RU101601U1 - ADAPTIVE MODELING OF FILTRATION OF RANDOM PROCESSES - Google Patents

Abstract

 Adaptive modeling system for filtering random processes, which contains a random signal generator, an ADC, a digital white noise generator, a white noise digital sequence generator, an addition unit, a digital delay unit, first and second adaptive filters, a subtraction unit and a DAC, and having the following connections: the output of the random signal generator is connected to the ADC input, the output of which is connected to the first input of the addition unit and, through the digital delay unit, to the first input of the unit to subtraction; the output of the digital sequence generator generating white noise is connected to the second input of the addition unit; the output of the addition unit is connected to the first input of the first digital adaptive filter, and the output of the digital sequence generator of modeling white noise is connected to the first input of the second digital adaptive filter, the output of the subtraction unit by the OS bus is connected to the second inputs of both adaptive filters, and the output through the DAC is the system output .

Description

The utility model relates to radio electronics and can be used in modeling broadband communication, radio and sonar systems, for example, in adaptive digital control that combines the problems of control, filtering and identification.

The construction of dynamic stochastic models from a sample of the studied process is described in detail in [1]. Often, deterministic processes are represented by deterministic models using a deterministic function of time, and stationary random processes can be represented as autoregressive functions of the corresponding order. However, a single model that could adapt to the parameters of a random signal does not exist. Therefore, it seems relevant to develop an algorithm for generating a random process model that could dynamically adapt to the parameters of the input signal, while not being parametric.

The adaptive filter in modeling unknown physical dynamical systems has been used for a long time and is fairly well researched. The input of an unknown system and adaptive filter is the same signal. The adaptive filter is adjusted so that its output signal matches the output signal of an unknown system [2].

However, until now, the possibility of using an adaptive filter as a model of a random signal has not been investigated.

The technical task of the utility model is to expand the functionality of adaptive filters in the field of a random signal model.

To solve these problems, a adaptive modeling process for filtering random processes is proposed, which contains a random signal generator, an ADC, a digital white noise generator, a white noise digital sequence generator, an addition unit, a digital delay unit, the first and second adaptive filters, a subtraction unit and a DAC , and having the following compounds:

the output of the random signal generator is connected to the ADC input, the output of which is connected to the first input of the addition unit and through the digital delay unit with the first input of the subtraction unit; the output of the digital sequence generator generating white noise is connected to the second input of the addition unit; the output of the addition unit is connected to the first input of the first digital adaptive filter, and the output of the digital sequence generator of modeling white noise is connected to the first input of the second digital adaptive filter, the output of the subtraction unit by the OS bus is connected to the second inputs of both adaptive filters, and the output through the DAC is the system output .

The synthesis algorithm for the formation of an adaptive model of a random process (Fig. 2) is based on the Wiener estimation device, from which the Wiener-Hopf equation for the vector of weighting coefficients follows by the criterion of the least standard deviation of the modeled random process and its estimation.

Figure 2 shows: 12 is a block of white noise generation, 13 is a block of modeling white noise, 14 is a delay block, 15 and 17 are the first and second adaptive filters, respectively, 16 is a subtraction block, 18 is an addition block.

The main difference of the presented algorithm for the formation of adaptive models is the forced "mixing" of white noise into the original process. In turn, the white noise generating in the mixture excites the N-dimensional space of weighting coefficients [3], and the adaptive filter 1 is adjusted so that its output signal corresponds to an unknown initial signal by the criterion of the best mean-square approximation.

As a result, the impulse response of the adaptive filter 1 will tend to match the useful message. In order to form a model of the initial process, it is necessary to rewrite the weight coefficients in adaptive filter 2, at the input of which a modeling white noise acts. Thus, at the output of such an adaptive filter, a model of the initial process is generated.

Suppose that a discrete signal acting at the input of an adaptive filter 1 is represented as:

y (n) = Ao · x (n) + Vo · υ (n)

where x (n) is a centered real stationary random process;

υ (n) - forming white noise with a single dispersion. First, we write in a general form the cross-correlation vector P and the autocorrelation matrix R for the adaptive filter input signal [2]:

Where

k _x is the normalized autocorrelation function of the processes x (n).

And the optimal vector of weight coefficients from [2] is found as

or

We consider 2 cases: 1) q >> 1 and 2) when q << 1.

For large values of the ratio of the power of the simulated signal to the power of white noise, the normalized matrix R _{q is} obtained as

We write (2.4) in the form

RW = P

or

The solution to this equation is obvious, as a result of which we obtain the optimal vector:

This result indicates the impossibility of creating a model with large signal-to-noise ratios at the input of the adaptive filter, because in this case, white noise is generated at the output of the simulation circuit.

