Hindawi Publishing Corporation International Journal of Antennas and Propagation Volume 2013, Article ID 261230, 21 pages http://dx.doi.org/10.1155/2013/261230

Research Article Array Processing for : Achievements and Challenges

Ulrich Nickel

Fraunhofer Institute for Communication, Information Processing and Ergonomics (FKIE), Fraunhoferstrasse 20, 53343 Wachtberg, Germany

Correspondence should be addressed to Ulrich Nickel; [email protected]

Received 26 March 2013; Accepted 26 July 2013

Academic Editor: Hang Hu

Copyright Β© 2013 Ulrich Nickel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Array processing for radar is well established in the literature, but only few of these algorithms have been implemented in real systems. The reason may be that the impact of these algorithms on the overall system must be well understood. For a successful implementation of array processing methods exploiting the full potential, the desired radar task has to be considered and all processing necessary for this task has to be eventually adapted. In this tutorial paper, we point out several viewpoints which are relevant in this context: the restrictions and the potential provided by different array configurations, the predictability of the transmission function of the array, the constraints for adaptive beamforming, the inclusion of monopulse, detection and tracking into the adaptive beamforming concept, and the assessment of superresolution methods with respect to their application in a radar system. The problems and achieved results are illustrated by examples from previous publications.

1. Introduction between array processing and preceding and subsequent radar processing. We point out the problems that have to be Array processing is well established for radar. Publications of encountered and the solutions that need to be developed. We this topic have appeared for decades, and one might question start with spatial sampling, that is, the antenna array that has what kind of advances we may still expect now. On the other to be designed to fulfill all requirements of the radar system. hand,ifwelookatexistingradarsystemswewillfindvery A modern radar is typically a multitasking system. So, the few methods implemented from the many ideas discussed in design of the array antenna has to fulfill multiple purposes in the literature. The reason may be that all processing elements a compromise. In Section 3,webrieflyreviewtheapproaches of a radar system are linked, and it is not very useful to simply implement an isolated algorithm. The performance for deterministic pattern shaping which is the standard and the property of any algorithm will have an influence on approach of antenna-based interference mitigation. It has the thesubsequentprocessingstepsandontheradaroperational advantage of requiring little knowledge about the interference modes. Predictability of the system performance with the new scenario, but very precise knowledge about the array trans- algorithms is a key issue for the radar designer. Advanced fer function (β€œthe array manifold”). Adaptive beamforming array processing for radar will therefore require to take (ABF) is presented in Section 4. This approach requires little these interrelationships into account and to adapt the related knowledge about the array manifold but needs to estimate processing in order to achieve the maximum possible the interference scenario from some training data. Superres- improvement. The standard handbooks on radar1 [ , 2]donot olution for best resolution of multiple targets is sometimes mention this problem. The book of Wirth3 [ ]isanexception also subsumed under adaptive beamforming as it resolves and mentions a number of the array processing techniques everything, interference and targets. These methods are con- described below. sidered in Section 5. We consider superresolution methods In this tutorial paper, viewpoints are presented which are here solely for the purpose of improved parameter estimation. relevant for the implementation of array processing methods. In Section 6, we briefly mention the canonical extension We do not present any new sophisticated algorithms, but for ofABFandsuperresolutiontospace-timearrayprocessing. the established algorithms we give examples of the relations Section 7 is the final and most important contribution. Here 2 International Journal of Antennas and Propagation we point out how direction estimation must be modified if parameter. More important is that, if the element patterns adaptive beams are used, and how the radar detector, the arereallyabsolutelyequal,anycross-polarcomponentsofthe tracking algorithm, and the track management should be signal are in all channels equal and fulfill the array model in adapted for ABF. thesamewayasthecopolarcomponents;thatis,theyproduce no error effect. 2. Design Factors for Arrays 2.1.2. Arrays with Unequal Array Patterns. Thisoccurstypi- Array processing starts with the array antenna. This is hard- cally by tilting the antenna elements as is done for conformal ware and is a selected construction that cannot be altered. arrays. For a planar array, this tilt may be used to realize an It must therefore be carefully designed to fulfill all require- array with polarization diversity. Single polarized elements ments. Digital array processing requires digital array outputs. are then mounted with orthogonal alignment at different The number and quality of these receivers (e.g., linearity and positions. Such an array can provide some degree of dual number of ADC bits) determine the quality and the cost of polarization reception with single channel receivers (contrary the whole system. It may be desirable to design a fully digital to more costly fully polarimetric arrays with receivers for array with AD-converters at each antenna element. However, both polarizations for each channel). weightandcostwilloftenleadtoasystemwithreduced Common to arrays with unequal patterns is that we have number of digital receivers. On the other hand, because to know the element patterns for applying array processing 4 of the 1/𝑅 -decay of the received power, radar needs antennas methods. The full element pattern function is also called the with high gain and high directional discrimination, which array manifold. In particular, the cross-polar (or short π‘₯-pol) means arrays with many elements. There are different solu- component has a different influence for each element. This tions to solve this contradiction. means that if this component is not known and if the π‘₯-pol component is not sufficiently attenuated, it can be a signifi- (i) Thinned arrays: the angular discrimination of an cant source of error. array with a number of elements can be improved by increasing the separation of the elements and thus 2.2. Thinned Arrays. To save the cost of receiving modules, increasing the aperture of the antenna. Note that the sparse arrays are considered, that is, with fewer elements than thinned array has the same gain as the corresponding the full populated πœ†/2 grid. Because such a β€œthinned array” fully filled array. spans the same aperture, it has the same beamwidth. Hence, (ii) Subarrays: element outputs are summed up in an the angular accuracy and resolution are the same as the fully analog manner into subarrays which are then AD- filledarray.Duetothegaps,ambiguitiesoratleasthighside- converted and processed digitally. The size and the lobes may arise. In early publications like [1], it was advocated shape of the subarrays are an important design cri- to simply take out elements of the fully filled array. It was terion. The notion of an array with subarrays is very earlyrecognizedthatthiskindofthinningdoesnotimply general and includes the case of steerable directional β€œsufficiently random” positions. Random positions on a πœ†/16 array elements. grid as used in [3, Chapter 17] can provide quite acceptable patterns. Note that the array gain of a thinned array with 𝑁 2.1. Impact of the Dimensionality of the Array. Antenna ele- elements is always 𝑁, and the average sidelobe level is 1/𝑁. ments may be arranged on a line (1-dimensional array), on a Today, we know from the theory of compressed sensing that plane (a ring or a 2-dimensional planar array), on a curved a selection of sufficiently random positions can produce a surface (conformal array), or within a volume (3D-array, unique reconstruction of a not too large number of impinging also called Crow’s Nest antenna, [3,Section4.6.1]). A 1- wave fields with high probability5 [ ]. dimensional array can only measure one independent angle; 2D and 3D arrays can measure the full polar coordinates in 2.3. Arrays with Subarrays. If a high antenna gain with low 3 R . sidelobes is desired one has to go back to the fully filled Antenna element design and the need for fixing elements array. For large arrays with thousands of elements, the large mechanically lead to element patterns which are never omni- number of digital channel constitutes a significant cost factor directional. The elements have to be designed with patterns and a challenge for the resulting data rate. Therefore, often that allow a unique identification of the direction. Typically, subarrays are formed, and all digital (adaptive) beamforming a planar array can only observe a hemispherical half space. and sophisticated array processing methods are applied to To achieve full spherical coverage, several planar arrays can the subarray outputs. Subarraying is a very general concept. be combined (multifacetted array), or a conformal or volume Attheelements,wemayhavephaseshifterssuchthatall array may be used. subarraysaresteeredintoagivendirectionandwemayapply some attenuation (tapering) to influence the sidelobe level. 2.1.1. Arrays with Equal Patterns. For linear, planar, and vol- The sum of the subarrays then gives the sum beam output. ume arrays, elements with nearly equal patterns can be The subarrays can be viewed as a superarray with elements realized.Thesehavetheadvantagethattheknowledgeofthe having different patterns steered into the selected direction. element pattern is for many array processing methods not The subarrays should have unequal size and shape to avoid necessary. An equal complex value can be interpreted as a grating effects for subsequent array processing, because the modified target complex amplitude, which is often a nuisance subarray centers constitute a sparse array (for details, see [6]). International Journal of Antennas and Propagation 3

wwwwwwwwwwww w (1) Analogue

(2) Digital

(3)Β·Β·Β·(n) mm m

βˆ‘ ADC with bandpass/ matched filtering

Figure 1: Principle of forming subarrays.

The principle is indicated in Figure 1, and properties and options are described in [6, 7]. In particular, one can also combine new subarrays at the digital level or distribute the desired tapering over the analog level (1) and various digital levels (2)β‹…β‹…β‹…(𝑛). Beamforming using subarrays can be mathematically described by a simple matrix operation. Let the complex array element outputs be denoted by z.Thesubarrayforming operation is described by a subarray forming matrix T by 𝐻 which the element outputs are summed up as Μƒz = T z.For𝐿 subarrays and 𝑁 antenna elements, T is of size 𝑁×𝐿. Vectors and matrices at the subarray outputs are denoted by the tilde. Supposewesteerthearrayintoalookdirection(𝑒0, V0)=: 𝑗2πœ‹π‘“0(π‘₯𝑖𝑒0+𝑦𝑖V0)/𝑐 u0 by applying phase shifts π‘Žπ‘–(u0)=𝑒 and apply additional amplitude weighting 𝑔𝑖 at the elements (real vector of length 𝑁)forasumbeamwithlowsidelobes, then we have a complex weighting π‘”π‘–π‘Žπ‘–(u0) which can be included in the elements of the matrix T.Here,𝑓0 denotes the centre frequency, π‘₯𝑖, 𝑦𝑖 denote the coordinates of the 𝑖th array Figure 2: 2D generic array with 902 elements grouped into 32 element, 𝑐 denotes the velocity of light, and 𝑒, V denote the subarrays. components of the unit direction vector in the planar antenna (π‘₯, 𝑦)-coordinate system. The beams are formed digitally with the subarray outputs by applying a final weighting Figure 2 shows a typical planar array with 902 elements on π‘šΜƒ (𝑖=1,...,𝐿) 𝑖 as atriangulargridwith32subarrays.Theshapeofthesubarrays 𝐻 was optimized by the technique of [6]suchthatthedifference 𝑦=mΜƒ Μƒz. (1) beams have low sidelobes when a βˆ’40 dB Taylor weighting is applied at the elements. We will use this array in the sequel Μƒ In the simplest case, m consists of only ones. The antenna for presenting examples. patternofsuchasumbeamcanthenbewrittenas An important feature of digital beamforming with subar- 𝐻 𝐻 𝐻 rays is that the weighting for beamforming can be distributed 𝑓 (u) = mΜƒ T a (u) = mΜƒ Μƒa (u) , (2) between the element level (the weighting incorporated in the matrix T) and the digital subarray level (the weighting mΜƒ). 𝐻 where a(u)=(π‘Žπ‘–(u))𝑖=1⋅⋅⋅𝑁 and Μƒa(u)=T a(u) denotes This yields some freedom in designing the dynamic range of theplanewaveresponseatthesubarrayoutputs.Allkinds amplifiers at the elements and the level of the AD-converter of beams (sum, azimuth and elevation difference, guard input. This freedom also allows to normalize the power ofthe 𝐻 channel, etc.) can be formed from these subarray outputs. We subarray outputs such that T T = I.Aswillbeshownin can also scan the beam digitally at subarray level into another Section 4,thisisalsoareasonablerequirementforadaptive direction, [7]. interference suppression to avoid pattern distortions. 4 International Journal of Antennas and Propagation

