TELSIKS'99 z z zyxwvu zy zyxw 13-15. October 1999, NiS, Yugoslavia Minimax Envelope Constrained Filter for Radar and Communications Applications zyxwvutsrqponm zyx zyxwvuts Andrija PetroviC, Aleksa J. Zejak, Bojan ZrriiC A deterministic discrete finite sequence of length Abstruct- The Envelope Constrained Filter (ECF) is a useful approach to filter design. The authors of the original concept have proposed algorithms for solving the ECF design problem as well. These algorithms use the tools of nonlinear programming. They are based on a feasible solution assumption. In other words, they perform well when the solution can be found. In most radar applications, and in a great number of communications applications, this is not the case. The paper presents novel algorithms for ECF design, based on minimax algorithms: the minimax algorithms are modified in order to solve the ECF design problem. The main feature of the new approach is a better error control in the infeasible solution case. S E [ s1 s2 ... s,,, ] (2) is fed t o the filter input. While the sequence passes through the filter, the filter passes through its states. The first filter state is: u(1) = [ 0 . ' . 01, s1 (3) 12 while the last one is Keywords- Filter design, radar, sonar, spread spectrum communications u ( n + m - 1) = [ 0 0 '.. Sn, 1. n (4) If all the states were arranged "one under the other", a matrix would be obtained, commonly reffeered to as thc signal matrix, or the input matrix. This is a Toeplitz matrix by structure: I. INTRODUCTION In 111, [2] the concept of Envelope Constrained Filter (ECF) was introduced and the nonlinear programming approach was proposed for E C F design. One possible application of the approach is the mismatched filter design for sidelobe suppression in radar techniques. Our tests have shown that in this application the case of a infeasible solution is usually encountered; nonlinear programming approach is not a t it best then, because it is based on a feasible solution assumption. Minimax algorithms [4],[6], [7], [8], [9] are based on an opposite assumption - that the solution can not be found. This paper presents a modification of these algorithms in order t o make them capable of solving the E C F design problem. The paper is organised as follows: - Chapter 11 presents some elementary terms and definitions in filter design. - Chapter 111 presents the ECF concept. - Chapter IV presents the ECF modification of the minimax algorithms. - Chapter V presents numerical results. - Chapter VI gives the conclusion and the areas for further resear ch. 11. VL s=( :. .: .: The filter response $' E [ $1 .: II, is given by the product: $2 ... T A $wL+~-I ] = SW. (6) The filter response shaping problem is a problern of how t o find a filter w that will yield a most desirable response Sw. In order to assess the response obtained, the desired response is predefined: zyxwvutsrq zyxwvutsr zyxwvu and the error vector is evaluated as Filter design can be also defined as a proccss of lowering the error measure obtained. The most comrriorily used error measure is the rnean square error (MSE): S O M E ELEMENTARY TERMS A N D DEFINITIONS I N FILTER DESIGN (9) This paper deals with the filter design; the filter is considered to be the digital transversal filter with a finite impulse response (FIR) of order n. The filter is uniquely described by a vector of its weight coefficients: w = [ w1 A filter designed with the MSE measure is tlie Least, Squares (LS) filter. Another possible error measure is the maximal absolute error (MAE) value: M A E E SUP {IQ! w2 .'. w, IT1 where (.)T stands for the transposition operation. 0-7803-5768-X/99/$10.000 1999 IEEE (1) I i E [ l l n+~ 71 - I\} (10) A filter designed with the MAE nieasure is the miriiriiax filter. 