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Audio Engineering Society
Convention PaperPresented at the 119th Convention 2005 October 710 New York, New York USAThis convention paper has been reproduced from the author's advance manuscript, without editing, corrections, or consideration by the Review Board. The AES takes no responsibility for the contents. Additional papers may be obtained by sending request and remittance to Audio Engineering Society, 60 East 42nd Street, New York, New York 10165-2520, USA; also see www.aes.org. All rights reserved. Reproduction of this paper, or any portion thereof, is not permitted without direct permission from the Journal of the Audio Engineering Society.
Parametric control of filter slope versus time delay for linear phase crossoversDavid McGrath, Justin Baird, and Bruce Jackson Lake Technology, A Dolby Company Surry Hills, New South Wales, 2010, Australia www.lake.com
ABSTRACT Linear phase crossover filters are a powerful tool for sound system designers. They deliver a near-ideal response with ruler-flat pass-band, steep transition slopes and adjustable stop-band rejection all with zero phase shift. Transition slopes can be matched to a target response, for example 24 dB or 48 dB per octave, and can also be arbitrarily specified while still retaining a perfect-reconstruction characteristic. Practical application of linear phase crossovers requires manipulation of cutoff frequency, transition slope and stopband rejection. A graphical user interface is described which gives users new degrees of freedom in defining linear phase filter parameters. By setting bounds for parameters such as delay, a user can continuously vary other parameters while the graphical user interface optimizes the resulting filter. This paper presents new parameters for optimization of a target transition slope within a bounded delay parameter, providing fast and efficient user controls for working with and adjusting the crossover filters in real time.
1.
INTRODUCTION
In their 1983 JAES paper [4], Lipshitz and Vanderkooy summarized the following key attributes of the ideal crossover: 1. Flatness in the magnitude of the combined outputs Adequately steep cutoff rates of the individual low and high pass filters Acceptable phase response for the combined output
Linear phase crossovers are relatively well-known in the AES literature [1, 2 & 3] but are not prevalent in practical application. Most of the previous work with these more advanced crossover solutions has been for specific fixed implementations, since the filters are difficult to program and present in a useful way to an end user. The linear phase crossover family has a number of benefits as compared to existing crossover technologies.
2.
3.
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4.
Acceptable polar response for the combined output
Conventional crossovers typically sacrifice 2, 3, and 4 in the pursuit of 1. The linear phase crossover fulfills all four of these attributes and extends parametric control of additional parameters to provide the end user with an intuitive interface for system design. The linear phase attribute also lends itself to improved subjective response, by removing the affects of phase distortion introduced by classical crossover filter implementations. Linear phase crossover filters do not suffer from frequency dependent phase distortion, regardless of the transition band slope specification. Another unique attribute of the linear phase crossover is that the transition slope can now be arbitrarily specified. Sound system designers are no longer limited to transition slopes quantized in 6 dB or 12 dB increments. The removal of this limitation provides further optimization due to the reduction of additional equalization required on each output channel. Equalization filters are typically required to force a loudspeakers transition band response into a shape that will work with a fixed slope per octave classical crossover shape. We introduce a new control parameter, beyond the cutoff frequency and transition slope parameters typically provided for traditional crossover technologies. This control is called Alignment Delay and allows the end user to optimize the pure-delay component of the linear phase crossover filterbank. By adjusting this parameter, the user is able to choose the best tradeoff between steepness of transition slope versus the time delay acceptable for the particular application. In order to build an ultra-steep crossover, the amount of delay required is inversely proportional to frequency: the delay requirement becomes much larger as you move to lower frequencies. If the steepness constraint is relaxed, then you dont need as much delay. The Alignment Delay parameter allows the end user to specify a maximum amount of delay, and then this defines the steepness of the filters available at a specified cutoff frequency. This tradeoff between transition slope and time delay is discussed and illustrated.
