below 20 Hz, and play it repeatedly, asking the human subject if he can hear any difference. If you wish, you can double check the validity of the human subject's responses by various methods, for example by wiring a switch so that, when you tell the human subject that you are (or he is) switching to half the frequency for the filter, in reality the filter is staying at the same frequency.
       Continue this progression, cutting the filter frequency in half, so long as the human subject can reliably detect a difference in the sonic qualities of the step test signal transient. As this progression goes lower in filter frequency, you will reach a point where the human subject can no longer detect a difference in the sonic qualities of the step transient, when you cut the filter frequency in half. This point is the key frequency.
       This point tells us that the low frequency bass limit of human perception extends at least this low, in perceiving bass quality of real program material transients (as accurately simulated here by our step test signal transient), most of which actually have spectral energy down to DC. This point tells us that subwoofers must extend at least this low in bass frequency reproduction, in order to fully satisfy human bass perception capability, on real program material transients. This point tells us that, if a subwoofer extends at least this low, then its output should be sonically indistinguishable (to humans) from a subwoofer that extends even lower, so it should be good enough.
       We performed this experiment years ago, and the results seemed shocking at first. We found that, when listening to this highly relevant bass transient test signal (instead of the irrelevant conventional sine wave test signal), we could perceive a difference in bass quality all the way down to about .1 Hz (!!!). And that was using conventional loudspeakers to perform this test, which partially masked the true measure of human bass perception capability, since they interposed an additional steep filter at about 40 Hz. When we get a chance to repeat this same test using the TRW subwoofer, whose response is essentially flat to DC, we might well find that human bass perception ability on transients actually extends even below .1 Hz.
       But, for purposes of our review and discussion here, it's fine to use .1 Hz as a working approximation for the lower limit of human bass perception. No conventional subwoofer gets even close to this frequency, and the TRW extends down to .1 Hz and below with ease, so for purposes of our discussion here it doesn't matter whether the true human limit is .1 Hz or .01 Hz. In either case, only the TRW covers the entire bass range of human perception with ease, whereas conventional subwoofers don't even come close.
       Most importantly, this experiment proves that humans can indeed easily perceive the sonic benefits of a subwoofer that extends far below the commonly accepted human hearing bass limit of 20 Hz. This proves that the TRW's ability to cover this spectral region, from .1 Hz to 20 Hz, and to do so accurately and effortlessly, is highly relevant to human perception abilities in this spectral region, and it places the TRW at a crucial advantage over all conventional subwoofers, which cannot cover this relevant spectral region at all.

E.3. New Major Role for Subwoofer like TRW

       This finding also causes us to rethink the whole role of a subwoofer in your system, and its importance to your system. Old fashioned thinking, based on conventional subwoofer technology and capability, has a subwoofer adding merely one octave, 20-40 Hz, to a high quality loudspeaker array employing full range systems that capably cover 9 octaves, 40-20,000 Hz (satellite system abominations are not worthy of serious inclusion here). Thus, with the limitations of conventional subwoofers, the subwoofer has a relatively minor spectral span to cover, and a relatively minor supporting role to play.
       But this picture completely changes with a true subwoofer like the TRW, which can finally cover the full human spectral perception capability at low frequencies. There might be 9 octaves for your full range loudspeakers to cover, as they cover virtually all of what used be thought of as human hearing spectral range and virtually all of what used to be thought of as program material spectral range. But we have now discovered that both program material and human perception extend at least down to .1 Hz. And the spectral range from 40 Hz down to just below .1 Hz also happens to span 9 octaves. This means that the job of a competent subwoofer is fully as important as the full range loudspeaker's job, since the spectral range it must cover accurately is just as wide.
       This also explains some of our sonic findings discussed below, where we found that the sonic benefits from the TRW subwoofer not only were dramatic where we expected them to be, in better quality and quantity of bass, but also were surprisingly far reaching, dramatically improving other aspects of system sound that we had never suspected could or would be improved by a subwoofer. These further improvements, in unexpected sonic aspects of overall system sound, make good sense in light of this new fact, that fully half of the spectral range from the whole system is actually contributed by a competent subwoofer, such as the TRW.

E.4. Conceptual Analysis of Human Bass Sensing Ability

       The fact that human bass perception actually extends down below .1 Hz might seem shocking, to those who have been schooled to think only in sine wave and frequency domain terms. But this fact makes good sense if we think about this in the time domain instead of the frequency domain, and use relevant transients instead of irrelevant sine waves as our model.
       To understand this intuitively, picture first the vision of a hummingbird. As you know, hummingbirds live their lives at a very fast pace, and have a very short attention span. Thus, if you stand near a hummingbird and move about, the hummingbird can't see your actual motion, and thinks you are standing still. Similarly, we humans have a visual attention span that is longer than a hummingbird's, but still limited. Our visual motion detection threshold is just a bit faster than the sun moves across our sky, so the sun looks to us as though it were stationary in the sky, even though it is in fact continually moving slowly (we can indirectly infer this unsensed motion only when the sun is setting at the horizon, and keeps altering its position relative to the horizon).
       A similar phenomenon occurs with human perception of acoustic energy. We can sense the amplitude change or decline, i.e. amplitude "motion", only when it occurs reasonably rapidly.

