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.
This is the final part of my loudspeaker "trilogy".

July 25, 2024

Although directivity does affect reverberation in a room, neither T30 (Reverberation Time) nor EDT (Early Decay Time, 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. (Note: T30 is measured between -5 dB and -35 dB, EDT is measured between 0 dB and -10 dB.)

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 that the spectral change and directivity index (DI) is encoded in the energy decay curves. The difference between the decay of a point source and the decay of the actual system (measured after the transition) corresponds to the sound power loss or normalized sound power. (Why? Because the frequency spectrum of late reflections is the power response, if we ignore absorption.)

The transition in the high directivity curve lasts only a few milliseconds. This means that in domestic rooms DI issues can be corrected with traditional equalizers without introducing audible time-domain artifact. (The filters in the EQ must have short impulse response (low Q type filters).)

Some notes on directivity & equalization in small rooms. When loudspeakers with different directivity responses are equalized to the same steady state response, they will sound different because the generated sound field is different (different HRTF responses, different decay curves) and the spectrum of the reverb tail is also different. However, equalization doesn't ruin the impulse response, boosting the treble with a high directivity speaker won't make the attacks or transients "bright".

Note: When a pulse test signal is generated in a small room, we actually hear a short noise with exponential decay. The reflected pulses act as maskers, significantly reducing the audibility of group delay distortion compared to anechoic conditions.

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). Both EDT and T30 can describe absorption related effects with a low directivity source.

Since the spectral change due to frequency dependent absorption is a gradual process, correction with a graphic equalizer or standard FFT filter might introduce audible time-domain artifacts.

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 speech, 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 (impulsive 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)

For the sake of comparison, decay rates for guitar and different playing techniques. (Meaning of "stopped note": when playing a melody or solo, the notes must be stopped before the new note is started, otherwise the result is a chord-type sound.)

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/