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Linear Phase Brick Wall Crossovers

Conventional crossover design methods utilize traditional frequency selective networks to combine multiple transducers into a single full-bandwidth system. These traditional networks, whether they are implemented in analogue or digital form, exhibit large transition bands and suffer from phase distortion. These characteristics result in poor frequency, impulse and polar responses. A practical crossover implementation is presented that removes the detrimental effects of transition bands and phase distortion. This method implements linear phase crossovers whose transition bands approach a theoretical ideal brick wall response. Comparisons to conventional crossovers will be presented. Applications to large scale array optimization are also discussed and presented.

Introduction

This paper presents a practical method for optimization of multi-way loudspeaker systems through the use of a new type of crossover, readily available with current digital signal processing technology. This new method implements linear phase crossovers, whose transition bands approach a theoretical ideal brick wall response (LPBW crossover type). Typical transition slopes exceed 180 dB per octave. The LPBW crossover will be presented, and its attributes will be discussed. A similar analysis as has been previously described in available literature for conventional crossovers is offered. Through this comparative analysis, the improved response of the LPBW crossover is shown.

Conventional Crossovers

In 1976, Siegfried Linkwitz published a paper on a new active crossover network for multi-way loudspeaker systems. The crossover network described in his paper became affectionately known as a Linkwitz-Riley crossover. Russ Riley was a friend and fellow engineer of Siegfried’s who contributed to the research. Linkwitz had found that traditional filter implementations providing constant voltage and allpass crossover characteristics introduced off-axis errors in the radiation pattern1. Lipshitz and Vanderkooy introduced the term “lobing error” to describe this problem2. The Linkwitz-Riley (L-R) crossover optimizes the radiation pattern through the crossover transition region, resulting in a response that does not suffer from lobing error.

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Simple Source Analysis

Simple sources are typically used in the theoretical analysis of crossover responses. The acoustic source that is most simple from the standpoint of analysis is the pulsating sphere. A pulsing sphere simple source has radius r = a that varies sinusoidally with time. The pulsating sphere simple source produces an outgoing spherical wave in a medium that is infinite, homogeneous and isotropic3. Since we are interested in the far-field radiation pattern of simple sources, we will evaluate the pressure at a distance r >> a. Figure 1 represents the simple source analysis as presented in the original paper by Linkwitz. This analysis requires the combination of two simple sources with a crossover network in order to obtain the desired far-field radiation pattern, where θ1 = θ2 = θ.

Figure 1 – Radiation from coplanar sources H and L:


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In order to compute the desired radiation pattern, a mathematical basis must be established for the simulation model. The mathematics behind the combination of simple and not-so-simple sources is well known4 so these mathematical formulations will not be derived here. Instead, we will simply state that the radiation pattern of two simple sources can be directly calculated by finding the complex pressure:

P = r1A1 eikr1+ r2

A2 eikr2

Where the calculation of r1 and r2 is dependent upon the geometry. For the geometry as defined in figure 1, these radii are calculated by:

r1= d2+R2

- 2dRcos (i)

r2= d2+R2

- 2dRcos (r- i)

Although this formulation provides us with the capability of simulating the radiation pattern at any radius R we choose to perform the calculation in the far field. In order to review the radiation pattern resulting from this combination, the magnitude of this function is calculated and presented on a logarithmic polar plot.

Linkwitz-Riley Crossover Simulation Figures 2 and 3 represent the basis for an extended analysis of those presented in the original paper by Linkwitz. Figure 2 shows the complex frequency response of the L-R crossovers applied, and figure 3 illustrates the resulting radiation pattern at the crossover frequency. Linkwitz’s paper evaluated the radiation pattern at the crossover center frequency only. To investigate further, we will extend this analysis throughout the crossover transition region.


