Signal-to-Error Ratio
With a binary number system, the word length determines the number of quantizing intervals available; this can be computed by raising the word length to the power of 2. In other words, an n-bit word would yield 2n quantization levels. The number of levels determined by the first n = 1 to 24 bits are listed in TBL. 1. For example, an 8-bit word provides 28 = 256 intervals and a 16-bit word provides 2^16 = 65,536 intervals. Note that each time a bit is added to the word length, the number of levels doubles. The more bits, the better the approximation; but as noted, there is always an error associated with quantization because the finite number of amplitude levels coded in the binary word can never completely accommodate an infinite number of analog amplitudes.
It’s difficult to appreciate the accuracy achieved by a 16-bit measurement. An analogy might help: if sheets of typing paper were stacked to a height of 22 feet, a single sheet of paper would represent one quantization level in a 16-bit system. Longer word lengths are even more impressive. In a 20-bit system, the stack would reach 352 feet. In a 24-bit system, the stack would tower 5632 feet in height-over a mile high. The quantizer could measure that mile to an accuracy equaling the thickness of a piece of paper. If a single page were removed, the least significant bit would change from 1 to 0. Looked at in another way, if the driving distance between the Empire State Building and Disneyland was measured with 24-bit accuracy, the measurement would be accurate to within 11 inches. A high-quality digital audio system thus requires components with similar tolerances-not a trivial feat.
TBL. 1 The number (N) of quantization intervals in a binary word is N = 2n,
where n is the number of bits in the word.
At some point, the quantizing error approaches inaudibility. Most manufacturers have agreed that 16 to 20 bits provide an adequate representation; however, that does not rule out longer data words or the use of other signal processing to optimize quantization and thus reduce quantization error level. For example, the DVD and Blu-ray formats can code 24-bit words and many audio recorders use noise shaping to reduce in-band quantization noise.
Word length determines the resolution of a digitization system and hence provides an important specification for evaluating system performance. Sometimes the quantized interval will be exactly at the analog value; usually it will not.
At worst, the analog level will be one-half interval away- that is, the error is half the least significant bit of the quantization word. For example, consider Fgr. 5.
Suppose the binary word 101000 corresponds to the analog interval of 1.4 V, 101001 corresponds to 1.5 V, and the analog value at the sample time is unfortunately 1.45 V.
Because 101000 1/2 is not available, the quantizer must round up to 101001 or down to 101000. Either way, there will be an error with a magnitude of one-half of an interval.
FGR. 5 Quantization error is limited to one-half of the least significant
bit.
FGR. 6 Signals can be quantized in one of two ways. A.
A midtread quantizer. B. A midrise quantizer. Q (or 1 LSB) is the quantizer step size.
Generally, uniform step-size quantization is accomplished in one of two ways, as shown in the staircase functions in Fgr. 6. Both methods provide equal numbers of positive and negative quantization levels. A midtread quantizer ( Fgr. 6A) places one quantization level at zero (yielding an odd number of steps or 2n - 1 where n is the number of bits); this architecture is generally preferred in many converters. A midrise quantizer ( Fgr. 6B), with an even number of steps 2n, does not have a quantization level at zero. A/D converter architecture is described in Section 3.
FGR. 7 The amplitude value is rounded to the nearest quantization step. Quantization
error at sample times is less than or equal to 1/2 LSB.
Quantization error is the difference between the actual analog value at sample time and the selected quantization interval value. At sample time, the amplitude value is rounded to the nearest quantization interval, as shown in Fgr. 7. At best (sample points 11 and 12 in the figure), the waveform coincides with quantization intervals. At worst (sample point 1 in the figure), the waveform is exactly between two intervals. Quantization error is thus limited to a range between + Q/2 and - Q/2, where Q is one quantization interval (or 1 LSB). Note that this selection process, of one level or another, is the basic mechanism of quantization, and occurs for all samples in a digital system.
Moreover, the magnitude of the error is always less than or equal to 1/2 LSB. This error results in distortion that is present for an audio signal of any amplitude. When the signal is large, the distortion is proportionally small and likely masked. However, when the signal is small, the distortion is proportionally large and might be audible.
In characterizing digital hardware performance, we can determine the ratio of the maximum expressible signal amplitude to the maximum quantization error; this determines the signal-to-error (S/E) ratio of the system. The S/E ratio of a digital system is similar, but not identical to the signal-to-noise (S/N) ratio of an analog system. The S/E relationship can be derived using a ratio of S/E voltage levels.