User control of these advanced filtering algorithms for practical application is highlighted. An end user can now adjust linear phase crossover filterbanks in real time through a graphical interface. 2. GROUP DELAY, LINEAR PHASE VERSUS CLASSICAL CROSSOVERS
Linear phase filters are recognized as having larger group delay than their minimum phase counterparts, but this is an implementation specific phenomenon. In reality, a linear phase filter can achieve an equivalent group delay as compared to its minimum phase counterpart, as shown in figure 2.1. In this instance, the 510 Hz Linkwitz-Riley low pass filter exhibits group delay that varies across frequency, with a maximum group delay of approximately 2.4 milliseconds. Note also that a crossover constructed using this minimum phase filter will have 1.6 milliseconds of added group delay in the low-frequency region (below crossover), as compared to the high frequency region (above crossover). In contrast, the linear phase filter achieves a constant group delay of 2.5 milliseconds that is approximately the same as the maximum group delay of the LinkwitzRiley minimum phase implementation. Furthermore, the linear phase crossover maintains this group delay across all frequencies. This bulk delay, also known as pure delay, avoids the problem of time domain dispersion (or smearing of the impulse response). To simplify the implementation of the linear phase crossover filter, and therefore simplify the end users interface to this complex system, the length of linear phase filters is quantized to a power of two, resulting in a quantized group delay (0.625 ms, 1.25 ms, 2.5 ms, 5 ms and 10 ms). Figure 2.2 illustrates the group delay quantization, showing that a Linkwitz-Riley filters worst case group delay varies smoothly as a function of the crossover cutoff frequency, whilst the linear phase crossover group delay is quantized in steps. While figure 2.2 shows group delay versus cutoff frequency, the third variable in the equation is filter slope. In general, we can say: (1) Equation 1 introduces several relationships:
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For a given slope, group delay varies inversely with cutoff frequency (as shown in figure 2.2) For a given group delay, the achievable transition band steepness varies proportionally with cutoff frequency For a given group delay, there is a minimum cutoff frequency, below which the slope of the transition band degrades to the point where the crossover filter fails to achieve acceptable performance
allowable group delay for their particular application. This user selectable delay parameter is applied across all bands of a multi-way crossover, and for this reason we choose the term Alignment Delay. 3.1. Brickwall Filter Realization
Figure 2.3 illustrates this tradeoff between the three parameters. As cutoff frequency lowers, the allowable group delay must increase to maintain maximum slope. 3. CONTROL PARAMETERS
Once the Alignment Delay is selected by the user, the choice of allowable linear phase filters is limited to fit within that constraint. In the example shown in figure 3.1, the chosen alignment delay is 2.5 milliseconds, resulting in a crossover network that achieves a steep transition slope for filter cutoff frequencies above 2 kHz. Cutoff frequencies below 2 kHz will degrade in steepness as they approach the lowest allowable cutoff frequency of 250 Hz, at which point the crossover has degraded to a steepness of approximately 24 dB per octave. Once the user chooses a particular alignment delay, the user interface displays the frequency response of the linear phase crossover, allowing the user to observe the variation in steepness as cutoff frequency is varied. This is illustrated in figure 3.2. 3.2. Linkwitz-Riley Emulation
The challenge of a well implemented user interface is to manage this complex signal processing system in a way to allow a user to achieve maximum performance within the constraints of their system. For example, a crossover filter used in a stage monitor is generally required to maintain a short group delay, whereas a large-scale front-of-house system does not require such a short group delay, thus allowing more group delay for increased crossover filter performance. In such a large-scale system, it is not uncommon to require a bulk delay of 20 milliseconds or more in order to align the loudspeaker system to the acoustic origin of the group of musicians being reinforced. The acoustic origin is typically taken to be the drummer or back line in a traditional popular music stage configuration. The user interface provides two primary mechanisms for control of linear phase crossovers: Brickwall based on the maximum allowable group delay, the crossover is implemented with maximum achievable transition band slope. Linkwitz-Riley Emulation based on a user specified choice of 24 or 48 dB per octave transition band slope, the crossover is implemented to achieve a magnitude response that emulates a Linkwitz-Riley crossover. This implementation is also constrained by the maximum group delay parameter.
In practical application, it is often desirable to upgrade an existing crossover implementation with the improved linear phase implementation, without having to reoptimize the rest of the signal processing elements in the crossover system. In this case, only the phase response of the system is changed, and the magnitude response is kept the same. Based on this requirement, the user interface allows the end user to choose between 24 dB and 48 dB LinkwitzRiley magnitude responses, to emulate their existing crossover systems. As shown in figure 3.3, once the choice of emulation slope has been chosen, the transition slope no longer changes as a function of cutoff frequency. It should also be noted that the 24 and 48 dB per octave transition slopes are chosen for convenience, and are not fixed. The implementation of the linear phase filter does not preclude the option of providing a continuously variable transition slope parameter. We have described a methodology that enables the end user to choose amongst three degrees of freedom through easily accessible parametric controls. These
In both of these crossover design methods, the approach we take is to allow the user to specify the maximum
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controls allow for the specification of transition slope, Alignment Delay, and cutoff frequency. Figure 3.3 illustrates these choices diagrammatically. 4. USER INTERFACE PARAMETERS AND CONTROL
5.
