Noise quantuvannya. Quantum beyond the level of digital systems

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With the correct selection of sampling frequencies, in accordance with Kotelnikov's theorems, the accuracy of the conversion of the analog SZ to a digital format starts with the size of the quantization croc. The hitch is a reconfiguration of the less, less than the croc of the quantum. The difference between the output and quantized values of the signal of discrete moments is called the noise of the quantization for an hour (by the mercy of quantification).

The noise is quantitative in response to fluctuating noise, in case of fall, to have a non-flat character. That is more correct to talk about the creation of a signal during its analog-to-digital conversion. When fixed to the maximum The input analog ZS noise is quantized by the number of quantized values - the size of the analog-to-digital conversion (ADC).

When coded with double numbers and with a code word in m rows, set the number of double code words r (separate building). So for m = 16, r = 65536.

The number of code words on the ADC outputs is characterized by the speed of transmission - the number of bits of information transmitted in 1 second. Transmission speed - addition of the number of code word bits per sampling rate (in hertz). The memory volume is necessary for obtaining information about the implementation of the ZS triviality, so that it will be necessary to provide the flow of money for the triviality of the signal.

With line impulse-code modulation (ICM), tobto. in case of a normal quantization stage, the pressure to quantize noise starts only as a quantized one:

de - zagalny dynamic range of the signal.

The effective value of the quantifying pardon:

Quantum noise є, in case of linear ІKM, a kind of process with equal expansion at the boundaries, from the gap in the distance. The spectrum of the noise quantized is equal to the smooth frequencies.

Noise quantized to be manifested only because it is obvious to the signal. For the duration of the signal at the input of the ADC, the output of the ADC will be quantized in the youngest row of the ADC. Explain the price of the apparent thermal noise of the input analog parts of the ADC, the instability of the voltage supply, the drift of the post-storage warehouse post-struma and other reasons. At the outputs of the DAC (digital-to-analog conversion), the quantization of the input is converted to noise, which is called the pause noise. The pause noise is less normal, less loud noise, characteristic of analog attachments, and this is often called granular. Pause for pause noise:

4.7 dB more than quantized noise.

Oscillations do not lie at the level of the input signal, due to the increased pressure of the input signal, it grows linearly until quietly, as long as there is no noise around it. Rivne obmezhennya ADC input is set to the maximum input working power of the ADC. The noise between the signals is called the difference between the incoming and outgoing signals. The ADC system rozrahovuєtsya in such a way, which is interchangeable, tobto.

here is the R-peak factor to the signal; S SR - mean square value signal.

The number of crocs can be based on the amount of money:

de - the maximum and minimum value of the signal at the ADC inputs;

Croc Quantum.

With urahuvannyam viraziv (9.6), (9.9), (9.10) we can use viraz for pressure to noise

Signal demand on support 1 Ohm, tod

abo in decibels

With m-bit coded, todi

The harmonic signal has a peak factor,

For signals changing the peak factor, lay down in the genre of programs. Yaksho in the middle rahuvati R = 13 dB then

As long as the sensitivity of the hearing person's hearing to the warehouse noise of low frequencies is not the same, then the output signal / noise quantization decreases by 8.5 dB for the signal in smoothies of frequencies up to 15 kHz and to become

The dynamic range of the digital signal should be estimated by the value, dB from the level of what is recognizable

From viraz (9.15) it can be seen that an increase in the number of outlets per unit is to produce up to a 6 dB increase in the signal-to-noise ratio.

Figure 9.2. Shown is the presence of signal-to-noise ratio for 3V signals at different values of m from the input signal (9.17).

With a 16-bit quantized value for a harmonic signal D = 90 dB, S-W = 98 B (s 9.15, 9.18). Vidnoshennya S-W when rocking according to the formula (9.17) should be equal to 80 dB when the maximum signal is encoded after equal. When coded weak signals appointee S-Sh less by the amount dynamic range the signal і appears even more malim at D = 50 ... 60 dB.

80 -70 -60 -50 -40 -30 -20 -10 0

9.2. Signal-to-noise ratio at ІКМ

In the preceding paragraphs of the discrete power filters, the inevitable misbehavior of the conversion of the input signal from the analog form to the digital is not visible. Possibility of winning when quantizing a signal to a kintsev, surrounded by a number of rivnivs. Let's find out the nature of the trick, turning to the structural diagram in fig. 12.1 There are two attachments visible from it: ADC and DAC.

It is easy to understand the connection to the spilny robot of the cich attachments without the need for a digital filter for the hour of feeding to the input of the ADC post-lay springs iznogo rivnya (Fig. 12.28, a). The main parameter of the ADC is the number of outputs, which can be used for coding the input voltage. With a two-way code, the number of rows starts with the number of two elements (for example, triggers), which can be in one of two stations: from a zero or a non-zero voltage to the outputs. One of the tsikh stanіv is mentally attributed to zero, to іnshom - odinitsa. With the number of two elements on the ADC outputs, enter a combination (code number) of symbols, skin from which one can accept one or two values (zero or one).

