PROCESOR PRE POTLÁČANIE IMPULZOVÉHO ŠUMU V TELEKOMUNIKAČNÝCH KANÁLOCH PROCESSOR FOR IMPULSIVE NOISE SUPPRESSION IN TELECOMMUNICATION CHANNELS PROCESOR PRE POTLÁČANIE IMPULZOVÉHO ŠUMU V TELEKOMUNIKAČNÝCH KANÁLOCH PROCESSOR FOR IMPULSIVE NOISE SUPPRESSION IN TELECOMMUNICATION CHANNELS

64 K O M U N I K Á C I E / C O M M U N I C A T I O N S 2 / 2 0 0 0 V ostatnom čase bolo mnoho lineárnych filtrov nahradených nelineárnymi filtrami, obzvlášť v prípade poškodenia signálu impulzovým šumom. Trieda kompozičných filtrov, ktorá patrí medzi nelineárne filtre, je zaujímavá kvôli ľahkej hardvérovej realizácii a vlastnosti nakladania. Rozvoj digitálnej HDTV podporil technologický vývoj v tejto oblasti. V mnohých aplikáciách je vhodné použitie veľmi účinných poriadkovo-štatistických filtrov, napr. mediánového alebo vyhladzovacieho LUM filtra. Tieto filtre môžu byť realizované ako kompozičné filtre, ktorými sa zaoberáme v tomto krátkom prehľade.

In recent years, many linear filters have been replaced by nonlinear filters, especially in case of impulse noise distorting. Stack filters, the class of non-linear filters are very interesting for easy hardware realisation and threshold decomposition property. The development of digital HDTV gives technological push in this area. Various of very efficient order-statistics filters such as median or LUM smoother are useful in many applications. In addition, these filters can be realised as stack filters. This paper gives a short review of the stack filter class.

Introduction
The majority of the present systems in communication, control and signal processing are developed under the assumption that the interfering noise is Gaussian [2]. However, many physical environments are more accurately modelled as impulsive non-Gaussian distributions [16][17][18]. In practice, sources of impulse noise include atmospheric noise, such as lightning spikes and spurious radio emission in radio communication, and relay switching noise in telephone channels. In addition to these natural non-Gaussian noise sources, there is a great variety of man-made sources such as automatic ignition, neon lights, and other electronic devices. Impulse noise is highly dependent on the physical environment and may be relatively infrequently occurring and non-stationary, which often renders it impossible to obtain an accurate statistical description. The non-Gaussian nature of impulse noise dictates that the suppression filter should be non-linear, and due to the presence of impulses, it must be robust. For these properties, the stack filters are widely used.
The paper is organized as follows. In the first section, impulsive noise types are described. The second section outlines the basic principle of the processor. The following sections describe the two-dimensional median filtering, LUM smoothers, fundamentals of stack filters and neural stack filters. Conclusions are drawn in the last section.
Two impulse noise models are usually used. The first one is the variable valued impulse noise, simply called impulse noise, where some of the signal elements are replaced by random value with uniform distribution from interval ͗0, 255͘ in case of 8bitquantized signal. It is defined as follows: where y is the distorted signal, x is the original, noise-free signal, rnd random value, p rnd probability of impulse occurrence and i is position of the signal elements.
The second noise model (2) is the fixed value impulse noise, or so-called salt and pepper noise, which distorts the signal by outlier. The p 0 and p 255 are the probabilities of "0" and "255" impulses, respectively.

The proposed processor
From the impulse noise model is clear that only few elements of the signal, usually up to 30%, are distorted. Therefore, only distorted signal elements should be filtered and the rest, noise-free elements should be left without filtering. However, the majority of present filters process every elements of the signal even the not distorted signal elements, too. The proposed processor consists of two parts (Fig. 1).
Its principle is simple [7,8,[10][11][12]14]. The first part of the processor, the detector, investigates the input data y i . In the case of noise-free detection, the processed sample is sent to the output without change. The amount of false detection influences the detector performance. In the case of impulse detection, the distorted signal element is sent to the estimator that tries to correct this element. This paper is focused on non-linear estimators. The following sections describe several type of estimators belong to class of stack filters.

Median filter
The simplest filter belonging to the class of stack filter is the median filter [1]. This filter is preferred in application because of its performance. The MF well suppresses impulse noise and maintains image edges at same time. Moreover, it is generally the most robust filter.
Výstup vyhladzovacieho LUM filtra je definovaný Its principle is following. An operation window (OW) is moved over the signal. Usually, the OW size varies from three up to seven. The data from OW are sent to the median operator (4) that calculates the filter output. The median operator arranges the data of the OW in ascending order of magnitude. Let x 1 , x 2 , …, x 3 are the data of the OW, then the arranged data are The x (i) is called the i-th order statistic, and the maximum and minimum of the OW are denoted by x (n) and x (0) , respectively. Usually, in the case of image processing the size of the OW is odd.
x ϩ x OW size is even The main drawback of the MFis that it blurs signal edges and removes fine details of the signal. On that account another filters based on order statistic trying to eliminate those disadvantages were developed.