For q → 0, the normalized autocorrelation matrix R _q turns into the identity one, i.e.

R _q ≈I

Then the optimal vector of weights of the adaptive filter (for small q) is obtained from (3)

As a result, for small signal-to-noise ratios q, we conclude that the model of the original signal is described by its autocorrelation function.

To ensure the causality of the adaptive filter, a delay is introduced (Fig. 1), which provides the desired response in real time [2]. Then the cross-correlation vector and, accordingly, the optimal vector of weight coefficients take the form:

where L is the delay;

i = 0.1 ... N-1.

The selection of the adaptation algorithm is based on the conditions of use, as well as the characteristics of the original random process. In particular, for stationary processes, it is advisable to use the simple least-squares method - OLS [4], which does not require large computational costs and is simple in software implementation.

Figure 1 shows the structural diagram of a utility model, which shows: 1 - a random signal generator, 2 - analog-to-digital converter (ADC), 3 - a digital sequence generator for generating white noise, 4 - a digital sequence generator for modeling white noise, 5 - addition unit, 6 - digital delay unit, 7 and 8 - the first and second adaptive digital filters, respectively, 9 - subtraction unit, 10 - digital-to-analog converter (DAC). All these nodes and blocks can be made on separate ERE and IC, but can be performed on the same VLSI (for example, on a microcontroller with developed peripherals). In FIG. power buses and secondary control buses are not shown (starting the DAC and ADC, etc.).

The circuit has the following connections.

Adaptive modeling system for filtering random processes, containing a random signal generator 1, ADC 2, a digital sequence generator of generating white noise 3, a digital sequence generator of modeling white noise 4, addition unit 5, digital delay unit 6, the first and second adaptive filters 7 and 8, a subtraction unit 9 and a DAC 10, and having the following connections: the output of the random signal generator 1 is connected to the input of the ADC 2, the output of which is connected to the first input of the addition unit 5 and through the digital delay unit 6 from the first the input of the subtraction block 9; the output of the digital sequence generator generating white noise 3 is connected to the second input of the addition unit 5; the output of addition unit 5 is connected to the first input of the first digital adaptive filter 7, and the output of the digital sequence generator of modeling white noise 4 is connected to the first input of the second digital adaptive filter 8, the output of the subtraction unit 9 is connected by the OS bus to the second inputs of both adaptive filters 7 and 8 and moreover, the output through the CDL 10 is the output of the system.

The nodes and blocks of the structural diagram can be performed on the following ERE and IC. The analog-to-digital converter is made on the AD876 IC of the Analog Device company; the digital-to-analog converter is made on the AD7390 IC of the Analog Device company; white noise generators are made on IC K561IR2 and K561LP2; digital adaptive filters are made on a digital signal processor ADSP2181 from Analog Device; the operation “addition”, the operation “subtraction” and digital delay are implemented as part of the digital adaptive filter 1 on the ADSP2181.

The scheme works as follows.

To analyze the operation of the circuit in figure 1, an approach is proposed, which was highlighted in [3], when the adaptive filter weights (using the least squares algorithm - least squares method) are modeled as a deterministic time-averaged component E [W (n)] and fluctuating part with zero mean β (n) (figure 3).

The fluctuating component of the vector of weight coefficients is generated by an unsuppressed adaptive filter 2 (Fig. 3) generating white noise and its value determines the power of fluctuation noise at the output of the simulation circuit.

The ratio of the signal model power P _S to the fluctuation noise power P _β at the output of the Q _out simulation circuit is a criterion for optimizing the adaptive model and affects the adequacy of the random process model.

Fluctuation noise can be estimated using the method [3], namely

where B (n) is the column vector of fluctuation noise β (n);

N _M is a column vector of modeling white noise with a single dispersion.

Then we write the correlation matrix of fluctuation noise [3]

Where

Next, substituting (7) in (6) and using the transformations in [3], we find the power of fluctuation noise

where M is the coefficient of detuning [4], equal

Then we find the power of the model of the simulated signal, given (3)

where s (n) is the model of the modeled signal.

Substituting formulas (8) and (9) into (5), we obtain the signal-to-noise ratio at the output of the simulation circuit for q << 1:

Evaluation of the adequacy or quality of the model [5] is carried out according to the normalized mean square error calculated by the formula:

where x (t) and s (t) are compared between the original signal and its model,

In view of the fact that the modeled signal and its model are not correlated with each other, a block diagram for determining the variance of the modeling error in Fig. 4 is proposed, according to which the transmission coefficient of the so-called The “reference filter” is determined by the spectrum of the modeled signal x (t).