0 1 βˆ’5 0.9 βˆ’10 0.8 βˆ’15 0.7 βˆ’20 βˆ’25 0.6 (dB) βˆ’30 0.5 βˆ’35 0.4

βˆ’40 0.3 βˆ’45 0.2 βˆ’50 βˆ’1 βˆ’0.8 βˆ’0.6 βˆ’0.4 βˆ’0.2 0 0.2 0.1 0 510152025303540 sin (πœƒ)

Conventional BF βˆ’15dB Gaussian Conventional BF βˆ’15dB Gaussian βˆ’30dB Taylor n =5 35 dB Chebyshev βˆ’30dB Taylor n =5 35 dB Chebyshev (a) Low sidelobe patterns of equal beamwidth (b) Taper functions for low sidelobes

Figure 3: Low sidelobes by amplitude tapering.

2.4. Space-Time Arrays. Coherent processing of a time series is to impose some bell-shaped amplitude weighting over the ] 𝑧1,...,𝑧𝐾 can be written as a beamforming procedure as in aperture like 𝑔𝑖 = cos (πœ‹π‘₯𝑖/𝐴) + 𝛼 (for suitable constants ], 2 ,..., βˆ’]π‘₯𝑖 (1). For a time series of array snapshots z1 z𝐾,wehave 𝛼), or 𝑔𝑖 =𝑒 . The foundation of these weightings is quite therefore a double beamforming procedure of the space-time heuristical. The Taylor weighting is optimized in the sense =( ,..., ) data matrix Z z1 z𝐾 of the form that it leaves the conventional (uniformly weighted) pattern 𝐻 undistorted except for a reduction of the first 𝑛 sidelobes 𝑆=m𝑠 Zm𝑑, (3) below a prescribed level. The Dolph-Chebyshev weighting where m𝑠, m𝑑 denote the weight vectors for spatial and creates a pattern with all sidelobes equal to a prescribed level. temporal beamforming, respectively. Using the rule of Kro- Figure 3 shows examples of such patterns for a uniform linear necker products, (3) can be written as a single beamforming array with 40 elements. The taper functions for low sidelobes operation were selected such that the 3 dB beamwidth of all patterns is equal. The conventional pattern is plotted for reference 𝑆=( βŠ— )𝐻 { } , m𝑑 m𝑠 vec Z (4) showing how tapering increases the beamwidth. Which of where vec{Z} isavectorobtainedbystackingallcolumnsof these taperings may be preferred depends on the emphasis the matrix Z on top. This shows that mathematically it does on close in and far off sidelobes. Another point of interest is not matter whether the data come from spatial or tem- the dynamic range of the weights and the SNR loss, because at poral sampling. Coherent processing is in both cases a the array elements only attenuations can be applied. One can beamforming-type operation with the correspondingly mod- see that the Taylor tapering has the smallest dynamic range. ified beamforming vector. Relation4 ( ) is often exploited For planar arrays the efficiency of the taperings is slightly when spatial and temporal parameters are dependent (e.g., different. direction and Doppler frequency as in airborne radar; see The rationale for low sidelobes is that we want to min- Section 6). imize some unknown interference power coming over the sidelobes. This can be achieved by solving the following 3. Antenna Pattern Shaping optimization problem, [8]: 󡄨 󡄨2 Conventional beamforming is the same as coherent integra- ∫ 󡄨 𝐻 ( )󡄨 𝑝 ( ) 𝑑 minw 󡄨w a u 󡄨 u u tionofthespatiallysampleddata;thatis,thephasedifferences Ξ© of a plane wave signal at the array elements are compensated, 𝐻 subject to w a0 =1, or equivalently and all elements are coherently summed up. This results in a pronounced main beam when the phase differences minw wCw (5) match with the direction of the plane wave and result in sidelobes otherwise. The beam shape and the sidelobes can be 𝐻 s.t. w a0 = 1, influenced by additional amplitude weighting. Let us consider the complex beamforming weights 𝑀𝑖 = 𝐻 𝑇 with C = ∫ a (u) a(u) 𝑝 (u) 𝑑u. 𝑗2πœ‹π‘“r𝑖 u/𝑐 𝑔𝑖𝑒 , 𝑖 = 1⋅⋅⋅𝑁. The simplest way of pattern shaping Ξ© International Journal of Antennas and Propagation 5

∘ ∘ Main beam elevation +60 Main beam elevation βˆ’60 1 1

βˆ’20 0.5 0.5

βˆ’40

0 0 βˆ’60

βˆ’0.5 βˆ’0.5 βˆ’80

βˆ’1 βˆ’100 βˆ’1 βˆ’1 βˆ’0.50 0.5 1 βˆ’1 βˆ’0.50 0.5 1 u u

∘ ∘ Main beam elevation +2.87 Main beam elevation βˆ’2.87 1 1

βˆ’20 0.5 0.5

βˆ’40

0 0 βˆ’60

βˆ’0.5 βˆ’0.5 βˆ’80

βˆ’1 βˆ’100 βˆ’1 βˆ’1 βˆ’0.50 0.5 1 βˆ’1 βˆ’0.50 0.5 1 u u

Figure 4: Antenna patterns of planar array with reduced sidelobes at lower elevations for different beam pointing directions (by courtesy of W. B urger¨ of Fraunhofer (FHR)).

Ξ© denotes the angular sector where we want to influence the This is just the weight for deterministic nulling. To avoid 2 2 pattern, for example, the whole visible region 𝑒 +V <1,and insufficient suppression due to channel inaccuracies, one may 𝑝 is a weighting function which allows to put different empha- also create small extended nulls using the matrix C.Theform sis on the criterion in different angular regions. The solu- of these weights shows the close relationship to the adaptive tion of this optimization is beamforming weights in (11)and(17). βˆ’1 An example for reducing the sidelobes in selected areas = C a0 . where interference is expected is shown in Figure 4. This is an w 𝐻 βˆ’1 (6) a0 C a0 application from an airborne radar where the sidelobes in the negative elevation space have been lowered to reduce ground For the choice of the function 𝑝,weremarkthatforaglobal 2 2 2 clutter. reduction of the sidelobes when Ξ©={u ∈ R |𝑒 + V ≀1}, oneshouldexcludethemainbeamfromtheminimizationby 4. Adaptive Interference Suppression setting 𝑝=0on this set of directions (in fact, a slightly larger region is recommended, e.g., the null-to-null width) to allow Deterministic pattern shaping is applied if we have rough a certain mainbeam broadening. One may also form discrete knowledge about the interference angular distribution. In the 𝑀 nulls in directions u1,...,u𝑀 by setting 𝑝(u)=βˆ‘π‘˜=1 𝛿(u βˆ’ sidelobe region, this method can be inefficient because the uπ‘˜).Thesolutionof(5)thencanbeshowntobe antennaresponsetoaplanewave(thevectora(u))mustbe exactly known which is in reality seldom the case. Typically, βˆ’1 = Pa0 = βˆ’ ( 𝐻 ) 𝐻, much more suppression is applied than necessary with the w 𝐻 with P I A A A A a0 Pa0 (7) pricepaidbytherelatedbeambroadeningandSNRloss. Adaptive interference suppression needs no knowledge of the =( ( ),..., ( )) . A a u1 a u𝑀 directional behavior and suppresses the interference only as 6 International Journal of Antennas and Propagation much as necessary. The proposition for this approach is that are equivalent. The constrained optimization problem can be weareabletomeasureorlearninsomewaytheadaptive written in general terms as beamforming (ABF) weights. 𝐻 In the sequel, we formulate the ABF algorithms for minw w Qw subarray outputs as described in (1). This includes element (10) 𝐻 𝐻 space ABF for subarrays containing only one element. s.t. w C = k (or w c𝑖 =π‘˜π‘–, 𝑖=1β‹…β‹…β‹…π‘Ÿ),