Small. 12.28. Revision A-C ta Ts-A (a), characteristic of quantization (b) that grading of quantization (a)

As stated in § 12.1, the number of possible combinations is determined by the number of discrete equal parts, on which the range of input changes can be adjusted.

At the DAC, there is a ringing revision. Skin combination of zero and one unit, which fits on the DAC input, displays a discrete pattern of the output stress. As a result, at an equal stage of quantization, the abundance of them in the lamina line is shown in Fig. 12.28, b.

Pristіy representations in fig. 12.28, but I’ll give you a description, I’ll look like it’s not line, and the mark is like a pity, a quantification. It can be seen that there is the greatest amount of gratitude, for the absolute value I don’t perevit і zrostannya, їkh їkh to be unimportant (Fig. 12.28, c).

Progressively viewing for a harmonious input collector (Fig. 12.29 a). The number of steps to fill the stage of the formation, so that one can see from the input call (in Fig. 12.29, b, shown with a thin line), and the quantization of the number of functions

shown in Fig. 12.29, art.

With a change in the wide ranges of the amplitude of the frequency of the harmonic jitter, the frequency of the teeth will change less: the form becomes close to triangular in case of constant amplitude. The function can be called the transition noise of the quantization. It doesn’t matter virahuvati in the middle of the pressure of the noise of the quantiuvannya. If allowed tricot form teeth (Fig. 12.29 c) with an average amplitude for the triviality of one tooth to pull the door.

Small. 12.29. Signal at the inputs (a) and outputs (b) of the quantized attachment; transcript quantum (v)

So, as the value does not lie in the triviality of the teeth, it can be increased, but the average pressure on the noise of the quantization

(12.63)

The whole result, in the introduction for a harmonic signal, can be broadened and whether it is a signal, it can be more broadly and more broadly. It is less likely that the function q (t) will be a kind of process through a kind of triviality of teeth.

It is unimportant to count the signal-transition signal when quantized. When the number of signals is high, the ADC characteristics fall within the boundaries of L, the amplitude of the harmonic signal is not to blame for the change in value, and the average pressure on the signal is the value (that the signal is unnatural). From the same time, the signal-to-transient response when quantizing the harmonic signal

So, since the number of Rivniv L is tied with the number of two Rows in children, then the remaining Viraz can be paid in the form

(12.64)

Tse sp_vvidnoshennya is possible yak okremiy vypadok zagalny vyslovlyuvannya

de - peak factor to the signal, so that the ratio of the maximum value to the mean square.

With a harmonious colony, it is possible to produce up to viraza (12.64); in case of a bad signal with a normal law, the growth can be accepted (div. § 4.2, p. 3); I have a lot of problems, and a mean square wave is not to blame for a signal. Physical change in speed (12.65) is obvious: from the increase in the number of rows, the number of discrete rows grows rapidly, so that the range of changes and, moreover, the difference between two pairs decreases.

With a rough estimate, the signal is reassigned over the noise, the quantization is output from the

The current ADCs have a number of rows of ten or more. At the same time, the value that characterizes the dynamic range of the ADC is approximately 60 dB (6 dB per discharge).

Інshiy important characteristic Noise quantization є th spectral characteristic. With harmonic collation at the input of the ADC, the conversion code is quantized by the periodic function hour. Spectrum її lіnіychastim, so to avenge the frequency, multiples of the input frequency. Through the tooth-shaped form of the function (div. Fig. 12.29, c), the spectrum of noise is rich in great harmonics.

In the case of an input type, a drop-off process with a dispersion and a mean-square width of the spectrum of the statistical characteristics, the noise of the quantization does not only lie in the characteristics of the input process, but rather as a result of the process. Zokrema, with the width of the spectrum, the noise of the quantization is much larger than the width of the spectrum of the process

Small. 12.30. Until the date of the pardon quantuvannya

Introduced for a look at the sampling of the input signal.

In fig. 12.30 presented one of the implementation inappropriate signalі the number of vibrations taken from the croc T. In the ADC, the skin vibration is converted to a digital code, as described in § 12.1 on the tip of this paragraph for continuous stress.

Yak is obviously from the previous world, re-incarnation to come with a mercy, laid down no more. If vibrations are taken from a drop-off signal, and the change in function in an hour T will change, or even more than a drop, then gratuities in different moments of the hour can be varied by both independent and dynamic ones. Dispersion of a type of value, equal to one in the interval of roads (div. § 4.2, p. 1). The whole result will be reduced to viraz (12.62), we will reject the average exertion of the noise of the quantization for an hour. Broken food poached is equal to solidity, but the discrete aftermath of pardons is attributed to vibrations from uncorrected noise, so that noise with equal spectrum. The whole spectrum, which means whiske, is wider than the spectrum of the outward process. At the connection from the digital filter, the noise of the quantization can be interpreted as the noise of the digital filter (input).

