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Modeling and Delay-Equalizing Loudspeaker Responses

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This paper focuses on the modeling of the linear properties of loudspeakers. The impulse response of a generalized multi-way loudspeaker is modeled and delay-equalized using digital filters. The dominant features of a loudspeaker are its low- and high-frequency roll-off characteristics and its behavior at the crossover points. The proposed loudspeaker model also characterizes the main effects of the mass-compliance resonant system. The impulse response, its logarithm and spectrogram, and the magnitude and group-delay responses are visualized and compared with those measured from a high-quality two-way loudspeaker. The model explains the typical local group-delay variations and magnitude-response deviations from a flat response in the passband. The group-delay equalization of a three-way loudspeaker is demonstrated with three different methods. Time-alignment of the tweeter and midrange elements using a bulk delay is shown to cause ripple in the magnitude response. The frequency-sampling method for the design of an FIR group-delay equalizer is detailed and is used to flatten the group delay of the speaker model in both the whole and limited audio range. The full-band equalization is shown to lead to preringing in the impulse response. In contrast, group-delay equalization at mid- and high-frequencies only reduces the length of the loudspeaker impulse response without introducing preringing.

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JAES Volume 66 Issue 11 pp. 922-934; November 2018
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Frank Schultz
Frank Schultz


Comment posted November 21, 2018 @ 22:47:42 UTC (Comment permalink)

Thanks for the interesting article! I would like to indicate a related Ph.D. work from 1999, unfortunately available only in German, chapter 4 covers similar problems:

Swen Müller (1999): "Digitale Signalverarbeitung für Lautsprecher", doctoral thesis, RWTH Aachen, URL = http://sylvester.bth.rwth-aachen.de/dissertationen/1999/2/99_2.pdf

Best regards
Frank Schultz


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Tom Magchielse


Comment posted December 10, 2018 @ 16:38:38 UTC (Comment permalink)

In an important series of AES papers the late Richard C. Heyser ( Determination of Loudspeaker Signal Arrival Time, november /december 1971) argued that the concept of Group Delay is not a valid description of the actual delay of a signal through a network. Group delay is a measure of the delay of a narrow-band signal that passes through a medium with a non-linear phase characteristic. The term was originally intrduced by Lord Rayleigh in "Theory of Sound", pp 301-302. Heyser shows that even a simple minimum-phase network can have a negative group delay. If group delay was the true measure for the signal delay in the network, that would mean the network was non-causal. Incidentally, one can obtain a similar result by analysing the well-known equivalent network of a bass-reflex system. In the case of non-zero losses in the port, the analysis will show a negative group delay at very low frequencies. So perhaps group delay is not a very suitable parameter for system optimization?

The authors show a three-way speaker model (Fig 10) where a Linkwitz-Riley cross-over filter is used . The inputs for the higher cross-over part are not identical, the input for the highpass section being Hxo1LP and the input for the lowpass Hxo1HP. The unavoidable difference in phase at these two different inputs will compromise the performance of the Linkwitz-Riley filter as evidenced in the author's Fig 11. The introduction of Hsync does not fully alleviate this problem, as it is still connected to the wrong input..

It is not clear why the authors prefer to use FIR filters as cross-overs, especially as these seem to bring considerable problems with pre-ringing. One reason could be that the FIR filters could also be used to compensate for the irregularities in the response of real loudspeakers? It must be remembered however, that loudspeakers tend to be minimum-phase devices over most of their frequency range, so that a non-minimum-phase compensating filter might easily introduce even more problems with the transient response


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