Power Response (Listening Room)
Let’s begin with the important acoustic measurements we know of from a speaker in a listening room:
- Direct Sound, Early Reflections, Late Reflections: which constitute our reception of Sound Power in a room, which contributes to our perception of sound [Sound Reproduction: The Acoustics and Psychoacoustics of Loudspeakers and Rooms, Floyd E. Toole, Focal Press, 2008, pp.27-141].
We’ve seen the importance of the combination of direct sound and the ratio of reflections to our perception of mid and high frequency speakers in a car. What we discover is in a small room, such as a car, what we are receiving is Sound Power. We need a reliable way to measure sound power in a car, for any localized listening position.
This need was realized and solved by Earl Geddes and Henry Blind in “The Localized Sound Power Method”, J. Audio Eng. Soc., Vol. 34, Number 3pp. 167+ (1986).
Geddes began “ in order to … know exactly what the room’s frequency response is before it can be corrected, … single-point microphone measurements lack sufficient stability to be useful for sound system equalization. A new measurement procedure was developed which has proven to yield significantly better results. Using this technique one can obtain a one-third-octave measurement of the acoustic frequency response with a 90% confidence of having no greater than a 0.5-dB expected error. This level of accuracy is necessary for ‘unambiguous’ sound system equalization.”
Geddes continues, explaining the transition region we experienced with speakers in a room and why a single mic is not stable enough.
In 1954 a frequency point was defined using statistical room theory (the Schroeder frequency). Schroeder showed, as we experience, that the pressure response in a room is essentially a random variable.
The simple example to demonstrate this principle, and leads to the problem with a single mic measurement, considers a room being excited by a pure tone emitted by a source.
The room’s response to this tone will depend upon the summation of the complex pressures over all of the excited modes in the room. We can see this in a speaker in a room example. [Ibid., pp 141-159]
Each of these modes will contribute to the total sum with a different amplitude and phase. Above the Schroeder frequency (where the eigenmodes are sufficiently dense) there will usually be hundreds of these modes being excited by a pure tone.
With this many virtually random (in phase and amplitude) modes being added together, the net summation will have a Gaussian distribution in both the real and the imaginary parts (the central limit theorem). The net result will be an amplitude response to the pure tone excitation which must be dealt with as a random variable with a defined standard deviation.
Schroeder put it this way:
- The frequency response curve of a “large room” will lie 70 percent of the time in a strip 11 dB wide about the mean line. The only assumptions used were that the eigenfrequencies are “sufficiently” dense and that the microphone is placed sufficiently far from the sound source so as to be in the far field.
These are quite general assumptions – and they imply what we think we know: that the statistical properties of all rooms are virtually identical above the Schroeder frequency.
That is about 50 Hz in a typical auditorium and 150 to 200Hz in an automobile. The Schroeder frequency can be calculated from the equation:
F (Schroeder) = 2000 SQRT(T60 /enclosure volume)
Each of these modes will contribute to the total sum with a different amplitude and phase.
What Schroeder’s principle also implies is:
- A single point microphone measurement has only a 70% confidence of obtaining a pressure response measurement of a pure tone with an accuracy of better than 5.5 dB in any enclosure above its Schroeder frequency.
This is hardly a stable enough measurement to allow one to equalize an audio system’s sound pressure level to within 1 dB or so.
- Single-point microphone measurements in enclosed spaces lack sufficient stability to be useful for sound equalization (within Schroeder’s assumptions).
- Any desired accuracy and confidence can be obtained by spatially averaging a wide-bandwidth signal.
- The exact number of measurements and their bandwidth was hard to determine analytically and needed to be done empirically.
And that’s what Geddes did.
He had the advantage that he knew where the listeners head would be, and that the concern for ergonomics in automobiles had grown.
From ergonomics he could know the zone where 99% of a driver’s eyes would be located, which was know for instrument panel design. From the 99% eye ellipsoid data (ellipsoid = 99th percentile) he could calculate 99% ear ellipsoid.
The ellipsoid is used for spatial averaging, which is required for statistical stability of a measured response.
Measuring the spatially averaged pressure response over the volume of an interior, the total sound power in the vehicle could be calculated.
If the pressure is averaged over a smaller volume, the results are proportional to a “localized sound power measurement.”
By taking measurements over the 99% ear ellipsoid one can obtain the approximate sound power response of the system that would be perceived by the driver.
So, we have our theory.
The method followed the goal of achieving a ±0.5 dB stability.
Geddes started with 18 locations in the ear ellipsoid for point measurements.
He ran an optimization routine to determine the minimum number of points and their locations to obtain the accuracy.
The number of points was 6, and there was only one set of 6 points that worked. Those 6 points were at 30° angle through a rectangular space.
This microphone array should be placed at the approximate intended location of the listeners head, at a slight angle, approximately 30 °.
The output of each microphone can be immediately multiplexed together into one averaged measurement and recorded using a spectrum analyzer. Or, collected individually for a multiple combination of uses.