The underlying problem is that we have only low frequency information output from the subwoofer.
From the equation:
Δt =1/ Δf
where Δt is time resolution and Δf is frequency resolution, we can see that high frequency resolution (small value of Δf) will yield low time resolution (large values of Δt).
We need higher frequency output from the subwoofer (corresponding to higher Δf, lower frequency resolution) to increase the time resolution in order for us to know when to position the cluster.
If possible, we can bypass the low-pass filter on the subwoofer to get more high frequency content in the output signal.
This may help in more precisely determining the arrival time of energy from the subwoofer.
Let’s assume that we can’t do this or if we can that it still doesn’t give us sufficient time resolution.
What we need is a way to get precise time information without high frequency content. This is a seemingly impossible task, but only so in the time domain. In the frequency domain there is a metric available that yields quite precise relative timing information.
This is the group delay. The group delay is defined mathematically as the negative rate of change of the phase response with respect to frequency.
τg = −dφ / dω
Figure 3 and Figure 4 show different views (domains) of the same measurement for the individual pass bands. If we look at the group delay of this same data in Figure 7, we can derive some valuable information.
Figure 7 – Group delay of the cluster (red) and subwoofer (blue) with crossover filters in place. (click to enlarge)
The high frequency limit (plateau) of each curve indicates the arrival time of the signal from that device.
From this we see that the cluster arrival time is approximately 3.3 ms. This correlates very well with the ETC in Figure 4.
Don’t let the appearance of the subwoofer curve in the high frequency region be bothersome. This is due to the low signal-to-noise ratio of the measurement above 400 Hz. Referring to Figure 3 the output of the subwoofer is less than -24 dB at 200 Hz.
Our use of a fourth order filter would indicate a level of less than -48 dB at 400 Hz and decreasing rapidly. It’s no wonder there is a SNR problem at higher frequencies.
We can look at the subwoofer curve around 300 Hz to get an indication of the high frequency limit of its group delay. This turns out to be approximately 11.0 ms. The group delay of the cluster at this frequency is approximately 3.9 ms.
This is a bit different than the 3.3 ms at higher frequencies, and is caused by the phase shift of the high-pass filter and the natural high-pass response of the device. The low-pass filter being used on the subwoofer will have similar phase shift, and consequently, similar group delay differences in the high frequency region if our measurement SNR was good enough to see it.
Taking the difference in 11.0 ms and 3.9 ms we now have a value of 7.1 ms to use as our delay setting for the cluster. This yields the summation, along with the individual pass bands, shown in Figure 8 and Figure 9. This is almost the exact response we desire.
Figure 8 – Magnitude response of individual pass bands and the summed response with the cluster delayed 7.1 ms. (click to enlarge)
Figure 9 – ETC of individual pass bands and the summed response with the cluster delayed 7.1 ms. (click to enlarge)
There is less than 0.5 dB error in the vicinity of 150 Hz. This error is due to the output of the cluster and high pass filter not exactly matching the Linkwitz-Riley target (see Figure 2).
There is one more item that I think might be of interest in helping to see how a low-pass filter response affects apparent arrival time. I say apparent because it only appears that the arrival time is changing. Figure 10 and Figure 11 show the ETC and IR, respectively, of a 4th order Butterworth low-pass filter.
Figure 10 – ETC of low pass filter with different corner frequencies. (click to enlarge)
Figure 11 – IR of low pass filter with different corner frequencies. (click to enlarge)
The only difference in these curves is the corner frequency (-3 dB point) of the filter. The true arrival time for all of these filter curves is 5 ms. A complementary high pass filter with an arrival time of 5 ms will combine properly with its low pass counterpart in the graph.
If the high pass is delayed so as to place the arrival so that it occurs later than 5 ms there will be errors in the summation of the filters just as was illustrated in Figure 5 and Figure 6.
We have seen how the response of an electrical filter can combine with the response of a loudspeaker to yield the desired response (alignment) from the combined output. We have seen how the low pass behavior of a device may make it appear that its arrival time is later than it actually is.
We also demonstrated how to use group delay to determine the correct delay setting with relatively high precision when the high frequency output of a device is limited due to its low pass behavior.
I hope that some will find use for these techniques.
Charlie Hughes has been int he pro audio industry for over 20 years, having worked at Peavey and Altec Lansing. He currently heads up Excelsior Audio Design & Services, a consultation, design and measurement services company located near Charlotte, NC. He is also a member of the AES, ASA, CEA and NSCA, in addition to being an active member of several AES and CEA standards committees. width=b