4.1. Adaptive Beamforming Algorithms. Let us first suppose andithasthesolution that we know the interference situation; that is, we know the π‘Ÿ βˆ’1 βˆ’1 𝐻 βˆ’1 βˆ’1 interference covariance matrix Q.Whatistheoptimum w = βˆ‘πœ†π‘–Q c𝑖 = Q C (C Q C) k. (11) beamforming vector w?FromtheLikelihoodRatiotestcrite- 𝑖=1 rion, we know that the probability of detection is maximized if we choose the weight vector that maximizes the signal-to- Examples of special cases are as follows: 𝐻 noise-plus-interference ratio (SNIR) for a given (expected) (i) Single unit gain directional constraint: w a0 =1β‡’ signal a0, 𝐻 βˆ’1 βˆ’1 βˆ’1 w =(a0 Q a0) Q a0. This is obviously equivalent to the SNIR-optimum solution (9)withaspecific 󡄨 𝐻 󡄨2 󡄨 𝐻 󡄨2 󡄨w a0󡄨 󡄨w a0󡄨 normalization. 󡄨 󡄨 = 󡄨 󡄨 . maxw 󡄨 󡄨2 maxw 𝐻 (8) 𝐻 𝐻 σΈ€  𝐸{󡄨 𝐻 󡄨 } w Qw (ii) Gain and derivative constraint: w a0 =1, w a0 = 󡄨w n󡄨 βˆ’1 βˆ’1 σΈ€  0β‡’w =πœ‡Q a0 +πœ…Q a0 with suitable values of the Lagrange parameters πœ‡, πœ†. A derivative constraint is The solution of this optimization is added to make the weight less sensitive against mis- βˆ’1 𝐻 match of the steering direction. w =πœ‡Q a0 with Q =𝐸{nn }. (9) 𝐻 𝐻 (iii) Gain and norm constraint: w a0 =1, w w =𝑐 β‡’ βˆ’1 w =πœ‡(Q +𝛿I) a0. The norm constraint is added to πœ‡ is a free normalization constant and n denotes interference make the weight numerically stable. This is equivalent and receiver noise. This weighting has a very intuitive inter- βˆ’1 𝐻 to the famous diagonal loading technique which we pretation. If we decompose Q = LL and apply this weight 𝐻 𝐻 βˆ’1 𝐻 𝐻 𝐻 will consider later. w z = a Q z = a L Lz =(La0) (Lz) 𝐻 to the data, we have 0 0 . (iv) Norm constraint only: w w =1β‡’ w = min 𝐸𝑉(Q). This reveals that ABF does nothing else but a pre-whiten Without a directional constraint the weight vector z and match operation: if contains only interference, that is, produces a nearly omnidirectional pattern, but with 𝐸{ 𝐻}= 𝐸{( )( )𝐻}= zz Q,then Lz Lz I,theprewhiteningoper- nulls in the interference directions. This is also called ation; the operation of L on the (matched) signal vector a0 the power inversion weight, because the pattern restores just the matching necessary with the distortion from displays the inverted interference power. the prewhitening operation. This formulation for weight vectors applied at the array As we mentioned before, fulfilling the constraints may imply elements can be easily extended to subarrays with digital a loss in SNIR. Therefore, several techniques have been outputs. As mentioned in Section 2.3, a subarrayed array can proposed to mitigate the loss. The first idea is to allow a be viewed as a superarray with directive elements positioned compromise between power minimization and constraints atthecentersofthesubarrays.Thismeansthatwehaveonly by introducing coupling factors 𝑏𝑖 and solve a soft constraint 𝐻 𝐻 to replace the quantities a, n by Μƒa = T a, nΜƒ = T n.However, optimization there is a difference with respect to receiver noise. If the π‘Ÿ 2 󡄨 󡄨2 noise at the elements is white with covariance matrix 𝜎 I 𝐻 + βˆ‘ 𝑏 󡄨 𝐻 βˆ’π‘˜σ΅„¨ Μƒ minw w Qw 𝑖󡄨w c𝑖 𝑖󡄨 or it will be at subarray outputs with covariance matrix Q = 𝑖=1 𝜎2 𝐻 (12) T T.Adaptiveprocessingwillturnthisintowhitenoise. 𝐻 𝐻 +( 𝐻 βˆ’ ) ( 𝐻 βˆ’ ) Furthermore, if we apply at the elements some weighting for minw w Qw w C k B w C k low sidelobes, which are contained in the matrix T,ABFwill reverse this operation by the pre-whiten and match principle with B = diag{𝑏1,...,π‘π‘Ÿ}. The solution of the soft-constraint and will distort the low sidelobe pattern. This can be avoided 𝐻 optimization is by normalizing the matrix T such that TT = I.Thiscanbe 𝐻 βˆ’1 achieved by normalizing the element weight as mentioned in w = (Q + CBC ) CBk. (13) Section 2.3 (for nonoverlapping subarrays). Sometimes interference suppression is realized by min- One may extend the constrained optimization by adding imizing only the jamming power subject to additional con- inequality constraints. This leads to additional and improved 𝐻 straints, for example, w c𝑖 =π‘˜π‘–, for suitable vectors c𝑖 robustness properties. A number of methods of this kind and numbers π‘˜π‘–, 𝑖 = 1β‹…β‹…β‹…π‘Ÿ. Although this is an intuitively have been proposed, for example, in [9–12]. As we are only reasonable criterion, it does not necessarily give the maxi- presenting the principles here we do not go into further mum SNIR. For certain constraints however both solutions details. International Journal of Antennas and Propagation 7

0 20 0 20 20 βˆ’5 βˆ’10 βˆ’10 βˆ’20 βˆ’15 βˆ’20 βˆ’30

(dB) βˆ’25

SNIR (dB) SNIR βˆ’30 βˆ’40 βˆ’35

βˆ’50 βˆ’40 βˆ’45 20 βˆ’60 βˆ’50 20 20 βˆ’1 βˆ’0.8 βˆ’0.6 βˆ’0.4 βˆ’0.2 0 0.2 0.4 0.6 0.8 1 βˆ’1 βˆ’0.8 βˆ’0.6 βˆ’0.4 βˆ’0.2 0 0.2 0.4 0.6 0.8 1 u u

(a) Adapted antenna pattern (b) SNIR

Figure 5: Antenna and normalized SNIR patterns for a three jammer configuration and generic array.

The performance of ABF is often displayed by the adapted before the transmit pulse (leading or rear dead zone). On antenna pattern. A typical adapted antenna pattern with 3 the other hand, if we sample the training data before pulse jammers of 20 dB SNR is shown in Figure 5(a) for generic compression the desired signal is typically much below the array of Figure 2.Thispatterndoesnotshowhowtheactual interference level, and signal cancellation is negligible. Other jamming power and the null depth play together. techniques are described in [13]. The maximum likelihood Plots of the SNIR are better suited for displaying this estimate of the covariance matrix is then effect. The SNIR is typically plotted for varying target direc- 1 𝐾 tion while the interference scenario is held fixed, as seen Μ‚ = βˆ‘ 𝐻. QSMI 𝐾 zπ‘˜zπ‘˜ (14) in Figure 5(b). The SNIR is normalized to the SNR in the π‘˜=1 clear absence of any jamming and without ABF. In other words, this pattern shows the insertion loss arising from the This is called the Sample Matrix Inversion algorithm (SMI). jamming scenario with applied ABF. The SMI method is only asymptotically a good estimate. For The effect of target and steering direction mismatch is not small sample size, it is known to be not very stable. For matrix accounted for in the SNIR plot. This effect is displayed by invertibility, we need at least 𝐾=𝑁samples. According to the scan pattern, that is, the pattern that arises if the adapted Brennan’s Rule, for example, [1], one needs 2𝐾 samples to beam scans over a fixed target and interference scenario. Such obtain an average loss in SNIR below 3 dB. For smaller sample a plot is rarely shown because of the many parameters to size, the performance can be considerably worse. However, bevaried.Inthiscontext,wenotethatforthecasethatthe by simply adding a multiple of the identity matrix to the SMI training data contains the interference and noise alone the estimate, a close to optimum performance can be achieved. mainbeamoftheadaptedpatternisfairlybroadsimilarto This is called the loaded sample matrix estimate (LSMI) the unadapted sum beam and is therefore fairly insensitive to pointing mismatch. How to obtain an interference-alone 1 𝐾 covariancematrixisamatterofproperselectionofthe Μ‚ = βˆ‘ 𝐻 +𝛿⋅ . QLSMI 𝐾 zπ‘˜zπ‘˜ I (15) training data as mentioned in the following section. π‘˜=1 Figure 5 shows the case of an untapered planar antenna. The first sidelobes of the unadapted antenna pattern areat The drastic difference between SMI and LSMI is shown in βˆ’17 dB and are nearly unaffected by the adaptation process. Figure 7 for the planar array of Figure 2 for three jammers If we have an antenna with low sidelobes, the peak sidelobe of 20 dB input JNR with 32 subarrays and only 32 data snapshots. For a β€œreasonable” choice of the loading factor (a levelismuchmoreaffected;seeFigure 6.Duetothetapering 2 2 we have a loss in SNIR of 1.7 dB compared to the reference rule of thumb is 𝛿=2πœŽβ‹…β‹…β‹…4𝜎 for an untapered antenna) antenna (untapered without ABF and jamming). we need only 2𝑀 snapshots to obtain a 3 dB SNIR loss, if 𝑀 denotes the number of jammers (dominant eigenvalues) 4.2. Estimation of Adaptive Weights. In reality, the interfer- present, [13].Sothesamplesizecanbeconsiderablylower ence covariance matrix is not known and must be estimated than the dimension of the matrix. The effect of the loading from some training data Z =(z1,...,z𝐾).Toavoidsignal factor is that the dynamic range of the small eigenvalues is cancellation, the training data should only contain the inter- compressed. The small eigenvalues possess the largest statis- ference alone. If we have a continuously emitting interference tical fluctuation but have the greatest influence on the weight source (noise jammer) one may sample immediately after or fluctuation due to the matrix inversion. 8 International Journal of Antennas and Propagation

0 0 20 20 20 βˆ’5 βˆ’10 βˆ’10

βˆ’20 βˆ’15 βˆ’20 βˆ’30 βˆ’25 (dB)

SNIR (dB) SNIR βˆ’30 βˆ’40 βˆ’35 βˆ’40 βˆ’50 βˆ’45 20 20 βˆ’60 βˆ’50 20 βˆ’1 βˆ’0.8 βˆ’0.6 βˆ’0.4 βˆ’0.2 0 0.2 0.4 0.6 0.8 1 βˆ’1 βˆ’0.8 βˆ’0.6 βˆ’0.4 βˆ’0.2 0 0.2 0.4 0.6 0.8 1 u u

(a) Adapted antenna pattern (b) SNIR

Figure 6: Antenna and normalized SNIR patterns for a three jammer configuration for antenna with low sidelobesβˆ’ ( 40 dB Taylor weighting).