Consider a quantization system in which n is the number of bits, and N is the number of quantization steps. As noted:
N = 2n
Half of these 2n values are used to code each part of the bipolar waveform. If Q is the quantizing interval, the peak values of the maximum signal levels are ±Q2n-1. Assuming a sinusoidal input signal, the maximum root mean square (rms) signal Srms is:
The energy of the quantization error can also be determined. When the input signal has high amplitude and wide spectrum, the quantization error is statistically independent and uniformly distributed between the + Q/2 and - Q/2 limits, and zero elsewhere, where Q is one quantization interval. This dictates a uniform probability density function with amplitude of 1/Q; the error is random from sample to sample, and the error spectrum is flat.
Ignoring error outside the signal band, the rms quantization error Erms can be found by summing (integrating) the product of the error and its probability:
The power ratio determining the signal to quantization error is:
Expressing this ratio in decibels:
Using this approximation, we observe that each additional bit increases the S/E ratio (that is, reduces the quantization error) by about 6 dB, or a factor of two. For example, 16-bit quantization ideally yields an S/E ratio of about 98 dB, but 15-bit quantization is inferior at 92 dB.
Looked at in another way, when the word length is increased by one bit, the number of quantization intervals is doubled. As a result, the distance between quantization intervals is halved, so the amplitude of the quantization error is also halved. Longer word lengths increase the data signal bandwidth required to convey the signal. However, the signal-to-quantization noise power ratio increases exponentially with data signal bandwidth. This is an efficient relationship that approaches the theoretical maximum, and it’s a hallmark of coded systems such as pulse-code modulation (PCM) described in Section 3. The value of 1.76 is based on the statistics (peak-to-rms ratio) of a full-scale sine wave of peak amplitude; it will differ if the signal's peak-to-rms ratio is different from that of a sinusoid.
It also is important to note that this result assumes that the quantization error is uniformly distributed, and quantization is accurate enough to prevent signal correlation in the error waveform. This is generally true for high-amplitude complex audio signals where the complex distortion components are uncorrelated, spread across the audible range, and perceived as white noise. However, this is not the case for low-amplitude signals, where distortion products can appear.
Quantization Error
Analysis of the quantization error of low-amplitude signals reveals that the spectrum is a function of the input signal.
The error is not noise-like (as with high-amplitude signals); it’s correlated. At the system output, when the quantized sample values reconstruct the analog waveform, the in band components of the error are contained in the output signal. Because quantization error is a function of the original signal, it cannot be described as noise; rather, it must be classified as distortion.
As noted, when quantization error is random from sample to sample, the rms quantization error
Erms = Q/(12) 1/2.
This equation demonstrates that the magnitude of the error is independent of the amplitude of the input signal, but depends on the size of the quantization interval; the greater the number of intervals, the lower the distortion.
However, the relevant number of intervals is not only the number of intervals in the quantizer, but also the number of intervals used to quantize a particular signal. A maximum peak-to-peak signal (as used in the preceding analysis) presents the best case scenario because all the quantization intervals are exercised. However, as the signal level decreases, fewer and fewer levels are exercised, as shown in Fgr. 8. For example, given a 16-bit quantizer, a half-amplitude signal would be mapped into half of the intervals. Instead of 65,536 levels, it would see 32,768 intervals. In other words, it would be quantized with 15-bit resolution.
The problem increases as the signal level decreases. A very low-level signal , for example, might receive only single bit quantization or might not be quantized at all. In other words, as the signal level decreases, the percentage of distortion increases. Although the distortion percentage might be extremely small with a high level, 0 dBFS (dB Full Scale) signal, its percentage increases significantly at low amplitude , for example, -90 dBFS levels. As described in a following section, dither must be used to alleviate the problem.
FGR. 8 The percentage of quantization error increases as the signal level
decreases. Full-scale waveform A has relatively low error (16-bit resolution).
Half-scale waveform B has higher error (effectively 15-bit resolution). Low
amplitude waveform C has very high error.
