Quantization of a signal by level is the main operation of analog-to-digital signal conversion and consists of rounding its instantaneous values to the nearest allowed ones. With uniform quantization, the distance between the quantization levels is the same. When quantizing a signal, errors occur, the magnitude of which is random and has a uniform distribution, not exceeding half the quantization step. The signal after quantization is the sum of the original signal and the error signal, which is perceived as fluctuation noise.
Immunity from quantization noise for the weakest signals with uniform quantization:
– psosometric coefficient equal to 0.75 for the PM channel;
– dynamic range of the signal equal to 
, dB;
m – number of bits in binary code.
Table 5.2. Initial data
Signal levels:
Signal dynamic range:
Required number of digits:
– code bit depth for uniform quantization.
The number of steps for uniform quantization will be:
Conclusion: to encode with a uniform code with a given security, you will need a code with a bit depth of .
5.2.2. Uneven Quantization Noise
Real PCM systems use non-uniform quantization. Uneven quantization - reducing the slope of the characteristic by reducing the size of quantization steps for small instantaneous signal values by increasing steps for large values.
For uneven coding, 8-bit codes are used, i.e. the number of quantization levels is 256.
Compression of the dynamic range is carried out using the A - or m - compression characteristic. In our case, we use the compression characteristic, which is described by the following expression:
Rice. 5.2.2. Compression characteristic
The DSP uses segmented non-uniform quantization characteristics, because they can be implemented quite simply on a digital basis. The characteristic is symmetrical with respect to 0, its positive and negative branches consist of 8 segments, each segment is divided into 16 identical steps (within each segment the quantization is uniform).
The segments approximate a smooth type A compression curve. In the zero and first segments the step is minimal, and in each subsequent segment the step size is doubled relative to the previous one.
The expression for protection from quantization noise in the first two segments will be:
For 2–7 segments:
where i is the segment number.
The beginning of the graph - an inclined straight line - corresponds to the zero and first segments. This is a uniform quantization zone, so security increases in proportion to the increase in signal level. When moving to the second segment, the security decreases abruptly by 6 dB. When the upper limit of segment 7 is reached, an overload zone occurs.
With a correctly chosen sampling frequency, based on Kotelnikov’s theorem, the accuracy of converting an analogue signal into a digital one is determined by the size of the quantization step. The smaller the quantization step, the smaller the conversion error. The difference between the original and quantized signal values at discrete times is called quantization noise (quantization error).
Quantization noise, unlike fluctuation noise, is, in general, non-random in nature. Therefore, it is more correct to talk about signal distortion during its analog-to-digital conversion. At fixed maximum level The input analog GS quantization noise is determined by the number of quantization levels - the bit depth of the analog-to-digital converter (ADC).
When encoding with binary numbers and a codeword length of m bits, the number of binary codewords r (resolution) is. So with m=16, r=65536.
The stream of codewords at the output of the ADC is characterized by the data transfer rate - the number of bits of information transmitted in 1 second. The data transfer rate is the product of the number of bits of the codeword and the sampling frequency (in hertz). The amount of memory required to store information about the implementation of a signal with a duration is determined as the product of the data flow rate and the duration of the signal.
With linear pulse code modulation (PCM), i.e. with a uniform quantization step, the quantization noise power is determined only by the quantization step:
where is the total dynamic range of the signal.
Effective quantization error value:
Quantization noise is, with linear PCM, a random process with a uniform expansion within a probability density. The spectrum of quantization noise is uniform across the frequency band.
Quantization noise only appears when a signal is present. In the absence of a signal at the ADC input, quantization of oscillations in the least significant bit of the ADC will occur at the ADC output. This is explained by the presence of thermal noise in the input analog parts of the ADC, instability of the supply voltage, drift of the DC component in DC amplifiers and other reasons. At the output of the DAC (Digital to Analog Converter), this quantized fluctuation is converted into noise called pause noise. Pause noise is less uniform than the white noise found in analog devices and is often referred to as granular. Pause noise power:
4.7 dB more quantization noise.
Since it does not depend on the input signal level, as the input power increases, the ratio increases linearly until clipping noise occurs. The ADC input limit level is determined by the maximum input operating voltage of the ADC. Clip noise is the difference between the original and clipped signals. The ADC system is designed in such a way that no limitations arise, i.e.
here R is the peak factor of the signal; S SR – root mean square value of the signal.
The number of steps can be determined from the relationship:
where are the maximum and minimum signal values at the ADC input;
Quantization step.
Taking into account expressions (9.6), (9.9), (9.10), we obtain an expression for the noise power
The signal power at a resistance of 1 ohm, then
or in decibels
With m-bit coding, then
A harmonic signal has a crest factor, in this case
For broadcast signals, the crest factor depends on the genre of the program. If we assume on average R=13 dB then
If we take into account the unequal sensitivity of the listener's hearing to noise components of different frequencies, the signal-to-quantization noise ratio decreases by 8.5 dB for a signal in the frequency band up to 15 kHz and amounts to
Dynamic range digital signal estimated by value, dB, taking into account what we obtain
From expression (9.15) it is clear that increasing the number of bits by one leads to an improvement in the signal-to-noise ratio by 6 dB.
