DSP Project Two (Updated 30/3/02)

This is actually a revision of project one. For the moment I will just use this page to write about my latest thoughts on digital crossover design.

Revisions - second time around

To improve on the results of the first project, I wanted to revise the following aspects:

• Use a contoured frequency response target rather than aiming for totally flat frequency response. Because our hearing has increased sensitivity for direct field sound in midrange and treble frequencies, I have drawn the opinion that loudspeakers should be design with slightly recessed frequency response in this region. This result is also shown in technical papers by J. Gerhard of Audio Physic, and also the BBC research labs.
• Use separate design processes to create the "minimum phase section" and "linear phase section" of the filter transfer function. I expect this will allow for more accurate filters. There is also the opportunity to use optimisation routines, but this is probably not necessary for this application if the filter is implemented as FIR.
• At this stage I recommend using FIR filter implementation for the both minimum-phase, and linear-phase filters. This is a lot simpler and easier than IIR design of the same order, but has a high cost in processing requirements. With the latest DSP processors or Personal Computer with IBM compatible ATX platform, processing power is not a major limitation for hobbyist application.

Why minimum phase filters?

It is often anticipated that digital crossover implementations should equalise the drive units not only in frequency response but also in the time-domain. Before taking such an approach it is obviously important to recognise what can be achieved with minimum-phase filters, delay-equalisation, and excess-phase equalisers.

In the design of passive crossover networks, it is often assumed that drive units have minimum-phase characteristic. The assumption means that the phase response can be derived from the measured SPL magnitude response using the Hilbert-Bode transform. The minmum-phase assumption works very well, and in my experience using SoundEasy crossover design software, the models developed using this concept very accurately predict the response of the finished loudspeaker.

The minimum-phase characteristic is important for crossover design because it means that there exists a minimum-phase equalisation filter that is a perfect inverse of the drivers transfer function. So providing the drive unit is perfectly minimum-phase, a minimum-phase filter can be used to provide perfect equalisation in both frequency domain and time domains. For digital crossover design, minimum-phase filters have the additional advantage that they offer the shortest possible impulse response to achieve the required SPL equalisaion. This means there is minimum spread of energy either side of the initial impulse. Most importantly, minimum phase filters are always stable and causal so do not exhibit any pre-ringing on transients.

Drive unit measurements are generally not perfectly minimum-phase, and there are a couple of aspects to this:

1. When more than one drive unit is used in a multi-way speaker, there will be differences in the offsets of acoustic centers between any two drive units. This will result in a phase shift that increases linearly with frequency.
2. There may be delayed resonances and reflections that result in a delayed component of the drivers response.

Delayed resonances and reflections are normally small enough in magnitude that they do not affect the phase response of the drive unit in the crossover region. However, the relative acoustic offsets generally have a significant effect on the phase response that cannot be ignored. This causes headaches for analogue crossover design, but when using digital crossovers precise knowledge of the relative acoustic offset is much less important because it is easy to add in digital delay after the filters have been designed.

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