E.4.i. Change That's Too Slow to Detect

       Suppose for example that we play our step test signal, unfiltered, through the TRW subwoofer that can reproduce bass frequencies down to and including DC. The full amplitude of the step, after its transition up from zero amplitude, would last forever. But, could we tell via our perceptions that its full amplitude is staying the same forever? Of course not, since we won't be alive when forever arrives.
       Suppose next that we play the same step test signal through a very, very low frequency electrical bass filter, a filter which would make the step signal decline at a slow rate, so that the signal amplitude fell to a significantly lower level after say 2 minutes. That 2 minutes is surely well beyond our human attention span, so we could not detect the actual slow movement of the signal amplitude's downward sloping decline. In effect, the rate of change is too slow for our attention span capability to detect it.
       Then, suppose next that we set the electrical filter one octave higher, which would mean that the step signal's amplitude would decline (move downward) twice as fast, declining to a given lower amplitude level in half the time, 1 minute instead of 2 minutes.
       Now comes the key question: could we tell the difference in sonic quality, between the very, very slow motion of the decline that takes 2 minutes, vs. the very slow motion of the decline that takes 1 minute? Probably not, because even the 1 minute decline has a downward slope that is too shallow, a downward motional speed that is too slow, for us to detect. Our attention span is simply not long enough to sense this downward acoustic energy as a motional change in energy over time, in much the same way that our visual attention span is not long enough to see the actual, real motion of the sun in mid-sky.
       The only way that we could tell any difference between the bass filter set at 2 minutes vs. the bass filter set at 1 minute would be via an indirect inferential method, such as noting a time difference on the clock when the differing step signals finally seemed to fall silent (analogous to our indirectly, inferentially "seeing" the motion of the setting sun at the horizon, by noting the progressively smaller dimensions of its shape). Thus, it seems clear that human bass sensing ability does not require a subwoofer to extend so low in bass that it takes 2 minutes to decline instead of 1 minute (a one octave higher bass cutoff), for an input signal transient (be that transient a singular signal transition from program material, or a singular signal transition from a step test signal).

E.4.ii. Change That's Fast Enough to Detect

       So let's next change the time frame of this thought experiment. Let's set the electrical filter so that the step signal declines most of its amplitude within 2 seconds, not 2 minutes. This puts the filter's rolloff corner at a 60 times higher frequency. We can all agree that the human brain is capable of paying attention to the sonic quality of a big, whopping bass thump for the very short time of a mere 2 seconds. Moreover, humans can surely directly sense the difference in sonic quality when a bass thump lasts for only 1 second instead of 2 seconds. In effect, the step signal's amplitude decline is now fast enough so that we can now sense the actual difference in the sonic quality between a slower 2 second decline and a faster 1 second decline.
       When the electrical filter was set for a 1 minute or 2 minute decline, the acoustic amplitude in the room was actually changing all the time, but we could not directly sense this change as making any difference in the sonic quality of the bass thump. Moreover, we could therefore also not directly sense any difference between the 1 minute vs. 2 minute decline, so it was not important to our human sensing which of these very low frequencies this filter was set at. Analogously, even though the sun is actually moving across the sky, we cannot directly sense its motion, so it appears to be motionless, and, moreover, we could not directly sense any difference in the sun's actual motion if in fact it were to move twice as slowly across the sky.
       But now, with the electrical filter set for a 2 second decline, which is well within our human attention span, we can indeed directly sense the actual change over time in the acoustic bass signal. And, moreover, we can therefore also sense the difference in rate of decline, and the difference in sonic quality of the bass thump, if the electrical filter is set for a 1 second decline instead of a 2 second decline. To continue our analogy, if a propeller airplane moves across the sky, it is moving a lot faster than the sun, so we humans can indeed directly see its motion. Moreover, if a jet airplane then flies across the sky at twice the speed of the propeller airplane, we can easily directly sense the difference in the quality of speed between the two airplanes.
       The fact that we can and do sense a difference in sonic quality, between the bass thump having a 1 second decline vs. the bass thump having a 2 second decline, means that it becomes sonically important to set the filter at the lower frequency (the longer 2 decline), if we want our system to reproduce the full bass spectrum that humans can sense, and if we want the quality of the bass thumps to have the maximum high quality that humans can sense and benefit from.
       For argument's sake, let's now suppose that it turns out that we humans cannot directly sense any sonic difference between the bass thump that declines in 2 seconds vs. a bass thump that declines in 4 seconds (achieved by setting the electrical filter one octave lower in frequency than for the 2 second decline). This means that, so far as human bass sensing capability is concerned, it is not important to have a subwoofer extend so low in frequency that it can output a 4 second decline from an input step test signal. It is sufficient, for the intended human audience, to have the subwoofer extend only low enough in frequency to output a 2 second decline.
       Incidentally, note that we do not need to postulate here that the human brain can indeed pay attention to a bass transient for the full 2 seconds. That's because the 2 seconds represents the total time that it takes for a filtered step to decline most of the way toward zero. And this 2 seconds for the total measured decline implies a fairly rapid rate of declining change within that 2 second time period, instant by instant.
       What the human brain might well be focusing on, with its time domain perception and analysis capability, is this fairly rapid rate of declining change implied by the 2 second total time period, even if the brain has an attention span of say only a fraction of a second, and thus only looks at this rate of change for this fraction of a second. What counts here is that the rate of change is fast enough, when 2 seconds is the total measured decline time, for the brain to detect downward motion, perhaps instant by instant, in the declining amplitude of the filtered step.
       Meanwhile, when 4 seconds is the total measured decline time, the rate of decline is slower, so, in our posited case here, the brain cannot sense downward amplitude motion of the signal.
       Analogously, the human eye and brain can tell that an airplane is moving across the sky, with just a glance that lasts only a fraction of a second, because the rate of motion is fast enough, but the human eye and brain cannot directly see the sun actually move across the sky, even if it pays attention for a much longer time.

