forward, not in the frequency domain. And our ear/brain perceives these loudspeaker diaphragm and radiation events in the time domain, as they take place, not in the frequency domain. The frequency domain might be useful, commonly used tool for retroactively summarizing and abstractly conceptualizing a loudspeaker's performance, especially for design and engineering purposes. But the frequency domain is not the domain in which the loudspeaker's performance actually happens, nor the domain in which we actually perceive the loudspeaker's performance happening. That privilege is reserved for the time domain.
       Thus, the time domain response of a loudspeaker is arguably more important than any of the frequency responses (amplitude or phase), because signals themselves occur in the time domain, because the loudspeaker performance actually occurs in the time domain, and because the ear/brain hears these signals in the time domain. Also, the brain interprets and analyzes these heard signals in the time domain (especially at low frequencies, where the time domain waveform evolves more slowly, so the brain has plenty of time to interpret and analyze what the evolving waveform pattern sounds like and means, as it evolves in the time domain).

E.3.a. Inverse Relationship of Time and Frequency

       Now, it so happens that the abstract concept of frequency was invented to be the mirror inverse of time. So the abstract frequency domain is by definition the mirror inverse of the real, concrete time domain. This means that a loudspeaker's performance as characterized in one of these domains is related to, and indeed is the mirror inverse of, its performance as characterized in the other domain. For example, if we have characterized a loudspeaker's performance completely in the frequency domain (including both its amplitude frequency response and its phase frequency response), then we can thereby also predict what its performance will actually be in real time, i.e. in the time domain.
       Thus, the many physical handicaps and roadblocks that conventional subwoofer drivers face, which mess up the conventional subwoofer's performance as abstractly characterized in the frequency domain (via both amplitude and phase errors), as discussed above, also mess up the conventional subwoofer's actual concrete behavior in real time, in the time domain. The actual signal waveform produced by conventional subwoofers is a total mess.
       And an educated ear can easily hear what an ugly mess it in fact is. Moreover, even someone with an uneducated ear can instantly hear what a mess it is, if he merely hears it directly compared to the signal waveform from the TRW. It is night and day.
       There is a standard test signal which is used to directly test devices' performance in the time domain, called the step signal. This signal waveform looks just like a single step (of say a stairway). This step signal starts out at 0 level, then instantly (vertically) transitions to some higher level, and then stays flat at that higher level, for an indefinitely long period of time. This step is actually the simplest possible test signal, since it consists of just one single transition, and just two signal levels, to be reproduced. But it is very hard for any device to reproduce accurately even this simplest possible test signal, and most devices fail to do this rather miserably, especially conventional subwoofers.
       The actual performance of the device under test, as it actually happens in the time domain, in reproducing this test signal accurately or poorly, can be easily and directly viewed on a storage oscilloscope. Specific types of actual time domain waveform errors, made by the device in trying to reproduce this simplest test signal, can be visually interpreted, and can be correlated with specific shortcomings or failures in the device. Additionally, they can also be correlated with shortcomings as seen in the frequency domain.
       Since the concept of frequency was invented to be the mirror inverse of the real phenomenon (and dimension) of time, a simple inverse relationship holds between a device's performance as viewed indirectly in the frequency domain, and that same performance as viewed directly in the time domain. Performance errors that occur sooner in time, with a smaller amount of time elapsed, correlate with the inverse, a larger or higher frequency. And performance errors that occur later, with a larger amount of time elapsed, correlate with the inverse, a smaller or lower frequency.
       If you feel mentally disoriented from this inverse talk, it might help to think of this inverse relationship in a practical rather than merely a conceptual manner. If a device, say a loudspeaker, is cycling back and forth faster, each cycle obviously takes a smaller amount of time, but of course this means that there are more cycles per second occurring, and cycles per second is another word for frequency. Presto! Smaller amount of time is equivalent to larger frequency, an inverse relationship. Also, note that any errors committed by this faster cycling loudspeaker would have to be committed sooner, within its now shorter cycle period (presumably this same error would then be repeated each signal cycle), which means that an error at a higher (larger) frequency has to show up sooner, after a smaller amount of time, in the time domain waveform. Conversely, if a device, say a loudspeaker, is cycling back and forth slower, each cycle obviously takes a larger amount of time, but of course this means that there are less cycles per second occurring, and cycles per second is another word for frequency. Presto! Larger amount of time is equivalent to smaller frequency, an inverse relationship. Also, note that any errors committed by this slower cycling loudspeaker can be committed later, within its now longer cycle period, which means that an error at a lower (smaller) frequency can show up later, after a longer amount of time, in the time domain waveform.
       This inverse relationship means that, to examine and interpret a subwoofer's accuracy, at the low frequencies where it operates, we should look at the later (longer time) portions of its response to the standard step test signal. Because subwoofers do not reproduce large (high) frequencies, the beginning (small, low time period) portion of its reproduced step signal will fail to have an instant risetime, and will instead rise slowly. But that's perfectly fine. What we want to look at is a subwoofer's time domain response, in trying to reproduce this step test signal, later in time, i.e. after a higher period of time, since (thanks to the inverse relationship) this higher period of time shows the subwoofer's capabilities at lower frequencies, precisely the low bass frequencies where every subwoofer should be accurate.
       So, with the above discussion as preface, let's look at, and interpret, the time domain response of conventional subwoofers.

