Efficient Recursive Digital Filters with Variable Magnitude Characteristics Juha Yli-Kaakinen and Tapio Saram¨ aki Institute of Signal Processing Tampere University of Technology P. O. Box 553, FI-33101 Tampere, Finland email: {ylikaaki, ts}@cs.tut.fi ABSTRACT This paper considers designing in the minimax sense comple- mentary low-pass/high-pass recursive digital filters with variable magnitude characteristics. A filter structure based on the paral- lel connection of two variable fractional delay all-pass filters is proposed for implementing these filters. The filter optimization is performed in two basic steps. First, an initial filter is gener- ated using a simple design scheme. Second, this filter is used as a start-up solution for further optimization being carried out by an efficient constrained nonlinear optimization algorithm. Exam- ples are included for illustrating the efficiency of the proposed design scheme. In addition, the performance and the complex- ity of the proposed variable recursive digital filters are compared with those of the other variable recursive digital filters proposed in the literature. This comparison shows that the number of mul- tipliers for the proposed filters is less than 15 percent compared with other existing structures. 1. INTRODUCTION I N VARIOUS digital signal processing applications, there is a need for filter with variable frequency characteristics. These applications include, e.g., sampling rate conversion, echo cancel- lation, phased-array antenna systems, time delay estimation, tim- ing adjustment in all-digital receivers, modeling of music instru- ments, and speech coding and synthesis [1–4]. Recently, research on the optimal design and the efficient implementation of the re- cursive variable fractional delay filters has received considerable attention [5–8]. However, the implementation of recursive filters with variable magnitude characteristics have not gained as much attention and have been considered only by a few authors [9–11]. The purpose of this contribution is to propose a new class of magnitude-selective variable digital filters and an algorithm for their optimization. The methods for designing variable digital filters can be broadly classified into two categories, namely, frequency trans- formation methods [12–15] and spectral parameter approximation methods [3, 5–11, 16]. The disadvantage of the methods, belong- ing into the former class, is that the edge frequencies and the rip- ples of the various bands cannot be independently controlled. The second class of filters does not suffer from this restriction. In this technique, the coefficients of the variable filter are expressed as the polynomials of the adjustable parameter defining the desired filter characteristic. Variable digital filters can be constructed using either finite- impulse response or infinite-impulse response filters. From the This work was supported by the Academy of Finland, project No. 44876 (Finnish Centre of Excellence Program (2000–2005)). Juha Yli-Kaakinen was also financed by a postdoctoral research grant from the Academy of Finland, project No. 105823. x(n) y0(n) y1(n) 1/2 A1(z, μ) A0(z, μ) Fig. 1 Complementary low-pass/high-pass variable recursive filter pair implemented as a parallel connection of two variable fractional delay all- pass filters. implementation point of view, one of the best structures for re- cursive digital filters is a parallel connection of two all-pass fil- ter sections. These structures have some advantageous properties, such as a reasonably low coefficient sensitivity and a low roundoff noise level. This contribution proposes a highly efficient structure for im- plementing complementary low-pass/high-pass variable recursive digital filter pair. This filter structure is based on the parallel con- nection of two variable fractional delay all-pass filters. In addi- tion, an algorithm is proposed for optimizing the magnitude re- sponses of these filters. Furthermore, the performance and the complexity of these filters are compared with some other variable recursive digital filters proposed in the literature showing that the number of multipliers for the proposed filters are less than 15 per- cent compared with other existing structures. 2. VARIABLE RECURSIVE DIGITAL FILTERS The transfer functions of the variable digital filter pair as shown in Fig. 1 are given by H0,1(z, µ)= 1 2 [A0(z, µ) ± A1(z, µ)] . (1) Here, H0(z, µ) with the plus sign and H1(z, µ) with the mi- nus sign are the low-pass and high-pass filters, respectively, and A0(z, µ) and A1(z, µ) are variable fractional delay all-pass fil- ters [5, 6, 8, 17] of order N0 and N1, respectively, and are express- ible as A k (z, µ)= z -N k C k (z -1 ,µ) C k (z, µ) , (2a) where C k (z, µ)=1+ N k X n=1 a (k) n (µ)z -n =1+ N k X n=1 " P X p=0 c (k) pn µ p # z -n . (2b) Here, µ is an adjustable parameter in the range [−1, 1] and each coefficient in the overall filter is given as a polynomial of degree P in µ. In the case of low-pass filters, N0 = N1 − 1 or N0 = N1 +1 so that N0 + N1, the overall order of H0(z) and H1(z) is odd. An efficient implementation for the variable fractional delay all-pass filter based on the so-called gathering structure is shown in Fig. 1 in [5]. 1-4244-0413-4/06/$20.00 ©2006 IEEE 30 NORSIG 2006