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Digital crossovers with steep rolloff slopes


The commonly agreed requirements for loudspeakers crossover filters are


  1. Flat magnitude of the combined outputs

  2. Sufficiently steep rolloff slopes between the output bands

  3. Acceptable polar response

  4. Acceptable phase response


Increasing the rolloff slopes has a very positive impact on distortion and imaging: each driver operates in its own frequency range and the overlap region is reduced. There are many resonances
from woofer or mid range drivers that could be avoided by using crossovers with higher cutoff rate (individual speakers measurements from ASR are very revealing on that aspect)


It is quite easy to design digital filters with very steep slopes, like brickwall ones. If they additionally have linear phase property, they answer the 4 above mentioned objectives. So why are
they not so widely spread ? Let’s look at some filters properties and their audible impact.


In most cases the drivers are physically separated and there is only one position in the vertical plane where the listener is equidistant to both drivers. Any distance difference between each
driver and the listener creates a time delay and modifies the summed output in the crossover region.

The impact of such delay depends on the crossover characteristics and linear phase filters with steep slopes are very sensitive to any time delay b/w bands. This is because linear phase filters
have pre & post ringing that perfectly cancel out when low and high sections are “aligned” but become apparent when any slight delay is added to one path. The higher the slope the higher and
longer the pre and post oscillations (infinitely long if we consider a rectangular shape frequency response)


Here is an example for a 2 way system with a brickwall filter at 2500 Hz. When the listening position is at the exact same distance to the 2 drivers, the low pass and high pass section sum-up to
a perfect dirac pulse:


If the listener moves 20 cm in vertical direction the pre & post responses of each section become apparent:

While post ringing is usually less critical as masked by the musical message or room reverberation, pre-ringing is much more impactful as it is not present naturally (any emitted sound follows
the causality principle and sound events have a sharp onset) Any significant level of pre-ringing has an impact on rhythm perception as it smears the transients.


As our perception of sound level follows a logarithmic scale it is relevant to plot the decibel version of the same temporal response:

We can see that the pre response is exceeding the hearing threshold derived from different studies (threshold that we use when generating our mixed phase room correction filters)


The others and most popular filters are minimum phase as they can mimic analog filters. They are free of any pre-response but the phase of the summed signal is not linear. Phase variations are
inaudible under certain conditions (when similarity b/w left & right speakers is secured) but there is an agreed limit above which it can be detrimental especially on pulse like signals. As
the phase variations are increasing with the slope steepness there is a maximum slope that is reachable with such types of filters.


The conclusion is that both linear phase and minimum phase crossovers have limitations when we seek steep slopes. So what can we do ?


The idea is to push those limitations by designing mixed phase filters having the steepest slope considering constraints on pre-ringing and phase response. Mixed phase filters are somehow
intermediate b/w linear and minimum phase filters. Knowing the design parameters (distance b/w speakers, listening window) the objective is to design filters with the highest possible slope under
the constraints of maximum pre-ringing and phase variation.


Using the same 2500 Hz use case, we can compare our design with the Linkwitz Riley 24dB/octave (most popular topology) and brickwall linear phase. 


Let’s start with the attenuation level comparison at equal distance from the speakers:

We see that the overlapping range above  -20dB is much higher (2875 Hz) for LR24 compared to the proposed design (192 Hz)


Keeping the same color code, we can see the impact on the off axis response with broadband cancellation for the LR24. As our hearing system is less sensitive to narrow band notches, this is quite
advantageous for the reverberated sound that will be much more similar than the direct one (similarity is a criteria for good imaging)


Finally the time response for off axis listening position shows the better pre-response control vs brickwall filter:

Contact us if you’d like to use this new crossover filters in your system !

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