The Science of Subwoofing Page 3
Once the SPL requirement at a particular frequency is known, it is possible to calculate the corresponding woofer excursion. Fielder and Benjamin do this for 100dB at 20Hz in a typical listening room with a 2400ft3 volume and a reverberation time of 0.6 seconds. The required volume excursion is 41.8in2 which, for a 12" woofer with an effective piston diameter of 10", translates into a peak linear excursion of 0.53". With four 15" woofers, the linear excursion requirement is only 0.078"a much more reasonable figure.
Of dubious practical importance is the question of allowable amplitude deviations for subwoofers. Just how flat does the amplitude response need to be below 100Hz? Well, using the concept of "just noticeable difference" (JND) in level, available experimental data suggest that at an output level of 100dB the JND is about 1.0dB at 100Hz and about 1.5dB at lower frequencies. These JNDs were obtained, however, under idealized conditions: in an anechoic chamber or using headphonesroom effects did not enter the picture. In the real world, the problem of standing waves in small rooms is so severe that peaks and dips on the order of 8dB are quite common in the deep bass; generally speaking, the gross structure of room modes will overwhelm any inherent response flatness on the part of the subwoofer.
Response flatness is much more important in the upper bass and lower midrange, in the range from about 150 to 300Hz where room effects are less important. A 2dB or greater broad suckout here will definitely be audible as a lean and small-bodied balance. This is a range where typically very few commercial loudspeakers designed to be "free standing" exhibit flat response, this due mainly to a phenomenon dubbed by some as "diffraction loss."
Let's assume for the moment that the midrange/woofer puts out an equal amount of energy at each frequency in this range. At frequencies whose half wavelength is smaller than the dimensions of the front baffle, the baffle will keep the speaker's output "concentrated" in the forward half-space where the listener is located. At some frequency, the wavelength starts to wrap around the front baffle, and the speaker's output becomes less directional or more omnidirectional. Because now the same acoustic output is being radiated into a larger volume, the average intensity is reduced. Of course, as the wavelength increases further, the floor, back wall, ceiling, and side walls are eventually encountered and reflect acoustic energy toward the listener, which helps lift or boost the mid- and deep-bass response. The end result is a suckout located somewhere in this upper bass range. Diffraction loss can be combatted. Manipulating the midrange/woofer crossover (if it is in the diffraction-loss region), or using a dualvoice-coil woofer to provide a boost, are two possibilities.
The subject of phase-response errors for subwoofers is complicated by the fact that no perceptual studies have been performed on the audible effects of large group delays below 100Hz. Fielder and Benjamin note that subwoofers introduce phase distortion primarily from two sources. The direct source is the low-frequency cutoff of the woofer. Thus, the woofer may be viewed as a high-pass filter. A subwoofer with a 16Hz, Q=0.9, second-order, high-pass function has a 19ms group delay at 16Hz and 0.33ms delay at 100Hz. The second source of group delay is the inevitable crossover used to match the woofer to the upper-range speaker.
For example, a fourth-order, Linkwitz-Riley crossover would produce 6.2 and 3.4ms group delays at frequencies of 16 and 100Hz, respectively. This represents a worst-case situation; I'm not aware of any commercial audio crossovers using steeper slopes at the crossover point. Current experimental findings indicate a perceptual threshold of 2.5ms at 100Hz for non-reverberant conditions, and predict a doubling of the threshold under reverberant conditions. I therefore doubt that subwoofer- or crossover-induced phase deviations will be audible under typical listening conditions, and see no reason to avoid using a steep-sloped crossover in the bass.
A topic of great importance to the performance of subwoofers is the audibility of non-linear distortion. The ideal subwoofer does not have to produce zero distortionrather, its distortion products must be inaudible even to the most sensitive listener. Because distortion products are not produced in a vacuum, but instead accompany the music signal, the concepts of masking and critical bands are crucial for determining their audibility. Again, following Fielder and Benjamin's discussion, masking is the concealment of an otherwise audible tone or band of noise by the presence of a louder signal. The higher the masking signal is above the threshold of audibility at a given frequency, the more effective it is in concealing distortion. Because the threshold of hearing rises rapidly and is much higher in the bass octaves compared with the mids, bass signalsrelatively speakingare poor maskers.
Another characteristic of masking is that the closer in frequency the masking signal is to the lower-level signal, the more effective it is. As the masking signal moves away in frequency, its effectiveness diminishes. One way to widen the bandwidth over which a masking signal is effective is to increase its amplitude or volume. This means that the ear is more sensitive to distortion at lower volume levels because there is less masking going on. Certainly, this is a fortunate state of affairs because woofer distortion increases with playback level. The harder the woofer is pushed, the more it distorts, but we're more tolerant of distortion at these louder volume levels.
The critical band concept was first developed by Harvey Fletcher about 50 years ago. In its modern interpretation, the model essentially views the ear/brain as a 24-channel real-time analyzer with varying sensitivities and bandwidth. Above 300Hz, critical bands are about a fifth of an octave wide. Below 300Hz they are about 100Hz wide. A sound signal is detected only if the energy within a particular critical band exceeds a certain threshold. Each band is independent of the others, at least at sound levels near the threshold of hearing. It is possible then for distortion products that span a large number of critical bands to sound louder and more annoying than a larger signal that only triggers one band. Masking is greatest for sounds within the same critical band. More precisely, the maximum masking region is one-half of a critical bandwidth on either side of the masking signal. The subwoofer range spans one critical band, thus one can expect a masked threshold curve that parallels the hearing-acuity curve with its rapid rise at low frequencies.
On this basis, Fielder and Benjamin conclude that intermodulation distortion is not a very important consideration in subwoofer design. First of all, the difference IM products for subwoofers lie in the first critical band, with the masking signals, and are thus effectively masked. The sum products are no more audible than the harmonics of a single sinewave with equivalent level and average frequency. That this is so is apparent once you realize that the IM sum products are less than or equal in frequency to an appropriate harmonic product, and are lower in level because intermodulation divides the amplitude between components.
An important result of Fielder and Benjamin's analysis is that good reproduction of the bass octaves requires that the main speaker, the subwoofer, and the room all produce extremely small amounts of noise and distortion where hearing acuity is greatest: namely, in the midrange. They point out the need to look at the entire audio spectrum weighted by the masking effect and that previous analyses which considered only unweighted distortion measurements could not accurately predict sound quality due to all the non-linearities. Using this sort of analysis, they arrived at the surprising conclusion that the sensitivity to second-harmonic distortion is much less than to third-harmonic. They point out that this is important because speaker systems often have comparable amounts of second- and third-harmonic distortion, and that techniques aimed at reducing even-order distortion products may be of dubious value.
Manufacturers have spent much research and development money on such designs. An example is a dual-spider woofer that provides an even-harmonic canceling, push-pull piston action. The following simple distortion guidelines are offered: Harmonic distortion will not be audible if the second harmonic is below 3%, the third around 1%, and higher harmonics no greater than 0.10.3%.