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Figure 2 – Linkwitz-Riley crossover pair applied to simple sources H and L:

Figure 3 – Linkwitz-Riley radiation pattern evaluated at crossover center frequency:


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In the analysis of a conventional crossover, such as an L-R crossover, lobing error is defined to measure the severity of lobing in the response off-axis of the two simple sources used in the calculation of the radiation pattern. Lobing error therefore does not take into account the cancellations that occur in the radiation pattern off axis. Ideally speaking, a crossover applied between two sources should exhibit no lobing error and no cancellations off axis. Within the scope of this paper, we define the term “radiation error” to inclusively describe lobing and cancellation error. Figure 4 illustrates the application of the term radiation error to the radiation pattern shown in figure 3.

Figure 4 – Linkwitz-Riley radiation pattern, radiation error shown in gray:

The shaded area in the radiation pattern represents the total radiation error at the crossover center frequency.


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Transition Band Radiation Error For the purposes of this paper, the transition band is defined as the region where both sources contribute to the overall radiated pressure in the far-field. As with all conventional crossover implementations, the L-R crossover exhibits a large transition band. The most commonly used L-R crossover has a transition band slope of 24dB per octave. Figure 5 shows the frequency response of 24dB L-R high pass and low pass crossover filters (as first shown in figure 2), and highlights the transition band that is the subject of our radiation error analysis.

Figure 5 – Linkwitz-Riley crossover pair with transition region labeled for further analysis:


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It is necessary to extend the analysis of the radiation pattern response of the L-R crossover to extend throughout the transition band region in order to evaluate the radiation error. Figure 6 shows a series of representative radiation pattern responses of the L-R crossover through the transition band.

Figure 6 – Representative set of Linkwitz-Riley radiation patterns:


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The best way to represent radiation error through the transition band of the L-R crossover is to plot the magnitude of the error against angle and frequency. Figure 7 shows the radiation error for the 24dB L-R crossover. It is readily seen that a significant amount of radiation error is present in this crossover type.

Figure 7 – Linkwitz-Riley radiation pattern:


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Linear Phase Brick Wall Crossover

Lipshitz and Vanderkooy have summarized the following key attributes of the ideal crossover2:

1. Flatness in the magnitude of the combined outputs 2. Adequately steep cutoff rates of the individual low and high pass filters 3. Acceptable phase response for the combined output 4. Acceptable polar response for the combined output

Conventional crossovers typically sacrifice 2, 3, and 4 in the pursuit of 1. The Linear Phase Brick Wall crossover type being introduced fulfills all four requirements.

1. The LPBW crossover electrically combines to a perfectly flat magnitude response 2. The brick wall characteristic provides cutoff rates exceeding 180dB per octave 3. The linear phase characteristic provides for perfect reconstruction of the input at the combined output 4. The brick wall and linear phase characteristics provide an optimum polar response with the least amount of radiation error

Figure 8 shows the measured complex frequency response of the LPBW crossover.

Figure 8 – Linear Phase Brick Wall crossover pair:


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For this analysis, pure delay has been subtracted to illustrate the linear phase characteristic on a logarithmic frequency axis. Figure 9 shows a comparison between frequency responses of the L-R and LPBW crossovers.

Figure 9 – Linear Phase Brick Wall crossover comparison to Linkwitz-Riley crossover:


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LPBW Radiation Error Applying the same analysis as discussed in section 1.1 and 1.2, we move directly to evaluating the radiation error of the LPBW crossover. Figure 10 shows the same series of radiation pattern responses over the same range of frequencies as figure 6 depicts for the L-R crossover.

Figure 10 – Representative set of Linear Phase Brick Wall radiation patterns:

Through these polar plots it is shown that radiation error appears at the center frequency of the crossover, but the affected frequency range is dramatically smaller.


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Figure 11 shows the radiation error for the LPBW crossover.

Figure 11 – Linear Phase Brick Wall radiation error plot:

It is readily seen that radiation error is almost eliminated. Compared to the L-R crossover that exhibits radiation error across two octaves, the LPBW exhibits radiation error across less than a tenth of an octave.