The range of quantum noise is significant. Do not change the width of the noise spectrum without quantizing the roads. When sampling the noise, the quantization of a large amount of the resulting spectrum is a sum of partial spectra that are destroyed one and the same (div. § 2.17, Fig. 2.35). A special feature of the given type is those who have so little reckoning of the spectrum.

At the boundaries of the frequency interval, the tightness of the skin area of the spectrum. Ale the number of spectrіv, scho curving, dorіvnyuє. Resultuyucha pushing the noise of the quantization of the smoothies. This can be taken into account, but in the specified frequency interval the spectrum is equal (big noise) and road

For the type of analog-to-digital re-implementation, there can be a rounding signal (up to a singing discharge) or an increase in a signal.

Mathematical description

Model

Quantum noise can be an additive discrete signal $e (nT)$ , scho vrahovuє pompki kvantuvannya. Yaksho $d (nT)$ - is the input signal of the quantifier, and $F [\,]$ - yogo transfer function, I will step on the line model to the noise of the quantization:

$e (nT) = F - d (nT)$

The linear model is used for the analytical perception of the authorities to the noise of the quantum.

Determination of assessments

Determination of evaluations allow for the absolute difference between the noise of the quantization at the same level of quantification:

$| max | = frac (1) (m) 2 ^ (- b) = frac (1) (m) Q$ ,

de $b$ - The number of quantized razryadіv (signal $e (nT)$ ), $Q$ - Croc Quantum $m = 2$ - with rounded $m = 1$ - With an increase.

Іmovirnіsnі estimates

Іmovirnіsnі estimates runtuyutsya on the submitted pardons quantiuvannya (signal $e (nT)$ ) as a low noise-like process. Reducing the noise of the quantization:

Last $e (nT)$ є stationary video process
Last $e (nT)$ not correlated with quantized signal $d (nT)$
Be-like two examples of the last $e (nT)$ not correlated, that is, the noise is quantized є by the process of the "beep noise" type.
The range of quantitative grants is equal to the equal range of quantitative grants.

$M_e = -0.5Q$
$D_e = Q ^ 2/12$

Div. also

Write an update about the article "Quantum noise"

Literature

Goldenberg L.M., Matyushkin B.D. Digital processing Signals - M .: Radio that zvyazok, 1985.

Posilannya

(English)

Urivok, which characterizes the noise of a quantum

Princess Mar'ya was sounding everything.
All the same, Ale Vona encouraged and energized with words, in which she did not viril:
- Ale yak yogo wound? Vzagal, in what kind of wine is it?
- Vi, vi ... shake, - only Natalya could say.
The stench sat for a few days until the end of your room, just by the way of the posters and escape to the new in quiet guises.
- Yak has all the ailment gone? How long has it been for you? How long has it become? - was fed by Princess Marya.
Natasha suggested that a handful of bullets were not safe from the specotny camp and from the countrymen, but in Labor it had passed, and the lykar was afraid of one thing - Antonov in the fire. Ale y tsya nebezpeka passed. If they came to Yaroslavl, the wound began to fester (Natasha knew everything that she was suffering from suppuration), and she told me that she could be festering correctly. She became a feisty woman. Likar Kazav, the feisty tsya is not so uncomfortable.
- Ala two days ago, - Natalka said, - the rapt has become ... - I don’t know why, ale vie pachit, as vіn become.
- Weak? skinny? .. - the princess fed.
- Hi, not those, ale girshe. Wee shake it. Ah, Mary, Mary, win even garny, can't win, can't live ... more ...

If Natasha, with a vicious collapse, knocked down the door, letting the prince pass ahead of her, princess Maria saw the ready party in her throat. She didn’t get ready, she didn’t get cold, she knew that she wouldn’t be able to beat him without tears.
Princess Mary rose to mind, as Natasha said to her: we know the trapeze two days ago. Vona rasumіla, so tse meant those who wіn raptom pom'yakshav, і wіn hаνе tο mаkе, іt іѕ nοt іn thе way іn tе signs of death. Vona, walking to the door, was already bashing Andriyka in her face, as she knew of the child, now, lagid, zavorushena, as it was so rare in the new and so hungry for her. Vona knew that I’d say the quiet, lower words, like that, like saying to my father before he died, and that it’s not a winn’t thing, but to be over him. Ale early, it’s not enough buti, і won’t come to the room. Ridannya all the nearer and nearer stepped up to the throat, if, with their short-sighted eyes, she cleared and clearer the shape and rustled his rice, and the axis of the way shook him, revealing and staring at him.

However, it is logical to let go of the industrial files, as the digitizer has not been able to reliably visualize, it can not be so easy to know.

Shumi Quantum

Mіzh analog signal that yogo digital copy in your system I will write down the difference, how to be called pardons quantuvannya, abo noise quantum.