E.3.b. Conventional Subwoofer Region 1, Premature Decline

       The test input signal, a step, rises to some higher level above zero, and then stays at that same higher level indefinitely. But a conventional subwoofer's output signal fails to stay at that higher level indefinitely. Instead, a conventional subwoofer's output signal begins declining back down toward zero, much too soon, in the first region of its time domain response to the standard step test signal input. This too-soon declining means that a subwoofer robs all bass transients of their true full impact. All bass transients, be they bass drums or cannon shots or synthesized sound effects, depend, for their massive impact and weight, not only upon their loud initial level but also upon their duration in time. To see that this is so, consider as a reductio a signal that rises to full strength but then very quickly declines; it is known as an impulse, and it sounds like a mere tick, with no sonic weight or impact, simply because it lacks duration. Conventional subwoofers, by declining too quickly, wimp out before bass transients can convey their true, full impact and weight to you.
       Incidentally, this time domain error of conventional subwoofers, wimping out from full level too soon, and not lasting at full level for a high enough time period happens to correlate with a frequency domain error discussed above, wimping out from full amplitude level at too high a frequency, and not lasting (extending) at full level to a low enough frequency, i.e. rolling off in the low bass, below system resonance (note again the inverse relationship, in that failing to last at full level for a high enough time period correlates with failing to last at full level (extend) to a low enough frequency). But, when playing real music and films, with their real bass transients, it is the time domain premature declining error that we directly hear, while the corollary frequency response rolloff relates instead to steady tone sine wave frequency domain measurements, and to a few steady tone bass sounds (like pipe organs). Thus, for most music and most sound effects, it is the time domain error that is more significant than the corollary frequency domain error. And this fact, the predominance of time domain error, will become crucial when we later consider the limits of low bass that humans can hear and that real sounds produce.

E.3.c. Conventional Subwoofer Region 2, Negative Overshoot

       The second huge error that conventional subwoofers make, in their time domain performance, is overshoot below the zero axis (representing zero output signal level). After conventional subwoofers decline too soon from full level, they then commit a further blunder, in region 2 of their time domain response to the standard step test signal input. Instead of simply approaching the zero signal level axis, and settling down to zero signal level, they actually continue downward, overshooting below the zero signal level axis, and then they start putting out a negative signal.
       This negative signal output creates acoustic energy in the wrong direction, sucking instead of pushing. In effect, the conventional subwoofer changes its mind, and sucks back in some of the air, and some of the bass energy, that it had just previously put out.

E.3.c.i. Sucking vs. Pushing in Conventional Subwoofers

       As you can intuitively see, this overshoot error compounds the felony previously committed by the conventional subwoofer in wimping out too soon from full level. When the conventional subwoofer wimped out too soon, it robbed some of the true impact and weight from every bass transient. Now, by overshooting below the zero axis, and sucking back in some of the already-too-little bass energy it had just put out, the conventional subwoofer is reducing even further the impact and weight of each and every bass transient. Remember that the input test signal, a step, contains only positive pushing bass energy, so when the conventional subwoofer outputs negative sucking bass energy in its erroneous response to the input test signal, this is a major, major blunder. Note also that the standard step test signal, even though an artificial signal, does honestly represent and probe the bass reproduction requirements for actual bass transients, both musical and sound effects (as we'll discuss further below).
       The human ear/brain is a superb time domain analyzer in general, and here in particular is very good at two pertinent evaluation techniques, integrating bass energy over time, and performing a time slicing analysis of the bass waveform as it is output by a conventional subwoofer.
       As the ear/brain performs the first evaluation technique, integrating the bass energy over time, the negative sucking in by the conventional subwoofer's negative overshoot, even though occurring at a later time than the conventional subwoofer's initial positive foray, becomes integrated with that immediately preceding positive foray, and thus literally subtracts bass impact energy and weight away from the already-too-little amount in that preceding positive foray. In effect, the positive pushing bass energy sensed by the ear/brain over time, as the total positive area under the waveform curve, is lessened by the negative area under the negative overshoot portion of the conventional subwoofer's time domain response curve.
       The second evaluation technique, the time slicing analysis ability of the ear/brain, also comes into play here. The ear/brain is very good at being able to tell when a waveform of a familiar sound has the wrong polarity, and is thereby sucking air instead of pushing air. For example, in the spectral midrange we are very good at hearing when a trumpet or vocalist is sucking instead of pushing air, because it simply sounds wrong, since trumpets always push and never suck, as do singers when enunciating words (try saying or singing words yourself, while inhaling instead of normally exhaling, and note how different and absurdly wrong it sounds). Similarly, a trained or sensitive listener can tell when part of the bass waveform, put out by a conventional subwoofer, is suddenly sucking instead of pushing air, on what should still be the positive pushing impact portion of a bass transient. That especially true when the ear/brain, with its time slicing analysis capability, has just heard, for immediate direct comparison, part of the same bass transient sound put out in correct positive pushing polarity. The ear/brain hears this sucking blunder as making the bass transient puffy and hollow in quality, like a soft cotton ball, instead of being the solid hard impact punch to the stomach that an accurate bass transient, with only positive pushing, would be.

