Application of Linear-Phase Digital Crossover Filters to Pair-Wise Symmetric Multi-Way Loudspeakers Part 1: Control of Off-Axis Frequency Response

Various methods exist for crossing over multi-way loudspeaker systems. These methods include those loosely classified as Linkwitz-Riley filters, constant-voltage filters, and D’Appolito configurations. All these methods do not provide broad-band constant-beamwidth or constant-directivity operation because their vertical radiation patterns change shape as a function of frequency. This paper describes a simple, non-iterative linear-phase crossover filter design technique that provides uniform frequency responses vertically off-axis for a given multi-way loudspeaker. Distances between the individual drivers, and desired off-axis attenuation are prescribed as input parameters for the design process, the outcome of which is a set of crossover frequencies and unique filter frequency responses in each band. In order to obtain wide-band constant-beamwidth, a loudspeaker array configuration composed of a single central tweeter surrounded symmetrically by pairs of lower-operating-frequency transducers arranged in a vertical line is required. Practical implementation issues are outlined in the paper by means of various design examples. Two design methods are presented in in two-parts: Part 1: a general method which emphasizes flatness of arbitrary off-axis frequency responses and Part 2: a simplified method that emphasizes frequency uniformity of beam shape and coverage angle (vertical beamwidth) of the polar patterns. 0 INTRODUCTION Uniform and smooth off-axis responses are widely accepted as key features of a successful loudspeaker design [1]. Ideally, one wishes to achieve flat, frequency-independent amplitude responses at any measured point in space. A common design practice is to accept non-ideal behavior, such as interference due to path-length differences in multi-way speakers, as unavoidable, and then try to optimize the apparent sound quality by modifying crossover parameters – a process commonly known as voicing. There are efforts to circumvent the problem and approach the ideal more closely. Keele [2] has presented a novel array design that features excellent control over the radiated sound field. However, a high number of high-quality wide-band drive units are required, and a long curved array is not always suitable for domestic use. In [3] two different approaches are shown – a distributed mode loudspeaker, and a two-way system with large high-frequency waveguide and digital, brickwall crossover. Drawbacks are high distortion with the former, and remaining crossover artifacts such as preringing with the latter. Recently, Shaiek et al [4] have presented a high-end four-way full-bandwidth coaxial source. Despite of the very high complexity and cost of their design, the achieved off-axis responses appear very irregular and require iterative, sub-optimal equalization methods. Van der Wal’s article on logarithmic arrays [5] describes a design algorithm related to our method. Here, optimum zero-phase low pass filters provide frequency-dependent array aperture, in order to achieve constant-directivity. However, it is not a multi-way crossover design, since all drivers are required to reproduce the low-frequency band. In our new technique described here, a DSP-based crossover is designed to work with a loudspeaker array composed of a single central tweeter surrounded symmetrically by pairs of lower-operating-frequency transducers arranged in a vertical line. Each pair of drivers at the same distance from the center is driven by a separate crossover channel, including the single central tweeter. At any specific frequency, only one pair or at most two pairs of speakers are operating simultaneously (the single central tweeter is the sole exception which operates by itself at high frequencies). This feature of the new technique allows the design method to apply both to equallyand unequally-spaced pairs of drivers. In Part 1, the design procedure for the new technique is based on specifying a crossover frequency-response shape that forces a flat frequency response at a specified vertical off-axis angle. When thus specified, frequency responses at other vertical off-axis angles are found to be reasonable flat as well. In Part 2 of this paper we 1 Patent applied for May 6, 2005 (US 20060251272). AES 32nd International Conference, Hillerød, Denmark, 2007 September 21–23 1 Horbach and Keele Application of Digital Crossover Filters Part 1: Control of Off-Axis Response present a somewhat-simplified alternate design procedure that emphasizes frequency uniformity of beam shape and coverage angle (vertical beamwidth) of the polar patterns. This paper illustrates the two different design procedures by applying the former design technique to the design of multi-way loudspeaker monitors in Part 1, and the latter to the design of broad-band constant-beamwidth vertical line arrays in Part 2. In Part 1, we discuss the performance of some commonly used crossover alignments in section 1, then introduce our technique in section 2, briefly cover filter implementation issues in 3, and close with a summary in and section 4. In Part 2, we develop an alternate design method for constant-beamwidth line arrays. 1 TRADITIONAL CROSSOVER ALIGNMENTS We will employ a three-way loudspeaker design as depicted on the left in Fig. 1 throughout this section to illustrate the frequency responses of conventional crossovers. A small neodymium tweeter with low resonance frequency is used to minimize the distance to the midrange and allow a low crossover point. For a given crossover filter, we compute vertical off-axis responses by applying circular piston models for the transducers, and compute the complex sum of the respective terms after multiplication with the crossover transfer functions. In the asymmetric case, different sound pressure levels result for angles above and below the main axis. We also consider a symmetric arrangement of transducer pairs around the tweeter as illustrated on the right in Fig. 1, as proposed by d’Appolito [6]. We restrict our considerations to the far field, where the observation distance is large compared with the dimensions of the loudspeaker. The sound pressure of a single monopole xi at positive angles (upwards) is ) / 2 exp( 2 1 , λ π i i u d j C − = , (1) and negative angles (downwards) ) / 2 exp( 2 1 , λ π i i d d j C + = . (2) The sound pressure of a source pair is the sum of both: ) / 2 cos( , , λ π i i d i u i d C C C = + = (3) where (see Fig. 2) Figure 1: Speaker system configurations analyzed in this paper: conventional (left) and pair-wise symmetric threeway layout (right). X-axis (vertical) distances in meter. Figure 2: Source locations and path differences for two pairs of point sources located symmetrically about a center position at vertical positions ±x1 and ± x2. . 0 ; 2 , 1 , 0 sec / 346 , / , sin 0 = = = = ⋅ = x i m c f c x d i i λ α , (4) The far field sound pressure level of the three-way loudspeaker at angle α is 2 , 2 1 , 1 , 0 C H D C H D H D H LP BP HP α α α α + + = , (5) where Di,α is the attenuation of the i-th transducer at angle α, using the first order Bessel function J 1