E.4.iii. Calculation of Low Frequency Limit

       So let's run with a 2 second decline as a reasonable model. What bass frequency does this correspond to? In other words, how low in frequency does a subwoofer's response have to extend, in order to adequately cover a human's ability to pay attention for a mere 2 seconds, to the sonic qualities of a bass thump transient?
       A simple engineering formula is used to convert this 2 second decline, called a time constant, into the equivalent frequency. That formula is F=1/2T, where T is the time constant (2 seconds) and F is the equivalent frequency (in cycles per second, ie. Hz). Plugging our 2 second time constant into this formula shows that the equivalent frequency is (trumpets, please) .08 Hz!!
       This calculation means humans can directly sense bass quality down to .08 Hz, on all the real transients in our program material, transients which overwhelmingly outnumber steady (sine-wave-like) bass tones in our program material. This means that a subwoofer must be capable of extended, substantially flat bass response down to .08 Hz, if it is to adequately satisfy human bass sensing capability, in reproducing high quality transient bass sound (and, if the subwoofer does happen to have some rolloff at some very, very low bass frequency, then that rolloff point must be far below .08 Hz, in order that .08 Hz and above can be accurately reproduced in the time domain, so as to avoid incurring all the time domain problems discussed above for conventional subwoofers).
       This calculation of .08 Hz, which equals .1 Hz when rounded to one decimal place, was based on a conceptual analysis of transients in the time domain. Thus, our previous experimental finding, that we could actually perceive bass quality on transients down to at least .1 Hz, is not as shocking as it might have seemed at first. Indeed, the two approaches agree perfectly with each other. The conceptual analysis and the empirical experiment corroborate each other.
       By going to transients instead of sine waves, and the time domain instead of the frequency domain, we have gained a far more relevant and realistic picture, of what frequencies are actually contained in our program material, and of what frequencies humans can actually perceive. And it's a far different picture than the old-fashioned picture that embraces merely 20 to 20,000 Hz, which we can now see to be dreadfully wrong. It was wrong because it failed to understand the fundamental concept of frequency, and therefore used only a sine wave model, viewed only in the frequency domain.

E.5. Sensing Bass via Other Means

       It's worth noting that humans can sense low frequency information by means and mechanisms other than by listening with the ears to sine waves. We have just shown, by conceptual analysis corroborated by empirical experiment, that the human brain must also be able to perform some kind of time domain analysis upon transients, in order to perceive the sonic effects upon bass quality, of filtering at extremely low frequencies (.1 Hz), when applied to transients of the kind that fill our program material.
       Additionally, humans can obviously also sense high energy low bass by our sense of feeling. We all know that feeling of a kick in the stomach that a truly impactive bass transient can have and should have.
       And we also feel bass with our feet, perhaps subliminally. Transients with a lot of bass energy also shake a room's structure, in particular the floor, where our feet are planted. Indeed, when there is strong bass transient we actually feel it with our feet before we hear it with our ears, because the shock wave travels faster through solid materials like the floor than it does through air.
       In fact, a good portion of the thrill of truly impactive bass is the fact that our brain gets momentarily terrorized, in a primal caveman way, by the shock wave felt through our feet, before we can hear the explanation of this acoustic event arriving at our ears via the air, so for a split second our brain knows something is horribly wrong (e.g. an earthquake) and something big is coming, but it doesn't yet know what that something is.

(Continued on page 152)