Whitepaper: The Perfect XO for High End Stereo Systems?! definiteaudio GmbH Peter-Vischer-Str.2 D-91056 Erlangen Tel: 09131 758691 Fax: 09131 758691 e-mail: info@definiteaudio.de Web: http://www.definiteaudio.de Umsatzsteueridentnummer: DE254963094 HRB 11085 Fürth Seite1
Index Index...2 Looking for the perfect active XO...3 1.1 Requirements...3 1.2 Discussion...3 Consequences...4 Implementation...5 Seite2
Looking for the perfect active XO 1.1 Requirements Looking for the perfect active XO for High End audio speakers there are some basic requirements that have to be considered: 1. A perfect XO has good attenuation outside the pass band to protect the tweeters and suppress irregularities of the chassis outside their optimal operating range. 2. A perfect XO has no ringing. In the critical region (about 3000Hz) ringing is clearly audible. 3. A perfect XO is transient perfect. Most researchers argue today that transient imperfect XOs are only audible at low frequencies but there they definitely are audible. 4. A perfect XO has minimum delay. It should be usable for audio and video applications 5. A perfect XO has excellent polar response. This entire means the perfect XO seams to be impossible! At first view some of the requirements seem diametrically opposed to each other so fulfilling one requirement would violate the other. This means each real world XO always would be a tradeoff? 1.2 Discussion Well, let s find the perfect tradeoff that could be achieved for a High End Stereo playback system! First take a look at 3 and 4: Transient perfect XOs are in general realized with phase linear FIR-filters. This leads to an extensive delay compared to minimum phase filters. The lower the XO frequency the higher the delay in the phase linear FIR filter would be. This means a long linear phase XO could not be applicable for video playback because image and sound will become out of sync. It is really a tragedy: We might sacrifice linear phase at the higher XO points (linear phase is not that audible there) but we definitely need it at the bass XO and exactly here it leads to extensive delay. Today we know that transient perfect did not implicitly mean long filters! John Kreskovsky has shown that transient perfect filters could be realized with standard minimal phase filters using a Subtractive Delayed Approach (see: http://www.geocities.com/kreskovs/new_xo_rev1.zip). This approach leads to transient perfect XOs with a delay only as long as the group delay of the minimum phase filter from which the XO is derived. In real applications this delay time is lower than the time of one video frame which seems to be sufficient to keep image and sound in sync. Now let s face the points 1, 2 and 5: Tweeter protection is one reason why most new linear phase digital XOs often use very steep filters. The other is lobbing. In common speaker designs, the sound in the XO region is emitted by two chassis having a different acoustic center. This leads - in combination with the phase shift of a minimal phase filter - to an unwanted tilted main lobe. LR filters have quite good lobing behaviour. Therefore they are widely accepted in audiopro setups. Steep filters Seite3
(>200db/oct) dramatically reduce lobbing errors because the frequency region where two chassis are transmitting in parallel will be so small that it could be neglected. On the other hand steep filters produce a high amount of ringing. This is especially true with linear phase filters producing an audible preringing which is not masked by the subsequent signal and therefore audible. In theory the ringing of the two speakers working in the XO region would subtract to zero. This is true adding the two electrical signals at the output of the XO but no more after transmitting the signals in the air of our listening room. As we know steep filter slopes without preringing are impossible for linear phase filters so what is the tradeoff? Let s take a look at what we would like to achieve. Speaker protection especially tweeters is one point but this is much more important in PA systems than in High End Stereo systems. There are a lot of systems out in the market today which link the tweeter only by a capacitor. This means a filter of first order! The B&W 800D is such a candidate. They use a first order filter to get better transient response with their passive filters. Consequences This observation leads to the insight, that a second order filter should give enough tweeter protection for a High End Stereo system. Ringing with a second order slope is quite small but not coercive zero. On the other side it isn t a good idea to have a small second order slope to limit a bass or mid chassis towards higher frequencies. Bass and mid chassis often have audible break up resonances at higher frequencies so there should be quite a steep cut off outside the pass band. Browsing through the literature there is only ONE filter which is absolutely free of ringing. This is the Bessel filter with its very soft slope. Figure 1 shows a 40. order Bessel filter with fc=1000hz. Its high order guarantees a very good suppression of higher frequencies. At 2000Hz (one octave) the amplitude has fallen below -52db, quite a noticeable value for a flat Bessel filter! Sure, 40.order is quite much but easy to achieve using a FIR LP filter! Figure 1: Frequency response of Bessel LP 40. order Looking at the step response we see that there is no ringing at all and the transition is soft and smooth. Figure 2: Step response of Bessel LP 40. order Seite4