0 βŠ₯ PA is a projection on the space orthogonal to the columns of A. For strong jammers, the space spanned by the columns βˆ’5 of A will be the same as the space spanned by the dominant eigenvectors. We may therefore replace the estimated inverse βˆ’10 covariance matrix by a projection on the complement of the βˆ’15 dominant eigenvectors. This is called the EVP method. As the eigenvectors X are orthonormalized, the projection can

SNIR (dB) SNIR βŠ₯ 𝐻 βˆ’20 be written as P𝑋 = I βˆ’ XX . Figure 7 shows the performance of the EVP method βˆ’25 in comparison with SMI, LSMI. Note the little difference 20 20 between LSMI and EVP. The results with the three methods βˆ’30 20 are based on the same realization of the covariance estimate. βˆ’1 βˆ’0.8 βˆ’0.6 βˆ’0.4 βˆ’0.2 0 0.2 0.4 0.6 0.8 1 u ForEVP,wehavetoknowthedimensionofthejammer subspace (dimJSS). In complicated scenarios and with chan- LSMI/0 EVP/10 nel errors present, this value can be difficult to determine. LSMI/4 Optimum If dimJSS is grossly overestimated, a loss in SNIR occurs. 𝛿=4𝜎2 If dimJSS is underestimated the jammers are not fully sup- Figure 7: SNIR for SMI, LSMI ( ), and eigenvector projection pressed. One is therefore interested in subspace methods with with dim (JSS) = 3. low sensitivity against the choice of the subspace dimension. This property is achieved by a β€œweighted projection,” that is, by replacing the projection by One may go even further and ignore the small eigenvalue estimates completely; that is, one tries to find an estimate of the inverse covariance matrix based on the dominant 𝐻 = βˆ’ , (18) eigenvectors and eigenvalues. For high SNR, we can replace PLMI I XDX the inverse covariance matrix by a projection matrix. Suppose we have 𝑀 jammers with amplitudes 𝑏1(𝑑),...,𝑏𝑀(𝑑) in where D is a diagonal weighting matrix and X is a set directions u1,...,u𝑀. If the received data has the form z(π‘‘π‘˜)= 𝑀 orthonormal vectors spanning the interference subspace. βˆ‘π‘š=1 a(uπ‘š)π‘π‘š(π‘‘π‘˜)+n(π‘‘π‘˜),orshortzπ‘˜ = Abπ‘˜ + nπ‘˜,then PLMI does not have the mathematical properties of a projec- 𝐻 𝐻 tion. Methods of this type of are called lean matrix inversion 𝐸{zz }=Q = ABA + I. (16) (LMI). A number of methods have been proposed that can be Here, we have normalized the noise power to 1 and B = interpreted as an LMI method with different weighting matri- 𝐻 𝐸{bb }. Using the matrix inversion lemma, we have ces D.TheLMImatrixcanalsobeeconomicallycalculatedby an eigenvector-free QR-decomposition method, [14]. βˆ’1 βˆ’1 𝐻 βˆ’1 𝐻 Q = I βˆ’ A(B + A A) A One of the most efficient methods for pattern stabilization (17) while maintaining a low desired sidelobe level is the con- 𝐻 βˆ’1 𝐻 βŠ₯ strained adaptive pattern synthesis (CAPS) algorithm, [15], 󳨀→ I βˆ’ A(A A) A = PA. B β†’βˆž which is also a subspace method. Let m be the vector for International Journal of Antennas and Propagation 9 beamforming with low sidelobes in a certain direction. In full generality, the CAPS weight can be written as 100 1 βˆ’1 = Μ‚βˆ’1 βˆ’ ( 𝐻 ) wCAPS Q m XβŠ₯ XβŠ₯ CXβŠ₯ 80 m𝐻RΜ‚βˆ’1m SMI (19) 1 60 Γ— 𝐻 ( Μ‚βˆ’1 βˆ’ ), XβŠ₯ C Q m m 40 m𝐻RΜ‚βˆ’1m SMI 20 where the columns of the matrix XβŠ₯ span the space orthog- [ , ] 𝐿×𝑀 0 onal to X m and X is again a unitary matrix 6 5 with columns spanning the interference subspace which is 2 4 4 assumed to be of dimension 𝑀. C is a directional weighting 6 3 8 =∫ ( ) ( )𝐻𝑝( )𝑑 Ξ© 10 2 matrix, C Ξ© a u a u u u, denotes the set of direc- dimSS 12 1 tions of interest, and 𝑝(u) is a directional weighting function. If we use no directional weighting, C β‰ˆ I,theCAPSweight MDL plain WNT asymptotic vector simplifies to AIC plain WNT dload True target no. 1 MDL asymptotic = + ( Μ‚βˆ’1 βˆ’ ), MDL dload wCAPS m P[X,m] Q m m (20) m𝐻RΜ‚βˆ’1m SMI Figure 8: Comparison of tests for linear array with 𝑁=14elements, where P[X,m] denotes the projection onto the space spanned 𝐾=14snapshots, with and without asymptotic correction or 2 by the columns of X and m. diagonal loading (dload) of 1𝜎 .

4.3. Determination of the Dimension of Jammer Subspace (dimJSS). Subspace methods require an estimate of the and the decision is found if the test statistic is for the first time dimension of the interference subspace. Usually this is below the threshold: derived from the sample eigenvalues. For complicated scenar- 𝑖=1⋅⋅⋅𝑁 ios and small sample size, a clear decision of what constitutes a for do dominant eigenvalue may be difficult. There are two principle 2 if 𝐿 (π‘š) β‰€πœ’ : 𝑀=π‘š;Μ‚ STOP; (24) approaches to determine the number of dominant eigenval- 2𝐾(π‘βˆ’π‘š);𝛼 ues, information theoretic criteria and noise power tests. end. The information theoretic criteria are often based onthe sphericity test criterion; see, for example, [16], 2 2 The symbol πœ’π‘Ÿ;𝛼 denotes the 𝛼-percentage point of the πœ’ - π‘Ÿ (1/ (π‘βˆ’π‘š)) βˆ‘π‘ πœ† distribution with degrees of freedom. The probability to 𝑇 (π‘š) = 𝑖=π‘š+1 𝑖 , overestimate dimJSS is then asymptotically bounded by 𝛼. 𝑁 1/(π‘βˆ’π‘š) (21) (βˆπ‘–=π‘š+1πœ†π‘–) More modern versions of this test have been derived, for example, [17]. where πœ†π‘– denote the eigenvalues of the estimated covariance For small sample size, AIC and MDL are known for matrix ordered in decreasing magnitude. The ratio of the grossly overestimating the number of sources. In addition, arithmetic to geometric mean of the eigenvalues is a measure bandwidth and array channels errors lead to a leakage of of the equality of the eigenvalues. The information theoretic the dominant eigenvalues into the small eigenvalues, [18]. criteria minimize this ratio with a penalty function added; for Improved eigenvalue estimates for small sample size can mit- example, the Akaike Information Criterion (AIC) and Min- igatethiseffect.Thesimplestwaycouldbetousetheasymp- imum Description Length (MDL) choose dimJSS 𝑀̂ as the totic approximation using the well-known linkage factors, minimum of the following functions: [19], (π‘š) =𝐾(π‘βˆ’π‘š) [𝑇 (π‘š)] +π‘š(2𝑁 βˆ’) π‘š Μ‚ 1 𝑁 πœ†Μ‚ AIC log πœ†Μ‚ = πœ†Μ‚ βˆ’ πœ†Μ‚ βˆ‘ 𝑖 . 𝑖 𝑖 𝐾 𝑖 Μ‚ Μ‚ (25) π‘š 𝑗=1 πœ†π‘– βˆ’ πœ†π‘— (π‘š) =𝐾(π‘βˆ’π‘š) [𝑇 (π‘š)] +( ) (2𝑁 βˆ’) π‘š 𝐾. 𝑗 =𝑖̸ MDL log 2 log (22) More refined methods are also possible; see [16]. However, as explained in [16], simple diagonal loading can improve The noise power threshold tests (WNT) assume that the noise 2 AICandMDLforsmallsamplesizeandmakethesecriteria power 𝜎 isknownandjustchecktheestimatednoisepower robust against errors. For the WNT this loading is contained against this value, [16]. This leads to the statistic 2 inthesettingoftheassumednoiselevel𝜎 . Figure 8 shows 2𝐾 𝑁 an example of a comparison of MDL and AIC without any 𝐿 (π‘š) = βˆ‘ πœ†π‘–, (23) corrections, MDL and WNT with asymptotic correction (25), 𝜎2 2 𝑖=π‘š+1 and MDL and WNT with diagonal loading of πœ‡=1𝜎.The 10 International Journal of Antennas and Propagation threshold for WNT was set for a probability to overestimate and MUSIC method (Multiple Signal Classification): the target number of 𝛼 = 10%. The scenario consists of four βˆ’1 sources at 𝑒 = βˆ’0.7, βˆ’0.55, βˆ’0.31, βˆ’0.24 with SNR of 18, 6, 20, 𝑆 ( ) = ( ( )𝐻 βŠ₯ ( )) , MUSIC u a u P a u (27) 20.4 dB and a uniform linear antenna with 14 elements and βŠ₯ 𝐻 10% relative bandwidth leading to some eigenvalue leakage. with P = I βˆ’ XX ,andX spanning the dominant subspace. Empirical probabilities were determined from 100 Monte An LMI-version instead of MUSIC would also be possible. Carlo trials. Note that the asymptotic correction seems to The target directions are then found by the 𝑀 highest maxima work better for WNT than for MDL. With diagonal loading, of these spectra (𝑀 1- or 2-dimensional maximizations). all decisions with both MDL and WNT were correct (equal An alternative group of methods are parametric methods, to 4). which deliver only a set of β€œoptimal” parameter estimates A more thorough study of the small sample size dimJSS which explain in a sense the data for the inserted model by estimation problem considering the β€œeffective number of 𝑀 or 2𝑀 dimensional optimization [21]. identifiable signals” has been performed in [20], and a new Deterministic ML method (complex amplitudes are modified information theoretic criterion has been derived. assumed deterministic):

