Timing pulse code modulation as a tool for quantization noise reduction in special-purpose IT systems

The aim of the article is to examine the issue of eliminating quantization noise in the process of analogue-to-digital conversion by means of timing pulse code modulation. According to the results of the method proposed, it has established that the use of timing modulation significantly increases the number of sampling levels at a given interval without reducing the duration of individual code pulses, that, in its turn, significantly improves signal quality (readability) after the digital-to-analogue conversion on the receiving side.


Introduction
The mathematical model for measuring the signal to quantization noise relation. From the theory of communication, it is known about signal sampling, oversampling (interpolation, decimation etc) and quantization (pulse code modulation). The Nyquist-Shannon sampling theorem determines the sampling increment ≤ 1 2 , with a frequency FD ≥ 2Fmax, where Fmaxis the maximum frequency of the signal spectrum [1,2].
The higher the sampling frequency, the bigger the number of the quantization levels, and, thus, better quality of signal reproduction on the receiving side is to be attained for the "good" state of the channel (Hilbert channel model) [3]. In pulse code modulation (analogue-to-digital conversion) (figure 3a) a pulse code, the combination of which on the receiving side allows setting the amplitude of the signal at a certain point of time, is transmitted in each discrete as well as the clock synchronization pause (or pulse) required for making an appropriate decision on the received information and compensation of pulse shifts.
During a level quantization process, the value of each sample is replaced by the nearest allowed value (figure 3a), so the difference between the true value of the sample and its quantized value i.e. quantization error makes so-called quantization noise occur [4][5][6][7][8].
The standardized statistical characteristics of AD conversion are defined by the IEEE standards [9]. They have been examined in terms of reducing the quantization noise in a number of scientific papers, such as [4,5]. The other works [10, 11,12,13,14] investigate and define the dependences of the signal to quantization noise ratio (SINQ) for different pulse-code modulation systems (control and communication [6,7], radar [10], data/images digitization [11], effective signal circuits construction [12], metrology and measuring technology [13], acoustics [14], space [15] etc).
The types of dependences stated above have the same construct within the level quantization framework and determine that the change in the signal to quantization noise ratio is directly proportional to the pulse code length.
The statistical characteristics of digitized audio exchange signals in military telecommunication systems have been investigated in paper [6] with developing an adequate mathematical model for such exchange in view of certain initial data and limitations. To simplify the calculations, this model is considered to be appropriate for use with the corresponding given data.
Formula (16) from reference [6] is applicable to calculate the signal to quantization noise ratio: ( ) = 10 lg(3 · 2 2 2 ). (1) where: nnumber of bits in the digital code that determines the signal sampling level; σ 2 = 0,126 Vtvariance of the digitized signals accounting on signal sampling maximum values normalized in accordance with the rule [ ‫|‬ ± xmax ‫|‬ = 1 ] at a resistance of 1 Ohm based on the assumptions as follow: quantization is uniform; quantization noise is a stationary random process; quantization noise is not correlated with xi realization samples; quantization noise is not correlated within adjacent intervals. Herewith, figure 1 illustrates the regular dependence of the signal to quantization noise ratio on the number of bits in the bit-digital sampling code (BDC) [5]: Efficacy of some simple means (e.g. interpolation, companding etc) has been proved to solve the quantization noise reduction issue [6]. Increase in number of quantization levels and, consequently, in sampling frequency, appears necessary for quantization noise reduction. Therefore, in case of using a linear code, the code length must also be increased bringing about reduction of pulse duration, that, in its turn, leads to some undesirable consequences like increased computational complexity resulting in higher power inputs as well as expansion of code redundancy resulting in interference immunity deterioration.
Hence, in order to avoid derating of the synthesis parameters, it is necessary to solve the problem of coding the quantization levels so as not to reduce or increase the discrete period as well as not to reduce the pulse duration. To perform this task, it is proposed to examine the timing principles of signal structures having been investigated in the works of M. Zakharchenko and others [3, 16…]. 2. Timing pulse-code modulation as a method of reducing quantization noise. As noted in reference [16] reproduction moments values of a timing signal having been formed in time interval TC = mt0 (where: m  number of Nyquist elements; t0  their duration), in contrast to a bit-digital signal, are not multiple of 0 t , but of some basic element Δ (where Δ = t0 / S; S = 1, 2, 3, …, linteger numbers), which is conventionally called "timing element". Signal segments are transmitted to the channel with duration tTSC = t0 + xiΔ (де xi = 0, 1, 2 … S · (n -i)).