For the help of the awkward mathematical formulas It is possible to calculate the frequency of the quantized noise. The same kind of character can be simulated on purpose, as well as analyzing the visualization of the digitized graphics from the original sinusoid. On the baby on the right, the difference between the output and the digitized signal is shown.

The noise of quantum is the price for digital sound, the stench is at the moment of digitization. For minimizing the flow of noise to the sound, special filters are used in the design of the converters. Purchase digitization from more expensive characteristics (for example, 24 /192 ), it’s a lot of things that aren’t a beast of respect for the quality of cich filters, if they are deprived of the numerical characteristics of the spacing and sampling frequency.

CHIM VISCHI Converter Indicators, tim dorche povinny booty filter However, the virobniks themselves are encouraged to save money on them, so that they can save their own performance at a low price and that they protect their competitiveness.

Aliasing

Another unacceptable sound, as it can happen in the process of sampling (digitizing) the sound, is called aliasing. Aliasing- superimposition of two signals without interruption of a different frequency and one on one during sampling, as a result of which the sound appears in the sound.

We can visualize aliasing visually. Guess the wrapping up of the number of cars and trips from the old films. Have singing moments you can clearly see how the wheels are spinning at the turn of the wheel. It’s not deceiving, it’s effective to appear at the moment, if the frequency of the wrapping comes close to the frame rate of the movie camera (call the clock 24 frames per second, but if the value is set at 16-20). The skin point of the wheel, collapsing behind the year's arrow, getting up to go through the next one is wrapped in one frame, vocal side vikhіdnoї point, nіbi tsya point destroyed the opposite direction. I mi bachimo ringing wrap.

As a result of aliasing of the recordings, the signal is seen as being out of the picture.

As of Kotelnikov's theorem To update the signal without interruption, sampling is guilty at a frequency that doubles the frequency in the recorded spectrum.

So, say, if the maximum speed of wrapping is 10 revolutions per second, then in order to make an efficient aliasing the camera needs a frequency of at least 20 frames per second. And a movie camera is a perfect sampler, so only it will record not sound, but images. If the values are significant, if the wheel was spinning, the camera for the first revolution would start to rotate two samples, and the ringing wraps would not be overwhelming.

So, if we need to record the sound at the 20 kHz boundary (the upper threshold of frequencies, which is identified by the human sound), then the sampling can be taken at a sampling rate of at least 40 kHz.

At the same time, half of the sampling frequency is called Nyquist number(Nyquist and Kotelnikov - in fact, as one of one kind of one was busy with the progress of the problem).

However, we know that it’s not possible to navigate our vuho, which does not mean frequencies, it doesn’t mean that it’s dumb. And if it stinks є, then the sampler (digitizer) will try it zafiksuvati, if there is a lack of sampling frequency for recording the sampling frequency range. I vinikne aliasing.

If you have a negative effect on aliasing, when sampling, the sampling rate is required with a margin of more than two times... In addition, it is necessary at the inputs of the digitizer filter, which does not show the frequency and value of the higher value.

The very "standard" sampling frequency, which is vikorystyutsya in sound recording, even 40 kHz - 44.1 and 48 kHz: so sampling will provide a margin for usunennya for a song.

It is possible, in handwriting, almost "good" and "trash", to record a file-like injury at frequencies of 440, 880 and 1760 Hz. In the first version, the boules are stuck with filters, and in the other, there is a bit aliasing.

Today, I haven’t been able to reach 32 bits of 96–192 kHz. With skin rock, virobniks will increase the characteristics of the attachments. Ale oskilki, as I already Kazav, for filtering more high frequencies required more and more expensive filters, it is not easy to go, but the converter is working in 16 / 44.1 mode, yes more sound sound low converter 24/192. The noise of the quanta, the aliasing and the presence of good filters to correct their fairness. At the same time, there is a possibility of misbehavior, tied up with the help of assignments to the system for hours with robotic sound parameters.

As soon as the article turned out to be cinnamon, you can before paying the update to the blog, so that you can get new materials on e-mail... Abo join

Lecture number 9

"Effective Quantum and Noise in Digital Filters"

Have real annexes, which implements algorithms for digital processing of signals, it is necessary to ensure the efficiency, to zoom in the quanta of the input signals and to the fineness of all registries. Dzherelami of pardons in the processes of processing signals є rounding (or amplifying) the results of arithmetic operations, quantum noise, connections to analog-to-digital conversions of analog input signals, inaccuracy of the implementation of the characteristics of digital filters

For the analysis of the effects that have been knitted from the fineness of the given data, it is necessary to complete the deyakis of the statistical independency of the growing noise of the digital filter. The model is statistically used, like runting on the following steps:

1. If there were two indications of the noise from one of the same dzherel not correlated.

2. If there are two dzherela noise, there will be unrelated noises.

3. The noise of the cutaneous dzherel is not correlated due to the input signal.

The aim is to significantly support the analysis of processes associated with the noises of the quantization in digital filters, to reduce the amount of noise that is statistically unrelated to the noise, and give the possibility of performing an analysis for the skin condition. It is far from expecting to accept the righteousness. It is possible to set up a bezlich butt, except that the pripushennya is not valid. For example, if the input signal is permanent or sinusoidal, at the frequency of the multiple sampling frequency. The first one will have the same stench, but the other will have a stench of periodical last. In such a rank, in both cases the poached is wrong.