APPLICATION UTILIZING A DISSIMILAR MULTIWAY SPEAKER ARRAY
The linear phase crossover user interface provides access to all of the parameters described above. Real time adjustment of the parameters enables quick optimization. As shown in figure 4.1, all parameters are directly accessible from a single display: Alignment Delay Parameter Selection allows the end user to choose an acceptable bulk delay for the specific application Crossover Cutoff Frequency Parameter adjusts the crossover cutoff frequency in real time between output channel pairs Magnitude Displays each output channels magnitude response is displayed, calculated directly from the underlying crossover functions Crossover Filter Information provides details of the crossover channel pair, including cutoff frequency, transition band slope and crossover filter type Additional Filter Tools provides equalization tools for each crossover output, including low and high shelf functions, parametric sections and firstand second-order allpass functions Selected Crossover Highlighted the additional filter tools apply to the currently selected crossover channel
In order to illustrate the benefits of the linear phase crossovers and the parameters introduced, a practical example was devised and implemented. This practical application consisted of arraying two dissimilar loudspeakers in a common array configuration. 5.1. Array Configuration
Two loudspeakers were chosen from two different professional manufacturers, with each loudspeaker consisting of dissimilar loudspeaker components. A larger three-way device was selected, consisting of two low frequency transducers, a midrange cone transducer mounted with a phase plug, and a high frequency compression driver mounted in a horn with 35 degree beamwidth in both the horizontal and vertical planes. The second loudspeaker selected was a smaller device, consisting of a single low frequency transducer and a high frequency compression driver mounted in an asymmetrical horn that provided a 50 degree beamwidth in the horizontal plane, the plane of interest in our example. Both loudspeakers were arrayed in a typical sidefill configuration, where one loudspeaker is intended to cover a larger, more distant area, and the smaller loudspeaker is utilized to cover a smaller, closer area. In such a configuration, the intended goal of the sound system designer is to array these loudspeakers such that there is consistent coverage throughout the operating frequency range, whilst minimizing the destructive interference introduced by arraying multiple loudspeaker elements. 5.2. Beamwidth-Based Array Design Issues
The example crossover in figure 4.1 shows a linear phase crossover channel pair providing a 48 dB per octave Linkwitz-Riley emulation, and another crossover channel pair providing a Brickwall filter type. With an Alignment Delay of 10 milliseconds, and a cutoff frequency of 1.4 kHz, the Brickwall filter provides an 87.69 dB per octave transition band slope.
In order to minimize destructive interference, loudspeakers are typically oriented to provide overlap at the 6 dB down-points of their nominal polar coverage. The nominal 6 dB down-point is utilized to define the beamwidth of the loudspeaker system. Beamwidth in this application is defined to be a numerical average of the 6 dB down-points over the frequency range of interest, where the frequency response of the loudspeaker is relatively uniform compared to this average. Nominal beamwidths are typically reported in manufacturers specifications utilizing the numerical average of third-octave band SPL measurements obtained in a controlled (preferably anechoic)
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environment. By arraying two loudspeakers so that their beamwidths overlap at the intended listening position, the sound system designer can minimize destructive interference while (hopefully) retaining consistent coverage in the listening area. The common practice of arraying multiple loudspeakers based upon each loudspeaker manufacturers nominal beamwidth specification is an oversimplification of a much more complex situation. Loudspeakers do not maintain a consistent beamwidth across all frequencies of reproduction. Due to the geometric displacement between loudspeakers in an array, both constructive and destructive interference occurs in the combined acoustic response. Additionally, beamwidth-based system design ignores the complex frequency response, e.g. magnitude and phase response, of the loudspeakers in the array. Differences in phase response between loudspeakers placed in an array will cause both destructive and constructive interference in the overlap region (listening area), exactly where the system designer is intending to obtain the most consistent coverage. 5.3. Measurement and Comparison
and a linear phase implementation. The classical implementations use a traditional Linkwitz-Riley crossover design. The linear phase crossover design was created through the use of the aforementioned user interface control. The frequency response specification for the Linkwitz-Riley and linear phase designs was matched to provide the same magnitude-frequency response. The matched channel responses included gain, delay and parametric equalization filters, all of which were matched between the two different crossover implementations. 5.4. User Interface Controls
In order to improve upon simple beamwidth-based design, in-situ measurement becomes a necessity. Measuring and optimizing the complex frequency response of a loudspeaker array in the intended acoustic environment will improve the consistency and intelligibility of the sound system. In measuring a sidefill array configuration, a good place to start is to place a measurement microphone in the listening position where the two loudspeakers coverage patterns overlap (assuming of course that the two loudspeakers are well-behaved on-axis). Figure 5.1 presents a schematic view of this measurement and array configuration, and figure 5.2 displays the practical system setup (as seen from the measurement microphones perspective) utilized for the measurements performed. Frequency response and impulse response measurements were obtained at the overlap position for the two-way, the three-way and the acoustic combination of both loudspeaker systems. Two pairs of crossover configurations were utilized for this example: a classical crossover implementation
