Abstract:
The fuzzy filtering of a noise signal comprising a plurality of signal samples [s(t,k)] is carried out using as variables the variation of the signal in the considered window and the distance of the samples from a sample to be reconstructed, to distinguish the typical variations of the original signal from those due to the noise and to identify the signal fronts. The method comprises the steps of defining a current signal sample [s(t)] from among the plurality of signal samples, calculating a plurality of difference samples [D(t,k)] as the difference in absolute value between the current signal sample and each signal sample and defining distance values (k) between the current signal sample and each signal sample. The method further comprises determining weight parameters [P(k)] on the basis of the difference samples and the distance values by means of fuzzy logic and weighing the signal samples with the weight parameters so as to obtain a reconstructed signal sample [o(t)].

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The invention relates to a fuzzy filtering method and associated fuzzy filter. 
     2. Discussion of the Related Art 
     As is known, a large number of mass-market applications (such as hi-fi, telecommunications) requires the definition of dedicated methods and architectures for filtering the deterministic signals corrupted by noise. 
     Traditionally, the problems of noise reduction (“denoising”) are dealt with using linear filters (such as low-pass type filters with a fixed cut-off frequency) or with non-linear filters such as median filters. 
     The traditional linear filters are ideal for solving problems in which the frequency specifications are well defined. They do, however, have the disadvantage that, to reduce noise, they often eliminate data belonging to the original signal. As far as non-linear type filters are concerned, median filters have proved to be very efficient in eliminating pulse-type noise, but they are less efficient in the case of Gaussian noise. 
     For this reason, filters of a completely different type have been proposed: for instance, the article entitled “Fuzzy Rule-Based Signal Processing and Its Application to Image Restoration” by Kaoru Arakawa, IEEE Journal on Selected Areas in Communications, Vol. 12, No. 9, December 94, proposes a filter based on the processing of the signal by means of “fuzzy” logic, in which the signal is reconstructed by means of a filter which weighs local samples of the signal received and the weights are calculated using three rules whose variables are the difference between the signal samples, the time difference between those signal samples and the local variance of the signal. The filter also uses a learning signal to fix a number of filtering parameters. 
     This approach is burdensome in its calculations and is not always capable of supplying the desired accuracy of reconstruction of the signal. 
     SUMMARY OF THE INVENTION 
     An object of the invention is therefore to provide a method and a filter for signals affected by noise, in particular by white noise of Gaussian distribution, which is capable of performing an efficient reconstruction of the signal, with calculation work that is reduced or in any event acceptable as regards the intended applications. 
     According to one aspect of the invention, a fuzzy filtering method and associated fuzzy filter are provided. In practice, according to one aspect of the invention, the filtering is carried out by using, as variables, the variation of the signal in the considered window and the distance between the samples and a sample to be reconstructed. This distinguishes the typical variations of the original signal from those due to the noise, and allows an identification of the signal fronts. 
     According to one embodiment of the invention, a method of fuzzy filtering of a noise signal comprising a plurality of signal samples is disclosed, the method comprising the steps of: a) defining a current signal sample from among the plurality of signal samples, b) calculating a plurality of difference samples as the difference in absolute value between the current signal sample and each sample of the plurality of signal samples, c) defining distance values between the current signal sample and each sample of the plurality of signal samples, d) determining weight parameters on the basis of the difference samples and the distance values by means of fuzzy logic and e) weighing the signal samples with the weight parameters so as to obtain a reconstructed current signal sample. 
     The method further comprises the steps of: f) calculating a noise sample as the difference between the reconstructed current signal sample and the current signal sample, repeating steps (a)-(f) to obtain a plurality of reconstructed current signal samples and a plurality of noise samples, determining a noise maximum variation value on the basis of a maximum value and a minimum value of the noise samples, determining a signal maximum variation value on the basis of a maximum value and a minimum value of the signal samples and determining at least two first classes of membership for the plurality of difference samples, the first classes of membership being defined by functions with sections having as limits the noise and signal maximum variation values. The method comprises associating with each difference sample a respective difference level of truth in each of the first classes of membership; determining at least two second classes of membership for the distance values, associating with each distance value a respective distance level of truth in each of the second classes of membership, applying fuzzy rules to associate the difference and distance levels of truth with weight values and with respective third classes of membership and determining the weight parameters as a function of the weight values and with pre-determined parameters of the third classes of membership. 
     The step of determining at least two first classes of membership comprises the steps of determining a SMALL membership class having a horizontal section for values of the difference samples between zero and the noise maximum variation value and a section of constant gradient for values of the difference samples between the noise maximum variation value and the signal maximum variation value, determining a LARGE membership limitation value as the difference between the signal maximum variation value and the noise maximum variation value and determining a LARGE membership class having a constant gradient section for values of the difference samples between zero and the LARGE membership limitation value and a horizontal section for values of the difference samples between the LARGE membership limitation value and the signal maximum variation value. 
     The step of associating with each difference sample comprises the steps of: 
     i. determining a central point of the constant gradient sections of the SMALL and LARGE membership classes; 
     ii. setting the difference level of truth equal to a pre-determined value; 
     iii. comparing the difference sample with the central point; 
     iv. determining a comparison point on the left of the central point if the difference sample is less than the central point and a comparison point on the right of the central point if the difference sample is greater than the central point; 
     v. comparing the difference sample with the comparison point; 
     vi. if the difference sample is less than the comparison point on the left, modifying the difference level of truth according to a first direction of increment and modifying the comparison point on the left; or 
     vii. if the difference sample is greater than the comparison point on the right, modifying the difference level of truth according to a second direction of increment and modifying the comparison point on the right; and 
     viii. repeating the steps v. and vi. or vii. until the difference sample is greater than the comparison point on the left or less than the comparison point on the right. 