It is effective to quantify to the point of failure in the output signals of digital filters, and in some cases to unsteady modes of robots. Through the accepted pardon of the digital filter, a superposition of pardons is calculated as a superposition of pardons, encumbered with a skin square dzherel.

At the input of a digital filter with an impulse characteristic h (t ) get a signal x (t ), then the filter's outbound signal is triggered by a viraz

(9.1).

As a result of the quantization of the input signal, the quantization noise is assumed. e in (n ), which is superimposed on the input signal and injected into the filter. Through the linearity of the filter, it is possible to calculate the reaction of the filter e out (n) on input noise

(9.2).

At the same time, rely on respect, for all the numbering attachments and filter attachments, for remembering, for lacking incomplete distribution.

Similarly, you can know the signal at any point of the filter's structural circuit, surrounded by the noise of the input signal quantization e in (n).

(9.3),

de h i (n ) - the impulse characteristic of a part of the filter from the input to the point at which the grave is estimated.

When the input signal of the filter is quantized b in , then the quantization of the input signal at the victorian rounding is enclosed by the value

(9.4),

and the definition of the input signal of the filter, the measurement of the input signal can be estimated as

(9.5).

In such a rank, the upper limit of the grating, the wikklican of the input signal, lie in the size of the quantization and sum of the modules of the vibrok and the pulse characteristics of the filter.

Dispersion of input noise rounded

(9.6),

The dispersion of the quantized noise at the filter outputs is similar to (9.3)

(9.7).

As long as Parseval

(9.8)

you can write (9.7) at the viewer

(9.9),

de - Amplitude-frequency characteristic of the digital filter.

In such a rank, for the permissible value s out 2 that at home the frequency response or the impulse characteristic of the filter, it is possible by virtue of the permissible value of the variance of the input signal s in 2 , yak at my house, I need a size b in input signal quantization

Signal-to-noise performance at the output of the filter, when the signal starts to be pulled up to the noise in a logarithmic scale, it starts as

(9.10),

de s s 2 - Dispersion of the corian input signal, and b in - Distribution of input signal quantization. Also, with the increase in the size of the quantum, one discharge of the signal-to-noise ratio will increase by approximately 6 dB.

Yak butt is a clear digital filter of the first order, which can be described by the rules

(9.11).

Yogi structure diagram is shown in Figure 9.1.

Noise quantizing input signal low variance s in 2 ... The impulse characteristic of such a filter

(9.12).

According to (9.7) the dispersion of the noise of the input signal of such a filter, encumbered by the quanta of the input signal to the

(9.13).

For the style of the filter, you need to seei, from the same,, tobto. the pressure of the input noise is greater than the pressure of the input noise. Chim closerto one, it is more powerful than the input noise filter.

According to Parseval's theorems, it is possible to start the variance of the output noise of the filter with its frequency response. Let the filter tasks, the frequency response of which is presented in Fig. 9.2.

Todi, zgіdno (9.9) dispersion of the input noise of the filter, cyclic to the input signal quanta

(9.14).

The vibration of the optimal size of the input signal quantization is based on the necessary accuracy of the information provided, laid down in the input signal, which appears in the new input noise and the procedure, as the blockage is signaled for processing.

Noise, which should take place in the signals, is due to the upperbetween the number of rіvnіv quantiuvannya.Obviously, no senseu vikoristovuvati great number If there is a great noise in the signal, then, in spite of the great accuracy, the noise, and not the signal, will be quantized. There is enough vibration in the style of quantization, so that the contribution to the quantized noise is due to the noise, which can be found in the signals.

On the other hand, the minimum allowable number of quantized values can be used to prevent the appearance of the output signal. The loss of the quality of the input signal can be wicked into the flow of imperfections at the stage before the front processing of the signal (noise and encirclement) frequency characteristics the front scaled drives and analog filters).

Until the end of the day, the performance of the retail filter was set at an infinite precision. With the physical implementation of the filter, the efficiency is saved in the elements of the electronic memory (comics, which will not be stored), as it may interfere with the size. This means that the filter performance and the input signal are quantified.

The quantification of the performance of the filter is ordered by the very laws, as is the quantification of the input signal. As a result of the quantization of the filter characteristics, the values of the poles and zero of the transmission function of the filter change in a larger and smaller world, so that, at his side, to produce a general change in the frequency characteristics of the filter. So, the cantuvannya of the filter should be made before the pardon appears.