The error floor of a digital audio system differs from the noise floor of an analog system because in a digital system the error is a function of the signal. The nature of quantization error varies with the amplitude and nature of the audio signal. For broadband, high-amplitude input signals (such as are typically found in music), the quantization error is perceived similarly to white noise. A high-level complex signal may show a pattern from sample to sample; however, its quantization error signal shows no pattern from sample to sample. The quantization error is thus independent of the signal (and thus assumes the characteristics of noise) when the signal is high level and complex. The only difference between this error noise and analog noise is that the range of values is more limited for a constant rms value. In other words, all values are equally likely to fall between positive and negative peaks. On the other hand, analog noise is Gaussian-distributed, so its peaks are higher than its rms value.
However, the perceptual qualities of the error are less benign for low-amplitude signals and high-level signals of very narrow bandwidth. This is based on the fact that white noise is perceptually benign because successive values of the signal are random, whereas predictable noise signals are more readily perceived. For broadband high-level signals, the statistical correlation between successive samples is very low; however, it increases for broadband low-level signals and narrow bandwidth, high-level signals.
As the statistical correlation between samples increases, error initially perceived as benign white noise become more complex, yielding harmonic and intermodulation distortion as well as signal-dependent modulation of the noise floor.
Quantization distortion can take many guises. For example, the quantized signal might contain components above the Nyquist frequency; thus, aliasing might occur.
The components appear after the sampler, but are effectively sampled. The effects of sampling the output of a limiter or limiting the output of a sampler are indistinguishable. If the signal is high level or complex, the alias components will add to the other complex, noise-like errors. If the input signal is low level and simple, the aliased components might be quite audible. Consider a system with sampling frequency of 48 kHz, bandlimited to 24 kHz.
When a 5-kHz sine wave of amplitude of one quantizing step is applied, it’s quantized as a sampled 5-kHz square wave. Harmonics of the square wave appear at 15, 25, and 35 kHz. The latter two alias back to 23 and 13 kHz, respectively. Other harmonics and aliases appear as well.
The aliasing caused by quantization can create an effect called granulation noise, so called because of its gritty sound quality. With high-level signals, the noise is masked by the signal itself. However, with low-level signals, the noise is audible. This blend of gritty, modulating noise and distortion has no analog counterpart and is audibly unpleasant. Furthermore, if the alias components are near a multiple of the sampling frequency, beat tones can be created, producing an odd sound called "bird singing" or "birdies." A decaying tone presents a waveform descending through quantization levels; the error is perceptually changed from white noise to discrete distortion components. The problem is aggravated because even complex musical tones become more sinusoidal as they decay. Moreover, the decaying tone will tend to amplitude-modulate the distortion components.
Dither addresses these quantization problems.
Other Architectures
Quantization is more than just word length; it also is a question of hardware architecture. There are many techniques for assigning quantization levels to analog signals. For example, a quantizer can use either a linear or nonlinear distribution of quantization intervals along the amplitude scale. One alternative is delta modulation, in which a one-bit quantizer is used to encode amplitude, using the single bit as a sign bit. In other cases, oversampling and noise shaping can be used to shift quantization error out of the audio band. Those algorithm decisions influence the efficiency of the quantizing bits, as well as the relative audibility of the error. For example, as noted, a linear quantizer produces a relatively high error with low-level signals that span only a few intervals. A nonlinear system using a floating point converter can increase the amplitude of low-level signals to utilize the increase the amplitude of low-level signals to utilize the greatest possible interval span. Although this improves the overall S/E ratio, the noise modulation by-product might be undesirable. Historically, after examining the trade-offs of different quantization systems, manufacturers determined that a fixed, linear quantization scheme is highly suitable for music recording. However, newer low bit-rate coding systems challenge this assumption. Alternative digitization systems are examined in Section 4. Low bit-rate coding is examined in Sections 10 and 11.
Dither
With large-amplitude complex signals, there is little correlation between the signal and the quantization error; thus, the error is random and perceptually similar to analog white noise. With low-level signals, the character of the error changes as it becomes correlated to the signal, and potentially audible distortion results. A digitization system must suppress any audible qualities of its quantization error. Obviously, the number of bits in the quantizing word can be increased, resulting in a decrease in error amplitude of 6 dB per additional bit. This is uneconomical because relatively many bits are needed to satisfactorily reduce the audibility of quantization error. Moreover, the error will always be relatively significant with low-level signals.
Dither is a far more efficient technique. Dither is a small amount of noise that is uncorrelated with the audio signal.
