AET 2050 - Daw Production
4. Conversion Principles - Quantization
Chapter 2.4.3-2.4.6, Desktop Audio Technology - Rumsey
Sampling results in a series of time-discreet pulses. After sampling, the modulated pulse chain is quantized.
Quantization is the process of assigning a discreet amplitude value to each discreet pulse.
The pulse's amplitude is measured against a scale of discreet quantities. Its amplitude is assigned to the closest quantity. (See fig. 2.19, p. 19, text)
The value is then given a binary number that can be stored or processed.
Significance of Bit Depth
The assignment of the value is an approximation - the measurement is "rounded off" to the closest predefined quantity, or quantization interval (Q).
The quantizer's accuracy is defined by the size of Q. The size of Q is defined by the number of intervals available.
The number of intervals available is equal to 2n, where n is the number of bits in the word used to represent the sample.
24 = 16 steps, or levels
216 = 65,536 levels
224 = 16,777,216 levels
Each bit that is added doubles the number of available steps. (See fig 2.20, p. 20, text)
The maximum signal range is divided by the number of quantizing intervals which determines the size of each step.
The total range does not increase with extra bits, but the step size decreases.
A given system has a voltage range of 10v.
If 16-bit quantization is used, each step would be 10 / 65,536, or .15 mv per step.
If 24-bit quantization is used, each step would be 10 / 16,777,216, or 596 nV per step.
Here are some practical comparisons:
If a sheet of paper represents one quantization level, how high would a stack of paper representing the entire range be?
If 16 bits, the stack would be 22 ft.
If 20 bits, the stack would be 352 ft.
If 24 bits, the stack would be 5,632 ft. - over a mile!
Or, if the distance between Los Angeles and New York was measured with 24-bit accuracy, the measurement would be accurate within 9 inches.
There is error involved with quantization, due to the approximation of the level being quantized.
Error size will be a maximum of +/- half the amplitude of one quantizing interval (1/2 Q).
Since the quantizing interval is constant, the maximum error is also constant.
The quantization error level is independent of the signal amplitude.
As the signal level gets smaller, the signal-to-error ratio gets larger. Therefore, the error is more apparent with lower levels of audio. (See fig. 2.24, p. 23, text)
A system with a 20v range is quantized at 16 bits. Each step has a size of approximately .3 mv; therefore there is a maximum error of .15 mv.
If a signal were at 0dBu (2.2 v), the signal-to-error ratio would be approx .007%.
If a signal is at -60dBu (2.2 mv), the signal-to error ratio would be approx 6.8%!
Important points regarding quantization error:
Digital Signal-to-Noise and Dynamic Range
and Dynamic Range
With complex audio (music), error is spread over a large amplitude and dynamic spectrum, and manifests more as noise.
Unlike analog noise, quantization noise only exists when signal is present, since it is the result of quantization error.
Also, keep in mind that the smallest amplitude that can be quantized is 1/2 Q.
Therefore, a quantization signal-to-error ratio can be evaluated like a signal-to-noise ratio, or overall dynamic range. Analysis of the error noise level has given way to the following formula for digital signal-to-error(noise) ratio:
6.02n + 1.76 dB
This implies a dynamic range of just over 6dB per bit. 8 bits deliver S/E of around 50 dB, 16 bits give around 98 dB.
Error that is spread out over the frequency spectrum manifests more as noise, not distortion.
If wideband noise is quantized, then the resulting distortion is transformed into a random, noise-like signal as well.
Dither is just such a signal - a wideband noise (e.g. white noise) at a very low level (usually 1/2 the LSB level) that is combined with the signal before quantization. The dither has the effect of removing the quantization distortion by distributing it accross the frequency spectrum.
Using dither in the quantization process provides several results:
Distortion is the result of correlation between the signal and the error, and is subjectively annoying.
(see p 23, Rumsey)
The random noise signal effectively randomizes the error, making it noise-like as well.
(see p 25, Rumsey)
Since dither is full-spectrum, and low-level, its quantized result wil randomly change from one quantization level to another. It is not unlike a pulse train (only random instead of periodic). The chance of its landing on one level or another is approximatly equal and the average energy level over time is zero.
White noise near LSB before and after quantization. Note "pulse width" appearance.
40 samples of 1-bit quantization. Note random pattern.
As a signal with dither is quantized, the changing signal amplitude will skew the probability of the dither's quantization level. The result is like pulse width modulation.
(see p 26, Rumsey)
The modulation skews the energy of the dither up or down, according to the input signal. The resultant energy is then linear, following the signal.
Dither can encode signals whose levels are below the LSB. When a low-level signal is mixed with the dither, it causes pulse width modulation of the dithered signal. When the resultant signal is heard, the ear averages the modulations, and the small signal is perceived.
Two types of dither: analog and digital.
Dither is characterized by probability of the bit changing. There are three different types:
Consider throwing of dice:
1 die, all numbers have same probability. This is Rectangular Probability Distribution (RPDF)
2 dice, the numbers 6-8 have greater probability than2 or 12. This is Triangular Probability Distribution (TPDF).
Analog white noise has Gaussian probability.
Research has shown that the most suitable dither is TPDF with a p-p value of 2 Q. If Rectangular is used, the p-p value should be 1 Q.