Multichannel Sound Systems and Their Interaction with the Room (1998)

The ideal number and placement of drivers in small listening rooms has been controversial. Most research has examined only the pressure (loudness) as a function of frequency and source-listener positions. We believe that two additional properties of the soundfield, externalization and envelopment, are important to listener preference. Externalization is solely a low frequency problem, and the interaction between envelopment and loudspeaker position is particularly important at low frequencies. We proposed in a previous paper a mathematical method of quantifying these two perceptual properties given a measured or calculated binaural impulse response. The Average Interaural Time Difference (AITD) is the measure for externalization, and the Diffuse Field Transfer function (DFT) is the measure for envelopment. This paper will present a simple image model for rectangular rooms, and use it to study the interaction between multiple drivers and listening rooms. The image model is used to predict the values of pressure, AITD, and DFT for different combinations of room properties and driver locations. It is found that the low frequency pressure uniformity and the AITD can be increased in the prime listening area by using multiple low frequency drivers – especially at the sides of the listeners. When playing material where the bass energy is primarily monaural, the drivers on the left side of the room should lead or lag the drivers on the right side by a constant phase angle of 90 degrees. With musical signals we find DFT is also increased by multiple woofers. Once again a side placement is optimal. 1. IMAGE MODEL FOR SMALL RECTANGULAR ROOMS The work of developing measures is greatly aided by an efficient method of checking how they work in typical rooms. Unfortunately it is difficult to find a room which seems typical, and it is exceedingly tedious to measure a large number of binaural impulse responses for each new loudspeaker arrangement. We need an efficient computer model. Image models have the virtue of simplicity and computational speed. Unfortunately the image model assumes that the surfaces involved in the model behave as simple plane reflectors – or that the reflecting surfaces are large compared to the wavelength of the sound being reflected. This assumption is clearly violated when we study small rooms. However the principal error is due to diffraction at the boundary between two surfaces of different reflectivity. When all the surfaces of the room all have the identical reflectivity the model could give reliable results. In fact, we have tested the model against measurements in two rooms, and have found the model to predict the results surprisingly well. In spite of the absorbing ceiling in one of the rooms the model produces plausible results. Our conclusion – the image model may not be perfect, but for developing measures and concepts it is more than good enough! 2. A SIMPLE MODEL FOR THE HUMAN HEAD However we are looking for more than the sound pressure at a point in the room. We are looking for the interaural time difference for a human head placed at a given point. ITDs are influenced both by the distance between the ears of a listener and by sound diffraction around the head. Ideally we should sum delayed and attenuated head related transfer functions for each image. While not inconceivable, this procedure would be computationally expensive – and each HRTF would need to be quite long to accurately yield the ITD at low frequencies. Fortunately published HRTF data suggest that for frequencies below 125Hz the interaural delay can be accurately predicted if we model the head by two omnidirectional receivers, spaced by about 25cm. The model works fairly well above this frequency if the interaural spacing is reduced. This simple head model allows us to find the sound pressure at each ear by summing the pressure contribution from each image. We do not have to worry about the sound direction. This head model is an enormous simplification. It is valid only for low frequencies, but it makes our image model practical. 3. DETAILS OF THE ROOM/HEAD MODEL Our image model is written in MATLAB. The code is available from the author by email on request. The image model uses loops and conditionals, so the Matlab C compiler (with real number math) is highly recommended. 49 binaural receiver positions and two or four source positions can be evaluated in a few minutes using compiled versions of critical subroutines, something that takes hours without compilation. We use a recursive method to calculate the images that result from an arbitrary source position in a rectangular room. We first find the line of images formed by the side walls, out to the selected image order. We then reflect this line of images with the front and back walls to form a plane of images. This plane is then reflected with the floor and ceiling to produce a series of image planes. The strength of each image is found by multiplying the source strength by the reflectivity of each surface encountered. Once the images are found, the distance from each image to each receiver position is calculated. These distances are combined with the image strength to calculate the binaural impulse response for the particular source/receiver pair. To find the impulse response we use a sampling technique. The sampling process results in timing errors, which can be particularly important when studying ITDs at low frequencies. For the work on externalization we find a sample rate of 44100Hz gives good results, and allows the resulting impulse response to be convolved with recorded source material. The work on envelopment required an initial sample rate of 176400Hz to give consistent results. We found that splitting the pressure (not the energy) from each reflection linearly between adjacent samples gives the best results, and we sum pressures (not energy) from multiple reflections. The Fourrier transform of the impulse response gives the steady-state pressure response as a function of frequency. One can use the resulting response curve to estimate the number of images needed. As the number of images increases the length of the impulse response increases, as does the sharpness of the individual peaks and notches in the response. In practice one increases the order of the reflections until the pattern stabilizes. Ideally one would like to double the order between each test. Unfortunately the sharpness of the resonances increases approximately linearly with the reflection order, and the computation time increases as the cube of the order. There is a large payoff in using the minimum number of images. When the room surfaces have a reflectivity of 0.9 we need an order of at least 16 to approximate the room response. Fortunately we have not had to model a room which is that reflective. To model the rooms we have measured, a reflectivity of about 0.8 seems to work, and a reflection order of 11 seems sufficient. Our model calculates the contribution of each image to the total amplitude and phase at the receiver position. Although the method assumes that the surfaces have no net phase shift with each reflection, such a phase shift could be modeled over a narrow frequency range by simply altering the dimensions of the room. There have been studies that compare the pressure distribution measured in real rooms to results calculated with an image model. In general the accuracy of the model has been good. Although we have not made careful measurements of many different rooms, at least in two listening rooms the pressure distribution at several frequencies was checked with a sound level meter. The match to the predicted patterns from our image model was good. In another experiment the variation of pressure with frequency over a range of 30Hz to 100Hz was measured. The average absorption was then adjusted in the model to make the best match with the measured response. For the particular room – a 12’x15’x9’ listening room – the best match occurred with an average reflectivity of 0.8 for all the surfaces. Once this was chosen, the model and the measurement agreed within 2dB. 4. DFT IN AN ANECHOIC SPACE In the previous paper we presented a measure called the Diffuse Field Transfer function (DFT) for low frequency envelopment in small listening rooms. In the examples here we will plot results for the 63Hz octave band, although results at higher frequencies are likely to be equally interesting. We found in the section on calibration of the DFT that an optimum value would be approximately 0.24ms in the 63Hz octave band. This calibration depends on details of the bandwidth and time constants chosen, and has little real physical meaning. However the results we will show here will plot the DFT in dB relative to this value. As mentioned in the previous paper a single sound source in an anechoic space cannot produce envelopment, and the DFT value is consistently <-40dB. When there are two sound sources (stereo woofers) driven by decorrelated signals in an anechoic space the DFT can be significant. In fact, when we are directly between the two sources, we will get the calibrated value of 0dB. In conventional stereo listening the speakers are in front of us, at an angle of +-30 degrees. Figure 1 shows the DFT for conventional stereo in an anechoic space.