βˆ’1 5. Parameter Estimation and Superresolution 𝐹 (πœƒ) = ( βŠ₯Μ‚ ), βŠ₯ = βˆ’ ( 𝐻 ) 𝐻, det tr PARML with PA I A A A A The objective of radar processing is not to maximize theSNR A =(a (u1),...,a (u𝑀)) . but to detect targets and determine their parameters. For detection, the SNR is a sufficient statistic (for the likelihood (28) ratio test); that is, if we maximize the SNR we maximize also Stochastic ML method (complex amplitudes are complex the probability of detection. Only for these detected targets Gaussian): we have then a subsequent procedure to estimate the target parameters: direction, range, and possibly Doppler. Standard 𝐹 (πœƒ) = ( (πœƒ)) + ( (πœƒ)βˆ’1Μ‚ ), sto logdet R tr R RML (29) radar processing can be traced back to maximum likelihood estimation of a single target which leads to the , where R(πœƒ) denotes the completely parameterized covariance [21]. The properties of the matched filter can be judged by matrix. A formulation with the unknown directions as the thebeamshape(forangleestimation)andbytheambiguity only parameters can be given as function (for range and Doppler estimation). If the ambiguity function has a narrow beam and sufficiently low sidelobes, 𝐹 (πœƒ) = { (πœƒ) (πœƒ) 𝐻 (πœƒ) +𝜎2 (πœƒ) } sto det A B A I with the model of a single target is a good approximation as other targets are attenuated by the sidelobes. However, if we have 2 1 βŠ₯ 𝜎 (πœƒ) = tr {P RΜ‚ }, closely spaced targets or high sidelobes, multiple target mod- π‘βˆ’π‘€ A ML (30) els have to be used for parameter estimation. A variety of such βˆ’1 βˆ’1 (πœƒ) =( 𝐻 ) 𝐻 (Μ‚ βˆ’πœŽ2 (πœƒ) ) ( 𝐻 ) estimation methods have been introduced which we term B A A A RML I A A A here β€œsuperresolution methods.” Historically, these methods have often been introduced to improve the limited resolution for A = A (πœƒ) . of the matched filter. The deterministic ML method has some intuitive interpreta- 5.1. Superresolution. The resolution limit for classical beam- tions: forming is the 3 dB beamwidth. An antenna array provides 𝐾 𝐻 βŠ₯ 𝐾 βŠ₯ 2 (1) 𝐹det(πœƒ)=(1/𝐾)βˆ‘π‘˜=1 zπ‘˜ PAzπ‘˜ =(1/𝐾)βˆ‘π‘˜=1 β€–PAzπ‘˜β€– = spatial samples of the impinging wavefronts, and one may 𝐾 𝐻 βˆ’1 𝐻 2 define a multitarget model for this case. This opens the (1/𝐾) βˆ‘π‘˜=1 β€–zπ‘˜ βˆ’ A(⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟A A) A zπ‘˜β€– ,whichmeans possibility for enhanced resolution. These methods have been =bΜ‚ discussed since decades, and textbooks on this topic are avail- that the mean squared residual error after signal able, for example, [22]. We formulate here the angle param- extraction is minimized. 𝐾 𝐻 𝐻 βˆ’1 𝐻 eter estimation problem (spatial domain), but corresponding (2) 𝐹det(πœƒ)=πΆβˆ’βˆ‘π‘˜=1 zπ‘˜ A(A A) A zπ‘˜,whichcan versions can be applied in the time domain as well. In the be interpreted as maximizing a set of decoupled sum spatial domain, we are faced with the typical problems of 𝐻 𝐻 beams (a (u1)zk,...,a (u𝑀)zk). irregular sampling and subarray processing. 𝐹 (πœƒ)=πΆβˆ’ 𝐻 Μ‚ / 𝐻 = βŠ₯ ( ) From the many proposed methods, we mention here (3) det anullRanull anullanull with anull PAΜ† a u , only some classical methods to show the connections and wherewehavepartitionedthematrixofsteering Μ† relationships. We have spectral methods which generate a vectors into A =(a, A). This property is valid due to spiky estimate of the angular spectral density like. the projection decomposition lemma which says that βŠ₯ Capon’s method: for any partitioning A =(F, G) we can write PA = βŠ₯ βŠ₯ 𝐻 βŠ₯ βˆ’1 𝐻 βŠ₯ 𝐾 P βˆ’P F(F P F) F P .Ifwekeepthedirectionsin βˆ’1 1 G G G G 𝑆 ( ) = ( ( )π»Μ‚βˆ’1 ( )) Μ‚ = βˆ‘ 𝐻, Μ† 𝐢 u a u RMLa u with RML 𝐾 zπ‘˜zπ‘˜ A fixed,thisrelationsaysthatwehavetomaximizethe π‘˜=1 scan pattern over u while the sources in the directions (26) of AΜ† are deterministically nulled (see (7)). One can International Journal of Antennas and Propagation 11

80 70 20 60 15 50 10

40 (dB)

(dB) 30 5 20 0 10 0.5 0 0.5 1 12 12 0 βˆ’10 0 βˆ’0.9 βˆ’0.8 βˆ’0.7 βˆ’0.6 βˆ’0.5 βˆ’0.4 βˆ’0.3 u βˆ’0.5 u βˆ’0.5

(a) Simulated data (b) Measured real data

Figure 9: MUSIC spectra with a planar array of 7 elements.

now perform the multidimensional maximization by manifold has then to be taken at the subarray outputs as in alternating 1-dimensional maximizations and keeping (2). This manifold (the subarray patterns) is well modeled in the remaining directions fixed. This is the basis of the the main beam region but often too imprecise in the sidelobe alternating projection (AP) method or IMP (Incre- region to obtain a resolution better than the conventional. In mental MultiParameter) method, [22,page105]. that case it is advantageous to use a simplified array manifold model based only on the subarray gains and centers, called A typical feature of the MUSIC method is illustrated in the Direct Uniform Manifold model (DUM). This simpli- Figure 9. This figure shows the excellent resolution in sim- fied model has been successfully applied to MUSIC (called ulation while for real data the pattern looks almost the same Spotlight MUSIC, [25]) and to the deterministic ML method. as with Capon’s method. Using the DUM model requires little calibration effort and A result with real data with the deterministic ML method gives improved performance, [25]. is shown in Figure 10. Minimization was performed here with More refined parametric methods with higher asymptotic a stochastic approximation method. This example shows in resolution property have been suggested (e.g., COMET, particular that the deterministic ML-method is able to resolve Covariance Matching Estimation Technique, [26]). However, highly correlated targets which arise due to the reflection on application of such methods to real data often revealed no theseasurfaceforlowangletracking.Thebehaviorofthe improvement (as is the case with MUSIC in Figure 9). The monopulse estimates reflect the variation of the phase differ- ∘ ∘ reason is that these methods are much more sensitive to the ences of direct and reflected path between 0 and 180 .For ∘ signal model than the accuracy of the system provides. A 0 phase difference the monopulse points into the centre, for ∘ sensitivity with an very sharp ideal minimum of the objective 180 it points outside the 2-target configuration. function may lead to a measured data objective function The problems of superresolution methods are described where the minimum has completely disappeared. in [21, 23]. A main problem is the numerical effort of finding the 𝑀 maxima (one 𝑀-dimensional optimization or 𝑀 1- dimensional optimizations for a linear antenna). To mitigate 5.2. Target Number Determination. Superresolution is a com- this problem a stochastic approximation algorithm or the bined target number and target parameter estimation prob- IMP method has been proposed for the deterministic ML lem. As a starting point all the methods of Section 4.3 can method. The IMP method is an iteration of maximizations be used. If we use the detML method we can exploit that of an adaptively formed beam pattern. Therefore, the gener- the objective function can be interpreted as the residual error alized monopulse method can be used for this purpose, see between model (interpretation 2) and data. The WNT test Section 7.1 and [24]. statistic (23)isjustanestimateofthisquantity.ThedetML Another problem is the exact knowledge of the signal residual can therefore be used for this test instead of the sum model for all possible directions (the vector function a(u)). of the eigenvalues. The codomain of this function is sometimes called the array These methods may lead to a possibly overestimated manifold. This is mainly a problem of antenna accuracy or target number. To determine the power allocated to each calibration. While the transmission of a plane wave in the target a refined ML power estimate using the estimated (πœƒ) (πœƒ)=(𝐻 )βˆ’1 𝐻(Μ‚ βˆ’ main beam direction can be quite accurately modeled (using directions A canbeusedB A A A RML 𝜎2(πœƒ) ) ( 𝐻 )βˆ’1 𝜎2(πœƒ) = (1/(𝑁 βˆ’ 𝑀)) { βŠ₯Μ‚ } calibration) this can be difficult in the sidelobe region. I A A A with tr PARML as For an array with digital subarrays, superresolution has in (30). This estimate can even reveal correlations between to be performed only with these subarray outputs. The array the targets. This has been successfully demonstrated with 12 International Journal of Antennas and Propagation

1.0

0.6 Monopulse estimate with glint effect due 0.2 to multipath

βˆ’0.2 True elevation from optical

Elevation (BW) Elevation tracker βˆ’0.6 Superresolution with deterministic ML 2.57 3.08 3.58 4.08 4.59 5.09 5.60 6.10 6.61 7.11 7.62 8.12 8.62 Range (km)

Figure 10: Superresolution of multipath propagation over sea with deterministic ML method (real data from vertical linear array with 32 elements, scenario illustrated above).

81.1

71.0 0.5 0.4 60.8 0.3 0.2 50.7 0.1 0 40.6 βˆ’0.1 βˆ’0.2 30.4 βˆ’0.3 βˆ’0.4 20.3 βˆ’0.5 βˆ’0.5 0 0.5 u 10.1

SNR (dB) Test down (dB) Test up (dB)

Figure 11: Combined target number and direction estimation for 2 targets with 7-element planar array.

thelowangletrackingdataofFigure 10.Incasethatsome beamwidth separation. The directions were estimated by the target power is too low, the target number can be reduced stochastic approximation algorithm used in Figure 10.The and the angle estimates can be updated. This is an iterative test statistic for increasing the target number is shown by procedure of target number estimation and confirmation or the right most bar. The thresholds for increasing the number reduction. This way, all target modeling can be accurately are indicated by lines. The dashed line is the actually valid matched to the data. threshold (shown is the threshold for 2 targets). The target The deterministic ML method (28) together with the number can be reduced if the power falls below a threshold white noise test (24) is particularly suited for this kind of shown in two yellow bars in the middle. The whole estimation iterative model fitting. It has been implemented in an exper- and testing procedure can also be performed adaptively with imental system with a 7-element planar array at Fraunhofer changing target situations. We applied it to two blinking FHR and was first reported in [21, page 81]. An example of targets alternating between the states β€œtarget 1 on”, β€œboth the resulting output plot is shown in Figure 11.Theestimated targets on”, β€œtarget 2 on”, β€œboth targets on”, and so forth. directions in the 𝑒, V-plane are shown by small dishes having Clearly, these test works only if the estimation procedure has a color according to the estimated target SNR corresponding converged.Thisisindicatedbythetrafficlightintheright to the color bar. The circle indicates the 3 dB contour of the up corner. We used a fixed empirically determined iteration sum beam. One can see that the two targets are at about 0.5 number to switch the test procedure on (=green traffic light). International Journal of Antennas and Propagation 13

K range cells

M pulses

N channels

Figure 12: Symmetric auxiliary sensor/echo processor of Klemm [4], as an example for forming space-time subarrays.

