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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

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 =

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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Aki Makivirta

Comment posted December 23, 2018 @ 18:20:52 UTC (Comment permalink)

Dear Dr. Magchielse, Richard C. Heyser in his two-part paper you refer to correctly describes the nature of Group Delay as having the potential for expressing negative and positive time, and continues to point out how this relates to the requirement of causality for real-life systems. He does not say that Group Delay could not be a valid description of the actual delay of a signal in a system as long as the causality-related constant delay component is included. If such component is excluded, as seems to frequently happen, then the possibly resulting negative time values can create confusion. We are interested in particular in the variation of the group delay as a function of frequency, as this indicates that the signal content in different frequencies does not propagate through the system with the same delay, and this leads to shifting of the relative phases in the system for different frequencies, and to changes in the waveforms. Group delay is indeed the correct parameter to optimize when we are looking for systems that maintain the waveforms. At the same time, we must also consider the fact that many real-life systems, including loudspeakers and phase-equalizing filters, are dispersive in their response to signals.

The delays of the two crossover branches Hxo1,LP and Hxo2,HP are the same. The timing difference evident in Figure 11 is not produced by this crossover. The timing difference is produced by the total delay experienced by the three signals travelling through the crossover network, and the fact that the tweeter output does not experience the additional delay generated in the second crossover Hxo2 (having two outputs). This delay has been inserted by Hsync but could also be inserted by placing a second copy of the Hxo2,HP highpass filter in the tweeter channel. Both methods adequately model the typical three-way loudspeaker temporal characteristics.

All linear time-invariant filters work by convolving a signal with an impulse response, either finite or infinite in length. Using FIR filters in our paper is not of essential significance in terms of discussion. Further, FIR filters only present pre-ringing if they are not minimum phase in character, so any pre-ringing is a consequence of the filter design, not filter topology. Having a finite impulse response length has no causal connection to existence of pre-ringing. The focus of our paper is to describe systems that can present (at least approximately) constant delay within the selected band of frequencies, at least within the subjectively essential frequency range. What we consider is the system comprising the loudspeaker and equalizer filter designed to improve the loudspeaker’s characteristics.

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