In Fig.9.2. shows the dependence of the signal-to-noise ratio for 3V signals at different meanings m on the input signal level (9.17).
With 16-bit quantization, we have for a harmonic signal D = 90 dB, S-N = 98 B (from expressions 9.15, 9.18). The S-N ratio when calculated using formula (9.17) is equal to 80 dB when encoding a signal with maximum level. When coding weak signals, the ratio S-Sh less on the value of the dynamic range of the signal and turns out to be very small at D=50...60 dB.
80 -70 -60 -50 -40 -30 -20 -10 0
Fig.9.2. PCM signal-to-noise ratio
Effects of finite bit depth of digital filters
When analyzing LDS, it was assumed that the signals were sampled only in time and the sample samples and filter coefficients were represented with unlimited accuracy. However, in real or digital systems, the accuracy of calculations is limited and depends on the number of bits of the devices used: ADCs, registers, adders, multipliers. This circumstance leads to the following effects:
Quantization noise during analog-to-digital conversion;
Rounding the results of intermediate calculations;
Distortion of frequency characteristics due to quantization of digital filter coefficients;
Overflow of the bit grid during calculations;
Low-level limit cycles.
Quantization noise
Quantization noise refers to random errors between time-discrete signal samples and their digital representation with limited bit depth.
Adjacent quantization noise samples are assumed to be uncorrelated with each other. And the quantization noise is “white”.
The probability density of quantization noise corresponds to a uniform distribution law:
, (2.1)
where is the quantization interval by level.
The dispersion of quantization noise is determined by the distribution law:
. (2.2)
If the maximum value of the quantized signal is equal to , then the quantization interval is equal to:
,
where is the number of digits of a digital device.
The power spectral density of quantization noise is given by:
,
where DT is the sampling interval.
For example, the gain of the receiver upstream of the ADC input is selected so that the thermal level of the receiver exceeds the spectral density of the quantization noise.
The contribution of input quantization noise to the output signal of a digital filter is given by:
.
Accordingly, the variance of the output quantization noise is:
Literature
Markovich I.I. Digital signal processing in systems and devices: monograph / I.I. Markovich; South Federal University. – Rostov n/d: Southern Federal University Publishing House, 2012. – 236 p.
Fundamentals of Digital Signal Processing: tutorial/ Yu.A. Bryukhanov, A.A. Priorov, V.I. Dzhigan, V.V. Khryashchev; Yarosl. state University named after P.G. Demidova. - Yaroslavl: YarSU, 2013. – 344 p. (p. 152)
Kartashov V.G. Fundamentals of the theory of discrete signals and digital filters. – M.: Higher. school, 1982. – 109. (p. 86)
Invention in 1959….1961 coherent laser light sources marked the beginning of the development of optical communication lines, where light waves are the carrier of messages. For light waves in the range 10 14 – 10 15 Hz (0.5...10.6 microns), special guiding systems were created - light guides. The most promising of these have proven to be dielectric waveguides, or fibers, as they are called because of their small cross sections. The simplest light guide is a thin cylindrical fiber, which consists of a core and a cladding. Electromagnetic energy is transmitted through the core in the form of a light wave, so it is made from a material with the lowest optical losses (quartz, multi-component glass)
In the optical range, noise associated with the discrete nature of electromagnetic radiation—quantum noise (QN)—visibly manifests itself.
Communication channels in which CN limits the quality of message reception are called quantum channels.
For open space channels, the main promising areas have become space communications using artificial earth satellites (AES), short-range terrestrial communications through the atmosphere, and underwater communications. Gas, solid-state and semiconductor lasers of both visible and infrared ranges are used. They mainly use modulation with changes in radiation intensity - binary AM, binary BIM (bipulse signal), multi-position BIM. Polarization modulation (PM) is also successfully used. In multichannel systems, in addition to time, frequency division into subcarriers is also used. Intensity is modulated by harmonic subcarriers (usually in the microwave range), which are modulated in amplitude, phase, or frequency.
Demodulation is most often performed with direct detection. In the infrared range, heterodyne reception is also successfully used. In channels with a closed space, the optical signal is channeled either through pipes - light guides with discrete phase correctors (lenses, mirrors), or through dielectric fiber light guides.
In recent years, the main focus has been the development of fiber-optic channels with signal transmission in the near-infrared (wavelength approximately 1 micron) range. Semiconductor lasers and incoherent sources—LEDs—are used as radiation generators.
The main differences between optical channels, associated with the short wavelength and quantum nature of the radiation, are as follows:
Thermal noise may be negligible;
Due to quantum laws, the signal parameters are random, even in the absence of interfering factors.