All thresholds and iteration numbers have to be selected information about the targets after all processing. This means carefully. Otherwise, situations may arise where this adaptive that for implementing refined methods of interference sup- procedure switches between two target models, for example, pression or superresolution we have also to consider the effect between 2 and 3 targets. onthesubsequentprocessing.Togetoptimumperformance The problem of resolution of two closely spaced targets all subsequent processing should exploit the properties of the becomes a particular problem in the so called threshold refined array methods applied before. In region, which denotes configurations where the SNR or the particular it has been shown that for the tasks of detection, separation of the targets lead to an angular variance departing angle estimation and tracking significant improvements can significantly from the Cramer-Rao bound (CRB). The design be achieved by considering special features. of the tests and this threshold region must be compatible to 7.1. Adaptive Monopulse. Monopulse is an established tech- give consistent joint estimation-detection resolution result. nique for rapid and precise angle estimation with array These problems have been studied in [27, 28]. One way to antennas. It is based on two beams formed in parallel, a sum achieve consistency and improving resolution proposed in beam and a difference beam. The difference beam is zero at [27] is to detect and remove outliers in the data, which are thepositionofthemaximumofthesumbeam.Theratio basicallyresponsibleforthethresholdeffect.Ageneraldis- of both beams gives an error value that indicates the offset cussion about the achievable resolution and the best realistic of a target from the sum beam pointing direction. In fact, it representation of a target cluster can be found in [28]. can be shown that this monopulse estimator is an approxima- tion of the Maximum-Likelihood angle estimator, [24]. The 6. Extension to Space-Time Arrays monopulse estimator has been generalized in [24]toarrays with arbitrary subarrays and arbitrary sum and difference As mentioned in Section 2.4, there is mathematically no dif- beams. ference between spatial and temporal samples as long as the When adaptive beams are used the shape of the sum distributional assumptions are the same. The adaptive meth- beam will be distorted due to the interference that is to be ods and superresolution methods presented in the previous suppressed. The difference beam must adaptively suppress sections can therefore be applied analogously in the time or the interference as well, which leads to another distortion. space-time domain. Then the ratio of both beams will no more indicate the target In particular, subarraying in time domain is an important direction. The generalized monopulse procedure of24 [ ] tool to reduce the numerical complexity for space-time provides correction values to compensate these distortions. adaptive processing (STAP) which is the general approach for The generalized monopulse formula for estimating angles 𝑇 adaptive clutter suppression for airborne radar, [4]. With the (𝑒,Μ‚ Μ‚V) with a planar array and sum and difference beams 𝑇 formalism of transforming space-time 2D-beamforming of a formed into direction (𝑒0, V0) is data matrix into a usual beamforming operation of vectors 𝑒̂ 𝑒 𝑐 𝑐 𝑅 βˆ’πœ‡ introduced in (4), the presented adaptive beamforming and ( )=( 0)βˆ’( π‘₯π‘₯ π‘₯𝑦 )( π‘₯ π‘₯), Μ‚V V 𝑐 𝑐 𝑅 βˆ’πœ‡ (31) superresolution methods can be easily transformed into 0 𝑦π‘₯ 𝑦𝑦 𝑦 𝑦 corresponding subarrayed space-time methods. 𝑐 𝑐 πœ‡ =( π‘₯π‘₯ π‘₯𝑦 ) πœ‡ =( π‘₯ ) where C 𝑐𝑦π‘₯ 𝑐𝑦𝑦 is a slope correction matrix and πœ‡π‘¦ Figure 12 shows an example of an efficient space-time 𝐻 𝐻 subarrayingschemeusedforSTAPcluttercancellationfor is a bias correction. 𝑅π‘₯ = Re{dπ‘₯ z/w z} is the monopulse ratio formed with the measured difference and sum beam airborne radar. 𝐻 𝐻 outputs 𝐷π‘₯ = dπ‘₯ z and 𝑆=w z, respectively, with difference 7. Embedding of Array Processing into and sum beam weight vectors dπ‘₯, w (analogous for elevation Full Radar Data Processing estimation with d𝑦). The monopulse ratio is a function of the unknown target directions (𝑒, V).Letthevectorofmonopulse 𝑇 A key problem that has to be recognized is that the task of a ratios be denoted by R(𝑒, V)=(𝑅π‘₯(𝑒, V), 𝑅𝑦(𝑒, V)) .Thecor- radar is not to maximize the SNR, but to give the best relevant rection quantities are determined such that the expectation 14 International Journal of Antennas and Propagation of the error is unbiased and a linear function with slope 1 is approximated. More precisely, for the following function of 0.05 the unknown target direction: 0.04 M (𝑒, V) = C β‹… (𝐸 {R (𝑒, V)} βˆ’ πœ‡) , (32) 0.03 we require 0.02 0.01 M (𝑒0, V0)=0, 0 πœ• 1 πœ• 0 M (𝑒 , V )=( ), M (𝑒 , V )=( ) βˆ’0.01 πœ•π‘’ 0 0 0 πœ•V 0 0 1 or (33) βˆ’0.02 πœ•R πœ•R C ( )(𝑒0, V0)=I. βˆ’0.03 πœ•π‘’ πœ•V βˆ’0.04 These conditions can only approximately be fulfilled for suf- ficiently high SNR. Then, one obtains for the bias correction βˆ’0.05 for a pointing direction a0 = a(𝑒0, V0),[24] βˆ’0.05 0 0.05 𝐻 πœ‡ = { d𝛼 a0 } 𝛼=π‘₯,𝑦. u 𝛼 Re 𝐻 for (34) w a0 Figure 13: Bias and standard deviation ellipses for different target For the elements of the inverse slope correction matrix positions. 𝛼,β„Ž βˆ’1 (𝑐 ) π‘Ž=π‘₯,𝑦 = C ,oneobtains β„Ž=𝑒,V 0.05 { 𝐻 𝐻 + 𝐻 𝐻 } 𝐻 𝛼,β„Ž Re d𝛼 aβ„Ž,0a0 w d𝛼 a0aβ„Ž,0w w aβ„Ž,0 𝑐 = βˆ’πœ‡π‘Ž2 Re { } 0.04 󡄨 𝐻 󡄨2 𝐻 󡄨w a0󡄨 w a0 (35) 0.03 0.02 with 𝛼=π‘₯or 𝑦 and β„Ž=𝑒or V,andaβ„Ž,0 denotes the (πœ• /πœ•β„Ž)| 0.01 derivative a (𝑒0,V0). In general, these are fixed antenna determined quantities. For example, for omnidirectional 0 antenna elements, and phase steering at the elements we have 𝑇 a0 =𝐺𝑒(1,...,1) ,where𝐺𝑒 is the antenna element gain, and βˆ’0.01 𝑇 a𝑒,0 =𝐺𝑒(𝑗2πœ‹π‘“/𝑐)(π‘₯1,...,π‘₯𝑁). βˆ’0.02 It is important to note that this formula is independent of any scaling of the difference and sum weights. Constant βˆ’0.03 factorsinthedifferenceandsumweightwillbecancelled βˆ’0.04 by the corresponding slope correction. Figure 13 shows the- oretically calculated bias and variances for this corrected βˆ’0.05 generalized monopulse using the formulas of [24]forthe array of Figure 2. The biases are shown by arrows for different βˆ’0.05 0 0.05 possible single target positions with the standard deviation u ellipses at the tip. A jammer is located in the asterisk symbol Figure 14: Bias for 2-step monopulse for different target positions direction with JNR = 27 dB. The hypothetical target has a and jammer scenario of Figure 13. SNR of 6 dB. The 3 dB contour of the unadapted sum beam is shown by a dashed circle. The 3 dB contour of the adapted beam will be of course different. One can see that in the beam pointing direction (0, 0) thebiasiszeroandthevarianceis with only one additional iteration as shown in Figure 14.The small. The errors increase for target directions on the skirt of variances appearing in Figure 13 are virtually not changed themainbeamandclosetothejammer. with the multistep monopulse procedure and are omitted for The large bias may not be satisfying. However, one may better visibility. repeat the monopulse procedure by repeating the monopulse estimate with a look direction steered at subarray level into 7.2. Adaptive Detection. For detection with adaptive beams, the new estimated direction. This is an all-offline procedure the normal test procedure is not adequate because we have a with the given subarray data. No new transmit pulse is test statistic depending on two different kinds of random data: needed. We have called this the multistep monopulse proce- thetrainingdatafortheadaptiveweightandthedataunder dure [24]. Multistep monopulse reduces the bias considerably test. Various kinds of tests have been developed accounting International Journal of Antennas and Propagation 15 for this fact. The first and basic test statistics were the GLRT, receivedsignalpowerintheguardchannelisabovethepower [29], the AMF detector, [30], and the ACE detector, [31]. of the main channel, this must be a signal coming via the These have the form sidelobes. Such signals will be blanked. If the guard channel 󡄨 󡄨2 power is below the main channel power it is considered as a σ΅„¨Μƒπ»Μ‚βˆ’1 ̃󡄨 󡄨a0 Q z󡄨 detection. 𝑇 (Μƒ) = SMI , (36) GLRT z π»Μ‚βˆ’1 π»Μ‚βˆ’1 With phased arrays it is not necessary to provide an Μƒa Q Μƒa0 (1 + (1/𝐾) Μƒz Q Μƒz) 0 SMI SMI external omnidirectional guard channel. Such a channel 󡄨 󡄨2 σ΅„¨Μƒπ»Μ‚βˆ’1 ̃󡄨 can be generated from the antenna itself; all the required 󡄨a0 Q z󡄨 𝑇 (Μƒz) = SMI , (37) information is in the antenna. We may use the noncoherent AMF Μƒπ»Μ‚βˆ’1 Μƒ sumofthesubarraysasguardchannel.Thisisthesameasthe a0 QSMIa0 average omnidirectional power. Some shaping of the guard 󡄨 󡄨2 σ΅„¨Μƒπ»Μ‚βˆ’1 ̃󡄨 pattern can be achieved by using a weighting for the nonco- 󡄨a0 Q z󡄨 𝑇 (Μƒ) = SMI . (38) ACE z π»Μ‚βˆ’1 π»Μ‚βˆ’1 herent sum: Μƒa0 Q Μƒa0 β‹… Μƒz Q Μƒz SMI SMI 𝐿 󡄨 󡄨2 Μ‚ 𝐺=βˆ‘π‘”Μƒπ‘–σ΅„¨π‘§Μƒπ‘–σ΅„¨ . (41) The quantities Μƒz, Μƒa0, Q arehereallgeneratedatthesubarray 󡄨 󡄨 𝑖=1 outputs, Μƒa0 denotes the plane wave model for a direction u0. Basic properties of these tests are 𝑇 If all subarrays are equal, a uniform weighting gΜƒ =(1,...,1) 𝑇 𝑇 may be suitable; for unequal irregular subarrays as in Figure 2 (i) 𝑇 = AMF ,𝑇= AMF . GLRT 1+(1/𝐾) Μƒπ»Μ‚βˆ’1 Μƒ ACE Μƒπ»Μ‚βˆ’1 Μƒ the different contributions of the subarrays can be weighted. z QSMIz z QSMIz The directivity pattern of such guard channel is given by (39) 𝐿 2 𝑆𝐺(u)=βˆ‘π‘–=1 𝑔̃𝑖|π‘ŽΜƒπ‘–(u)| .Moregenerally,wemayuseacom- (ii) The AMF detector represents an estimate of the signal-to- bination of noncoherent and coherent sums of the subarrays D K noise ratio because it can be written as with weights contained in the matrices , ,respectively, 󡄨 󡄨2 ̃𝐻 𝐻̃ ̃𝐻 𝐻̃ 󡄨̃𝐻̃󡄨 𝐺=z KDK z,𝑆𝐺 (u) = a (u) KDK a (u) . (42) 󡄨w z󡄨 Μ‚βˆ’1 𝑇 = with wΜƒ = Q Μƒa0. (40) AMF ̃𝐻̂ Μƒ SMI w QSMIw Examples of such kind of guard channels are shown in Figure 15 for the generic array of Figure 2 with βˆ’35 dB Taylor This provides a meaningful physical interpretation. A com- weighing for low sidelobes. The nice feature of these guard plete statistical description of these tests has been given channels is (i) that they automatically scan together with in very compact form in [32, 33]. These results are valid the antenna look direction, and (ii) that they can easily be as well for planar arrays with irregular subarrays and also made adaptive. This is required if we want to use the SLB mismatched weighting vector. device in the presence of CW plus impulsive interference. A Actually, all these detectors use the adaptive weight of CW jammer would make the SLB blank all range cells, that the SMI algorithm which has unsatisfactory performance as is,wouldjustswitchofftheradar.Togenerateanadaptive mentioned in Section 4.2.Theunsatisfactoryfinitesample guard channel we only have to replace in (42)thedata performance is just the motivation for introducing weight vector of the cell under test (CUT) by the pre-whitened data estimators like LSMI, LMI, or CAPS. Clutter, insufficient Μƒ = Μ‚βˆ’1/2Μƒ zpre-𝑀 R z. Then, the test statistic can be written as adaptive suppression and surprise interference are the moti- 𝐻 βˆ’1 𝑇=𝑇 (Μƒz)/𝐺 (Μƒz),where𝐺 (Μƒz)=Μƒz RΜ‚ Μƒz for vation for requiring low sidelobes. Recently several more AMF adapt adapt 𝐺 (Μƒ) = 1 + (1/𝐾)Μƒπ»Μ‚βˆ’1Μƒ complicated adaptive detectors have been introduced with ACE and adapt z z R z for GLRT. Hence 𝐺 the aim of achieving additional robustness properties, [34– adapt isjusttheincoherentsumofthepre-whitenedsubarray 𝑇 38]. However, another and quite simple way would be to outputs; in other words, ACE can be interpreted as an AMF 𝑇 generalize the tests of (36), (37), (38) to arbitrary weight detectorwithanadaptiveguardchanneland GLRT the same vectors with the aim of inserting well known robust weights with guard channel on a pedestal. Figure 16 shows examples as derived in Section 4.1.Thishasbeendonein[39]. First, of some adapted guard channels generated with the generic βˆ’ we observe that the formulation of (40)canbeusedforany array of Figure 2 and 35 dB Taylor weighting. The unadapted weight vector. Second, one can observe that the ACE and patterns are shown by dashed lines. GLRT have the form of a sidelobe blanking device. In Thisistheadaptivegeneralizationoftheusualsidelobe particularithasalreadybeenshownin[35]thatdiagonal blankingdevice(SLB)andtheAMF,ACEandGLRTtestscan loading provides significant better detection performance. be used as extension of the SLB detector to the adaptive case, A guard channel is implemented in radar systems to [32],calledthe2Dadaptivesidelobeblanking(ASB)detector. eliminate impulsive interference (hostile or from other neigh- The AMF is then the test for the presence of a potential target boring ) using the sidelobe blanking (SLB) device. The andthegeneralizedACEorGRLTareusedconfirmingthis guard channel receives data from an omnidirectional antenna target or adaptive sidelobe blanking. element which is amplified such that its power level is above A problem with these modified tests is to define a suitable the sidelobe level of the highly directional radar antenna, but threshold for detection. For arbitrary weight vector it is nearly below the power of the radar main beam, [1,page9.9].Ifthe impossible to determine this analytically. In [39] the detection 16 International Journal of Antennas and Propagation