E.3.c.ii. Compressed Dynamics by Conventional Subwoofers

       This negative overshoot blunder by conventional subwoofers also engenders further losses of sonic quality, which may be the worst of all (depending on program material). It can reduce (compress) dynamics, and can cause sonic confusion, even outright distortion, of the program content that immediately follows a bass transient. How does this happen?
       First, let's discuss the dynamic compression, of the program content following a bass transient. Recall that our standard step test signal goes positive and stays positive, so the ideal subwoofer, to be correct, should only put out positive pushing airflow and pressure. But every conventional subwoofer spontaneously misbehaves and goes negative on its own, during this second region of its time domain response, sucking air back in even while the input signal is still at full positive level, and is thereby still instructing the subwoofer to keep pushing air out. Moreover, this misbehaving conventional subwoofer executes this entire negative sucking portion of its output on its own time schedule, without paying any attention to the input signal (as proven by the fact that it ignored the step input signal's command to stay positive).
       Now, our standard step test signal consists of one single signal transition, with no further signal change after that, so it is in effect a single bass transient. But, in all real program material, a single bass transient is followed by other program information, hence further new signal changes. These further new signal changes could be a second distinct bass transient, or they could be a midrange or treble transient, or they could be a positive going half cycle of say a bass drum note that continues after the initial whack. Whatever these further new signal changes are, they are sure to contain some positive polarity waveform information, i.e. some instruction to increase the positive pushing airflow and pressure in your room and hence at your ear, perhaps from your main loudspeaker (if in the midrange or treble), or perhaps from the subwoofer itself.
       However, whatever the specifics of this new instruction to increase the positive pushing airflow and pressure in your room and at your ear, the conventional subwoofer will simultaneously still be in its negative sucking second region of time domain response, from the earlier original bass transient. Thus, the conventional subwoofer will still be negatively sucking (decreasing) airflow and pressure out of the room and out of your ear, even as the new program information instructs the airflow and pressure to be positively pushed (increased) into the room and into your ear. The obvious consequence is that the conventional subwoofer, during its period of negative sucking spurious misbehavior, will subtract from the positive dynamic peak of positive pushing airflow and pressure commanded by the new signal. In short, the negative overshoot blunder by conventional subwoofers reduces the dynamics of succeeding new positive signal waveforms in the program that happen to arrive during the second region of the conventional subwoofer's misbehaving time domain response.
       If the succeeding new signal waveform represents a second positive pushing bass transient, to be reproduced by the subwoofer, its dynamics will be compressed and reduced by the negative sucking that the misbehaving conventional subwoofer is executing at the same time. If the succeeding new signal waveform represents the positive pushing of a bass drum head as it continues to sound after the initial whack, that too will have its dynamics compressed and reduced. And, even if the succeeding new signal waveform represents a positive push from some musical instrument or other sound source in the midrange or treble, to be reproduced wholly by your main loudspeaker and not by the subwoofer at all, the subwoofer's negative sucking misbehavior will still compress and reduce its dynamics, since the subwoofer will be reducing and sucking air pressure out of the room even as the main loudspeaker is trying to push and build up air pressure into the room, and since all you get to hear at your listening seat is the net air pressure change in the room.

E.3.c.iii. Sonic Confusion Caused by Conventional Subwoofers

       This negative overshoot blunder by the conventional subwoofer also causes sonic confusion, that extends far beyond its own butchering of the sonic quality of the initial, original bass transient. The bass drum doesn't sound right when its positive pushing excursions are attenuated, and the midrange or treble transients from your main loudspeaker don't sound right when their positive pushing efforts are attenuated. Moreover, the misbehaving subwoofer actually creates a kind of even order distortion, for all music and all sounds, which happen to occur during this second region, of negative overshoot misbehavior, in its time domain response. The negative signal pressure, spuriously created by the misbehaving subwoofer, will act as an acoustic signal bias on all other program signals that happen to occur during the time period of this second region, reducing the positive going peaks and increasing the negative going peaks. That simply amounts to even order distortion, and should degrade the overall sound in much the same way that an amplifier with excessive even order distortion would.

E.3.c.iv. Worst Negative Overshoot Offenders among Conventional Subwoofers

(Continued on page 148)