitive properties and the corresponding activities in a cognitive radar. What exactly are the potential benefits of a radar possessing cognitive capabilities? Type of radar Type of mission Operative Environment During this presentation some benefits will be highlighted. J. R. Guerci, Cognitive Radar: The Knowledge-Aided Fully Adaptive Approach, Artech House Remote Sensing Library. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

4 The Cognitive Radar Functional Elements and Characteristics of a Cognitive Radar Architecture Let us consider the basic block diagram of a conventional adaptive radar. Adaptivity is usually confined to the receiver and is based solely on the received data stream. In general, there is not provision for learning over time, feedback to the transmitter, or the integration of environmental exogenous sources such as Geographical Information Systems (GISs) or Digital Terrain Maps. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

5 The Cognitive Radar Functional Elements and Characteristics of a Cognitive Radar Architecture A cognitive radar architecture is characterized by the presence of an environmental dynamic database and, remarkably, the possibility of transmitter adaptivity. Otherwise stated, there are a number of advanced functionalities attempting to emulate those in a biologically cognitive systems. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

6 The Cognitive Radar Functional Elements and Characteristics of a Cognitive Radar Architecture The Environmental Dynamic Database (EDDB) contains knowledge of the environment and/or targets of interest gleaned from endogenous or exogenous information sources. This component permits a Knowledge-Aided Signal Processing. In addition to an adaptive receiver, the cognitive radar includes an adaptive transmitter based on the feedback from the receive chain and the interactions with EDDB. The multiple transmit degrees of freedom also give rise to the concept of Multiple Input Multiple Output (MIMO) radar. The joint TX and RX adaptivity entails the development of innovative signal processing techniques. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

7 Waveform Diversity Waveform Diversity Most radars have some forms of primitive transmit adaptivity, usually in the terms of mode selection, as for instance: medium range vs long range; track vs search; low resolution vs high resolution; staggering. Waveform diversity is already used by many echolocating mammals (bats, whales, and dolphins) as an inherent component of their normal behaviour. On the radar side, there is an attempt to mimic what Mother Nature has realized during the evolution of many species. Let us focus on the bats which exploit tongue clicking to produce waveforms with a variety of modulations that are transmitted via bone and muscle tissue to form an illuminating beam. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

8 Waveform Diversity Waveform Diversity in the Bat s Signal Waveform Diversity in the Bat s Signal Let us consider the signal emitted by a bat to understand how it uses diversity. Observations: presence of spatial, temporal, and waveform diversity. C. Baker, H. Griffiths, A. Balleri, Biologically Inspired Waveform Diversity, chapter 6 of the book Waveform Design and Diversity for Advanced Radar Systems, IET 2012 (Editors: F. Gini, A. De Maio, L. Patton). A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

9 Waveform Diversity Waveform Diversity in the Bat s Signal Waveform Diversity in the Bat s Signal The signal emitted by the Eptesicus Nilssoni bat. There are a Low Pulse Repetition Frequency (PRF) regime and a High PRF regime. Moreover, the signal shape changes from pulse-to-pulse. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

10 Waveform Diversity Waveform Diversity in the Bat s Signal Waveform Diversity in the Bat s Signal Approach phase Terminal phase The distance between the bat and the prey is low enough that even slight trajectory changes would produce large Doppler errors (hence large ranging errors) between consecutive pulses. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

11 Waveform Diversity Waveform Diversity in the Bat s Signal Avoiding Being Eaten by Bats: Countermeasures Some insects have developed evasive behaviours in response to signals from bats. For instance, in two ways: arctiids might disturb bat biosonar by simulating multiple targets; by interfering with range determination. The same dicothomy exists between Electronic- Counter-Measures (ECMs) and Electronic-Counter- Counter-Measures (ECCMs). L. A. Miller and A. Surlykkej, How Some Insects Detect and Avoid Being Eaten by Bats: Tactics and Countertactics of Prey and Predator, BioScience, Vol. 51 No. 7, July A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

12 Waveform Diversity Waveform Diversity in the Bat s Signal Limitations Why transmit adaptivity is not yet used in radar? Two fundamental reasons: 1 it requires advanced hardware (digital arbitrary waveform generators, solid state transmitters, etc.); 2 it necessitates the availability of new transmit adaptation algorithms. Next hardware generation and the recent technological progress in digital electronic. Potential benefits achievable through waveform diversity to motivate this new radar functionality. J. S. Bergin and P. M. Techau, High-Fidelity Site-Specific Radar Simulation: KASSPER 02 Workshop Datacube, ISL Technical Note, Vienna, May A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