     The method determines a sub-interval width of the constant gradient sections equal to the width of the constant gradient sections divided by a power of 2. The step of modifying the comparison point on the left comprises the step of decrementing the comparison point on the left by a quantity equal to the sub-interval width and wherein the step of modifying the comparison point on the right comprises the step of incrementing the comparison point on the right by a quantity equal to the sub-interval width. 
     According to another embodiment of the invention, a filter for implementing fuzzy filtering is disclosed. The filter comprises first store means for storing signal samples, second store means for storing noise samples, third store means for storing difference samples, subtractor means connected to the first and second store means, maximum/minimum value determination means connected to the first and second store means. The filter further comprises fourth store means for storing minimum and maximum values of signal samples and of noise samples, the fourth store means being connected to the subtractor means and to the maximum/minimum determination means, fifth store means for storing values of maximum variation of signal samples and of noise samples, connected to the subtractor means, a fuzzy calculation unit connected to the third and the fifth store means, sixth store means for storing levels of truth associated with the difference samples and with distance values between the signal samples, connected to the fuzzy calculation unit, weight calculation means for calculating weight samples, connected to the sixth store means, output calculation means connected to the first store means and to the weight calculation means to perform the weighted addition of the signal samples with the weight samples and a control unit connected to the first, second, third, fourth, fifth and sixth store means, to the subtractor means, to the maximum/minimum determination means, to the fuzzy calculation unit, to the weight calculation means and to the output calculation means. 
     The maximum/minimum value determination means comprise a comparison element having a first and a second input and an output, the first input of the comparison element being connected to the first store means a first selector element having a first and a second datum input, a selection input and an output, the first and second datum input of the first selector element being connected to the first and, respectively, the second input of the comparison element and the selection input of the first selector element being connected to the output of the comparison element and a first switching element having a datum input connected to the output of the first selector element, a control input, a first and a second output. 
     The maximum/minimum value determination means further comprises a first and a second store element connected to the first and, respectively, the second output of the first switching element, second selector element having a first and a second datum input, a selection input and an output, the first and second datum input of the second selector element being connected to the first and, respectively, the second store element and the output of the second selector element being connected to the second input of the comparison element, wherein the control input of the first switching element and the selection input of the second selector element receiving a minimum/maximum control signal. 
     The fuzzy calculation unit comprises a third, a fourth and a fifth store element, each having at least one datum input and an output, a third selection element having a first and a second datum input, a selection input and an output, the first datum input of the third selection element being connected to the output of the third store element, the second datum element of third selection element being connected to the output of the fifth store element, an adder element having a first and a second datum input, a control input and an output, the first datum input of the adder element being connected to the output of the third selection element, the second datum input of the adder element being connected to the output of the fourth store element and a second switching element having a datum input, a control input, a first, a second and a third output, the datum input of the second switching element being connected to the output of the adder element, the first, second and third output of the second switching element being connected to the datum input of each of the third, fourth and fifth store element, respectively. 
     According to another embodiment of the invention, a fuzzy logic filter for filtering a noise signal comprising a plurality of signal samples is disclosed. The filter comprises (a) means for defining a plurality of current samples from among the plurality of signal samples, (b) means for calculating a plurality of difference samples as the difference in absolute value between each of the plurality of current signal samples and each sample of the plurality of signal samples, (c) means for defining distance values between each of the plurality current signal samples and each sample of the plurality of signal samples, (d) means for determining weight parameters on the basis of each of the difference samples and each of the distance values by means of fuzzy logic, (e) means for weighing the samples with the weight parameters to obtain a reconstructed current signal sample and (f) means for calculating a noise sample as the difference between the reconstructed current signal sample and the current signal sample. Elements (a)-(f) process each of the plurality of signal samples to obtain a plurality of reconstructed current signal samples and a plurality of noise samples. 
     The filter further comprises means for determining a noise maximum variation value on the basis of a maximum value and a minimum value of the noise samples, means for determining a signal maximum variation value on the basis of a maximum value and a minimum value of the signal samples, means for determining at least two first classes of membership for the plurality of difference samples, the first classes of membership being defined by functions with sections having as limits the noise and signal maximum variation values and means for associating with each difference sample a respective difference level of truth in each of the first classes of membership. The filter further comprises means for determining at least two second classes of membership for the distance values, means for associating with each distance value a respective distance level of truth in each of the second classes of membership, means for applying fuzzy rules to associate the difference and distance levels of truth with weight values and with respective third classes of membership and means for determining the weight parameters as a function of the weight values and with pre-determined parameters of the third classes of membership. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     For an understanding of the invention, a preferred embodiment will now be described purely by way of non-exhaustive example, with reference to the accompanying drawings in which: 
     FIGS. 1 and 2 are graphs which show the plot over time of two signals, and specifically of a square-wave signal and a square-wave signal affected by white noise, respectively; 
     FIGS. 3,  4  and  5  are graphs which show the membership functions of three variables of the system, according to fuzzy logic; 
     FIG. 6 is a flow diagram which shows the general flow of the filtering method according to the invention; 
     FIG. 7 is a graph which shows the membership function of a first variable of the system, in discrete form; 
     FIG. 8 is a graph which shows the membership function of a second variable of the system, in shifted discrete form; 
     FIG. 9 is a flow diagram which shows a detail of the flow diagram of FIG. 6, with discrete system variables; 
     FIG. 10 is a part block, part schematic drawing which shows the architecture of an embodiment of the fuzzy filter according to the invention; 
     FIGS. 11 and 12 are part block, part schematic drawings which show more detailed block diagrams of the fuzzy filter of FIG.  10 . 
    
