How does directivity, listening distance and absorption affect energy decay in a room?

The effect of loudspeaker directivity on steady-state frequency response is well understood. However, the time domain behavior (spectral change during a transient) seems to be a somewhat less studied subject and therefore less clear what happens during a transient.
My original intention was to understand how directivity affects reverberation in the time domain. Analysis of distance and absorption was a logical step to complete the view.

July 25, 2024

Although directivity does affect reverberation in a room, neither reverberation time (T30) nor early decay time (EDT, see below) clearly characterizes non-diffuse reverberation. Directivity has a huge impact on the early part of the energy decay, as reflected in EDT measurements, but T30 is quite insensitive to the directivity and location of the source. On the other hand, the contribution of late reflections in EDT measurements is very low. What happens in a room can only be understood by analyzing energy decay curves in frequency bands.

On the energy decay curve, T30 is twice the time it takes for the signal to drop from -5 dB to -35 dB. Early decay time (EDT) is defined as the time interval required for the sound energy level to decay 10 dB after the excitation has stopped. The result is multiplied by 6.

There are several methods for analyzing energy decay in an acoustical system: Cumulative Spectral Decay (CSD), SFT analysis (short FFT analysis window is shifted through the impulse response) and energy decay vs time in a selected frequency band. Each has its own pros and cons.

Energy decay curves provide the best insight into what happens in a frequency band over time. Downside is that frequency spectrum is not directly visible and analysis of resonances is not possible due to the poor frequency resolution. In order to analyze the spectral change with energy decay curves, we need to plot different frequencies on the same graph. Nevertheless, the effect of directivity, distance and absorption can be easily studied with energy decay curves.

How does directivity affect energy decay?

The following graphs shows the energy decay of a small full range speaker (6 cm driver) in a room at 1 kHz and 8 kHz. Measurement was taken from two meters.

We can see that except for the first 5 ms, the rate of the decay is the same (curves are parallel). We can also see that the result of high directivity is a very rapid attenuation at the beginning of the curve (the result is similar to going closer to the source, see below). In the impulse response this short "transition region" corresponds to the shift from direct sound response to the power response.

(It's clear that directivity has minor effect on T30, and huge effect on EDT.)

What makes this type of representation special is how spectral change and directivity index (DI) is encoded in the energy decay curves. The difference between the decay of a point source (curve at 1 kHz in the graph above) and the decay of the actual system (measured after the transition) corresponds to the sound power loss or normalized sound power. (The frequency response of late reflections is identical to the power response, provided that sound absorption is constant over the measurement range.)

An age-old question is whether loudspeaker directivity issues in small rooms can be corrected without introducing time-domain artifacts. The transition in the high directivity curve lasts only a few milliseconds, which means that in a small room the shift from the direct/anechoic response to the room response is much shorter than the ear's integration time (fusion interval, 30 ms for speech). Provided that the EQ filters have short impulse responses (low Q type filters), equalization doesn't hurt, it doesn't ruin the temporal integrity of the sound.

It is worth mentioning that neither the radiation pattern nor the sound field resulting from radiation pattern can be modified by equalization (free field, diffuse field and in-between). The response of the reverb tail, which can be heard with short percussive sounds, can't be corrected either.

How does distance affect energy decay?

The following analysis is based on impulse response measurements of a concert hall in Pori, Finland [1, 2]. The critical distance in the empty concert hall at 1 kHz is approx. 2.5 m (critical distance: level of direct sound equals with the level of reflections).

Distance doesn't affect the rate of energy decay, only the curves are shifted up or down. The behavior is similar to directivity.

Decay curves run close in the far field (audience positions). One curve (most distant point) is "shifted" to the right with 100 ms. Neither T30, nor EDT describes this behavior, though due to the ear's integration time it has little importance if the delay doesn't exceed 50 ms.

Overview graph for distance

Increasing directivity is NOT the same as increasing absorption in the room. Increasing directivity is like moving closer to the source.

How does absorption affect energy decay?

We are still analyzing the Pori concert hall impulse responses. At high frequencies air absorption is greater, resulting in a higher average absorption coefficient. When sound-absorbing panels are placed only in the first reflection points in a small room, the effect will be similar to increasing directivity.

We can see that sound absorption changes the rate of energy decay (this is evident since each wall bounce reduces the reflected energy by a certain amount).

Perhaps there is no other room acoustic measurement that is as closely related to sound absorption as reverberation time. Increasing slightly the overall sound absorption of the room can greatly reduce reverberation time. If the room has frequency-dependent absorption, the reverberation time will also vary with frequency. Frequency-dependent absorption may also affect the frequency response. The spectral change due to frequency-dependent absorption can be corrected with a graphic equalizer or an FFT filter in small rooms only (standard living rooms).

Overview graph (all parameters)

Reproduction of recorded reverberation

The perception of reverberation depends on the relationship between the decay rate of reverberation and the decay rates found in music or speech. For example, in a living room reverberation can be heard with short percussive sound, but not with slowly fading notes.

The same applies to recorded reverberation. In order to properly reproduce the recorded reverberation, the energy decay in the room has to be faster (room's decay curve has to be steeper) than the decay of the reverberation in the recording. Otherwise, reverberation in the recording will be masked by the room's reverberation, at least for sounds with decay rate larger than the decay of a room reverb (short, percussive sounds: snare drum, plucked muted string, 'k' ad 't' consonants in speech, etc).

Addition of room and hall reverb - this can be modeled by convolution (meaning of the red curve: concert hall reverb is reproduced in a small room on a low DI speaker)

Concert hall reverberation reproduced in a room inherits the long decay of the hall reverberation and the strong early reflections of the room. The result is that C₈₀ and C₅₀ corresponding to the miking positions in the hall are slightly reduced (in this example, C₈₀ is reduced by one decibel, C₅₀ is reduced by two decibels). This is not a bad thing for listening to music, however, early reflections can trick the brain, giving the impression that we are not in a large hall.

For the sake of comparison, decay rates for guitar and different playing techniques.

Energy decay for acoustic guitar

Energy decay curve from DI and reverberation time (linear approximation)

Since energy decay obeys relatively simple laws, we can draw the energy decay curve of an existing system from DI and reverberation time. Steps required:

Calculate the rate of decay and critical distance.
Calculate the energy decay of a point source for a particular distance. Offset for the energy decay curve can be calculated from the ratio of distance and critical distance (this can be a table, polynomial approximation).
Correct the curve with DI.

By rearranging the Sabine formula, we can estimate the rate of decay:

DR = ( A / V ) ⋅ 373 (dB⋅m/sec)

where

V = the room volume [m³]

A = the effective absorption area [m²]

DR = rate of decay [dB/sec] (note: T_60dB = 60 dB / DR)

In concert halls (T30=2.5sec) the rate of decay is ~25 dB/sec. In a small furnished living room (20 m²) the rate of decay is ~120 dB/sec.

For example, a loudspeaker with DI of 10 dB set up in a concert hall with a reverberation time of 2.5 sec and a distance is twice the critical distance (~5 m). From reverberation time the rate of decay is ~25 dB/sec. For the distance correction is not required. The curve has to be shifted down with 10 dB (DI). The transition time is roughly 50 ms.

Csaba Horváth

References:

[1] Concert Hall Impulse Responses - Pori, Finland: Reference, Juha Merimaa, Timo Peltonen, and Tapio Lokki, 2005, link to the paper

[2] Pori Promenadikeskus concert hall impulse response files at http://www.acoustics.hut.fi/projects/poririrs/