In a system with a passive pause, neglecting thermal noise, the symbol 0 (no radiation) is accepted without error. Whereas symbol 1 (radiation pulse) is skipped with non-zero probability.
Due to the high frequency of carrier oscillations, coordinated filtering along the frequency spectrum and coordinated spatial selection of the signal are practically impossible to implement; orthogonal signals are not separated.
The energy of the electromagnetic field has a discrete nature - it is emitted and absorbed by quanta,
where: h = 6.624·10 -34 W·s / Hz – Planck’s constant, f – frequency.
Quantum noise is fluctuations in the measured signal parameters.
Quantum noise is not additive since it is correlated with the signal.
In the region of infrared and visible radiation, the photon energy increases, and spectral density the average power of thermal fluctuations decreases.
conclusions
1. Laser noise is quantum noise, as it manifests itself in fluctuations of signal parameters determined according to classical concepts.
2. Quantum noise is not additive, since it depends on the useful signal itself.
Conclusion
1. There is interference in communication channels that impairs the accuracy of message reception.
2. Interference can be additive or multiplicative.
3. Among additive noise, the most common are fluctuation, concentrated in the spectrum and pulse.
4. A convenient model of additive noise is white noise, which can be used to describe real processes occurring in communication channels.
5. Quantum noise is not additive, since it depends on the useful signal itself.
In the optical frequency range, thermal noise is very weak. However, in this range, with weak signals, “quantum noise” caused by the discrete nature of light radiation is of significant importance. According to the quantum theory of the electromagnetic field, its signal energy is emitted and absorbed by quanta, and the energy of one such quantum (photon) is equal to . In an elementary signal of duration with a highly stable carrier frequency (coherent single-mode radiation) and deterministic amplitude, only the average energy (proportional to ( is the average number of photons in the interval T) can be determined. A specific implementation of an elementary signal has energy where random number registered photons.
IN modern systems Optical communications mainly use AM optical carrier waves in amplitude or intensity (power).
An ideal optical communication system for isochronous transmission of binary messages (1 and 0) has the following characteristics:
1. The bit transmission time (clock interval) is constant and is therefore equal to the information transfer rate
2. When transmitting 1, the optical energy emitted in the form of pulses during the transmission of one bit, where the number of emitted photons,
The energy of one photon (quantum), and the optical energy when transmitting 0 is equal to zero. The optical energy at the reception site is equal to the value at the clock interval when transmitting 1 and zero when transmitting 0, respectively.
3. Probabilities of transmission. In this case, the received power averaged over a long time can be expressed in terms of the average power received during the transmission of a bit when sending 1. Thus,
A real optical communication system differs from an ideal one in the following ways:
1. The transmission time of a bit of information does not remain constant - this effect is called phase jitter of a digital signal.
2. The emitted optical energy does not remain strictly the same. When transmitting both code 1 and code 0, transmitter noise occurs, leading to random changes in amplitude from pulse to pulse. In addition, there is "laser noise" due to the statistical nature of the interaction between the laser excitation and the generated photon flux. Fluctuations in received energy increase even further due to changes in attenuation in the communication channel. In addition, energy fluctuations appear at individual clock intervals at the reception site, due to the statistical nature of the interaction between the photon flow (optical signal) and the flow of electron-hole pairs created by the photodetector (usually a photodiode). Conventionally, in this case we will talk about the noise of the photodetector.
3. It is very likely that when transmitting 0, a small but quite definite level of energy is emitted (laser noise), not counting the noise of the transmitter and channel. The ratio of the average energy received when transmitting 0 to the average energy when transmitting 1 is characterized by the coefficient. It is believed that in an ideal system, however, this is usually not the case, especially if the laser radiation source is biased near the lasing threshold.
4. The finite duration of the emitted pulses and the additional time dispersion (scattering) during their transmission over the channel lead to the fact that in practical communication systems there is an overlap of adjacent messages, i.e. intersymbol interference occurs.
The laser noise mentioned above is of a quantum nature. The probability of exactly photons appearing in an interval on the transmitting side is determined by the Poisson distribution (see § 2.76):
Thus, laser noise is “quantum noise”, since it manifests itself in fluctuations of the parameters of a signal determined according to classical concepts. This noise is not additive, since it depends on the wanted signal itself. Taking this into account, in the above formula it should be assumed that when transmitting a during transmission As mentioned above, when transmitting 0 (no laser excitation), a certain, albeit small, level of energy can be observed, due to the fact that the probability of the non-appearance of photons in this interval is where the average number noise photons in the interval in the absence of laser excitation. As the average power of the emitted signal Pper increases, the contribution of quantum noise compared to other noise in the transmission path decreases.
The noise of a photodetector is of a nature similar to that of a laser, since a stationary light flux incident on a photodiode generates electron-hole pairs of charge carriers as independent random events. If, over a period of time, an optical energy equal to, on average, falls on the photodiode, then we should expect that, on average, pairs of charge carriers will be created, and
