Analog-to-Digital Conversion The A/D conversion process consists of three steps:

1. sampling 2. quantizing 3. coding

The sampling process means the taking of a number of sample values at evenly spaced intervals across a continuous signal. As shown in Figure 5.2, a higher- repetition frequency of the sample-taking process will result in better assessment and, later, representation of that signal. To sample analog electric signals that change over time means to take as many samples in a unit of time as is workable. An insufficient number of samples leads to an inadequate assessment and pos- sible misinterpretation of the original form of the sampled signal, but too many samples can lead to an unnecessary accumulation of redundant data, thus wasting time and resources. For example, measuring the outside temperature two times a day, for example, at 8:00 a.m. and 8:00 p.m. and obtaining identical values in each sample could lead to an erroneous conclusion that the temperature was constant for 24 hours while in reality it was probably higher at noon and lower

(a) Analog signal (b) Sampling frequency (c) Output

during the night. The insufficient number of sample values results in the wrong assessment of the measured phenomena. On the other hand, if measurements are taken every 10 seconds, they will provide 8,640 values for one day, of which a good deal will be redundant as it takes longer that 10 seconds for there to be a noticeable change in the outside temperature. Subsampling leads to an erroneous interpretation of the observed phenomena, while oversampling results in a waste of time and resources and a lot of largely redundant data.

Neglecting the frequency at which a sampled signal changes over time and subsampling it could result in the appearance of undesired artifacts known as

aliasing effects. The most obvious example of an aliasing effect is the impression one

has in Western movies that stagecoach wheels rotate backwards. The technique of motion reproduction in movie-making is based on a temporal sampling of the action that takes place in front of the lens. The action is sampled with a frequency of 24 pictures per second, which is far below the rotational frequency of the wheels. Consequently, such a subsampling produces a visible aliasing effect of backward rotation of the wheels. Namely, between two samples (two frames) the wheels have achieved more than one full revolution and the second sample (frame) appears to show the wheel in a position that precedes the one captured in the first frame.

It is therefore necessary to define the optimum number of samples, or the optimum value for the sampling frequency, that would permit a good assessment of the sampled signal and, at the same time, avoid the accumulation of redundant data. That optimum value is defined by the theorem of the Swedish mathematician Nyquist, which could be summarized as: “The sampling frequency should be at least two times higher than the highest frequency of the sampled signal.” So, for example, if the observed phenomenon changes 10 times during a given time interval, we should sample it at least 20 times during that same time interval. That value of sampling frequency is frequently called the Nyquist limit, the Nyquist

criterion, or the Nyquist frequency.

Figure 5.2a represents an analog signal. The sampling of that signal with a selected sampling frequency (Figure 5.2b) will result in a number of discrete sam- ples or pulses, which have the chosen repetition rate (equal to the sampling frequency) and whose amplitudes correspond to the values of the amplitude of the sampled analog signal at the point of its intersection with the sampling frequency (Figure 5.2c). As stated previously, the precision of that first step in the digitization process depends on the selected sampling frequency. To ensure an optimum interpretation of the sampled signal, it is recommended to select a sampling frequency that is at least 20% higher than the theoretical minimum set by the Nyquist theorem. The highest frequency of a video signal is usually 5 MHz, which means that the sampling frequency for video signals must be higher than 10 MHz.

5.2 Analog-to-Digital Conversion 59

3 2 1 0

3 quantization levels loss of information

Figure 5.3 Quantizing.

Quantizing is the next step of the digitization process. In a way, it is a similar

process to sampling as it consists of interpreting the value of the amplitude of each sample with a set of predetermined equidistant steps. As can be seen in Figure 5.3, the amplitude of each sample is approximated to the nearest prede- termined level, so that any value that is higher than the previous predetermined level and lower than the next one will be disregarded as nonexistent. Therefore, it could be stated that an insufficient number of selected levels (only four in the illustrated example—from 0 to 3) unavoidably results in a considerable loss of valuable information or, in other words, in a misinterpretation of the real value of the quantized amplitudes. Also as with sampling, more levels will result in more data to handle, and for each application there is a limit to how much is technically and economically viable. The number of levels is determined by the nature of the sampled signal, the acuity of our senses, and the desired or needed level of transparency of the coupled A/D and D/A conversions.

In the third step, coding, each of the amplitude levels that coincides with the predetermined quantizing steps or levels will be represented with a given symbol composed of a set of binary values of 1 and 0 (bits), values that correspond to the two states of the constant amplitude pulse stream—the presence or absence of a pulse. Such a set of bits used for the description of amplitude levels is called a

binary group or binary word.

The example in Figure 5.3 shows four predetermined levels. These four levels can be described with different combinations (or sets) of only two bits:

Level 0 00 Level 1 01 Level 2 10 Level 3 11

This example shows that with two given binary values (0 and 1) and with two bits, it is possible to achieve four combinations, that is, to describe only four

levels (or, expressed mathematically, 22 = 4 combinations), which is adequate for our simplified explanation but not for coding complex signals. But from this simple case it is possible to draw a general rule that the number of levels that can be unequivocally described by different combinations of the bits constituting the selected binary could be expressed as 2 to the power of n, where n is the selected number of bits.

It is clear that the complexity of the video signal will require more than four levels for an adequate digital description of the analog signal. Theoretically there is no limit to the number of quantization levels that can be used, although more levels means more bits needed to describe them resulting in a higher overall bit rate to be handled. The higher the bit rate to be handled, the larger the necessary bandwidth of the transmission channel and the higher the complexity and cost of the equipment needed to handle such digital signals. Therefore, the exact defini- tion of the number of quantization levels needed for an acceptable description of a video signal has to take into account the criteria of the acuity of human senses, the required transparency of the A/D and D/A processes, and the always-present need to keep the bit rate at an acceptable level. After long studies and debates, eight-bit binary words were adopted as an optimum solution for video signals, allowing thus the description of 28_{, or 256 predetermined levels.}

The situation in the audio domain is slightly different. The highest sound frequency is considerably lower than the upper video frequency and is usually set at 20 or 22 kHz. On the other hand, the character of the sound signal and the psychophysical characteristics of the ear-brain combination require a considerably finer definition, thus a much larger number of quantizing levels. In view of that, it has been agreed to adopt 16 to 20 bits per sample for the definition of digital audio signals.

As mentioned above, in the digital domain the complexity of the necessary equipment and the required channel bandwidth are determined by the bit rate, or the quantity of bits per second that have to be transported or handled. That value can be simply obtained by multiplying the chosen sampling frequency, expressed in hertz, by the selected number of bits per sample.

Summing up, the result of an analog-to-digital conversion of a given analog signal, achieved in three steps (sampling, quantizing, and coding), is a stream of constant amplitude pulses of a given repetition rate that can be considered to be a mathematical description of the original signal. The quality and precision of that “description” will determine how faithful to the original analog signal will be the signal obtained at the end of the line through an inverse process of digital-to- analog conversion. It is clear that in broadcasting the parameters selected for the A/D conversion have to ensure a maximum transparency between the original analog signal and the analog signal at the output of the D/A converter that is at the end of the chain.