Now we can derive a symmetric HP filter by generating a dirac-signal and subtracting the impulse response of the Bessel filter from it. John Kreskovsky proved that all HP filters derived this way from a Bessel LP (regardless of which order) have a steepness of 2. order. The only remarkable thing about this derived XO is, that it didn t have its 6db XO point at 1000Hz but shifted a bit downwards to aprx. 700Hz. This is not really a drawback because the cut off frequency of the LP can easily be adjusted. Figure 3: Bessel LP and derived HP Deriving the HP from the LP by subtraction gives a perfect transient response by construction (blue line in Figure 4). Figure 4: Step response of LP (red), derived HP (green) and their sum (blue) With the XO shown, a lot of requirements to a perfect XO could be fulfilled. So we achieve a very good attenuation of the LP outside the passband. The HP gives an attenuation of 2. order outside the passband. This seems to be good enough for High End Stereo (not for PA!). There is absolutely no ringing and a transient perfect response could be achieved. Beside this Kreskovsky showed that this kind of XO delivers excellent polar response so we did not have to worry about the quite broad frequency range where both chassis are transmitting in parallel. The signals will be in phase resulting in a perfect addition of both wave fronts! Using the Bessel XO the only drawback remains is the 2. order attenuation of the HP outside its passband. The roll off rate is sacrificed for improved polar response but my guess is, that 2. order attenuation should give enough protection for tweeters in High End Stereo systems. This means the derived Bessel XO is really a candidate for the Perfect XO for High End Stereo Systems reward. Implementation Having learned about the theory of a Bessel derived XO the question arises how it could be realized with BruteFir? John Kreskovsky has published an EXCEL-Tool (http://www.pvconsultants.com/audio/tp/tpsdm.htm) to construct Transient Perfect Substractive Delay XOs. With this tool you could design Bessel LP filters up to 10. order and the derived HP filters. Figure 5 shows a 3-way XO design based on 10. order Seite5
Bessel LP filters. The group delay of this 10. order Bessel XO is only 8ms which means it is well suited for video applications. Figure 5: Tree way transient perfect XO based on 10. order Bessel LP Having constructed the XO you must save the data in a form suitable for BruteFir. In figure 6 you see how the attributes have to be selected to get a Rect Linear DSP File Type. Figure 6: Export settings Just select the sample frequency your BruteFir operates on and the number of samples your filter should have (here 96000kHz and 8192 Samples ). Export the data by clicking Save FRD response As a File Set. This generates a couple of Rect Linear.FRD files. These files are readable by any editor. They consist of the triples: frequency real- and imaginary part of the filter. Playing around with this EXCEL sheet I found out that there are sometimes undeterministic problems creating linear phase filters. For those filters the imaginary part (third value of the triples) should always be zero. Please check that to be sure the.frd files created were correct! The.FRD files did not consist of the filter coefficients needed for BruteFir. To get the coefficients I wrote a little tool (FFT) I will provide here (http://freerider.dyndns.org/anlage/mydrc/frdconv.zip). Just call frdconv giving it the name of the.frd file to convert as parameter. Based on this file frdconv creates a.dbl file (which is RAW double precision format) with the same name. It contains the coefficients as double precision values ready to be used by BruteFir. If you need the filters as WAV files you could use the SoX tool (http://sox.sourceforge.net), to convert from DBL to WAV. Loading the filters into BruteFir or any other engine will convolve the response of your speaker chassis with the response of the filters. This means what you will hear the Seite6
response of the filters polluted by the response of the chassis! This pollution will decrease the performance of the calculated filters dramatically. The more the transfer function differs from the ideal transfer function of the XOs the more the performance of the XO will be decreased. This pollution is quite natural and widely accepted by designers of conventional passive crossovers. To fight this pollution a speaker correction algorithm has to be run and convolved in an inverted manner with the XOs transfer function. References 1. John Kreskovsky, A New High Slope, Linear Phase Crossover Using the Subtractive Delayed Approach, Dec. 2002. Seite7