The use of the linear phase user interface controls allowed for a rapid creation of the new crossover implementation for this experiment. Alignment Delay and filter slope were appropriately chosen for this application, and the crossover frequencies were adjusted to correspond to the center frequencies of the classical crossover specification. With less than one hour of preparation and analysis, the linear phase crossover network was matched to the Linkwitz-Riley magnitude response specification. Previous implementations of linear phase crossover filter banks required intensive offline computations, which resulted in signal processing networks that did not allow for real-time variation. This relatively trivial matching exercise provides some notable results in both the frequency response and polar response of the sidefill configuration loudspeaker array. Figures 5.3 and 5.4 show the measured complex transfer function magnitude and phase responses of the classical and linear phase implementations for the two-way loudspeaker. Figures 5.5 and 5.6 show the classical and linear phase implementations for the three-way loudspeaker. In these figures, coherence is overlaid across the top of the magnitude plot. Note the similarity in magnitude response between the classical and linear phase implementations, and the differences in phase response of each implementation. All of these measurements were obtained at the same microphone position, at the point of overlap between the two loudspeakers beamwidth responses, which is why the magnitude responses are not as flat as would be expected in an on-axis measurement.
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5.5.
Loudspeaker Response Comparison
As these measurement-sets show, there is a significant difference between the phase responses of the classical and linear phase implementations. Although it has been argued that moderate phase shift through the frequency range of reproduction is not discernable subjectively [5], impulse response time data comparing classical and linear phase crossover implementations reveals the objective ramifications of the phase shift introduced by the classical crossover implementation. Figure 5.7 presents the impulse response time data for the classical crossover implementation for the two-way loudspeaker. Figure 5.8 presents the impulse response time data for the linear phase crossover implementation for the twoway loudspeaker. Figure 5.9 presents the impulse response time data for the classical crossover implementation for the three-way loudspeaker. Figure 5.10 presents the impulse response time data for the linear phase crossover implementation for the threeway loudspeaker. From these figures, it can be readily seen that the classical crossover implementations exhibit a larger distribution of energy across the time axis. It is also reasonably clear to see that the linear phase crossover implementations provide a smaller distribution of energy across the time axis, and that the resulting impulse response is more unidirectional. When the two types of crossovers (classical and linear phase) are implemented for each loudspeaker and acoustically summed in the sidefill configuration, contrasting results are obtained. 5.6. Classical Crossover Acoustic Combination Results
Figure 5.12 presents the impulse response time data for the classical two-way, the classical three-way and the combined classical response, and figure 5.13 shows the total averaged energy in each impulse response dataset. The total averaged energy statistic was obtained by numerically averaging the energy accumulated in thirdoctave bands from 100 Hz to 16 kHz in each impulse response dataset. This statistic shows that the combined classical response is similar in energy level to both the two-way and three-way loudspeakers, so there has not been a net-positive result throughout the frequency range of interest. Ideally we would obtain + 6 dB summation between adjacent speakers when measured in the overlap region. 5.7. Linear Phase Crossover Acoustic Combination Results
Figure 5.14 shows the linear phase two-way, linear phase three-way and combined linear phase magnitude and phase responses. Throughout the majority of the combined linear phase frequency response, constructive interference has resulted in an increase of energy and maintenance of coherence. Figure 5.15 presents the impulse response time data for the linear phase two-way, the linear phase three-way and the combined linear phase response, and figure 5.16 shows the total averaged energy in each impulse response dataset. Again the total averaged energy statistic was obtained by numerically averaging the energy accumulated in third-octave bands from 100 Hz to 16 kHz in each impulse response dataset. This statistic shows that the combined linear phase response is a significant improvement in energy level, and that there has been a net-positive result throughout the frequency range of interest. Ideally we would obtain + 6 dB summation between adjacent speakers when measured in the overlap region, and in the case of the linear phase implementation, we have come very close to this ideal. 6. CONCLUSION
In the case of the classical crossover implementations, interaction resulting in the occurrence of mostly destructive interference occurs. Figure 5.11 shows the classical two-way, classical three-way and combined classical magnitude and phase responses. Throughout the majority of the combined classical frequency response, destructive interference has resulted in a decrease of energy and coherence.