13 Optimizing Fast-Time Modulation Optimizing Fast-Time Modulation Let us analyze the benefits of tailoring the transmit waveform (fast-time modulation) to account for a colored noise RF interference source. The optimum waveform optimally redistributes the transmit energy in the frequency domain so as to maximizing the Signal-to-Interference-plus-noise-Ratio (SINR). Additional context-dependent constraints can be also forced to the radar waveform. This shaping technique can be also exploited to control the impact of radar on other communication systems. A. De Maio, S. De Nicola, Z.-Q. Luo and S. Zhang, Design of Phase Codes for Radar Performance Optimization with a Similarity Constraint, IEEE Transactions on Signal Processing, February A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

14 Cognitive Radar Waveform Design for Spectral Coexistence Spectral Coexistence Spectrally Crowded Environments Coexistence among radar and telecommunication systems is currently becoming one of the challenging research topics in both radar and communication communities. Basic electromagnetic considerations, such as good foliage penetration, and low path loss attenuation, push some communication and radar systems to coexist in the same frequency band (for instance VHF and UHF). It is thus mandatory the development of advanced radar signals ensuring compatibility with the surrounding electromagnetic radiators, namely keeping acceptable the mutual interference induced on frequency overlaid systems. A. Aubry, A. De Maio, M. Piezzo, and A. Farina, Radar Waveform Design in a Spectrally Crowded Environment via Nonconvex Quadratic Optimization, IEEE Transactions on Aerospace and Electronic Systems, April A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

15 Cognitive Radar Waveform Design for Spectral Coexistence Signal Model Let us consider a monostatic radar system transmitting a signal composed of N subpulses, and denote by c = [c(1),..., c(n)] T C N the N-dimensional fast-time radar code. Thus, the N-dimensional column vector v C N of the observations, from the range-azimuth cell under test, can be expressed as: v = αc + n. α is a complex parameter accounting for channel propagation and backscattering effects from the target within the range-azimuth bin of interest; n is the N-dimensional column vector containing the filtered disturbance echo samples: 1 it accounts for both white internal thermal noise as well as interfering signals with same frequencies as the radar of interest; 2 it is modeled as a complex, zero-mean, circular Gaussian random vector sharing the covariance matrix M. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

16 Cognitive Radar Waveform Design for Spectral Coexistence Cooperative Radiators & Produced Interference As to the cooperative radiators coexisting with the radar of interest, let us assume that each of them is working over a frequency band Ω k = [f1 k, f2 k ]. To guarantee spectral compatibility with K overlayed radiators, the radar has to control the energy produced on the shared frequency bands, namely the transmitted waveform has to comply with c R I c E I R I = K k=0 w kr k I ; RI k (m, l) = f k 2 f k 1 m = l e j2πf k 2 (m l) e j2πf k 1 (m l) j2π(m l) m l (m, l) {1,..., N} 2 ; w k 0, k = 0,..., K, are suitable weights related to the importance of a given radiator; E I is the amount of allowed interference level. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

17 Cognitive Radar Waveform Design for Spectral Coexistence Cognitive Spectrum Awareness Radio Environment Map (REM) represents the key to gain spectrum cognizance which is at the base of an intelligent and agile spectrum management. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

18 First Cognitive Waveform Design Approach Waveform Design: Objective Function & Constraints 1/2 Optimizing the detection performance, through the maximization of the Signal to Interference plus Noise Ratio (SINR),namely where R = M 1. SINR = α 2 c Rc, Ensuring desirable radar features to the transmitted waveform forcing an energy constraint and a similarity constraint with a prescribed waveform c 0, namely c 2 = 1 c c 0 2 ɛ (ɛ ruling the size of the similarity region, c 0 = 1), so as to indirectly control some relevant characteristics of the waveform. Providing a control on the interference energy produced on shared bands, in order to ensure spectral coexistence with overlaid wireless networks, enforcing the constraint c R I c E I. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

