Abstract
Quantization is the second main process in conversion. This chapter deals with the mathematical derivation and modeling of quantization in several resolution ranges. Quantization results in several specific parameters: integral and differential linearities and derived problems such as monotonicity. Various shapes of non-linearity are discussed and the relation to harmonic distortion is explained. The signal-to-noise ratio is also affected by quantization. The effect of dither and the relation between differential non-linearity and signal-to-noise are examined.
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Notes
- 1.
Truncation means that the additional fraction of the signal above the reference level is ignored.
- 2.
Other coding schemes will appear when discussing the implementation of complex converters, for instance, locally ternary (three-level) coding can be used in full-differential implementations. In error correction schemes, a base lower than 2 is applied.
- 3.
A group of 8 bits is referred to as 1 byte, irrespective of whether it represents a sample or other data. A speed of 1 GB/s is therefore equivalent to 8 Gb/s. Watch out for upper or lower case “b,B.”
- 4.
- 5.
Other binary code formats are discussed in paragraph 11.1.1.
- 6.
Naming the coefficients is a somewhat double-edged choice. When the LSB uses the index “0,” the mathematical link to the exponent is easy: b020. The consequence is that the MSB carries the index N − 1 as in bN−1, which is counter intuitive for a resolution N. Naming the MSB, bN requires to subtract in the exponent: bN2N−1, which is really confusing.
- 7.
N. Blachman has mathematically analyzed many processes around quantization. His publications from 1960 to 1985 form a good starting point to dive deeper in this field, for instance, [6].
- 8.
And for some smart students of course, who will notice that this DC-shift changes the truncation operation into rounding.
- 9.
Note that in a formal sense just voltage-squared is calculated, which lacks the impedance level and the time span needed to reach the dimension of power: Watt or V2/Ωsec. In quantization theory, “voltage-squared” power is only used to compare to another “voltage-squared” power, assuming that both relate to the same impedance level and the same time span. This quantization error becomes visible to an engineer (mostly) as part of a power spectrum. For that reason this book prefers the term quantization power instead of energy.
- 10.
The second law states that the entropy of a closed system cannot decrease, or the non-entropy energy will evolve to a minimum.
- 11.
As the habit is to call quantization errors “noise,” the “N” in SNQR is kept, the subscript “Q” indicates that quantization is meant.
- 12.
A THD of −43.8 dB is shorthand for 10−43.8∕10 = 4.167 × 10−5 power ratio. Use the exponential notation in complex calculations, as most modern engineers are not educated with logarithms.
- 13.
A manipulation like this is aversely coined: “specmanship”.
- 14.
Non-English speakers often confuse monotonic with monotonous which is synonymous to boring, dull, uninteresting.
- 15.
This is a bit of hand-waving statistics. A real formal derivation takes a page.
- 16.
There has been an extensive search for optimum dither signals in the 1960s–1970s. Wadgy and Goff [16] provides a theoretical background. Today, dither is actively used to mitigate conversion artifacts. The concept provides also valuable insight, for instance, sigma-delta modulators (Σ Δ) can be understood as low-resolution converters that generate their own dither.
- 17.
Although useful as a measure of performance, the importance of an F.o.M. should not be overemphasized. On the ISSCC in the same year, a span of over 100 × can be seen, which displays that other performance aspects dominate.
- 18.
The author of this book was educated with the notion that “log” operations can only be performed on dimensionless quantities. Obviously, this is not the case here.
- 19.
In the evolution of the F.o.M., the equation was inverted, leading to the counterintuitive behavior that a low F.o.M. number is better than a high one.
- 20.
The Signal-to-Noise Ratio is the ratio between signal power and noise power, for instance, 2000. In many notations not this number but its \(10\log ()\) is used, here: 33 dB. Whenever the SNR is used in calculations, the ratio number is used, not the dB value.
- 21.
Compared to Moore’s law for digital circuits from the previous decades, where speed doubles and area and power halve for every generation (2 years), this is a meager result.
- 22.
A memory designer once said “it is not a matter whether a memory device meets the specifications, the discriminating factor is how far a device can be used outside its specifications.”
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Pelgrom, M.J.M. (2022). Quantization. In: Analog-to-Digital Conversion. Springer, Cham. https://doi.org/10.1007/978-3-030-90808-9_9
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