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Application of LPBW to Loudspeaker Arrays

One area of considerable research interest is phase distortion in loudspeaker crossover networks. In previous work, including the references in this paper, the discussion has been focused upon the audibility of phase distortion within a single loudspeaker system. Through subjective and empirical tests, it has been determined that the phase distortion introduced by a conventional crossover network is insignificant. However, there are other implications that are worth discussion regarding phase distortion when multiple loudspeaker systems using conventional crossover networks are arrayed. Let us take the simple array example of a primary loudspeaker system and an auxiliary loudspeaker system. Such a system is commonly in use in professional applications that require consistent coverage across an audience seating area. In most scenarios, the auxiliary loudspeaker’s enclosure dimensions, acoustic output power, coverage and required operating frequency range is different than the requirements of the primary loudspeaker system. Using a conventional crossover network for both of these different loudspeaker systems will introduce problems. Figure 12 illustrates the problem.

Figure 12 – Main and auxiliary loudspeaker schematic diagram:

The primary loudspeaker is pointed straight ahead towards the audience area. The secondary loudspeaker is pointed down towards the lower audience area. The midpoint of the transition region between the two loudspeaker systems is also shown.


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Figure 13 shows the complex frequency response of the primary loudspeaker comprised of two simple sources gain and delay optimized measured at the midpoint of the transition region.

Figure 13 – Main loudspeaker complex frequency response using Linkwitz-Riley crossover:

Figure 14 shows the complex frequency response of the secondary loudspeaker comprised of two simple sources gain and delay optimized measured at the midpoint of the transition region.

Figure 14 – Auxiliary loudspeaker complex frequency using Linkwitz-Riley crossover:


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Figure 15 shows the combined response of both loudspeaker systems as measured at the midpoint of the transition region. Figure 15 – Main and auxiliary loudspeaker combined complex frequency response using Linkwitz-Riley crossover:

Due to the difference in center frequencies between the crossovers in the two loudspeaker systems, there will always be cancellations within the transition region. Gain, delay, equalization and directivity can be used to reduce the problem, but the phase distortion introduced by the conventional crossover cannot be removed by these methods.


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The LPBW crossover solves this problem. The linear phase characteristic of the LPBW crossover allows for a seamless transition between different loudspeaker systems. Figures 16 and 17 show the complex frequency responses of the primary and secondary loudspeaker systems gain and delay optimized, now using the LPBW crossover as measured at the midpoint of the transition region.

Figure 16 – Main loudspeaker complex frequency response using LPBW crossover:

Figure 17 – Auxiliary loudspeaker complex frequency response using LPBW crossover:


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Figure 18 shows the combined response of both loudspeaker systems as measured at the midpoint of the transition region.

Figure 18 – Main and auxiliary loudspeaker combined complex frequency response using LPBW crossover:

Using gain, delay and the LPBW, the transition region between loudspeaker systems can be optimized for a seamless transition between the loudspeaker systems.

Conclusion

A new type of crossover has been introduced that provides a new opportunity for the advancement of loudspeaker crossover technology. This crossover exhibits a magnitude response whose transition slopes approach a brick wall response. The linear phase characteristic of the crossover has been shown to provide significant benefits for individual loudspeaker system optimization as well as for multiple loudspeaker system arrays. We have shown that the LPBW crossover achieves the attributes of an ideal loudspeaker crossover.

References

[1] S. H. Linkwitz, “Active Crossover Networks for Noncoincident Drivers,” J. Audio Eng. Soc., vol. 24, pp. 2-8 (1976 Jan/Feb). [2] S. P. Lipshitz, J. Vanderkooy, “A Family of Linear-PhaseCrossoverNetworks of High Slope Derived by Time Delay,” J. Audio Eng. Soc., vol. 31, pp. 2-20 (1983 Jan/Feb). [3] L. Kinsler, A. Frey, Fundamentals of Acoustics (John Wiley and Sons, New York, 2000). [4] D. G. Meyer, “Computer Simulation of Loudspeaker Directivity,” J. Audio Eng. Soc., vol. 32, pp. 294-315 (1984 May).

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