367 zyxwvutsr zyxwvutsrqponm zyxwvutsrqponmlkjih zyxwvuts zyxwvutsrqponmlkjihg zyxwvuts In conventional minimax algorithms, the error is defined as the difference of the obtained response .IC, to the desired ECF introduces a new approach in defining the desired response d, e = d - $J. response and error evaluation. Besides the desired response In ECF design a predefined envelope { € - , e + } is introvector, defined as in Eq(7), the tolerance vector is introduced instead of the desired response d. The basic interduced as vention in ECF modification of a minimax algorithm is the E z [ €1 E2 . * . Efn+f,-l (11) reformulation of t.he error: if the response is inside the envelope, the error is zero per definitionem. Otherwise, the where E & 2 0:VZ. error is the difference of the response to the closer boundary The obtained response at the i-th point is satisfactory if of the envelope: ( 4 - $'I L € 1 , (12) ei - ci, if e; > 111. THEECF CONCEPT zyxwvutsr zyxwv that is. if it deviates from the desired response by not more tlinn the allowed tolerance. An equivalent definition can be made by introducing the upper and the lower boundary vector, as The boundary vectors forin a particular envelope as a coiistraint to the filter design process. Hcrice the name ECF. The obtained response is then satisfactory if it*fits inside the envelope { E + , E - } : €- eye' = { ei '€6: i: (16) :;:i:- The above redefinition of tlie error was applied to three minimax algorithms known for giving the best results in niinimlzx filtcr design - the Itcrntivc Reweighted Least Squares (IRLS), tlie Minirnas Modification of the Recursive Least, Squares (MM RLS), and the Minimax Modification of the Least Mean Squares (MM LMS). The MM RLS and the hlM LNS became capable of solving the ECF design problem with just the above error reforniula.tion and IIO further intervention. However; IRLS still used t o run into numeric problems and diverge in some particular ECF design scenarios. Since this feature proved t o be problem-dependent, the solution was found in letting t,he algorithm adapt the desingularising constant, crucial for its convergence, to a needed value. By introducing tlie adaptive sub-algorithm for the desingularising constant, ECF IRLS also became s t a l k and capable of 1 all cases it was m t e d solving the ECF design p I ' O b ~ ~ i Tin in. zyxwvutsrqponmlkjih 5 qj 5 E + . (14) t V. NUMERICAL RESULTS Novel ECF minimax (ECF M M ) algorithms were compared t o t,he nonlinear programming ECF algorithm, named the optimal ECF algorithm by the authors, for its tendency to minimise the norm of tlie weight vector Ilwll' = It is obvious that 1 d = -(E+ 2 Iv. ECF + E-), and e= 1 -(E+ 2 - E-). CWZl (17) i Pig. 1. Filter response popping out of the given cnvelope Ji), and inside the givcn envelope (ii) (15) MODIFICATION O F T H E MINIMAX ALGORITIIMS When the nonlinear programming approach from [I], [2] is used, you soon firid yourself seeking for a tightest ellwlope that still yields a feasible solution. This process is a trial-and-error one. Minimax approach to ECF design introduces a new feature: if the response "pops out" of the envelope, it will still respect, the envelope shape frorn the outer side, curve (i) in Fig 1. So, if the error happens to be unavoidable, its maximal value will be minimised. Tlicreby, error control is introduced. which enables an optimal behaviour of the ECF designed in a noisy enviroiiment. ECF MM RLS was taken as a representative ECF MM algorithm: it seems to make a good trade-off between the computational complexity and the convergence speed. The ECF IRLS showed to be too cornplex, and the ECF Mhsl LhG was too slow in convergence. Comparison will be presented in a case of a bell-shaped envelope, made of two Gaussian shaped functions set c apart, so that E:. - E ; = c,Vi. If c is largc cnough, the envelope is loose, and the solution is feasible. In that case, we found that the performance of the novel ECF MM algorithms and the nonlinear programming ECF optimal (ECF OPT) algorithm is equal. As an example, Fig 2 shows the case of c = .5, for the Barker 13 code at the input. Filter length was 13. If c is not. large enough, the envelope is tight and the solution is not feasible. In that case ECF OPT can riot 368 -14 zyxwvutsrqpon I -15 - -18 - zyxwvutsrqponm time time Fig. 2. Case of a feasible solution: E C F OPT (solid), E C F MMRLS (dash), envelope (dot) Fig. 4. Error level in a case of an infeasible solution, ECF OPT (solid), E C F MM RLS(dash) zyxwvut - In the feasible solution case, the new algorithms corivergc t o the same solution as the optimal ECF algorithm, based on the nonlinear programming approach. ...'.-',, 1- : -. .* - REFERENCES time Fig. 3. Case of an infeasible solution: E C F O P T (solid), E C F MMIUS (dash), envelope (dot) R. J . Evans, T. E. Fortman, A. Cantoni, "Envelope-Constrained Filters, part I, Theory and Applications", IEEE Trans. IT, Vol. IT-23, NO. 4, pp. 421-434, July 1977. R. J . Evans, T. E. Fortman, A. 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