Linear phase crossovers are complex and difficult to design, presenting implementation difficulties for the end user. Through the definition of parametric controls and the implementation of a user interface that provides realtime interaction, the benefits of linear phase crossovers are now more accessible to the sound system designer.
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Brickwall and Linkwitz-Riley Emulation crossover filter design functions provide further enhancements to address real-world implementation issues. Additionally, we have demonstrated a practical implementation that exhibits an improved response in a real-world situation. The use of linear phase crossovers has been shown to improve energy levels and coherence as compared to traditional crossover implementations. 7. ACKNOWLEDGEMENTS
The authors would like to thank Scott Willsallen of Auditoria Pty. Ltd., Chris Kennedy of Norwest Productions Pty. Ltd., Morset Sound Development and David Cooper. 8. REFERENCES
[1] G. Berchin, Perfect Reconstruction Digital Crossover Exhibiting Optimum Time Domain Transient Response in All Bands, presented at 107th AES Convention, paper 5010, New York, USA (1999 September). [2] U. Zolzer & N. Fliege, Logarithmic Spaced Analysis Filter Bank for Multiple Loudspeaker Channels, presented at 93rd AES Convention, paper 3453, San Francisco, USA (1992 October). [3] S. Azizi, H. Hetzel & H. Schopp, Design and Implementation of Linear Phase Cross Over Filters using the FFT, presented at 98th AES Convention, paper 3991, Paris, France (1995 February). [4] S. P. Lipshitz & J. Vanderkooy, A Family of Linear-Phase Crossover Networks of High Slope Derived by Time Delay, J. Audio Eng. Soc., vol. 31, pp. 2-20 (1983 Jan/Feb). [5] S. P. Lipshitz, M. Pocock & J. Vanderkooy, On the Audibility of Midrange Phase Distortion in Audio Systems, J. Audio Eng. Soc., vol. 30, pp. 580-595 (1982 September).
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Linear Phase Linkwitz-Riley
Figure 2.1 Group delay response (as a function of frequency) of a 48 dB/Oct Linkwitz-Riley low pass filter compared to a 48 dB/Oct linear phase low pass filter. Both filters have a cutoff frequency of 510 Hz.
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Linear Phase
Linkwitz-Riley
Figure 2.2 Maximum group delay response of 48 dB/Oct Linkwitz-Riley low pass filter compared to the group delay of linear phase filter implementation
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Figure 2.3 As cutoff frequency lowers, the allowable group delay increases to maintain maximum slope
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Figure 3.1 User selection of the Alignment Delay parameter limits the available choice of linear phase crossover filters, which will also create a lower limit for the cutoff frequency of the crossover filter
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Figure 3.2 Illustration of filter steepness as a function of Alignment Delay and cutoff frequency
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Figure 3.3 Three degrees of freedom: transition slope, Alignment Delay, and cutoff frequency
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Figure 4.1 Linear phase crossover user interface components
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Figure 5.1 Schematic representation of measurement configuration
Figure 5.2 Practical system setup, from the perspective of the measurement microphone
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COHERENCE
MAGNITUDE
Figure 5.3 Magnitude and phase response of classical crossover two-way loudspeaker
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COHERENCE
MAGNITUDE
Figure 5.4 - Magnitude and phase response of linear phase crossover two-way loudspeaker
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COHERENCE
MAGNITUDE
Figure 5.5 Magnitude and phase response of classical crossover three-way loudspeaker
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COHERENCE
MAGNITUDE
Figure 5.6 Magnitude and phase response of linear phase crossover three-way loudspeaker
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Figure 5.7 Impulse response of classical crossover two-way loudspeaker
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Figure 5.8 Impulse response of linear phase crossover two-way loudspeaker
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Figure 5.9 Impulse response of classical crossover three-way loudspeaker
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Figure 5.10 Impulse response of linear phase crossover three-way loudspeaker
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COHERENCE
MAGNITUDE
Figure 5.11 Magnitude and phase response of classical two-way, classical three-way and acoustic combination of both loudspeakers. Destructive interference results in a decrease of energy and coherence.
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Figure 5.12 Impulse responses of classical two-way, classical three-way and combined systems
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Figure 5.13 Total averaged energy of classical two-way, classical three-way and combined systems
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COHERENCE
MAGNITUDE
Figure 5.14 Magnitude and phase response of linear phase two-way, linear phase three-way and acoustic combination of both loudspeakers. Constructive interference results in an increase of energy and coherence.
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Figure 5.15 Impulse responses of linear phase two-way, linear phase three-way and combined systems
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Figure 5.16 Total averaged energy of linear phase two-way, linear phase three-way and combined systems
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