Dither is added to the audio signal prior to sampling. This linearizes the quantization process. With dither, the audio signal is made to shift with respect to quantization levels.
Instead of periodically occurring quantization patterns in consecutive waveforms, each cycle is different.
Quantization error is thus decorrelated from the signal and the effects of the quantization error are randomized to the point of elimination. However, although it greatly reduces distortion, dither adds some noise to the output audio signal. When properly dithered, the number of bits in a quantizer determines the signal's noise floor, but does not limit its low-level detail. For example, signals at -120 dBFS can be heard and measured in a dithered 16-bit recording.
Dither does not mask quantization error; rather, it allows the digital system to encode amplitudes smaller than the least significant bit, in much the same way that an analog system can retain signals below its noise floor. A properly dithered digital system far exceeds the signal to noise performance of an analog system. On the other hand, an undithered digital system can be inferior to an analog system, particularly with low-level signals. High-quality digitization demands dithering at the A/D converter. In addition, digital computations should be digitally dithered to alleviate requantization effects.
Consider the case of an audio signal with amplitude of two quantization intervals, as shown in Fgr. 9A.
Quantization yields a coarsely quantized waveform, as shown in Fgr. 9B. This demonstrates that quantization ultimately acts as a hard limiter; in other words, severe distortion takes place. The effect is quite different when dither is added to the audio signal. Fgr. 9C shows a dither signal with amplitude of one quantization interval added to the input audio signal. Quantization yields a pulse signal that preserves the information of the audio signal, shown in Fgr. 9D. The quantized signal switches up and down as the dithered input varies, tracking the average value of the input signal.
FGR. 9 Dither is used to alleviate the effects of quantization error. A.
An undithered input sine wave signal with amplitude of two LSBs. B. Quantization
results in a coarse coding over three levels. C. Dither is added to the input
sine wave signal. D. Quantization yields a PWM waveform that codes information
below the LSB.
Low-level information is encoded in the varying width of the quantized signal pulses. This encoding is known as pulse-width modulation, and it accurately preserves the input signal waveform. The average value of the quantized signal moves continuously between two levels, alleviating the effects of quantization error. Similarly, analog noise would be coded as a binary noise signal; values of 0 and 1 would appear in the LSB in each sampling period, with the signal retaining its white spectrum. The perceptual result is the original signal with added noise-a more desirable result than a quantized square wave.
Mathematically, with dither, quantization error is no longer a deterministic function of the input signal, but rather becomes a zero-mean random variable. In other words, rather than quantizing only the input signal, the dither noise and signal are quantized together, and this randomizes the error, thus linearizing the quantization process. This particular technique is known as nonsubtractive dither because the dither signal is permanently added to the audio signal; the total error is not statistically independent of the audio signal, and errors are not independent sample to sample. However, nonsubtractive dither techniques do manipulate the statistical properties of the quantizer, statistically rendering conditional moments of the total error independent of the input, effectively decorrelating the quantization error of the samples from the signal, and from each other. The power spectrum of the total error signal can be made white. Subtractive dithering, in which the dither signal is removed after requantization, theoretically provides total error statistical independence, but is more difficult to implement.
John Vanderkooy and Stanley Lipshitz demonstrated the remarkable benefit of dither with a 1-kHz sine wave with a peak-to-peak amplitude of about 1 LSB, as shown in Fig. 10. Without dither, a square wave is output ( Fgr. 10A).
When wideband Gaussian dither with an rms amplitude of about 1/3 LSB is added to the original signal before quantization, a pulse-width-modulated (PWM) waveform results ( Fgr. 10B). The encoded sine wave is revealed when the PWM waveform is averaged 32 times ( Fgr. 10C) and 960 times ( Fgr. 10D). The averaging illustrates how the ear responds in its perception of acoustic signals; that is, the ear is a lowpass filter that averages signals. In this case, a noisy sine wave is heard, rather than a square wave.
FGR. 10 Dither permits encoding of information below the least significant
bit. A. Quantizing a 1-kHz sine wave with peak-to-peak amplitude of 1 LSB without
dither produces a square wave. B. Dither of 1/3 LSB rms amplitude is added
to the sine wave before quantization, resulting in PWM modulation. C. Modulation
conveys the encoded sine wave information, as can be seen after 32 averagings.