To study the timing properties from this point of view, the timing signal design (see figure 2) with following attributable parameters has been modeled: m = 5duration of a single code structure in the Nyquist elements (considering the clock synchronization pause duration of 1 Nyquist element/bit); n = 4duration of the code word active zone; S = 7 -number of basic elements Δ per 1 Nyquist element (t0 = 7Δ); i = 3number of timing segments and significant reproduction moments. The formation of all possible variants of the design must suit the conditions of the equations set (x1+ x2+ x3) Δ = 7; x1+ x2+ x3 ≤ 7; x1+ x2+ x3 = xn = 0÷7, at x1 = 0÷7, x2 = 0÷7, x3 = 0÷7. (2) An important aspect of timing signal transmission is that each subsequent pulse must change the voltage to fix uniquely a significant modulation moment on the receiving side. Therefore, the transmission of a binary linear code by two-level pulses in the channel shall look like a change of pulses from 1 to 0 and vice versa. As for two-level transmission, realization of two 0-pulses or two 1pulses (see figure 2c and figure 2d) transmitted sequentially by the same voltage proves impossible.
To check elementarily the validity of the assumption, it is possible to assign odd sums of additional elements Δ: xn = 1; 3; 5; 7 for odd numbers {1, 3, 5,… 119} and start the transmission with a 0-pulse of a negative voltage value e.g. U = -1V after the synchronization clock pause (see figure 2c), and also assign even sums of additional elements Δ: xn = 0; 2; 4; 6 for even numbers {0, 2, 4,… 120} and start the transmission with a 1-pulse of a positive voltage value e.g. U = +1V (see figure 2d). Also this approach appears applicable in detecting a multiple error of the significant modulation moment front shift as an odd number kΔ = 1; 3; 5; 7. On the contrary, assignment "0" for the even sum xn = 0; 2; 4; 6 and assignment "1" for the odd sum xn = 1; 3; 5; 7 enables detection of a multiple error of the significant modulation moment front shift as an even number kΔ = 2; 4; 6… Considering the problem of information concealment, such an exchange (discrete interchange of "0" and "1") enables an Electronic Signal Monitoring (interception) station to receive an entropy-free pulse series of 0 and 1 voltage levels (see figure 2c and figure 2d). Additional uncertainty of signal decoding by the interception station may be introduced with the aid of encryption algorithms (see table 1).   Table 1 shows all possible variations of redundancy sum compositions Δ in timing signal constructions (TSC) with set-up parameters. To simplify the examination procedure, one may assume that the ordinal combination numbers in table 1 correspond to quantization levels (figure 3b). Figure 2c and figure 2b illustrate forming of linear TSC for Nq = 9 and Nq = 50 quantization levels encoding.
The number of such TSC realization is calculated by the expression [3,16]: When using TSC with different number i in each code word [3,16]: According to the accepted conditions (2) and expression (3) the total of possible realizations is 120 i.e. 120 quantization levels instead of 16 (figure 3) appear discernible considering the step (duration) of the pulses (elements) remains uncut and the necessary pause of clock synchronization lasting one element (pulse) remains maintained due to the fact that the entropy measured of log2120 = 6,907 elements is obtainable over a period of log216 = 4 information positional pulses.