βˆ’35 n =32 dB Taylor, i βˆ’35 dB Taylor, ni =32 0 0

βˆ’10 βˆ’10

βˆ’20 βˆ’20

βˆ’30 βˆ’30 (dB) (dB)

βˆ’40 βˆ’40

βˆ’50 βˆ’50

βˆ’60 βˆ’60 βˆ’1 βˆ’0.8 βˆ’0.6 βˆ’0.4 βˆ’0.2 0 0.2 0.4 0.6 0.8 1 βˆ’1 βˆ’0.8 βˆ’0.6 βˆ’0.4 βˆ’0.2 0 0.2 0.4 0.6 0.8 1 u u

(a) Uniform subarray weighting (b) Power equalized weighting

βˆ’35 dB Taylor, ni =16 0

βˆ’10

βˆ’20

βˆ’30 (dB)

βˆ’40

βˆ’50

βˆ’60 βˆ’1 βˆ’0.8 βˆ’0.6 βˆ’0.4 βˆ’0.2 0 0.2 0.4 0.6 0.8 1 u

(c) Power equalized + difference weighting

Figure 15: Guard channel and sum beam patterns for generic array of Figure 2. margin has been introduced as an empirical tool for judging a 7.3. Adaptive Tracking. A key feature of ABF is that overall good balance between the AMF and ASB threshold for given performance is dramatically influenced by the proximity of jammer scenarios. The detection margin is defined as the the main beam to an interference source. The task of target difference between the expectation of the AMF statistic and tracking in the proximity of a jammer is of high operational the guard channel, where the expectation is taken only over relevance. In fact, the information on the jammer direction the interference complex amplitudes for a known interference canbemadeavailablebyajammermappingmode,which scenario. In addition one can also calculate the standard determines the direction of the interferences by a background deviation of these patterns. The performance against jammers procedure using already available data. Jammers are typically closetothemainlobeisthecriticalfeature.Thedetection strong emitters and thus easy to detect. In particular, the margin provides the mean levels together with standard Spotlight MUSIC method [25]workingwithsubarrayoutputs deviations of the patterns. An example of the detection is suited for jammer mapping with a multifunction radar. margin is shown in Figure 17 (same antenna and weighting Let us assume here for simplicity that the jammer direc- as in Figures 15 and 16). tion is known. This is highly important information for the Comparing the variances of the ACE and GLRT guard tracking algorithm of a multifunction radar where the tracker channels in [39]revealedthattheGLRTguardperforms determines the pointing direction of the beam. We will use significantly better in terms of fluctuations. The GLRT guard for angle estimation the adaptive monopulse procedure of channel may therefore be preferred for its better sidelobe Section 7.1.ABFwillformbeamswithanotchinthejammer performance and higher statistical stability. direction. Therefore one cannot expect target echoes from International Journal of Antennas and Propagation 17

βˆ’35 n =32 dB Taylor, i βˆ’35 n =16 0 0 dB Taylor, i 34 34 βˆ’10 βˆ’10

βˆ’20 βˆ’20

βˆ’30 βˆ’30 (dB) (dB)

βˆ’40 βˆ’40

βˆ’50 βˆ’50

βˆ’60 βˆ’60 βˆ’1 βˆ’0.8 βˆ’0.6 βˆ’0.4 βˆ’0.2 0 0.2 0.4 0.6 0.8 1 βˆ’1 βˆ’0.8 βˆ’0.6 βˆ’0.4 βˆ’0.2 0 0.2 0.4 0.6 0.8 1 u u

(a) Weighting for equal subarray power (b) Difference type guard with weighting for equal subarray power

∘ Figure16:Adaptedguardpatternsforjammerat𝑒 = βˆ’0.27 (βˆ’15.7 ) with JNR of 34 dB for generic array.

