Investigation of Computationally Efficient Complementary IIR Filter Pairs with Tunable Crossover Frequency Jelena D. Ćertić a , Ljiljana D. Milić b a School of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia b Mihajlo Pupin Institute, University of Belgrade, Volgina 15, 11000 Belgrade, Serbia Abstract This paper concentrates on the computationally efficient implementation of three classes of complementary IIR filter pairs with tunable crossover frequency. Class I denotes the IIR filter pair implemented as a parallel connection of two all-pass filters, whereas Class II (Class III) stands for the tapped cascaded interconnection of two distinct all-pass filters. We propose new tuning formulae for shifting the crossover frequency of the existing filter pair to a new location. The new tuning scheme includes the previously introduced tuning scheme as a special case. The new closed-form expressions representing the sensitivity functions for the amplitude frequency characteristics to the filter constants are developed. The sensitivity functions show that the realization structures of Class II and Class III filter pairs inherently contribute to the low pass-band and the low stop-band sensitivities of those filter classes. We present the approximation of tuning formulae and display the measured amplitude responses obtained on the fixed-point DSP. We show the multiplierless implementation of tap coefficients in Class II and Class III filter pairs. Keywords: Complementary filter pairs, IIR filters, sensitivity analysis, variable filters 1. Introduction Complementary IIR filter pairs are frequently used as computationally efficient building blocks in multi-channel filter banks [1]–[3], and in frequency-response masking techniques [3], [4]. Furthermore, complementary IIR filter pairs may be used for constructing digital crossovers for loudspeaker applications [5], and also for the non-uniform channel separation in a multi-band audio system [6]. Design and implementation of IIR filters with a variable cutoff frequency has been considered in several papers, see e.g. [7]–[12]. It has been shown that in the case off an IIR filter, the tuning process suffers from the high sensitivity to the filter constants, and the finite-precision arithmetic inevitably produces an error when computing the new constant values from the existing ones. Complementary IIR filter pairs with variable crossover frequency have been considered in [13][15]. The simple tuning scheme has been developed that shifts the crossover frequency of the start-up half-band filter pair to the desired location. The tuning scheme mentioned above has been developed for the following three classes of complementary filter pairs [15]. Class I denotes the power complementary/all-pass complementary IIR filter pair implemented as a parallel connection of two all-pass filters. Class II (Class III) filter pair is a power *Corresponding author. Tel.: +381113218348; Fax: +381113248681. E-mail addresses: certic@etf.rs (J. Ćertić), milic@kondor.imp.bg.ac.rs (L. Milić). complementary (magnitude/all-pass) complementary filter pair implemented as a tapped cascaded interconnection of identical copies of two distinct all-pass filters. This paper concentrates to the computationally efficient implementation of the tuning scheme for three filter classes mentioned above. We develop new tuning formulae that extend the application of the tuning scheme from [13][15] in a manner that the start-up filter pair can be an arbitrary Class I filter pair. This tuning scheme preserves the pass- band/stop-band properties of the start-up Class I filter pair. The tuning scheme introduced earlier in [13][15] can be considered as a special case. In order to gain an insight into the sensitivity characteristics of three different filter classes, we develop the closed-form expressions for the sensitivity of the magnitude responses of Class I, Class II, and Class III filter pairs. In the sequel, we present the approximate procedure that is suitable for the fixed-point implementation of the tuning formulae. Finally, we illustrate the tuning range by exposing the experimental results obtained on the fixed-point DSP, and propose the multiplierless implementation of the tap coefficients in Class II and Class III filter pairs. This paper is organized as follows. Section 2 presents a brief description of Class I complementary filter pairs. The simple tuning formulae that convert a start-up Class I filter pair to a new filter pair with a desired crossover frequency are given in Section 3. Section 4 presents the basics concerning the Class II and Class III complementary filter pairs, and the tuning procedure for those two classes. In Section 5, the closed-form expressions for the magnitude- response sensitivity functions for three filter classes are