Digital Dharma of Audio A/D Converters By Dennis A. Bohn

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Among the many definitions for the wonderful word "dharma" is the essential function or nature of a thing. That is what this note is about: the essential function or nature of audio analog-to-digital (A/D) converters.

Like everything else in the world, the audio industry has been radically and irrevocability changed by the digital revolution. No one has been spared. Arguments will ensue forever about whether the true nature of the real world is analog or digital; whether the fundamental essence, or dharma, of life is continuous (analog) or exists in tiny little chunks (digital).

Seek not that answer here. Here we shall but resolve to understand the dharma of audio A/D converters.

Data Conversion
It is important at the onset of exploring digital audio to understand that once a waveform has been converted into digital format, nothing can inadvertently occur to change its sonic properties.

While it remains in the digital domain, it is only a series of digital words, representing numbers. Aside from the gross example of having the digital processing actually fail and cause a word to be lost or corrupted into none use, nothing can change the sound of the word. It is just a bunch of "ones" and "zeroes." There are no "one-halves" or "three-quarters".

The point being that sonically, it begins and ends with the conversion process. Nothing is more important to digital audio than data conversion. Everything in-between is just arithmetic and waiting.

That's why there is such a big to-do with data conversion. It really is that important. Everything else quite literally is just details. We could go so far as to say that data conversion is the art of digital audio while everything else is the science, in that it is data conversion that ultimately determines whether or not the original sound is preserved (and this comment certainly does not negate the enormous and exacting science involved in truly excellent data conversion.)

Since analog signals continuously vary between an infinite number of states and computers can only handle two, the signals must be converted into binary digital words before the computer can work. Each digital word represents the value of the signal at one precise point in time. Today's common word length is 16-bits or 32-bits. Once converted into digital words, the information may be stored, transmitted, or operated upon within the computer.

In order to properly explore the critical interface between the analog and digital worlds, it is necessary to review a few fundamentals and a little history.

Binary Numbers
Whenever we speak of "digital," by inference, we speak of computers (throughout this paper the term "computer" is used to represent any digital-based piece of audio equipment).

And computers in their heart of hearts are really quite simple. They only can understand the most basic form of communication or information: yes/no, on/off, open/closed, here/gone - all of which can be symbolically represented by two things - any two things. Two letters, two numbers, two colors, two tones, two temperatures, two charges - it doesn't matter. Unless you have to build something that will recognize these two states - now it matters.

So, to keep it simple we choose two numbers: one and zero ... a "1" and a "0." Officially this is known as binary representation, from Latin bini two by two. In mathematics this is a base-2 number system, as opposed to our decimal (from Latin decima a tenth part or tithe) number system, which is called base-10 because we use the ten numbers 0-9.

In binary we use only the numbers 0 and 1. "0" is a good symbol for no, off, closed, gone, etc., and "1" is easy to understand as meaning yes, on, open, here, etc. In electronics it is easy to determine whether a circuit is open or closed, conducting or not conducting, has voltage or doesn't have voltage.

Thus the binary number system found use in the very first computer, and nothing has changed today. Computers just got faster and smaller and cheaper, with memory size becoming incomprehensibly large in an incomprehensibly small space.

One problem with using binary numbers is they become big and unwieldy in a hurry. For instance, it takes six digits to express my age in binary, but only two in decimal. But, in binary, we better not call them "digits" since "digits" implies a human finger or toe, of which there are ten, so confusion reigns.

To get around that problem John Tukey of Bell Laboratories dubbed the basic unit of information (as defined by Shannon - more on him later) a binary unit, or "binary digit" which became abbreviated to "bit." A bit is the simplest possible message representing one of two states.

So, I'm 6-bits old ... well, not quite. But it takes 6-bits to express my age as 110111. Let's see how that works. I'm fifty-five years old. So in base-10 symbols that is "55," which stands for 5-1s plus 5-10s. You may not have ever thought about it, but each digit in our everyday numbers represents an additional power of 10 beginning with 0.

Figure 1. Number Representation Systems

That is, the first digit represents the number of 1s (100), the second digit represents the number of 10s (101), the third digit represents the number of 100s (102), and so on. We can represent any size number by using this shorthand notation. Binary number representation is just the same except substituting the powers of 2 for the powers of 10 [any base number system is represented in this manner]. Therefore (moving from right to left) each succeeding bit represents 20 = 1, 21 =2, 22 =4, 23 =8, 24 = 16, 25 =32, etc. Thus, my age breaks down as 1-1, 1-2, 1-4, 0-8, 1-16, and 1-32, represented as "110111," which is 32+16+0+4+2+1 = 55 ... or double-nickel to you cool cats. Fig. 1 shows the two examples. Now let's take a brief look at how all this came about.

