ROUNDING THE CORNERS
You have just settled down in your plush, row-K seat in Carnegie Hall for an evening of classical music. Halfway through the Bruckner, you suddenly realize that the direct sound from the clarinets is arriving 9 mS after that of the violins, and whoa-the tympani is a good 36 mS late! Fortunately, a little quick thinking on your part restores peace of mind; since the violins are seated 10 feet in front of the clarinets and 40 feet in front of the tympani, the delay in arrival times is a natural consequence.
Indeed, it is that delay which imparts depth to live (and well-recorded) music. But this unsettling experience puts another strange notion in your head.
What if, instead of time delays according to instrument placement, there were delays according to frequency? In other words, what if an orchestra were set up so that all the low notes arrived sooner than the high notes? That would sound pretty strange! Well, it turns out that the phenomenon of delay with respect to frequency is a distinctly real one, and in fact has been given the name of group delay.
Moreover, when you listen to a Compact Disc, you're most likely hearing group delay. Fortunately, engineers are hard at work on a solution to this problem in digital recordings.
As we've discussed in the past two months, every digital audio system is preceded and followed by a low-pass filter. The input filter prevents aliasing from occurring, and the output filter removes ultrasonic images from the signal. To maximize the usable band width of the system (the highest usable audio frequency is half the sampling frequency), sharp-cut filters are employed. However, such brick-wall filters introduce massive phase shift in the audio signal. One solution, as we've seen, is the use of digital over-sampling output filters. The filtering is performed largely in the digital domain before the D/A converter, and phase shift is negligible. As a result, a properly designed CD player is a triumph of phase linearity, with error of perhaps 0.5° or less. That has never been achieved in the history of audio.
The problem is that phase shift is still there-not in the player, but in the discs themselves. You see, when a disc is originally recorded or transcribed from analog, it necessarily passes through an input anti-aliasing filter. And that filter is an analog brick-wall design, with frequency-dependent phase shift measurable as group delay. Your CD library is a repository of it.
Some kinds of phase shift are perfectly tolerable. For example, when you play a 30-year-old recording, that is a lot of time delay or phase shift. How ever, because all of the information at all frequencies on the recording has been delayed evenly, the delay is perfectly acceptable. Indeed, a recording (at least conceptually) is only a delay; without that delay, there could be no recordings. The better the delay in terms of frequency response, noise, and distortion, the better the fidelity.
Unfortunately, recordings can also add delay that is unequal for all frequencies. Analog devices are guilty of this, and digital systems (because of their analog filters) are guilty of it too.
As noted earlier, brick-wall analog filters introduce phase shift which is frequency-dependent-in other words, they introduce group delay. Specifically, the high frequencies lag behind the lower frequencies. In many cases, there is no delay up to 5 kHz, then it steadily increases to 300 uS or more by 20 kHz (in other words, a 20-kHz signal arrives 300 uS late). A CD player with oversampling is free of this de lay, but the digital recorder (with analog input filter) used to record the performance is not. The discs are, unfortunately, encoded with it.
Given its presence, the foremost question concerns the audibility of group delay. It has been explored, but-like so many other questions concerning aural perception--it has not been resolved. While some researchers contend that the ear is deaf to treble phase shift, others surmise that the objections to digital's "high-frequency sound" could be attributed to high-frequency group delay. The jury is still out. However, using experience as a guide, we might propose that group delay is like other aural phenomena: Often, what is first dismissed as inaudible or inconsequential later turns out to be both clearly audible and con sequential. Even if only from a purely academic standpoint, group delay in digital recordings is undesirable.
What is to be done? The problem, of course, is the analog input filter. The multiple poles used to. achieve a sharp cutoff each contribute delay; when combined, the delay is considerable.
Unfortunately, that result is a natural, inevitable consequence of steep analog filters. While we could obtain flat net response by introducing a compensating phase shift, an ideal approach would prevent the shift from ever occurring. If only we could use phase-perfect, digital oversampling filtering, as with the output filter. But for oversampling, you need a digital signal, and to get a digital signal you must first convert the analog signal, and to do that you must have an analog filter.
It's a cart-before-the-horse problem.
Fortunately, a clever design engineer can indeed put the cart before the horse and come up with an oversampling A/D converter which outputs 16-bit samples at a 44.1-kHz rate. Only a little sleight of hand is required. The trick is the use of a very high initial sampling rate followed by a reduction to a lower sampling rate. In that way, the requirements of the Nyquist (half-sampling) theorem can be met without resorting to a brick-wall filter.
The higher the sampling rate, the gentler the required cutoff of the anti aliasing filter, and the less severe the resulting phase shift will be. In an over-sampling A/D, the initial sampling rate might be, for example, 3,175.2 kHz.
Two important benefits result: Only a very gentle analog input filter (with negligible phase shift) is required, and fewer quantization bits are needed. In deed, each sample can be represented by only one bit-a form of digitization known as delta modulation. Its advantage over PCM encoding is simplicity.
Next, the intermediate delta signal must be converted into a 16-bit PCM signal with a sampling rate of 44.1 kHz, compatible with Compact Disc standards. This is accomplished with a decimating filter, which digitally performs two functions: The low-pass filtering formerly done with an analog filter, and sampling-rate reduction. The decimating filter suppresses frequencies above the audio band; one practical choice is a transversal-filter architecture such as is used in output oversampling circuits. Samples are input to a delay line, the delayed signal is tapped off and multiplied by a coefficient, and the products are summed and output. Since the signal is only one bit and represents only + 1 or -1, the number-crunching is simple indeed. In fact, all the possible results can be precalculated, stored in a ROM, and read out using the input signal as an address for the memory location.
The architecture of the transversal filter can be designed to achieve sampling-rate reduction. The original sampling rate of 3,175.2 kHz is exactly 72 times 44.1 kHz, the desired output rate. This makes operation of the filter even easier. There is no need to generate a new output after every input bit.
Rather, one output is generated for every 72 input bits by calculating the output sample in 72 steps and then adding the 72 results together successively to form the ROM address. The sampling rate is thus reduced by a factor of 72. Finally, the output samples are rounded off to 16 bits (alternatively, one could choose 18 bits, or more).
The result is 16-bit samples taken at a 44.1-kHz rate, properly low-pass filtered--but digitally--so there is little phase distortion. Other methods are being explored (the one described above is a favorite of Philips), but the bottom line will be identical.
Not only have phase shift and group delay been avoided, but the resulting A/D is a better bit of technology as well. Conventional anti-aliasing filters and A/D converters are primarily analog devices; they suffer from all the frailties of analog, including difficult-to-acquire precision and higher cost.
With an oversampling design, both operations are done digitally--with greater precision and lower cost. A single integrated circuit replaces a lot of analog components and yields linear phase response as well.
Additional Reading
Meyer, J., "Time Correction of Anti Aliasing Filters Used in Digital Audio Systems," Journal of the Audio Engineering Society (JAES), Vol. 32, No. 3 (March 1984), p. 132.
Adams, R., "Design and Implementation of an Audio 18-Bit Analog-to-Digital Converter Using Oversampling Techniques," JAES, Vol. 34, No. 3 (March 1986), p. 153.
Van der Kam, J. J., "A Digital 'Decimating' Filter for Analog-to-Digital Conversion of Hi-Fi Audio Signals," Philips Technical Review 42, No. 6/7 (April 1986).
(adapted from Audio magazine, Feb. 1988)
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