    
     DETAILED DESCRIPTION 
     The filtering according to the invention is applied in particular in the case of signals with constant sections, and with rapid transitions, like the signal of FIG.  1 . Let us suppose that, following transmission, for example, the signal is corrupted by white Gaussian-type noise and therefore has the form of FIG.  2 . If we call the corrupted signal of FIG. 2, s(t), the non-corrupted signal of FIG. 1 it is desired to reconstruct g(t) and the noise n(t), and if we allow the signal to be sampled at times t= 0  . . . T, this will give: s(t)=g(t)+n(t). 
     The purpose of filtering, therefore, is to localize and filter the zones disturbed by the noise by means of a local analysis of the signal, moving on windows of pre-determined size. In particular, the signal s(t) is sampled so as to respect the constraints of the sampling theorem. The signal is analyzed in work windows which contain 2*N+1 points [s(t−N), . . . , s(t−1), s(t), s(t+1), . . . , s(t+N)] and which shift whenever centered on successive points of the signal; the size 2*N+1 of the window thus tells us how much of the signal&#39;s history is taken into account at time t. 
     The variables used with this method are the variables k, D(k) and P(k), in which k is an index which assumes whole values between −N and +N; D(k) (also denoted as D below) is given by the absolute value of the difference between different samples of the signal s(t), and P(k) are the weights with which the signal samples s(t−k) have to be weighed so as to obtain the filtered output signal, according to the equations (which define the antecedents of the rules): 
     