19 First Cognitive Waveform Design Approach Code Design: Objective Function & Constraints 2/2 The waveform design problem can be formulated as the following optimization problem: P 1:QCQP max c Rc c s.t. c c = 1 c R I c E I c c 0 2 ɛ Problem P 1 is a non-convex optimization Quadratically Constrained Quadratic Problem (QCQP). A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

20 First Cognitive Waveform Design Approach Code Design: Feasibility of the Optimization Problem Not all the pairs (E I, ɛ) produce a feasible problem P 1. As a consequence, it is mandatory characterizing the feasibility of P 1 as function of E I and ɛ, for any given similarity code c 0. This observation leads to the following definition of the so-called I/S achievable region associated to the radar code c 0: F = {(E I, ɛ) : E I λ min (R I ), 0 ɛ 2, problem P 1 is feasible} By studying the set F, it is possible to show that: 1 the I/S achievable region is a convex set (from a practical point of view we can control its accuracy description); 2 each point on its boundary region can be computed in a polynomial time. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

21 First Cognitive Waveform Design Approach Code Design: Solution to the Optimization Problem 1/3 Let us observe that an optimal solution to P 1 can be obtained from an optimal solution to the following non-convex Enlarged Quadratic Problem (EQP) P 2: P 1:QCQP max c Rc c s.t. c c = 1 c R I c E I c c 0 2 ɛ Equivalent P 2:EQP max c Rc c s.t. c c = 1 c R I c E I c c 0c 0 c δɛ where δ ɛ = (1 ɛ/2) 2. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

22 First Cognitive Waveform Design Approach Code Design: Solution to the Optimization Problem 2/3 Exploiting the equivalent matrix formulation of P 2 and neglecting the rank-one constraint, we obtain the following convex SDP Enlarged Quadratic Problem Relaxed (EQPR) P 3: P 2:EQP max c Rc c s.t. c c = 1 c R I c E I c c 0c 0 c δɛ Relaxation P 3:EQPR max tr (CR) C s.t. tr (C) = 1 tr ( ) C R I EI tr ( ) C C 0 δɛ C 0 Problem P 3 is solvable, since its feasible set is compact and its objective function is continuous. Problem P 2 is hidden convex, namely the relaxation of P 2 into P 3 is tight. It is possible to construct a rank-one optimal solution c c to P 3, starting from an arbitrary rank optimal solution C. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

23 First Cognitive Waveform Design Approach Code Design: Solution to the Optimization Problem 3/3 Algorithm 1 summarizes the procedure leading to an optimal solution to P 1. The computational complexity connected with the implementation of the algorithm is polynomial as both the SDP problem and the decomposition can be performed in polynomial time. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

24 First Cognitive Waveform Design Approach Code Design: Example 1/4 The radar designer can choose the pair (E I, ɛ) to suitably trade off spectral coexistence, desirable radar waveform characteristics and achievable SINR. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

25 First Cognitive Waveform Design Approach Code Design: Example 2/4 A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

26 First Cognitive Waveform Design Approach Code Design: Example 3/4 A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

27 First Cognitive Waveform Design Approach Code Design: Example 4/4 A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

28 Extension of the Cognitive Waveform Design Approach Extension: Energy Modulation The previous design technique can be further improved through a suitable modulation of the transmitted waveform energy, which is no longer kept fixed. The energy modulation can be accounted for through the constraint 1 η c c 1 where η (0 η 1) is a design parameter which rules the maximum allowable decrease of the radar transmit power (hence it can be set based on radar range equation argumentations or radar maximum operation range). A. Aubry, A. De Maio, Y. Huang, M. Piezzo, and A. Farina, A New Radar Waveform Design Algorithm with Improved Feasibility for Spectral Coexistence, accepted for publication on IEEE Transactions on Aerospace and Electronic Systems. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

29 Extension of the Cognitive Waveform Design Approach Code Design: Objective Function & Constraints 1/2 The considered waveform design problem can be formulated as the following non-convex optimization Quadratically Constrained Quadratic Problem (QCQP) P 1: P 1:QCQP max c Rc c s.t. 1 η c c 1 c R I c E I c c 0 2 ɛ Equivalent P 2:QCQP Reformulation max tr (Q x 0X ) s.t. 1 η tr (Q 1X ) 1 tr (Q 2X ) E I tr (Q 3X ) 0 tr (Q 4X ) = 1 X = xx, x = [c T, t] T where [ R 0 Q 0 = 0 0 [ Q 3 = ] [ I 0, Q 1 = 0 0 I c 0 1 ɛ c 0 ] [ RI 0, Q 2 = 0 0 ], Q 4 = [ ] ], A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