D. The encoded sine wave information is more apparent after 960 averagings.
(Vanderkooy and Lipshitz, 1984)
The ear is quite good at resolving narrow-band signals below the noise floor because of the averaging properties of the basilar membrane. The ear behaves as a one-third octave filter with a narrow bandwidth; the quantization error, which is given a white noise character by dither, is averaged by the ear, and the original narrow-band sine wave is heard without distortion. In other words, dither changes the digital nature of the quantization error into white noise, and the ear can then resolve signals with levels well below one quantization level.
This conclusion is an important one. With dither, the resolution of a digitization system is far below the least significant bit; theoretically, there is no limit to the low-level resolution. By encoding the audio signal with dither to modulate the quantized signal, that information can be recovered, even though it’s smaller than the smallest quantization interval. Furthermore, dither can eliminate distortion caused by quantization by reducing those artifacts to white noise. Proof of this is shown in Fgr. 11, illustrating a computer simulation performed by John Vanderkooy, Robert Wannamaker, and Stanley Lipshitz.
The figure shows a 1-kHz sine wave of 4 LSB peak-to peak amplitude. The first column shows the signal without dither. The second column shows the same signal with triangular probability density function dither (explained in the following paragraphs) of 2 LSB peak-to-peak amplitude. In both cases, the first row shows the input signal. The second row shows the output signal. The third row shows the total quantization error signal. The fourth row shows the power spectrum of the output signal (this is estimated from sixty 50% overlapping Hann-windowed 512-point records at 44.1 kHz). The undithered output signal ( Fgr. 11D) suffers from harmonic distortion, visible at multiples of the input frequency, as well as inharmonic distortion from aliasing. The error signal ( Fgr. 11G) of the dithered signal shows artifacts of the input signal; thus, it’s not statistically independent. Although it clearly does not look like white noise, this error signal sounds like white noise and the output signal sounds like a sine wave with noise. This is supported by the power spectrum ( Fgr. 11H) showing that the signal is free of signal-dependent artifacts, with a white noise floor. The highly correlated truncation distortion of undithered quantization is eliminated. However, we can see that dither increases the noise floor of the output signal.
Types of Dither
There are several types of dither signals, generally differentiated by their probability density function (pdf).
Given a random signal with a continuum of possible values, the integral of the probability density function describes the probability of the values over an interval. The probability of where the dither signal falls within an interval defines the area under the function. For example, the dither signal might have equal probability of falling anywhere over an interval, or it might be more likely that the dither signal is in the middle of the interval. An interval , for example, might be 1 or 2 LSBs wide. For audio applications, interest has focused on three dither signals: Gaussian pdf, rectangular (or uniform) pdf, and triangular pdf, as shown in Fgr. 12.
For example, we might speak of a statistically independent, white dither signal with a triangular pdf having a peak-to peak level or width of 2 LSB. Fgr. 13 shows how triangular pdf dither of 2-LSB peak-to-peak level would be placed in a midrise quantizer.
FGR. 11 Computer-simulated quantization of a low-level 1-kHz sine wave without
and with dither. A. Input signal. B. Output signal (no dither). C. Total error
signal (no dither). D. Power spectrum of output signal (no dither). E. Input
signal. F. Output signal (triangular dither). G. Total error signal (triangular
dither). H. Power spectrum of output signal (triangular dither). (Lipshitz, Wannamaker,
and Vanderkooy, 1992)
FGR. 12 Probability density functions are used to describe dither signals.
A. Rectangular pdf dither. B. Triangular pdf dither. C. Gaussian pdf dither.
FGR. 13 Triangular pdf dither of 2-LSB peak-to-peak level is placed at the
origin of a midrise quantizer.
Dither signals may have a white spectrum. However , for some applications, the spectrum can be shaped by correlating successive dither samples without modifying the pdf; for example, a highpass triangular pdf dither signal could easily be created. By weighting the dither to higher frequencies, the audibility of the noise floor can be reduced.
All three dither types are effective at linearizing the transfer characteristics of quantization, but differ in their results. In almost all applications, triangular pdf dither is preferred. Rectangular and triangular pdf dither signals add less overall noise to the signal, but Gaussian dither is easier to implement in the analog domain.
Gaussian dither is easy to generate with common analog techniques; for example, a diode can be used as a noise source. The dither noise must vary between positive and negative values in each sampling period; its bandwidth must be at least half the sampling frequency. Gaussian dither with an rms value of 1/2 LSB will essentially linearize quantization errors; however, some noise modulation is added to the audio signal. The undithered quantization noise power is Q2/12 (or Q/(12) 1/2 rms), where Q is 1 LSB.
