Time Dilation, Part 2:
Page 3
I had hoped that figs.8 and 9 would evince further changes due to the addition of stand resonances, but in the event there is no more than a hint of additional resonance activity. (Perhaps I should have chosen a less rigid and well-damped stand as an example.) Even with a 42ms time window, the frequency resolution in these measurements is no better than 24Hz, which may not be sufficient to clarify all the speaker/stand resonant behavior at lower frequencies. This is confirmed by fig.12, which shows the result of processing the simulated impulse response used in Part 1 with a 42ms time window. Clearly, the lower four of the six simulated cabinet resonances are not well resolved.

Fig.12 The simulated impulse response from Part 1 analyzed using a 42ms time window, as achieved in the sports hall. Even this isn't enough to clearly resolve the lower four resonances.

If this is the situation even in such a large measurement space, the outcome is mildly depressing. There is, for instance, no prospect of performing insightful measurements on the behavior of subwoofer cabinets, where the frequencies of concern are even lower.

Deliverance?
As I mentioned in passing toward the close of Part 1, a method of improving the accuracy of bass measurements within small measurement windows was suggested over two decades ago by KEF. The underlying principle is simple: If the impulse response of the speaker has not decayed to a sufficiently small value by the end of the measurement window, then truncation of the impulse response will cause the bass response to be inaccurately recorded. To avoid this, the speaker's impulse response must be shortened such that it fits within the available time window. This can be achieved using shelf filtering, the effect of which is to confer on the speaker a higher corner (–3dB) frequency and hence a faster-decaying impulse response.

The KEF method is illustrated in detail in fig.13. Complementary filter sections are used, the second undoing the effect of the first so as to restore the correct frequency response. These are placed either side of the time-windowing process so that truncation effects are minimized, in this case with the first shelf filter used to modify the test signal sent to the loudspeaker. The downside of this arrangement is that it reduces the LF energy radiated by the loudspeaker, and hence the signal/noise ratio at low frequencies. This may not have been a problem in KEF's purpose-built reverberant measurement chamber, but it certainly can be in rooms that are pressed into service as measurement spaces, as a result of traffic noise, air-conditioning hum, and anything else that contributes to the raised levels of low-frequency environmental noise typically encountered.

Fig.13 Block diagram of the method proposed by KEF in the 1980s for improving the accuracy of bass frequency-response measurements obtained using short time windows.

When Part 1 of this article was published, Eric Benjamin of Dolby Labs contacted me to bring to my attention a preliminary paper he had presented at last October's AES Convention that investigates this method more closely and suggests the superior test arrangement depicted in fig.14 (footnote 1). Here the test signal remains unmolested, so LF signal/noise ratio is not compromised. All the filtering is now applied downstream of the measurement microphone, immediately to either side of the time windowing. By performing this filtering digitally using floating-point arithmetic, the measurement resolution remains that of the A/D converter used to capture the impulse response (in the case of MLSSA, a 12-bit auto-ranging device).

Fig.14 Building on the KEF idea, this improved filtering scheme was recently suggested by Eric Benjamin of Dolby Labs.

To see how effective this ploy is, let's take a simple example of a closed-box loudspeaker with a system resonant frequency of 50Hz and a Q of 0.707. This alignment gives a frequency response the equivalent of a second-order Butterworth (maximally flat) high-pass filter with a corner (–3dB) frequency of 50Hz and an ultimate rolloff of 12dB/octave. The impulse response of this filter is shown in fig.15a and, zoomed on the amplitude axis, fig.15b. Let's assume that the speaker is measured in my listening room, where the maximum available time window is 6ms. Clearly, the impulse response has not decayed sufficiently by the end of this period, so we should expect the measured frequency response to be in error. Fig.16 shows just how badly wrong the result is by comparing the response obtained using the 6ms window with that from a longer 200ms window.

Fig.15 Simulated impulse response of a closed-box speaker with a corner frequency of 50Hz (Q=0.707), complete in (a, top) and zoomed in amplitude in (b, bottom graph).

Fig.16 Frequency response obtained by time-windowing the impulse response of fig.15 to 6ms (blue trace), with the correct response for comparison (red).

Now let's use the filtering technique of fig.14 to improve matters, using ahead of the windowing process a second-order minimum-phase LF shelf-down filter with corner frequencies of 50Hz (to match the speaker rolloff) and 300Hz. This has the effect of converting the impulse response from that of a closed-box speaker with a corner frequency of 50Hz to one with a corner frequency of 300Hz, thereby shortening the impulse response by a factor of 6. If we truncate this new impulse response at 6ms as before and then apply inverse filtering before calculating the frequency response, the result is as shown in fig.17—the response we obtain is now much closer to the correct one, with an error of less than 0.6dB at 20Hz.

Fig.17 Same as fig.16, but with complementary shelf filtering applied before and after the time windowing, prior to FFT analysis. Now the frequency response, measured using the same 6ms time window, is accurate to within 0.6dB at 20Hz.

This finding is no more than a confirmation of those already described in the KEF and Dolby papers, but let's not underplay its importance: it shows that, with the application of appropriate filtering, accurate measurements of speaker LF response can be achieved in quite small rooms without resorting to the uncertainties of performing a nearfield measurement and attempting to knit this together with the freefield curve. Anyone who measures loudspeakers will like the sound of that prospect, although there is the not-inconsiderable practical problem that pre- and post-filtering of this type are not provided within MLSSA or, so far as I'm aware, any other measuring system. So if you can't write your own software to implement it, you're stymied. Moreover, to achieve the best results, the filtering must be designed correctly, with reference to the particular speaker under test—so the procedure is not a simple one.

It would be welcome, to put it mildly, if this same approach could also help out in resolving cabinet resonances. Sadly, it can't. Whereas the filtering method applied to bass rolloff genuinely shortens the impulse response by increasing its rate of decay, this method can reduce the effects of cabinet resonances only in amplitude: the decay of those components of the impulse response associated with the resonances remains exactly the same. We could genuinely curtail them by applying appropriate notch filtering and its inverse on either side of the time windowing, but to do this we would need prior knowledge of the resonant frequencies and their Qs—which is what we're trying to measure! Values might be guesstimated in various ways (for instance, from finite-element analysis of the enclosure), but this will never be a practicable option in review circumstances.

Musings
Where does all this leave us? The first item of good news is that, using the filter method if necessary, accurate measurement of speaker bass response can be obtained without resorting to the nearfield technique and its uncertainties. Second, by using a combination of a larger measurement space and the ground-plane or multiplane setups, it is feasible to dilate the time window sufficiently to obtain significantly better insight into speaker-cabinet and -stand resonance behavior than is usually the case, without the need to contrive a means of raising speakers high off the floor.

But the undeniable fact remains that, even with the 42ms time window I've been able to achieve, the resolution of this resonant behavior is still short of ideal. The only solution, so far as I can see, is to extend the time window still further by braving the elements and performing windowed ground-plane measurements in wide, flat, quiet spaces outdoors. This won't be music to the ears of JA, who lives in New York City, but I'm a country bumpkin nowadays, domiciled in the county of Kent on—it so happens—the 25,000 acres of Romney Marsh, the flattest area of land in southeast England. There are large planar fields all about here, and a very tempting small airport with lots of flat asphalt only a mile down the road. What isn't generally available, as readers who have visited this neck of the woods will well know, is the kind of benign meteorology that encourages performing speaker measurements alfresco. But if that's what it takes . . .

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Footnote 1: Eric Benjamin, "Extending Quasi-Anechoic Electroacoustic Measurements to Low Frequencies," Audio Engineering Society 117th Convention, San Francisco, November 2004.