30 Extension of the Cognitive Waveform Design Approach Code Design: Objective Function & Constraints 2/2 Problem P 2 can be relaxed into the following problem P 3 P 2:QCQP Reformulation P 3:SDP Relaxation max tr (Q x 0X ) s.t. 1 η tr (Q 1X ) 1 tr (Q 2X ) E I tr (Q 3X ) 0 tr (Q 4X ) = 1 X = xx, x = [c T, t] T Relaxation max tr (Q 0X ) X s.t. 1 η tr (Q 1X ) 1 tr (Q 2X ) E I tr (Q 3X ) 0 tr (Q 4X ) = 1 X 0 Problem P 2 is hidden convex, namely the relaxation of P 2 into P 3 is tight. An optimal solution c to problem P 1 can be obtained from an arbitrary rank optimal solution X to problem P 3. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

31 Spectral Coexistence in Signal-Dependent Interference Extension: Signal Dependent Interference The previous design techniques can be further extended to the case of a radar operating in a highly reverberating environment. The additional signal-dependent clutter environment disturbance contribution can be accounted for through the term i = N 1 k= N+1,k 0 α k J k c which is the superposition of returns from the range cells adjacent that under test, with covariance matrix Σ i (c), where J k (l, m) = 1 if l m = k, 0 elsewhere, and α k CN (0, β k ). The covariance matrix of the clutter Σ i (c) depends on the radar code c and the m.s.v. of the clutter amplitude returns β k. The radar system exploits a dynamic environmental database to predict the actual scattering scenario. A. Aubry, A. De Maio, M. Piezzo, M. M. Naghsh, M. Soltanalian, and P. Stoica, Cognitive Radar Waveform Design for Spectral Coexistence in Signal-Dependent Interference, Proceedings of the 2014 IEEE Radar Conference (RADARCON),Cincinnati, OH, USA, May A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

32 Spectral Coexistence in Signal-Dependent Interference Signal Model The N-dimensional column vector v C N of the fast-time observations, can be expressed as: v = αc + i + n. α is a complex parameter accounting for channel propagation and backscattering effects from the target within the range-azimuth bin of interest; n CN (0, M int ) accounts for white internal thermal noise as well as interfering (licensed and unlicensed) radiators. Hence, the SINR can be expressed as SINR = w c 2 w [Σ i (c) + M int ] w A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

33 Spectral Coexistence in Signal-Dependent Interference Problem Formulation The main goal is to optimize the radar detection performance, through the maximization of the SINR, ensuring the spectral coexistence with overlaid licensed radiators. The developed optimization procedure is based on the joint design of the radar code c and the receive filter w, which can be formulated as the following constrained optimization problem: P max c,w w c 2 w [Σ i (c) + M int ] w s.t. c c = 1 c R I c E I c c 0 2 ɛ Problem P is a non-convex optimization problem, since the objective function is a non-convex function and the constraint c 2 = 1 defines a non-convex set. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

34 Spectral Coexistence in Signal-Dependent Interference SINR Sequential Optimization Procedure 1/2 Good Quality Solution to problem P 1 Given w (n 1), we search for an admissible radar code c (n) at step n improving the SINR corresponding to the receive filter w (n 1) and the transmitted signal c (n 1). 2 Whenever c (n) is found, we fix it and search for the filter w (n) which improves the SINR corresponding to the radar code c (n) and the receive filter w (n 1), and so on. w (n) and c (n) are used as starting points at step n + 1. To trigger the procedure, the optimal receive filter w (0), for an admissible code c (0), can be considered. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

35 Spectral Coexistence in Signal-Dependent Interference SINR Sequential Optimization Procedure 2/2 P w (n) : Filter Design max w w c (n) 2 w [ Σ i ( c (n)) + M int ] w Problem P (n) w is solvable, and a closed form optimal solution w (n) can be found for any c (n). w (n) = [Σ i ( c (n)) + M int ] 1c (n) c (n) [ Σ i ( c (n)) + M int ] 1 c (n) P c (n) : Code Design w (n 1) c 2 max c w (n 1) [Σ i (c) + M int ] w (n 1) s.t. c 2 = 1 c R I c E I c c 0 2 ɛ Problem P (n) c is a hidden-convex optimization problem. Hence, an optimal solution can be obtained: relaxing the problem into a fractional SDP; applying the Charnes-Cooper transformation to get an Equivalent SDP. exploiting a suitable rank-one matrix decomposition procedure to get an optimal solution to the original non-convex optimization problem. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

