Quantization

Resolution properly refers to spatial phenomena; see page 65. In my view, it is a mistake to refer to

a sample as having “8-bit resolution”: Say quantization or precision instead. To make a 100-foot-long fence with fence posts every 10 feet, you need 11 posts, not 10! Take care to distinguish levels (in the lefthand portion of Figure 4.1, 11) from steps or risers (here, 10).

Eq 4.1

A signal whose amplitude takes a range of continuous values is quantized by assigning to each of several (or several hundred or several thousand) intervals of amplitude a discrete, numbered level. In uniform quantization, the steps between levels have equal amplitude. Quantization discards signal information lying between quantizer levels. Quantizer performance is characterized by the extent of this loss. Figure 4.1 shows, on the left, the transfer function of a uniform quantizer.

Linearity

Electronic systems are often expected to satisfy the principle of superposition; in other words, they are expected to exhibit linearity. A system g is linear if and only if (iff) it satisfies both of these conditions:

Figure 4.1 A Quantizer transfer function is shown on the left. The usual 0 to 255 range of quantized R’G’B’ components in computing is sketched on the right.

The function g can encompass an entire system:

A system is linear iff the sum of the individual responses of the system to any two signals is identical to its response to the sum of the two. Linearity can pertain to

255

STEP (riser)

LEVEL (tread)

Bellamy, John C. (2000),

Digital Telephony, Third Edition (New York: Wiley): 98–111 and 472–476.

Eq 4.2 Power ratio, in decibels:

Eq 4.3 Power ratio, with respect to a reference power:

steady-state response, or to the system’s temporal response to a changing signal.

Linearity is a very important property in mathematics, signal processing, and video. Many electronic systems operate in the linear intensity domain, and use signals that directly represent physical quantities. One example is compact audio disc (CD) coding: Sound pressure level (SPL), proportional to physical intensity, is quantized linearly into 16-bit samples.

Human perception, though, is nonlinear, and in applications where perceptual quantities are being encoded or transmitted, the perceptual nonlinearity can be exploited to achieve coding more efficient than coding the raw physical quantity. For example, audio for digital telephony is nonlinearly coded using just

8 bits per sample. (Two coding laws are in use, A-law and µ-law; both of these involve decoder transfer functions that are comparable to bipolar versions of Figure 4.1.) Image signals that are captured, recorded, processed, or transmitted can similarly be coded in

a nonlinear, perceptually uniform manner in order to optimize perceptual performance.

Decibels

In the following sections, I will describe signal amplitude, noise amplitude, and the ratio between these – the signal to noise ratio (SNR). In engineering, ratios such as SNR are usually expressed in logarithmic units. A power ratio of 10:1 is defined as a bel (B), in honour of Alexander Graham Bell. A more practical measure is one-tenth of a bel – a decibel (dB), which represents a power ratio of 100.1, or about 1.259. The ratio (expressed in decibels) of a power P1 to a power P2 is

given by Equation 4.2. Signal power is often given with respect to a reference power PREF, which must either be specified (often as a letter following dB) or implied by the context; the computation is expressed in Equation 4.3. An increase of 3 dB in power represents very nearly a doubling of power (100.3 = 1.995). An increase of +10 dB multiplies power exactly tenfold;

a change of -10 dB reduces power to a tenth. Consider a cable conveying a 100 MHz radio

frequency signal. After 100 m of cable, power has diminished to some fraction, perhaps 1⁄8, of its original

Eq 4.4 Power ratio, in decibels, as a function of voltage:

Table 4.1 Decibel examples

In photography, a stop is taken to be a ratio of 2. For scientific and engineering purposes it is more convenient to define a stop as exactly three tenths of a density unit, that is, 100.3, or about 1.995.

value. After another 100 m, power will be reduced by the same fraction again. Rather than expressing this cable attenuation as a unitless fraction 0.125 per 100 m, we express it as 9 dB per 100 m; power at the end of 1 km of cable is -90 dB referenced to the source power.

