A. W. Potchinkov, Der Entwurf digitaler FIR-Filter mit Methoden der semiinniten konvexen Optimierung, Ph.D. Thesis, Technische Universit#t Berlin, Berlin, Germany, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Hence, a least squares behavior in the stopband of the cascaded system consisting of the analog prelter and the digital lter to be designed is desirable [123] A Chebyshev behavior is useful in the passband of the cascaded anti aliasing lter [38] A similar design example has been presented in [89]. We use the design specications as given in [95] The analog prelter is chosen to be a third order Chebyshev lowpass lter with an equiripple passband and a monotonic stopband response. The passband ripple is 0.5 dB, and the attenuation is 30 dB at 3 p , where p is the passband edge. The ....

A. W. Potchinkov, Der Entwurf digitaler FIR-Filter mit Methoden der semiinniten konvexen Optimierung, Ph.D. Thesis, Technische Universit#t Berlin, Berlin, Germany, 1994.

....the last two functions are de ned on P D only. Details are found in Section 2. Then four central design problems for digital lters can be stated. They are known to specialists for a long time (e.g. 13] 20] but seem to have been formulated as mathematical problems for the rst time in [16], probably since they had not been approached directly as approximation problems before. With some closed and possibly nite sets D and P P D and prescribed norms (typically the least squares or the maximum norm) these are 1. the approximation of D by HN (x; on 2. the ....

.... [5] 11] 15] 1] and see [19] for the relations between these) Also a linearization of the group delay response of the lter has been used in order to control the group delay in connection with the frequency response problem ( 5] and to approximately solve problem 4 for small phase errors ([16], 8] Typically those of the mentioned design methods, which have proven convergence, are based on characterizations of solutions of the respective problem which were derived by arguments from approximation theory, and hence these normally are applicable to unconstrained problems only. ....

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A. Potchinkov. Der Entwurf digitaler FIR-Filter mit Methoden der konvexen semi-inniten Optimierung. PhD thesis, Technische Universitat Berlin, Berlin, Germany, 1994.

.... types of design problems which consist in (simultaneously) approximating a desired frequency response D( a desired magnitude response jD( j, a desired phase response arg(D( and a desired group delay response d d arg(D( by the corresponding functions of the lter (e.g. 13] 5] [15]) For linear phase lters the desired phase response arg(D( and the phase response arg(HN (x; of the lter coincide (by (4) and (3) they equal N ( so that zero phase response and group delay errors can be reached in this case. Moreover, the frequency and magnitude response errors are ....

A. Potchinkov. Der Entwurf digitaler FIR-Filter mit Methoden der konvexen semiin niten Optimierung. PhD thesis, Technische Universitat Berlin, Berlin, Germany, 1994.

....KN , and KN satisfies Assumption 4. 7 with K : fF 2 CC( Omega Gamma j fi fi Imfe Gammaifi D ( F ( g fi fi Gamma tan(U fi ( Refe Gammaifi D ( F ( g 0; 2 Omega P g: Hence, what is quite remarkable, bounds on the phase response error can be expressed by linear constraints ([4]) An equivalence with linear constraints as used here is not known for the similarly looking constraints in the corresponding semi infinite program when the magnitude and phase response errors are simultaneously minimized (cf. 6] and x6 below) In [6] and [7] problem (5.1) was considered for the ....

....programming problem. Replacing Omega and Omega P by finite sets, the authors of [2] and [1] have solved the resulting finite program, but only for problems with rather small N . As far as we now, alternatively only a (coarse) linear approximation of the group delay response was employed ( 3] [4]) Thus a satisfying numerical solution for problem (7.3) has not yet been given. Analogously to Lemma 6.4 we can derive kG p;N (x)k p [2 max(L Omega ; L Omega P ) 1=p kG1;N (x)k 1 ; x 2 K N ; 7.4) which in particular implies p;N G p 1;N for 1 p 1 where G p is the constant in ....

A. W. Potchinkov, Der Entwurf digitaler FIR-Filter mit Methoden der konvexen semiinfiniten Optimierung, PhD thesis, Techn. Univ. Berlin, 1994.

....one has x 2 R N (with F : 0; but x 2 C N (with F : Gamma; also is possible. With closed sets (possibly finite sample sets) Omega Omega D and Omega P Omega P D , the four main digital filter design problems in the frequency domain then are the following (e.g. 18] [20]) 1. the approximation of the frequency response D on Omega Gamma 2. the approximation of the magnitude response jDj on Omega Gamma 3. the simultaneous approximation of the magnitude response jDj on Omega and the phase response arg(D( Delta) on Omega P , 4. the simultaneous approximation of ....

....S D . The attenuation or the amplification of a signal at a frequency is measured in decibel and given by 20 log 10 jHN (x; j. It was only recently observed that, in case of the maximumnorm, the above four problems can be formulated as semi infinite programming (briefly: SIP) problems ( 2] [20], 22] where, for FIR filters, the first problem is convex (at least in the unconstrained case) and the others are nonlinear nonconvex and, for IIR filters, all problems are strongly nonlinear. The use, for example, of the L 2 resp. l 2 norm leads likewise to a SIP problem in a natural way ....