36 Target Classification Target Classification Let us denote by h 1(n) and h 2(n) the impulsive responses of targets 1 and 2. It is reported H 1(v) H 2(v) ; The modulus of the optimum pulse spectrum. The optimal waveform places more energy in those spectral regions where the two frequency responses (H 1(ν) and H 2(ν)) are significantly different. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

37 Transmit Beamforming Diversity Beamforming at the Transmitter end While fast-time modulation has been used in the previous example, other degrees of freedom can also be exploited at the transmitter end: 1 transmitting azimuth and elevation beampattern; 2 polarization; 3 slow-time coding. Transmit beampattern can be optimized to reduce as much as possible the effects of strong sidelobe unwanted targets or clutter discretes. Note the presence of nulls in transmit pattern along directions of competing targets/discrete clutter elements. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

38 Knowledge-Aided (KA) Radar Signal Processing Knowledge-Aided (KA) Radar Signal Processing Real-time exploitation of the a- priori knowledge about the radar operating environment. KA Phylosophy: The radar knows what it sees The environmental context is the key to an efficient adaptivity. Many different knowledge sources can be available. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

39 Knowledge-Aided (KA) Radar Signal Processing A-Priori Information and GIS Representation Geographic map of the experiment site. Intensity of clutter returns: strongest returns are represented in red, weakest returns are in blue. GIS representation of the considered dataset: blue cells indicate sea, orange cells indicate land, red cells transition interfaces land-sea. A. De Maio, A. Farina, G. Foglia, Design and experimental validation of knowledge-based constant false alarm rate detectors, IET Radar Sonar & Navigation, A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

40 Knowledge-Aided (KA) Radar Signal Processing Cell Averaging CFAR (CA-CFAR) Decision Rule r cut 2 K i=1 r i 2 H 1 > < H 0 T CA 621 false alarms over tests. Only one target is over the threshold. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

41 Knowledge-Aided (KA) Radar Signal Processing Knowledge-Aided CFAR System Decision Rule Knowledge-Aided Training Data Selection plus CA-CFAR 82 false alarms over tests. All the targets are over the threshold. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

42 Knowledge-Aided Techniques for Covariance Matrix Estimation Knowledge-Aided Techniques for Covariance Matrix Estimation Conventional adaptive detectors require an estimation of the interference covariance matrix. They achieve satisfactory detection performance when the size K of the homogeneous sample support complies with K 2N. In real environments the number of data where the interference is homogeneous is very limited. Indirect Approach: indirect exploitation of prior knowledge sources (for instance secondary data selection) Direct Approach: a-priori information used directly in the adaptive receiver design process. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

43 Knowledge-Aided Techniques for Covariance Matrix Estimation Knowledge-Aided STAP: Indirect Approach The data selector chooses secondary range-doppler cells that have the same terrain as the test clutter cell. National Land Cover Data (NLCD) are used to classify the ground environment illuminated by the radar. Conventional Adaptive Processing Terrain map for MCARM flight 5, acquisition 151. Knowledge-Aided Processing C. T. Capraro, G. T. Capraro, A. De Maio, A. Farina, and M. Wicks, Demonstration of knowledge-aided space-time adaptive processing using measured airborne data, IEE Proceedings Radar, Sonar & Navigation, Vol 153, Issue 6, A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

44 Knowledge-Aided Techniques for Covariance Matrix Estimation Multiple Models Exploitation: Direct Approach Resorting to reflectivity/spectral clutter models, meteorological data, and previous scans/experiences, multiple models for the interference covariance matrix can be conceived. These a-priori models can be used to devise KA detection algorithms, where slowtime covariance estimation is performed forcing the inverse interference covariance X to belong to the uncertainty set { } H A = X 0 : X = t i X i, X I σ, t 2 i R, i = 0,..., H, with X i, i = 0,..., H the inverse of the a-priori models. i=0 A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

