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.
In their 1983 JAES paper , 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.
McGrath et al
Parametric Control of Linear Phase Crossovers
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:
AES 119th Convention, New York, New York, 2005 October 710Page 2 of 25
McGrath et al
Parametric Control of Linear Phase Crossovers
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 cro