Gaussian dither contributes noise power of Q2/4 so that the combined noise power is Q2/3 (or Q/(3) 1/2 rms). This increase in noise floor is significant.
Rectangular pdf dither is a uniformly distributed random voltage over an interval. Rectangular pdf dither lying between ±1/2 LSB (that is, a noise signal having a uniform probability density function with a peak-to-peak width that equals 1 LSB) will completely linearize the quantization staircase and eliminate distortion products caused by quantization. However, rectangular pdf does not eliminate noise floor modulation. With rectangular pdf dither, the noise level is more apt to be dependent on the signal, as well as width of the pdf. This noise modulation might be objectionable with very low frequencies or dynamically varied signals. If rectangular pdf dither is used, to be at all effective, its width must be an integer multiple of Q.
Rectangular pdf dither of ± Q/2 adds a noise power of Q2/12 to the quantization noise of Q2/12; this yields a combined noise power of Q2/6 (or Q/(6) 1/2 rms).
It’s believed that the optimal nonsubtractive dither signal is a triangular pdf dither of 2 LSB peak-to-peak width , formed by summing (convolving the density functions) of two independent rectangular pdf dither signals each 1 LSB peak-to-peak width. Triangular pdf dither eliminates both distortion and noise floor modulation. The noise floor is constant; however, the noise floor is higher than in rectangular pdf dither. Triangular pdf dither adds a noise power of Q2/6 to the quantization noise power of Q2/12; this yields a combined noise power of Q2/4 (or Q/2 rms). The AES17 standard specifies that triangular pdf dither be used when evaluating audio systems. Because all analog signals already contain Gaussian noise that acts as dither, A/D converters don’t necessarily use triangular pdf dither.
In some converters, Gaussian pdf dither is applied.
Using optimal dither amplitudes, relative to a nondithered signal, rectangular pdf dither increases noise by 3 dB, triangular pdf dither increases noise by 4.77 dB, and Gaussian pdf dither increases noise by 6 dB. In general, rectangular pdf is sometimes used for testing purposes because of its expanded S/E ratio, but triangular pdf is far preferable for most applications including listening purposes, in spite of its slightly higher noise floor.
Clearly, Gaussian dither has a noise penalty. Because rectangular and triangular pdf dither are easily generated in the digital domain, they are always preferable to Gaussian dither in requantization applications prior to D/A conversion. When measuring the low-level distortion of digital audio products, it’s important to use dithered test signals; otherwise, the measurements might reflect distortion that is an artifact of the test signal and not of the hardware under test. However, a dithered test signal will limit measured noise level and distortion performance. In practical use, analog audio signals contain thermal (Gaussian) noise; even when otherwise theoretically optimal dither is added, nonoptimal results are obtained.
FGR. 14 Input/output transfer characteristic showing the effects of dither
of varying amplitudes. A. Gaussian pdf dither of 1/2 LSB rms linearizes the
audio signal. B. Rectangular pdf dither of 1 LSB linearizes the audio signal.
(Vanderkooy and Lipshitz, 1984)
The amplitude of any dither signal is an important concern. Fgr. 14 shows how a quantization step is linearized by adding different amplitudes (width of pdf) of Gaussian pdf and rectangular pdf dither. In both cases, quantization artifacts are decreased as relatively higher amplitudes of dither are added. As noted, a Gaussian pdf signal with an amplitude of 1/2 LSB rms provides a linear characteristic. With rectangular pdf dither, a level of 1 LSB peak-to-peak provides linearity. In either case, too much dither overly decreases the S/N ratio of the digital system with no additional benefit.
The increase in noise yielded by dither is usually negligible given the large S/E ratio inherent in a digital system, and its audibility can be minimized , for example, with a highpass dither signal. This can be easily accomplished with digitally generated dither. For example, the spectrum of a triangular pdf dither can be processed so that its amplitude rises at high audio frequencies. Because the ear is relatively insensitive to high frequencies, this dither signal will be less audible than broadband dither, yet noise modulation and signal distortion are removed. Such techniques can be used to audibly reduce quantization error , for example, when converting a 20-bit signal to a 16 bit signal. More generally, signal processing can be used to psychoacoustically shape the quantization noise floor to reduce its audibility. Noise-shaping applications are discussed in Section18.