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A. W. Potchinkov, Der Entwurf digitaler FIR-Filter mit Methoden der konvexen semiinfiniten Optimierung, PhD thesis, Techn. Univ. Berlin, 1994.

....small N since the number of chosen frequencies should be and has been at least 10N ( 17, p. 89] 41] Potchinkov and Reemtsen suggested to view lter design problems as special optimization problems and to solve them by general optimization methods for smooth convex and nonlinear programs ([20], 21] 22] 23] 24] 25] 26] 27] 28] 35] In particular discretized and continuous (constrained) least squares and maximum norm approximation problems can be straightforwardly converted into smooth nite and semi in nite programming (SIP) problems respectively, where a SIP problem ....

....problems. Potchinkov and Reemtsen have shown that, for the least squares and the maximum norm, problem 1 is a convex SIP problem if added constraints are convex (as those in (6) and (7) and that under certain conditions problems 3 and 4 can be approximated by convex SIP problems suciently well ([20], 23] 26] 27] 28] 31] 32] and Section 5 below) They have furthermore solved such problems with two cutting plane methods for convex SIP problems which they had developed in [33] 26] and [27] These methods are very reliable and able to yield highly accurate solutions of convex ....

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A. Potchinkov. Der Entwurf digitaler FIR-Filter mit Methoden der konvexen semi-inniten Optimierung. PhD thesis, Technische Universitat Berlin, Berlin, Germany, 1994. 27

....computational power at low costs, in combination with intelligent and high speed interfaces. 3 about complex Chebyshev approximation [16, 17] design of linear phase FIR filters [14] and the problems of simultaneous approximation of magnitude and phase [18] or magnitude and group delay responses [13]. Convex optimization fits exactly the design of linear phase FIR filters and complex approximation. The nonlinear simultaneous approximation problems can be solved by that in good approximation. Furthermore, Potchinkov and Reemtsen developed a method which has been used successfully for numerous ....

....to those with D( 0; and transition bands to nonspecified D( Stopbands require magnitude response approximations only. Phase and group delay responses naturally cannot be specified in stopbands. Further time domain constraints are formulated w.r.t. the discrete sampling time instants [13, 20]. Error valuations Assuming here an error function of the frequency domain, the error valuations are the maximum norm (L1 norm) k (h; k 1 = max 2B j (h; j ; the L p norm, 1 p 1; k (h; k p = Z B j (h; j p d 1=p ; the bound by a function U( j (h; j U( 2 ....

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A. Potchinkov. Der Entwurf digitaler FIR-Filter mit Methoden der konvexen semi-infiniten Optimierung. PhD thesis, Techn. Univ. Berlin, 1994.

....in filter design often are preferably measured by the maximum norm. It was only recently observed that, in case of employment of this norm and for typical kinds of constraints, the four described approximation problems can be formulated as smooth nonlinear semi infinite programming (SIP) problems ([19], 22] Similarly the use of the least squares norm leads to an SIP problem in a natural way when functional constraints on the filter coefficients are included in the program. Remarkably, application of the L 1 resp. l 1 norm has not been investigated much, which should be favored when ....

.... yield highly accurate solutions of convex design problems also for large filter lengths (see also [20] 21] 24] In this context it is important to note that, for FIR filter design, some of the relevant problems either are convex SIP problems or can be approximated by such sufficiently well ([19], 21] 24] 26] and Section 5 below) However, in particular, the problem of approximating a desired magnitude response is nonlinear and does not seem to be properly representable by a convex optimization model. In Section 2 we list our notations. In Section 3 we provide quite general ....

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A. Potchinkov. Der Entwurf digitaler FIR-Filter mit Methoden der konvexen semi-infiniten Optimierung. PhD thesis, Technische Universitat Berlin, Berlin, Germany, 1994.

....problems, the maximum norm often is the requested measure of distance. It was only recently observed that, in case of employment of this norm and for typical kinds of constraints, these approximation problems can be formulated as smooth nonlinear semi infinite programming (SIP) problems ( 1] [18], 21] Similarly the use of the least squares norm leads to a SIP problem in a natural way when functional constraints on the filter coefficients are included in the program. Remarkably, application of the L 1 resp. l 1 norm has not been investigated much, which especially is of interest ....

.... of best approximation of a desired frequency response with respect to the maximum norm results in a convex SIP program (at least in the unconstrained case) and that, for relevant situations, some of the nonconvex nonlinear problems can be sufficiently well approximated by convex SIP problems (see [18], 20] 23] 25] and Section 5 below) But, in particular, the problem of approximating a desired magnitude 2 response is nonlinear and does not seem to be properly representable by a convex optimization problem in an approximative way. In Section 2 we list our notations, which allow the ....

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A. Potchinkov. Der Entwurf digitaler FIR-Filter mit Methoden der konvexen semi-infiniten Optimierung. PhD thesis, Technische Universitat Berlin, Berlin, Germany, 1994.

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