Figure 4 shows: 18 - block modeling of white noise, 19 - adaptive filter 2, 20 - reference filter.

From [6, 7] it is known that if the input of two linear systems with intersecting frequency characteristics -K ₁ (ω) and K ₂ (ω) is affected by a stationary random process with zero mean value and spectral density S _ξ (ω), then the function the mutual correlation of the signals s (t) and x (t) at the outputs of these systems is calculated:

We turn to the spectral densities in formula (11) and obtain:

Find the optimal transfer function of the adaptive filter in the steady state. According to [8]

where S _x (ω) is the spectral density of the simulated signal;

S _y (ω) is the spectral density of the forming white noise.

Considering that for stationary input signals S _x (ω) and _Sυu (ω) are non-negative and real functions [9], the transfer coefficient of the adaptive filter is also non-negative and real. Then K ₂ (jω) can be taken equal to the square root of the spectral density of the simulated signal so that the estimate of the spectral density of the simulated signal (according to Fig. 3) is equal to

Therefore, substituting (14) and (15) in (13), we finally find

Using two criteria for assessing the adequacy of models (5) and (16), we can find the optimal q value at which a high-quality simulation of the initial signal is achieved.

The optimal choice of the delay L allows one to slightly reduce the variance of the modeling error and improve the signal-to-noise ratio at the system output. The optimal delay value according to [2] can be taken equal to half the length of the adaptive filter.

We define the Markov process through a linear transformation of white Gaussian noise υ (t) with unit dispersion in the form of a stochastic differential equation of the first order [7]

with correlation function

and spectral density

We write the optimal vector of weight coefficients (5) of the Markov process in discrete time:

Where

And the signal-to-noise ratio at the output of the simulation circuit (10):

Then, we express the spectral density of the continuous Markov process (17) in discrete time in terms of the normalized (to the sampling frequency) frequency those

Then the optimal transfer function of the adaptive filter (14) is represented

Substituting (19) and (20) into expression (16), we obtain the variance of the modeling error of the Markov process

First, find the power estimates of the source signal and the model of the source signal

Then we calculate the integral in the expression (21)

Next we will transform

Substituting expressions (22), (23) and (24) in (21) we obtain

We find the minimum of expression (25) by differentiating it with respect to the parameter q and equating the resulting expression to zero

Solving the obtained expression by numerical methods / we determine the optimal signal-to-noise ratio q at which the signal model is most adequate:

where Δf is the signal bandwidth in Hz,

fs is the sampling frequency in Hz.

To study the effectiveness of the presented modeling algorithm, the program "Adaptive models of random processes" was developed. It is designed to create signal models in accordance with the formation algorithm shown in Fig. 1, adaptive filters were implemented in a non-recursive scheme with the OLS adaptation algorithm. The Gaussian-Markov process with a spectral density width of 3354 Hz (α = 0.1) was used as the initial random process.

As variable input parameters were set:

1) q is the ratio of the powers of the simulated signal and the shaping noise from minus 40dB to 10dB;

2) N - the number of weighting factors equal to 256;

The following parameters were observed as observed:

one) - variance of modeling error (f.16);

2) Q _out - the ratio of the power of the model of the modeled signal to the power of fluctuation noise at the output of the simulation circuit (f.5).

The results of theoretical research and computer modeling of the Markov random process are shown in Figs. 5 and 6, the analysis of which indicates the coincidence of theoretical calculations and experimental data, which proves the efficiency of the proposed algorithm for the formation of adaptive models of Gaussian stationary random processes.

The algorithm for the formation of adaptive models can be used:

1) for the formation of signal-like interference,

2) in the identification of stationary random processes,

3) when developing new algorithms for adaptive interference compensation,

4) in the development of instruments by which the sound pressure of acoustic radiation sources is evaluated.

Bibliography

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Claims (1)

Adaptive modeling system for filtering random processes, containing a random signal generator, ADC, digital white noise generator, white noise digital sequence generator, addition unit, digital delay unit, first and second adaptive filters, subtraction unit and DAC and having the following connections: the output of the random signal generator is connected to the ADC input, the output of which is connected to the first input of the addition unit and, through the digital delay unit, to the first input of the unit to subtraction; the output of the digital sequence generator generating white noise is connected to the second input of the addition unit; the output of the addition unit is connected to the first input of the first digital adaptive filter, and the output of the digital sequence generator of modeling white noise is connected to the first input of the second digital adaptive filter, the output of the subtraction unit by the OS bus is connected to the second inputs of both adaptive filters, and the output through the DAC is the system output .