The decibel is defined as a power ratio. If a voltage source is applied to a constant impedance, and the voltage is doubled, current doubles as well, so power increases by a factor of four. More generally, if voltage (or current) into a constant impedance changes by

a ratio r, power changes by the ratio r2. (The log of r2 is 2 log r.) To compute decibels from a voltage ratio, use Equation 4.4. In digital signal processing (DSP), digital code levels are treated equivalently to voltage; the decibel in DSP is based upon voltage ratios. In historical analog systems it was common to use a reference of 1 mV (dBmV); in digital systems, the reference is usually the “full scale” range from reference black to reference white (dBfs), equivalent to 219 codes at 8-bit interface levels. Beware: Historical 8-bit computer graphics processed 8-bit signals with no footroom and no headroom, and that practice found its way into PSNR calculations in the MPEG community, where it is common to have full scale interpreted as 0–255 instead of 0–219.

Table 4.1 gives numerical examples of decibels used for voltage ratios.

A 2:1 ratio of frequencies is an octave, referring to the eight whole tones in music, do, re, me, fa, sol, la, ti, do, that cover a 2:1 range of frequency. When voltage halves with each doubling in frequency, an electronics engineer refers to this as a loss of 6 dB per octave. If voltage halves with each doubling, then it is reduced to one-tenth at ten times the frequency; a 10:1 ratio of quantities is a decade, so 6 dB/octave is equivalent to 20 dB/decade. (The base-2 log of 10 is very nearly 20⁄6.)

A stop in photography is a 2:1 ratio of light power. As mentioned above, a decibel is a power ratio of 100.1, or about 1.259. Sensor and camera engineers prefer to use units that are equivalent between the optical and electrical domains: They treat digital code level as signal (like voltage), and they describe an optical power of 2 as 6 dB. It is a numerological coincidence that 100.3 is very nearly equal to 2; so 6 dB corresponds to one stop, and 2 dB corresponds to 1/3 stop.

Figure 4.2 Peak-to-peak, peak, and RMS values are measured as the total excursion, half the total excursion, and the square root of the average of squared values, respectively. Here, a noise component is shown.

Noise, signal, sensitivity

Analog electronic systems are inevitably subject to noise introduced from thermal and other sources. Thermal noise is unrelated to the signal being processed. A system may also be subject to external sources of interference. As signal amplitude decreases, noise and interference make a larger relative contribution.

Processing, recording, and transmission may introduce noise that is uncorrelated to the signal. In addition, distortion that is correlated to the signal may be introduced. As it pertains to objective measurement of the performance of a system, distortion is treated like noise; however, a given amount of distortion may be more or less perceptible than the same amount of noise. Distortion that can be attributed to a particular process is known as an artifact, particularly if it has

a distinctive perceptual effect.

In video, signal-to-noise ratio (SNR) is the ratio of the peak-to-peak amplitude of a specified signal, often the reference amplitude or the largest amplitude that can be carried by a system, to the root mean square (RMS) magnitude of undesired components including noise and distortion. (It is sometimes called PSNR, to emphasize peak signal; see Figure 4.2.) SNR is expressed in units of decibels. In many fields, such as audio, SNR is specified or measured in a physical (intensity) domain. In video, SNR usually applies to gammacorrected components R’, G’, B’, or Y’ that are in the perceptual domain; so, SNR correlates with perceptual performance.

Sensitivity refers to the minimum source power that achieves acceptable (or specified) SNR performance.

Quantization error

A quantized signal takes only discrete, predetermined levels: Compared to the original continuous signal, quantization error has been introduced. This error is correlated with the signal, and is properly called distortion. However, classical signal theory deals with the addition of noise to signals. Providing each quantizer step is small compared to signal amplitude, we can consider the loss of signal in a quantizer as addition of an equivalent amount of noise instead: Quantization