(9.15),

de A (w ) - frequency response of the filter with non-quantized coefficients, A d (w ) - AFC of the filter with quantized performance. The value is not guilty of changing the permissible value, but it should start from the mind, to see the real frequency response from the ideal boule in the permissible boundaries.

Determining the structure of filters may be sensitive to the change in terms of performance. This is a universal method of assigning the necessary number of razryadiv quantized performance for all types of filters proponutizing is unwise. The need for a number of rows for quantized filter rates can be calculated by way of calculation for the last increasing number of rows for the performance codes until you decide .

It is possible and practical to use these methods, the growing methods, based on the forward sensitivity of the characteristics of a particular type of filter to the change of efficiency.

The yak butt is clearly a square block, which is described by the transfer function.

(9.16),

a structural diagram of this is shown in Figure 9.3.

If you designate the poles of the transfer function (9.16) through, then it is easy to roll over, so

(9.17).

Todi for malikh zmin a 1 ma a 2 the coordinates of the poles fluctuate in value

(9.18),

similarly (9.19).

You can pomititi, scho D r r close to one, tody yak D q to be sharp at values q close to zero.

Sensitivity of the frequency characteristics of the filters to change the value of the efficiency to lay down in the structure, which is used for the implementation of the filter.

When implementing the digital filter algorithm, the operation of the folding and multiplication at the function is determined. The folding of numbers with a fixed point with the size of the sumator, not less for the size of the submission of the documents, do not bring the rounded submission of sumi to the point of pardon.

The operation of the operation is often tied with rounding off notes. Dobutok two numbers with a fixed point z b 1 ta b 2 by discharges from one place to another up to b 1 + b 2 rozryad_v. With the last week of operation, it is necessary to interconnect the number of creations. In addition, the size of the advancing creatures is not limited. To that, for the sake of creation, call on to bring in a few times b 1 + b 2 ... Thus, the result of the multiplicity is rounded. As a result of the rounding off of the creation, the filter algorithm is not realized exactly, and the outgoing signal is counted with mercy.

The multiplying model with the number of rows is represented as the last time the ideal multiplying machine (with not interchangeable number of rows) and the summer, on the input of any order due to the exact values, the noise of the quantization comes. On the outputs of the sumator, go to quantize the value of the b mul discharges (Figure 9.4).

A rounded graveyard can be assessed by its upper cordon

(9.20),

de Q mul - Krok quantiuvannya to the creature. It can be seen as a discrete stationary vypadkovy process with a steady rate of change, with a zero average and a dispersive level

(9.21).

Having accepted such a linear model for a skin university, it is multiplied on the structural circuit of the filter, it is possible to calculate the pardon at the outgoing signal of the filter as a supposition of pardons, which are summed up by the djerels to the rounding noise. At the same time, if you do not have the necessary impulse characteristics g i (n ) parts of the structure of the filter from the skin i -go dzherela noise (tobto vikhodu i th multiply) before the filter goes out and count the warehouse near the filter noise, i -m dzherel noise yak

(9.22).

Todi the noise is rounded at the input, the noise of the usima L you can count the yak with dzherel noise

(9.23).

In such a rank, the outrageous noise of the filter, i -m rounding dzherel I do not change the size

(9.24).

Todi is the maximum value of the input noise, zoomed in L dzherelami rounded (while the size of the number of the same is the same)

(9.25).

On display (9.7), it is possible to estimate the variance of the resulting noise, rounded off as

(9.26).

The level of the outward noise of the filter, enveloped by the quanta of the creations, is to be found in the peculiarities of the structure, which is designed for the implementation of the filter. It should be noted that the impulse characteristic of the filter filter is based on the input of a particular multiplying device until the input of the filter is deposited from the fixed structure. When the structure of the filter is vibrated, it is necessary to fill in the grants in order to quantify the creations from the grants to the quantized values.

All the noise was quantifying the creation to reduce the amount of input from the resulting noise.

Yak butt can be easily assessed for the input noise of the quantization of creations in a bicadrat block, which has a powerful impulse characteristic h (n ). The noise model of the analyzed structure is shown in Figure 9.5.

It can be seen from the presented model that the filter's structure was less than the noise of the quantized creation. Dzherela e mul .4 and e mul .5 pass through the lancet, as the input signal. Tse means, who s impulse characteristics g 4 (n) and g 5 (n ) are based on the impulse characteristics of the filter h (n). Dzherela e mul.1, e mul.2, e mul.3 be sure to add a pardon to the filter's input, as long as you can't be able to do it with the filter. Хні impulse characteristics рівні d (n ). As of (9.7) and (9.26), it is possible to estimate the additions of the

(9.27).

Dispersion of the total noise quantization at the filter outputs according to (9.26)

(9.28).

Zagalnaya grave quantum, pummeled with the quanta of the input signal and the quanta of the creations, which begin with the sum of assessments of the received grants.