LUM smoother
In many applications, the median introduces too much smoothing that in the practice means the blurring of edges and details. The LUM smoother [13,15], a subclass of a lower-uppermiddle (LUM) filters, achieves the best balance between noise smoothing and signal-detail preservation. A structure of LUM smoother is based on tuning parameter k for the smoothing. Varying this parameter changes the level of the smoothing from no smoothing (i.e. identity filter for kϭ1, where yϭx) to the maximum amount of smoothing (i.e. median, k ϭ (N ϩ 1)/2).
Thus, the smoothing function is created by a simply comparing of the processed sample to the lower-and upper-order statistics. If x 4 lies in a range formed by these order statistics, it is not modified. If x 4 lies outside this range it is replaced by a sample that lies closer to the median.
The output of LUM smoother is given by where 1 Յ k Յ {N ϩ 1}/2, x (k) a x (NϪkϩ1) are lower and upper order statistics of the ordered set. Fig. 2 LUM filter: a)
Prahová dekompozícia má teoretický aj praktický význam v oblasti mediánovej filtrácie. Používa sa na popísanie štatistických rozdelení výstupu a na nájdenie koreňov mediánového This definition is equivalent to the centre-weighted median that is given by the median over a modified set of observations containing multiple processed samples. However, the implementation of the LUM smoother as shown in (5) requires fewer operations than that of (6), since fewer elements must be sorted.
In (6) w is the weight of the central sample and is assumed to be an odd positive integer. The relationship between the parameter in the centre-weighted median and the parameter k in the LUM smoother is The process of arranging the data needs high computational demand. The next section describes a filter whose VLSI implementation is fast.

Stack filter
Median filtering of binary signals is relatively easy and fairly well understood. Its computation can be reduced to counting the 1's inside the OW. If their number is greater than or equal to (N ϩ 1)/2, the output of the median is 1, otherwise is 0. The filtering of the binary signal is attractive from both practical and theoretical point of view.
The binary signal can be obtained by the operator of threshold decomposition. Let x be an M -valued signal, in case of 8bit quantized signal M ϭ 256: 0 Յ x Յ ⌴ Ϫ 1 and consider the M Ϫ 1 thresholds: 1 Յ j Յ M Ϫ 1. The signal x can be decomposed to M Ϫ 1 binary valued signals x (j) , by using a threshold decomposition function I j (x) as follows: These M Ϫ 1 binary valued signals can be filtered independently for example by MF.
The output of the whole filter can be reconstructed by summing the binary output signals.
The architecture of stack filter is shown in Fig. 3.
The binary median filters at each level can be replaced by Boolean functions. The outputs of these binary filters possess what is referred to as the stacking property if the binary output signals y (j) 1 Յ j Յ k Ϫ 1, are piled and the pile of y at time i consists of a column of 1's having a column of 0's on top. The filters that support the stacking property are called stack filters [1,5]. The output of a stack filter is given by: Stack filters constitute a broad class of non-linear filters having median filters and LUM filters as special cases. The replacement of MF by Boolean function greatly facilitates the VLSI implementation of the MF based on threshold decomposition.
The class of stack filters is very large. Their number, when the filter window is n, grows faster than 2 2 n/2 . The problem of finding them reduces to the problem of finding stackable Boolean functions, or so-called positive Boolean functions (PBF). It has been established that a Boolean function is stackable if and only if it contains no complements of the input variables.
The main task of designers is to find a suitable PBF functions that well suppress the impulsive noise and concurrently well preserves signal edges and fine details. However, the amount of PBF functions is very high. In case of OW of size five, there are 7581 functions and in case of size seven, the number of PBF is tremendously high Ͼ 2.4 10 12 ! Therefore, it is necessary to find methods that find the optimal PBF from such great set.

Neural stack filters
Neural networks (NN) [9] greatly simplified the design of the stack filters. In this case, the binary median filters at each level are replaced by neural networks. Similarly as in previous case the stack filter can be homogenous, at each level are identical filters with identical parameters, or non-homogenous, at each level the parameters of the filters can vary. Usually, it a sufficient result can be obtained by the homogenous representation.
The stack filter structure is useful in case of NN, because the input data are {0,1} so it is not necessary to normalize the input for the NN. The output of the NN should be also 0 or 1. Such output can be obtained only by a hard limiter, or also called threshold activation function. y ϭ f act (x) ϭ Ά (12) where x is the input of the neuron and y is the neuron output.

Recenzenti: V. Moucha, L. Schwartz
where the NN processes the input data. In this case the NN weights are fixed. However, the activation function of the neurons cannot be hard limiter in training mode. Therefore, it must be changed to sigmoidal function, also called the soft limiter.
It was shown [3,4] that the NN with only one neuron gives sufficient results. The architecture of the NN is shown in Fig. 4.
The output of the NN is obtained by weighted summing the input data and applying the appropriate activation function on result. Such a simple structure can greatly improve the performance of the stack filter. Moreover, the neural filter can be trained for an arbitrary task, impulse noise suppression, edge enhancement, etc. However, the VLSI realization of the NN is more difficult than the realization of the Boolean functions.

Conclusion
This paper gave a short review of the class of stack filters. The simplicity of hardware realisation and the performance of impulse noise suppression make this class widely useful in many applications. The well-known median filter and its improvement the LUM smoother can be realised as the stack filters. The design of stack filters with optimal Boolean function is very computation demanding. An alternative way of design, the neural stack filters were shown. The performance of the neural stack filters is very interesting because of simple architecture of neural network. However the VLSI implementation is more complicated than the implementation of the PBF. Therefore, the future research should be oriented to accelerate the search of suitable PBF. Permutation theory gives one way of solving the problem and another way, the genetics algorithm [6].

Acknowledgements
The work presented in this paper was supported by the Grant Agency of the Ministry of Education and Academy of Science of the Slovak Republic VEGA under Grant No. 1/5241/98.