AMF/ACE detection margins/Guard=uniform LSMI/load=3 track file. The width of the jammer notch defines the uncer- 50 tainty of this pseudo measurement. Moreover, if one knows 40 the jammer direction one can use the theoretically calculated variances for the adaptive monopulse estimate of [24]asapri- 30 ori information in the tracking filter. The adaptive monopulse can have very eccentric uncertainty ellipses as shown in 20 Figure 13 which is highly relevant for the tracker. The large 10 bias appearing in Figure 13, which is not known by the (dB) tracker, can be reduced by applying the multistep monopulse 0 procedure, [24]. All these techniques have been implemented in a tracking βˆ’10 algorithm and refined by a number of stabilization measures βˆ’20 in [41]. The following special measures for ABF tracking have been implemented and are graphically visualized in Figure 18. βˆ’30 βˆ’1 βˆ’0.8 βˆ’0.6 βˆ’0.4 βˆ’0.2 0 0.2 0.4 0.6 0.8 1 (i) Look direction stabilization: the monopulse estimate u may deliver measurements outside of the 3 dB con- tour of the sum beam. Such estimates are also heavily ACE-detect. margin Conf. bound(adG) biased, especially for look directions close to the jam- νœ‚ AMF pattern/ ACE Conf. bound(adG) mer, despite the use of the multistep monopulse pro- νœ‚ Conf. bound(AMF) Ant. patt. clear/ ACE cedure. Estimates of that kind are therefore corrected Guard clear Conf. bound(AMF) by projecting them onto the boundary circle of sum Adapted guard νœ‚ /νœ‚ AMF ACE beam contour. Figure 17: Detection margin (gray shading) between AMF and (ii) Detection threshold: only those measurements are nonweighted adapted guard pattern with confidence bounds for 3𝜎2 considered in the update step of the tracking algo- ACE with LSMI (64 snapshots and diagonal loading). Two CW rithmwhosesumbeampowerisaboveacertain interferences of 40 dB INR are present at 𝑒 = βˆ’0.8, βˆ’0.4. detection threshold (typically 13 dB). This guarantees useful and valuable monopulse estimates. It is well known that the variance of the monopulse estimate directions close to the jammer and therefore it does not make decreases monotonically with this threshold increas- sensetosteerthebeamintothejammernotch.Furthermore, ing. in the case of a missing measurement of a tracked target inside (iii) Adjustment of antenna look direction: look directions the jammer notch, the lack of a successful detection supports in the jammer notch should generally be avoided due theconclusionthatthisnegativecontactisadirectresultof to the expected lack of good measurements. In case jammer nulling by ABF. This is so-called negative informa- that the proposed look direction lies in the jammer tion [40]. In this situation we can use the direction of the notch, we select an adjusted direction on the skirt of jammer as a pseudomeasurement to update and maintain the the jammer notch. 18 International Journal of Antennas and Propagation

Projection of monopulse estimate Adjustment of antenna look direction QuadSearch . Antenna look direction Jammer notch Adjusted Jammer notch Diff look (·) Direction Directions Projection 4 3 Predicted ( ) ( ) → → → Direction Jammer Jammer (1) (2) Elevation Elevation Elevation BW 훼·BW BWJ BWJ Azimuth u→ Azimuth u→ Azimuth u→

(a) (b) (c)

Figure 18: Illustration of different stabilization measures to improve track stability and track continuity.

(iv) Variable measurement covariance: a variable covari- 130 12 ance matrix of the adaptive monopulse estimation according to [24] is considered only for a mainlobe 110 10 jammer situation. For jammers in the sidelobes, there is little effect on the angle estimates, and we can use 90 8 the fixed covariance matrix of the nonjammed case. North (km) North

70 (km) Altitude 6 (v) QuadSearch and Pseudomeasurements: if the pre- dicted target direction lies inside the jammer notch 50 4 and if, despite all adjustments of the antenna look 10 30 50 70 90 110 130 0 120 240 direction, the target is not detected, a specific search East (km) Time (s) pattern is initiated (named QuadSearch) which uses Sensor Standoff jammer look directions on the skirt of the jammer notch to Target obtain acceptable monopulse estimates. If this proce- Target in jammer notch dure does not lead to a detection, we know that the Figure 19: Target tracking scenario with standoff jammer: geo- target is hidden in the jammer notch and we cannot graphic plot of platform trajectories. see it. We use then the direction of the jammer as a pseudobearing measurement to maintain the track file. The pseudomeasurement noise is determined by the width of the jammer notch. as unbiased measurements. To avoid track instabil- ities, an acceptance region is defined for each mea- (vi) LocSearch: in case of a permanent lack of detections surement depending on the predicted target state and (e.g., for three consecutive scans) while the track the assumed measurement accuracy. Sensor reports position lies outside the jammer notch, a specific lying outside this gate are considered as invalid. search pattern is initiated (named LocSearch) that is similar to the QuadSearch. The new look directions In order to evaluate our stabilization measures we considered lieonthecircleofcertainradiusaroundthepredicted a realistic air-to-air target tracking scenario [41]. Figure 19 target direction. provides an overview of the different platform trajectories. In this scenario, the sensor (on a forward looking radar (vii) Modeling of target dynamics: the selection of a platform flying with a constant speed of 265 m/s) employs suitable dynamics model plays a major role for the ∘ the antenna array of Figure 2 (sum beamwidth BW = 3.4 , quality of tracking results. In this context, the so- ∘ field of view 120 , scan interval 1 s) and approaches the target called interacting multiple model (IMM) is a well- (at velocity 300 m/s), which thereupon veers away after a known method to reliably track even those objects short time. During this time, the target is hidden twice in whose dynamic behavior remains constant only dur- the jammer notch of the standoff jammer (SOJ)β€”first for ing certain periods. 3sandthenagainfor4s.TheSOJisonpatrol(at235m/s) (viii) Gating: in the vicinity of the jammer, the predicted andfollowsapredefinedracetrackatconstantaltitude. target direction (as an approximation of the true Figure 20 shows exemplary the evaluation of the azimuth value) is used to compute the variable angle measure- measurements and estimates over time in a window where ment covariance. Strictly speaking, this is only valid thetargetfirstpassesthroughthejammernotch.Thedifferent exactly in the particular look direction. Moreover, the error bars of a single measurement illustrate the approxi- 𝜎SIM tracking algorithm regards all incoming sensor data mation error of the variable measurement covariance: π‘’π‘˜ International Journal of Antennas and Propagation 19

βˆ’0.08 (i) Interference suppression by deterministic and adap- βˆ’0.12 tive pattern shaping: both approaches can be rea-

u sonably combined. Applying ABF after deterministic βˆ’0.16 sidelobe reduction allows reducing the requirements βˆ’0.2 on the low sidelobe level. Special techniques are Azimuth available to make ABF preserve the low sidelobe level. βˆ’0.24 9095 100 105 110 115 (ii) General principles and relationships between ABF Time (s) algorithms and superresolution methods have been discussed, like dependency on the sample number, Jammer notch 휎FILT uν‘˜ Target in notch robustness, the benefits of subspace methods, prob- Pseudomeasurement lems of determining the signal/interference subspace, Azimuth (truth) No detection 휎SIM and interference suppression/resolution limit. uν‘˜ Gating Measurement (update) (iii) Array signal processing methods like adaptive beam- Figure 20: Exemplary azimuth measurements and models for a forming and superresolution methods can be applied specific time window for tracking scenario of Figure 19. to subarrays generated from a large fully filled array. This means applying these methods to the sparse superarray formed by the subarray centers. We have pointed out problems and solutions for this special array problem. denotes the true azimuth standard deviation (std) which is 𝜎FILT generated in the antenna simulation; π‘’π‘˜ corresponds to the (iv) ABF can be combined with superresolution in a std which is used in the tracking algorithm. More precisely, canonical way by applying the pre-whiten and match thetrackingprogramcomputestheadaptiveanglemeasure- principle to the data and the signal model vector. ment covariance only in the vicinity of the jammer with a ∘ diameter of this zone of 8.5 . Outside of this region, the (v) All array signal processing methods can be extended tracking algorithm uses a constant std of 0.004 for both to space-time processing (arrays) by defining a corre- components of the angle measurement. The constant std for sponding space-time plane wave model. the other parameters are 75 m and 7.5 m/s for range and range-rate measurements. The signal-to-noise and jammer- (vi) Superresolution is a joint detection-estimation prob- to-noise ratios were set to 26 dB and 27 dB at a reference range lem. One has to determine a multitarget model which of 70 km. From Figure 20 the benefits of using pseudobearing contains the number, directions and powers of the tar- measurements become apparent. gets. These parameters are strongly coupled. A practi- From these investigations, it turned out that tracking only cal joint estimation and detection procedure has been with adaptive beamforming and adaptive monopulse nearly presented. always leads to track loss in the vicinity of the jammer. With additional stabilization measures that did not require the (vii) The problems for implementation in real system have knowledge of the jammer direction (projection of monopulse been discussed, in particular the effects of limited estimate, detection threshold, LocSearch, gating) still track knowledge of the array manifold, effect of channel instabilities occurred culminating finally in track loss. An errors, eigenvalue leakage, unequal noise power in advanced tracking version which used pseudomeasurements array channels, and dynamic range of AD-converters. mitigated this problem to some degree. Finally, the additional (viii) For achieving best performance an adaptation of the consideration of the variable measurement covariance with processing subsequent to ABF is necessary. Direction a better estimate of the highly variable shape of the angle estimation can be accommodated by using ABF- uncertainty ellipse resulted in significantly fewer measure- monopulse; the detector can be accommodated by mentsthatwereexcludedduetogating.Inthiscaseallthesta- adaptive detection with ASLB, and the tracking algo- bilization measures could not only improve track continuity, rithms can be extended to adaptive tracking and track but also track accuracy and thus track stability, [41]. This tells management with jammer mapping. us that it is absolutely necessary to use all information of theadaptiveprocessforthetrackertoachievethegoalof detection and tracking in the vicinity of the interference. With a single array signal processing method alone no significant improvement will be obtained. The methods have tobereasonablyembeddedinthewholesystem,andall 8. Conclusions and Final Remarks functionalitieshavetobemutuallytunedandbalanced. This is a task for future research. The presented approaches In this paper, we have pointed out the links between array constitute only a first ad hoc step, and more thorough studies signal processing and antenna design, hardware constraints are required. Note that in most cases tuning the functionali- and target detection, and parameter estimation and tracking. ties is mainly a software problem. So, there is the possibility More specifically, we have discussed the following features. to upgrade existing systems softly and step-wise. 20 International Journal of Antennas and Propagation

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