When numbers are summed with a fixed point, the rounding is rounded off (just like the sumator, the size is less than the size of the numbers). However, when the numbers are summarized with a fixed distribution, it is possible to re-recalculate, if the result, as a result, does not fit into a number of rows, but it is possible to determine the size of the previous ones. In case of a re-adjustment, if the algorithm of the filter's function is found to be broken, the sum is guilty of being bridged with the urahuvannya of the sign at the equal of the maximum value, so that the given number of lines in the result will be matched. With the software implementation of the filter, it is possible to use the functions of the algorithm for the function, and the hardware implementation of the filter is included before the filter circuit of special attachments for the analysis of the replacement of the sum and the exchange. However, to find the realization of the meanings, not all the problems associated with the re-upgrading, so as, due to the obviousness, the re-framing of the filter will be re-matched to the rather incongruous adaptation of the flavor to the taste. Therefore, for a normal filter robot, it is necessary to implement special entries, so that the situation is re-enrolled to be eliminated.

One of the inputs for the re-adjustment of the field at the introduced scale, as it is brought up to the right (which is equivalent to the right) two codes are added on all the inputs of the summers. If the changes are normalized at the level of 1.0, then when the number of skin changes is equal to two numbers, to enable the possibility of renewal, the skin needs to be damaged by one discharge to the right, but it is equivalent to the same number for the second period of the second. , їх sum does not overwrite 1.0. As the sumator has more than two inputs, then there are more ways to blame for the destruction. This method is called automatic scaling.

As a result of such a scale, the scale-up of the scale is tied together with a young generation (or, if broken, there are more than one distribution) of warehouses, which will be destroyed, destroyed, and the resulting pardon will break down. So, in the event of a damaged amount of money for one category, the maximum value of a grant is scaled

(9.29),

de b - number of distributions from the submitted supplement. If it is shifted by є by the number of zi by the sign of the direct code, then the value of the pardon rіvnі 2 - b, -2 - b, 0. Take

(9.30),

then the amount of noise can be represented as low noise from average values equal to 0 and variance

(9.31).

If dodanok is the number of the pre-dodat code, then the scale punch can be taken as -2 - b or 0 with 0.5. With a large noise, the scale of the average value is -2 - b / 2 and variance

(9.32).

In such a rank, scale grants can be included in the filter model similar to quantum grants.

The best way to save the power of re-upgrading is to scale up the input signals in the filter anyway warehouse parts... If the impulse characteristic of the filter or the other part of the road h i (n ), then the output signal of the filter (either part) y i (n ) enclosed by the value

(9.33),

de is the upper boundary of the filter input to the signal. Yaksho then necessary mind changeover rate є

(9.34).

Yaksho performance filter set (to set h i (n )), then, schob bulo perevnen, tobto. If the output signal of any summator without changing one, it is necessary to intercept the value of the input signal and output signals in multiplying. At the same time, the same scale is introduced, and the signals

(9.35),

de g i - Coefficiencies, scale up.

Scale multiplying switches on at the filter inputs or at multiplying inputs. Yaksho, then with a sufficient mind in the daytime, reapply є right away (9.35), vibrate scalable performance from the mind

(9.36).

Coefficiencies g i Vibrate, like and at the end of the automatic scale, trickle down to the equal steps of the two, and the scale is built up to zsuv_v. At the same time, similarly to the drop in automatic scaling, scaling noise, which reduces the signal-to-noise ratio at the filter outputs.

When there is a significant change in the amplitude of the signals passing through the filter, the signal-to-noise ratio at the filter outputs will change. The calculation of scalable performance for the formula (9.36) often leads to enviable results and, also, a decrease in the efficiency of the filter robot. Besides, with the folding structures of the filter, the sum of the uninterrupted number of indications of the impulse characteristics of the filter can be realized. In addition, the design of large-scale performance is often implemented according to the same methodology, based on the analysis of the spectrum of the input signal and the frequency powers of the filter.

The structure of the filter is to revenge m sumator, then the output signal of the i-th adder vi (n ) can be seen at the viglyadі

(9.37),

de x (n ) - filter input signal, h i (n ) - impulse characteristic of a part of the filter from input to output of the i-th combiner.

Z - reversing the signal v i (n ) you can write yak

(9.38),

de H i (z ) - the transfer function of the filter part from the input to the output of the i-th combiner.

Frequency response of the signal v i (n ) (for a stable filter) you can remove the damage from the viraz (9.38) replace the changes

(9.40).

Todi itself is the output signal of the sumator v i (n ) it is possible to visually revert to Fur'є vіd V i (e j w T)

(9.41).

Yakscho zrobiti pripushennya, scho module to the spectrum of the input signal x (n C , then it is possible to estimate the maximum value of the totalizer output module

(9.42).

Filter input signal x (n ) to be scaled up to the front l i , then the rest of the viraz nabuva viglyadu

(9.43).