Designers have observed that the amplitude of a dither signal can be decreased if a sine wave with a frequency just below the Nyquist frequency, with an amplitude of 1 or 1/2 quantization interval, is added to the audio signal. The added signal must be above audibility yet below the Nyquist frequency to prevent aliasing. It alters the spectrum of quantization error to minimize its audibility and overall does not add as much noise to the signal as broadband dither. For example, discrete triangular pdf dither might yield a 2-dB penalty, as opposed to 4.77 dB. However, a discrete dither frequency might lead to intermodulation products with audio signals. Wideband dither signals alleviate this artifact.
FGR. 15 An example of a subtractive digital dither circuit using a pseudo-random
number generator. (Blesser, 1983) An additive dither signal necessarily decreases
the S/E ratio of the digitization system. A subtractive dither signal proposed
by Barry Blesser that would preserve the S/E ratio is shown in Fgr. 15. Rectangular
noise is a random valued signal that can be simulated by generating a quickly
changing pseudo-random sequence of digital data. This can be accomplished with
a series of shift registers and a feedback network comprising exclusive or
gates. This sequence is input to a D/A converter to produce analog noise which
is added to the audio signal to achieve the benefit of dither. Then, following
A/D conversion, the dither is digitally subtracted from the audio signal, preserving
the dynamic range of the original signal. A further benefit is that inaccuracies
in the A/D converter are similarly randomized.
Other additive-subtractive methods call for two synchronized pseudo-random signal generators, one adding rectangular pdf dither at the A/D converter, and the other subtracting it at the D/A converter. Alternatively, in an auto-dither system, the audio signal itself could be randomized to create an added dither at the A/D converter, then re-created at the D/A converter and subtracted from the audio signal to restore the dynamic range.
Digital dither must be used to decrease distortion and artifacts created by round-off error when signal manipulation takes place in the digital domain. For example, the truncation associated with multiplication can cause objectionable error. Digital dither is described in Section 17.
For the sake of completeness, and although the account is difficult to verify, one of the early uses of dither came in World War II . Airplane bombers used mechanical computers to perform navigation and bomb trajectory calculations. Curiously, these computers (boxes filled with hundreds of gears and cogs) performed more accurately when flying on board the aircraft, and less well on terra firma. Engineers realized that the vibration from the aircraft reduced the error from sticky moving parts. Instead of moving in short jerks, they moved more continuously. Small vibrating motors were built into the computers, and their vibration was called dither from the Middle English verb "didderen," meaning "to tremble." Today, when you tap a mechanical meter to increase its accuracy, you are applying dither, and dictionaries define dither as "a highly nervous, confused, or agitated state." At any rate, in minute quantities, dither successfully makes a digitization system a little more analog in the good sense of the word.
Summary
Sampling and quantizing are the two fundamental elements of an audio digitization system. The sampling frequency determines signal bandlimiting and thus frequency response. Sampling is based on well-understood principles; the cornerstone of discrete-time sampling yields completely predictable results. Aliasing can occur when the sampling theorem is not observed. Quantization determines the dynamic range of the system, measured by the S/E ratio. Although bandlimited sampling is a lossless process, quantization is one of approximation. Quantization artifacts can severely affect the performance of a system.
However, dither can eliminate quantization distortion, and maintain the fidelity of the digitized audio signal. In general, a sampling frequency of 44.1 kHz or 48 kHz and a dithered word length of 16 to 20 bits yields fidelity comparable to or better than the best analog systems, with advantages such as longevity and fidelity of duplication. Still higher sampling frequencies and longer word lengths can yield superlative performance. For example, a sampling frequency of 192 kHz and a word length of 24 bits is available in the Blu-ray disc format.
Notes:
A special note on high sampling frequencies: Before ending our discussion of discrete time sampling, consider a hypothesis concerning the nature of time. Time seems to be continuous. However, physicists have suggested that, just as this guide consists of a finite number of atoms, time might come in discrete intervals. Specifically, the indivisible period of time might be 10^-43 second, known as Planck time. One theory is that no time interval can be shorter than this because the energy required to make the division would be so great that a black hole would be created and the event would be swallowed up inside it. If any of you are experimenting in your basements with very high sampling frequencies, please be careful.