The Story of Harry & Claude
The French mathematician Fourier unknowingly laid the groundwork for A/D conversion in the late 18th century. All data conversion techniques rely on looking at, or sampling, the input signal at regular intervals and creating a digital word that represents the value of the analog signal at that precise moment. The fact that we know this works lies with Nyquist.

Harry Nyquist discovered while working at Bell Laboratories in the late `20s and wrote a landmark paper [1] describing the criteria for what we know today as sampled data systems. Nyquist taught us that for periodic functions, if you sampled at a rate that was at least twice as fast as the signal of interest, then no information (data) would be lost upon reconstruction.

And since Fourier had already shown that all alternating signals are made up of nothing more than a sum of harmonically related sine and cosine waves, then audio signals are periodic functions and can be sampled without lost of information following Nyquist's instructions.

This became known as the Nyquist frequency, which is the highest frequency that may be accurately sampled, and is one-half of the sampling frequency. For example, the theoretical Nyquist frequency for the audio CD (compact disc) system is 22.05 kHz, equaling one-half of the standardized sampling frequency of 44.1 kHz.

As powerful as Nyquist's discoveries were, they were not without their dark side: the biggest being aliasing frequencies. Following the Nyquist criteria (as it is now called) guarantees that no information will be lost; it does not, however, guarantee that no information will be gained.

Although by no means obvious, the act of sampling an analog signal at precise time intervals is an act of multiplying the input signal by the sampling pulses. This introduces the possibility of generating "false" signals indistinguishable from the original. In other words, given a set of sampled values, we cannot relate them specifically to one unique signal. As Fig. 2 shows, the same set of samples could have resulted from any of the three waveforms shown ... and from all possible sum and difference frequencies between the sampling frequency and the one being sampled. All such false waveforms that fit the sample data are called "aliases."

In audio, these frequencies show up mostly as intermodulation distortion products, and they come from the random-like white noise, or any sort of ultrasonic signal present in every electronic system. Solving the problem of aliasing frequencies is what improved audio conversion systems to today's level of sophistication. And it was Claude Shannon who pointed the way.

Shannon is recognized as the father of information theory: while a young engineer at Bell Laboratories in 1948, he defined an entirely new field of science. Even before then his genius shined through for, while still a 22-year-old student at MIT he showed in his master's thesis how the algebra invented by the British mathematician George Boole in the mid-1800s, could be applied to electronic circuits. Since that time, Boolean algebra has been the rock of digital logic and computer design. [2]

Shannon studied Nyquist's work closely and came up with a deceptively simple addition. He observed (and proved) that if you restrict the input signal's bandwidth to less than one-half the sampling frequency then no errors due to aliasing are possible. So bandlimiting your input to no more than one-half the sampling frequency guarantees no aliasing. Cool ... only it's not possible.

In order to satisfy the Shannon limit (as it is called - Harry gets a "criteria" and Claude gets a "limit") you must have the proverbial brick-wall, i.e., infinite-slope filter. Well, this isn't going to happen, not in this universe. You cannot guarantee that there is absolutely no signal (or noise) greater than the Nyquist frequency. Fortunately there is a way around this problem. In fact, you go all the way around the problem and look at it from another direction.

If you cannot restrict the input bandwidth so aliasing does not occur, then solve the problem another way: Increase the sampling frequency until the aliasing products that do occur, do so at ultrasonic frequencies, and are effectively dealt with by a simple single-pole filter. This is where the term "oversampling" comes in. For full spectrum audio the minimum sampling frequency must be 40 kHz, giving you a useable theoretical bandwidth of 20 kHz -- the limit of normal human hearing.

Sampling at anything significantly higher than 40 kHz is termed oversampling. In just a few years time, we have seen the audio industry go from the CD system standard of 44.1 kHz, and the pro audio quasi-standard of 48 kHz, to 8-times and 16-times oversampling frequencies of around 350 kHz and 700 kHz respectively. With sampling frequencies this high, aliasing is no longer an issue.

Okay. So audio signals can be changed into digital words (digitized) without loss of information, and with no aliasing effects, as long as the sampling frequency is high enough. How is this done?

Quantization
Quantizing is the process of determining which of the possible values (determined by the number of bits or voltage reference parts) is the closest value to the current sample - i.e., you are assigning a quantity to that sample.

Quantizing, by definition then, involves deciding between two values and thus always introduces error. How big the error, or how accurate the answer, depends on the number of bits. The more bits, the better the answer. The converter has a reference voltage which is divided up into 2n parts, where n is the number of bits. Each part represents the same value. Since you cannot resolve anything smaller than this value, there is error. There is always error in the conversion process. This is the accuracy issue.

The number of bits determines the converter accuracy. For 8-bits, there are 28 = 256 possible levels as shown in Fig. 3. Since the signal swings positive and negative there are 128 levels for each direction. Assuming a ±5 V reference [3], this makes each division, or bit, equal to 39 mV (5/128 = .039).