45 Knowledge-Aided Techniques for Covariance Matrix Estimation Multiple Models Exploitation The constrained Generalized Likelihood Ratio Test (GLRT) shares the form max [det(x )] K+1 exp { tr [ X ((r αp)(r αp) + RR ) ]} α C, X A (rr + RR ) ]} max [det(x )] K+1 exp { tr [ X X A H 1 > < H 0 η r C N primary data (data from the cell under test). R = [r 1,..., r K ] C N,K secondary data matrix. p unitary norm steering vector. α unknown parameter accounting for target response. A covariance matrix uncertainty set: the inverse covariance matrix is expressed as linear combination of the inverse of the available a-priori models, also accounting for a lower bound on the power of the white disturbance term. The unknown parameters appearing in optimal decision statistics are replaced with their constrained Maximum Likelihood (ML) estimates under each hypothesis. A. Aubry, V. Carotenuto, A. De Maio, G. Foglia, Exploiting Multiple A-Priori Spectral Models for Adaptive Radar Detection, IET Radar Sonar & Navigation, A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

46 Knowledge-Aided Techniques for Covariance Matrix Estimation Multiple Models Exploitation The constrained ML estimates of the unknown parameters under the hypotheses H 0 and H 1 are optimal solutions to optimization problems P(S H0 ) and P(S H1 ) min α,x P(S Hk ) subject to { tr [ X SHk ] log det(x ) } X A α Θ Hk k = 0, 1 Θ H0 = {0}. Θ H1 = C. S H0 = 1 K+1 (rr +RR ). S H1 = 1 K+1 (Rα+RR ). R α = (r αp)(r αp). Under H 0 The ML estimate ˆX ML H 0 of X, under H 0, is the optimal solution to P(S H0 ) and can be efficiently computer in polynomial time using interior point methods. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

47 Knowledge-Aided Techniques for Covariance Matrix Estimation Multiple Models Exploitation Under H 1 The ML estimates ˆα ML of α and ˆX ML H 1 of X, under H 1, are obtained resorting to the following alternating optimization procedure: 1 Initialize the algorithm considering α (0) = p r p 2, ML (0) = 0, and R ˆα (0) = ( r α (0) p ) ( r α (0) p ) 2 set n = n estimate ˆX (n) H 1 solving the MAXDET problem P(S (n 1) H 1 ) 4 estimate the target complex amplitude ˆα (n) = (n) p ˆX H 1 r (n) p ˆX H 1 p 5 compute R ˆα (n) = ( r ˆα (n) p ) ( r ˆα (n) p ) { [ )]} 6 compute ML (n) = [det( ˆX (n) H )] K+1 exp tr ˆX (n) 1 H (R 1 ˆα (n) + KS 7 if ( ML (n) ML (n 1) ζ), set ˆα ML = ˆα (n) and ˆX ML H 1 else, return to the step 2. = ˆX (n) H 1 ˆX (n) f (α, X ) maximize over X given ˆα (n 1) maximize over α given ˆX (n) ˆα (n) A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

48 Knowledge-Aided Techniques for Covariance Matrix Estimation Multiple Models Exploitation Results for H = 20 Gaussian shaped a-priori models and a bimodal, exponential shaped, interference PSD. SINR values achieving P d = 0.9 GLRT-1 GLRT-2 Optimum AMF Kelly s GLRT K = 10, H = K = 20, H = A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

49 Knowledge-Aided Techniques for Covariance Matrix Estimation Multiple Models Exploitation Black curve: Actual PSD; Red curve: PSD using the proposed constrained covariance matrix estimator; Magenta curve: PSD using the sample covariance matrix. Also with very small training data size (K = 1) the proposed algorithm is able to track the actual PSD. The estimation accuracy is better and better as the sample support increases. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

50 Conclusions Conclusions The cognitive radar concept has been discussed based on two fundamental ingredients: 1 transmit diversity; 2 Knowledge-Aided signal processing. Understanding how and why biological systems exploit diversity is the key to improve sensing in synthetic systems. Even though the presentation is focused on radar, other sensors and communication systems could benefit of the cognitive paradigm. A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51

51 End THANK YOU FOR THE KIND ATTENTION A. De Maio (University of Naples) Cognitive Radar Signal Processing July 7, / 51