For zapobigannya re-assignment to sumator outputs, tobto. for a visitor, get enough vibrati the value of the normal multiplier l and so,

(9.44).

Yakshcho pripustit, what is the module of the frequency response H i (e j w T ) surrounded by a deyakoy value D , then it is possible to calculate the maximum value of the modulus of the output signal of the summer in the other way, but

(9.45).

Has a normal multiplier l i for switching on the re-assignment to the sumator outputs, you can use the same way as

(9.46).

Nareshti, stuck up to viraz (9.41), the inconsistency of Koshy-Bunyakovsky ( ) it is possible to negate such an inconsistency

(9.47).

Just let it go, but the energy of the spectrum of the input signal (other than the root of the viraz at the nerves (9.47)) is surrounded by a value E , then the normal multiplier l i you can get out of the way from the offensive viraz

(9.48).

All three options for the choice of the multiplier are based on the presence of reliable information about the spectral characteristics of the filter input to the signal. If the information is not absolutely reliable, then the value of the reassignment to the summator outputs is not zero.

To enable re-assignment on the outputs of all summers, before entering the filter's structural diagram, it is necessary to update the evaluation of the performance l i for skin summers and vibrates the residual value of the normal performance at the input of the filter

(9.49).

Yak and at the time of automatic scaling, functionality l Vibrate to the equal steps of the number 2, which will re-implement the operation of the scaled multiplication from the input signal code to the appropriate number of rows to the right.

Scalable multiplying device, as if it were the one who multiplies the filter's structural circuits, є the noise of the quantized noise, which can be added to the output signal in the same way as the noise of the other multiplying devices.

Obviously, in vapors, since a summer in the structural filter circuitry of a warehouse is more than two days old, it is possible to navigate for the day-to-day redefinition in a small sum, but perhaps in small industrial sums. The whole fact was not borne by the forerunners. However, if the input and industrial digital signals of the filter are presented in a pre-existing code, then all the guidance methods and standards will become unjustified;

The foregoing analysis is based on the pre-set, but the signals are statistically independent from the vibration to the vibration and from the dzherel to the dzherel. The price is fair, as the difference between two separate signals of the input signal is significantly greater for the quantization croc. Zrozumіlo, scho at bagatokh vipadkah (zokrema, if the input signal is permanently downgraded to zero) it is also unfair. For a number of furnishings, quantiuvannya can be strongly correlated. You can make the filter robotic until it breaks down, when the filter becomes unstable, and on its way, periodic calls are generated, as soon as they stand up. Tse appearances to be called dead zone effect, and periodic calls on the input are called by means of a boundary cycle. The out-of-the-box analysis of a non-linear effect є can be completed easily. This will be carried out for the simplest digital filters.

The filter of the first order, which can be described by the retailers

(9.50).

The transfer function of such a filter is mau viglyad

(9.51).

The block diagram of the filter is shown in Fig. 9.6.

The impulse characteristic of such a filter is

(9.52).

Yaksho kofіtsіnt a 1 door 1 or -1, then the filter becomes unstable and has an impulse characteristic

(9.53).

Table 9.1 presents exact values impulse characteristics (9.52) at b 0 = 10, a 1 = 0.9.

	h (n)	H Q (n)



	7.29
	6.561
	5.9049
	5.31441

	2.65614*10 -4

Now, it is permissible, that a filter of dozens of multipliers from a fixed point, in a kind of skin a 1 * y (n -1) round up to the nearest whole

(9.54).

The third storehouse of table 9.1 presents indications of the impulse characteristics of such a filter. It can be seen that when the filter is displayed, it becomes permanent, and when the filter is quantized, it becomes unstable.

Just let it go, but the profit margin (9.50) is not fair for an unstable filter, which is not effective... As soon as the output signal is not in the range [- k, k ], called dead zone... If it becomes, the filter mode will become unstable. Be it the reason for the zooming of the magnitude of the k to bring up to the renewal of stiffness. However, due to the appearance of the input signal, it will again be extinguished to a value that will cause the dead zone to appear.

In such a rank, the filter is transferred to the boundary cycle mode with the amplitude of the output signal equal to k ... Oscillations are not effective a 1 road 1 for a 1> 0 abo -1 for a 1 <0, то частота такого предельного цикла равна 0 или w s / 2.

(9.60).

Tsei viraz can be used for the selection of the minimum number of rods in the outboard attachment for changing the amplitude and the amount of the boundary cycle at a given level.

An analysis of the dead zone effect is carried out for a filter of a different order, which is described by the dorіvnyuvatime 1. At the same time

(9.66).

From the same, like and before I can think of a non-smart filter robot, I can

(9.67).

Yaksho k - the whole, then the magnitude a 2 z ranges

(9.68)

until dead zones appear [-1,1], [-2,2], ..., [- k, k ] for sure.

Yaksho at the filter vikoristovuyutsya two-way multiplying with a crocus quantized result, equal q , that umova show kolyvan boundary cycle ma viglyad