Hence, an 8-bit system cannot resolve any change smaller than 39 mV. This means a worst-case accuracy error of 0.78 percent. Table 1 compares the accuracy improvement gained by 16-bit, 20-bit and 24-bit systems along with the reduction in error. (Note: this is not the only way to use the reference voltage.

Many schemes exist for coding, but this one nicely illustrates the principles involved.) Each step size (resulting from dividing the reference into the number of equal parts dictated by the number of bits) is equal and is called a quantizing step (also called quantizing interval -- see Fig. 4).

Originally this step was termed the LSB (least significant bit) since it equals the value of the smallest coded bit, however it is an illogical choice for mathematical treatments and has since be replaced by the more accurate term quantizing step.

Table 1. Quantization Steps For ±5 Volts Reference

Figure 4. Quantization -- 3-Bit, 5V Example

The error due to the quantizing process is called quantizing error (no definitional stretch here). As shown earlier, each time a sample is taken there is error. Here's the not obvious part: the quantizing error can be thought of as an unwanted signal which the quantizing process adds to the perfect original. An example best illustrates this principle. Let the sampled input value be some arbitrarily chosen value, say, 2 volts. And let this be a 3-bit system with a 5 volt reference. The 3-bits divides the reference into 8 equal parts (23 = 8) of 0.625 V each, as shown in Fig. 4. For the 2 volt input example, the converter must choose between either 1.875 volts or 2.50 volts, and since 2 volts is closer to 1.875 than 2.5, then it is the best fit.

This results in a quantizing error of -0.125 volts, i.e., the quantized answer is too small by 0.125 volts. If the input signal had been, say, 2.2 volts, then the quantized answer would have been 2.5 volts and the quantizing error would have been +0.3 volts, i.e., too big by 0.3 volts.

These alternating unwanted signals added by quantizing form a quantized error waveform, that is a kind of additive broadband noise that is generally uncorrelated with the signal and is called quantizing noise. Since the quantizing error is essentially random (i.e. uncorrelated with the input) it can be thought of like white noise (noise with equal amounts of all frequencies). This is not quite the same thing as thermal noise, but it is similar. The energy of this added noise is equally spread over the band from dc to one-half the sampling rate. This is a most important point and will be returned to when we discuss delta-sigma converters and their use of extreme oversampling.

Successive Approximation
Successive approximation is one of the earliest and most successful analog-to-digital conversion techniques. Therefore, it is no surprise it became the initial A/D workhorse of the digital audio revolution. Successive approximation paved the way for the delta-sigma techniques to follow.

The heart of any A/D circuit is a comparator. A comparator is an electronic block whose output is determined by comparing the values of its two inputs. If the positive input is larger than the negative input then the output swings positive, and if the negative input exceeds the positive input, the output swings negative.

Therefore if a reference voltage is connected to one input and an unknown input signal is applied to the other input, you now have a device that can compare and tell you which is larger.

Thus a comparator gives you a "high output" (which could be defined to be a "1") when the input signal exceeds the reference, or a "low output" (which could be defined to be a "0") when it does not. A comparator is the key ingredient in the successive approximation technique as shown in Figures 5A & 5B.

The name successive approximation nicely sums up how the data conversion is done. The circuit evaluates each sample and creates a digital word representing the closest binary value. The process takes the same number of steps as bits available, i.e., a 16-bit system requires 16 steps for each sample. The analog sample is successively compared to determine the digital code, beginning with the determination of the biggest (most significant) bit of the code.

Figure 5B. Successive Approximation A/D Converter

The description given in Daniel Sheingold's Analog-Digital Conversion Handbook (see References) offers the best analogy as to how successive approximation works. The process is exactly analogous to a gold miner's assay scale, or a chemical balance as seen in Figure 5A. This type of scale comes with a set of graduated weights, each one half the value of the preceding one, such as 1 gram, 1/2 gram, 1/4 gram, 1/8 gram, etc. You compare the unknown sample against these known values by first placing the heaviest weight on the scale. If it tips the scale you remove it; if it does not you leave it and go to the next smaller value.

If that value tips the scale you remove it, if it does not you leave it and go to the next lower value, and so on until you reach the smallest weight that tips the scale. (When you get to the last weight, if it does not tip the scale, then you put the next highest weight back on, and that is your best answer.) The sum of all the weights on the scale represents the closest value you can resolve.

In digital terms, we can analyze this example by saying that a "0" was assigned to each weight removed, and a "1" to each weight remaining -- in essence creating a digital word equivalent to the unknown sample, with the number of bits equaling the number of weights. And the quantizing error will be no more than 1/2 the smallest weight (or 1/2 quantizing step).

As stated earlier the successive approximation technique must repeat this cycle for each sample. Even with today's technology, this is a very time consuming process and is still limited to relatively slow sampling rates, but it did get us into the 16-bit, 44.1 kHz digital audio world.