Thus, when using the timing method under given conditions (1) the signal to quantization noise ratio is to be calculated as ( ) = 34,91 , instead of ( ) = 19,86 for a bit-digital code (BDC) (see Figure 1). The codeword entropy (H) represents the numerical value of information quantity contained within the time interval of the codeword (the number of Nyquist elements) under equally probable characters at the input: For a given binary linear bit-digital code the codeword entropy is equal to H=log216=4 bits. The information capacity of a Nyquist element (bit) IН is defined as = (6) For the given binary linear bit-digital code the information capacity of a solitary element is equal to IH=4/4=1. Table 2 from reference [3] represents the number of realizations, codeword entropy, and information capacity for TSC-related parameters in comparison with BDC ones having been calculated per a solitary Nyquist element.     Contents of table 2 bring out the following regularities: a) a binary channel (even without changing the alphabet) provides for more information carrying capacity than the entire codeword interval due to a significant increase in the number of realizations Nr ˃˃ 2 n ( figure 4); b) according to (3) and (4), if i = n, the number of realizations on timing and positional encoding is Nr TSC = Nr BDC, and if i > n, the number of signal constructs is equal to Nr = 0. Indicator n in expression (1) is to be designated as the code batch entropy which value is equal to the code length (1 digit = 1 bit) in a linear bit-digital code BDC, but, as for a TSC, it can be higher or lower. Using expression (1) one can define the dependence of the signal to quantization noise ratio on TSC parameters for military control systems by substituting n for the corresponding entropy value. Figure 5 shows the dependences of the SINQ on the TSC parameters demonstrating PS/Pnq ratio subject to the number of TSC realizations and entropy value on the interval of one discrete lasting n = 4 BDC elements (ADC bitness) while assuming the number of timing signals S variable within the range 1 ÷ 7 and the number of significant reproduction moments i = const = 3 to replicate a military control telecommunication audio exchange system. Figure 5 indicates that the dependence of PS/Pnq ratio on the number of TSC realizations varies according to the exponential distribution law, whereupon, at the TSC parameters n = 4; S = 3; and i = 3, the code discrete entropy H = 4,322 is almost equal to the BDC entropy (n = 4 → H = 4). The number of significant reproduction moments i may also be increased to attain even greater entropy on a given interval for the "good condition" of the channel. Is it advisable to increase infinitely the codeword entropy by means of timing? According to [3,17], the maximal information quantity per element (IH max) depends on the selected number of segments within the codeword interval. In accordance with table 2, it is considered to be normally attainable provided transmitting a message with a signal structure length TC = 5t0 (n = 5) for i = 3 ( Figure 6) under given conditions.  It is also necessary to take into account transmission channel characteristics when selecting the timer pulse-code modulation parameters, in particular, the fluctuation-noise and signal-interference constituents. In references [3,16] the dependences of TSC's parameters on channel's ones have been determined in accordance with the "code element error probability" criterion for an urban telephone exchange network channel. The research findings have deducted the theoretical propositions as follow: Δ-value assures the error probability having been set at the receiving side to determine the duration of the segments at the output of a channel with specified interference parameters for each signal to noise value there is a Δ interval zone at which the bandwidth attains its maximum.
From the results having been deducted in references [3,16] it follows that despite the fact the energy interval between TSC is S = kΔ times less than for BDC, the error probabilities experience minor differences under significant rise of the entropy value provided approximately the same codeword length in an urban telephone exchange network channel. As for a radio channel this correlation proves worse to some significant extent.
Considering that with the number of realizations Nr increase the error probability of their reception also increasing, each channel has its own Δ-value which assures the maximum system capacity.

Conclusion.
The timing pulse code modulation significantly improves the quality of AD/DA conversions and reduces quantization noise. Nevertheless, its use is expedient solely when transmitting signals via channels whose Hilbert "good state" duration prevails over "bad" one (e.g. fiber optics systems or photo-video-audio data digitizing). When transmitting signals via radio channels (radio relay, satellite), it is necessary to apply additionally noise-resistant timing signal processing technologies having been analyzed in sufficient detail in a number of works, such as: Direct-sequence spread spectrum (DSSS) [18], orthogonal frequency and code division multiplexing (OFDM/OCDM) [19,20], adaptive synthesis based on the channel state analysis [21], operating frequency-hopping spread spectrum (FHSS) [22] etc., so in future studies, it is recommended to investigate in more detail the dependence of the efficiency of timer pulse modulation on the characteristics of the transmission channel: the type of signal, the presence of interference and fading of the signal, etc.