       
         D(t, k)=|s(t−k)−s(t)| where k=−N, . . . , N and k≠0  (1) 
       
     
     k=the time distance between s(t−k) and s(t) 
     and according to the equation (which constitutes the consequent of the rules): 
     P(k)=weighing coefficients between [0,1] inclusive. 
     The value of the filtered signal, called o(t), is given by the following equation:                  o        (   t   )       =             ∑   k            P        (   k   )       *     s        (     t   -   k     )               ∑   k          P        (   k   )                         k     =     -   N         ,   …              ,   N           (   2   )                                
     in which the relative weight at the point s(t), P( 0 ), is always set equal to 1. 
     This method uses an adaptive type of scheme to define the membership functions of the variable D, by evaluating the maximum excursion of the signal in the considered window and the local estimate of the noise n(t) in a window of size [2*W+1], centered on the point t−W−1 (noise estimated for the signal samples preceding the one considered). In particular, two values V x  and V n  are determined, defined as: 
     
       
         V x =max [s(t)]  (3) 
       
     
     which represents the maximum value of the universe of the variable D, and 
     
       
         V n =max [err(t,j)]  (4) 
       
     
     where err(t,j) is given by: 
     
       
         err(t,j)=s(t−j)−o(t−j)  (5) 
       
     
     and defines a vector of dimensions 2*W+1. 
     For every time t, therefore, two membership functions shown in FIG. 3 are given for the variable D, in which V n  and V n  have the meaning and are calculated as indicated above and μ represents the confidence level or truth level of the variable D for each membership function. Consequently, the membership functions of the variable D are parametrized with respect to the values V x  and V n , so as to distinguish the signal from the noise. 
     In contrast, the variable k has two fixed membership functions, shown in FIG. 4, which provide respective confidence levels η. 
     According to this method, the following rules are applied for every value of the variable k (and hence of the relative value of the variable D): 
     R 1 : IF D is SMALL and k is SMALL THEN p 1  is LARGE; 
     R 2 : IF D is SMALL and k is LARGE THEN p 2  is MEDIUM; and 
     R 3 : IF D is LARGE and k is LARGE THEN p 3  is SMALL; 
     to determine corresponding membership functions of values p i  used to calculate the weights P(k). 
     In practice, on the basis of such rules, if the variation D between the sample s(t) considered and the sample s(t−k) is large and k is large, then a noise zone has been identified and thus the weight of the point s(t−k) in the reconstruction of the signal must be small; if the variation D is small, p i  must be large if k also is small, whereas it is medium if k is large. Furthermore, the case of large variation D and small k is not taken into account as it has no influence. 
     In fact, only a very large window causes a loss of accuracy on the progress of the signal and tends to make the output function “smooth”. The variable k enables this smoothing effect to be limited, in that in the reconstruction of the sample s(t) it acts so that the size of the sample s(t−k) is smaller, the greater the distance between the two samples considered. 
     The membership functions of the weight values p i  are shown in FIG.  5  and are represented by curves of constant area, so as to permit a simplification of the counts in the defuzzing phase. In FIG. 5, C 1 , C 2  and C 3  represent the centroids respectively for the cases where p i  is SMALL, MEDIUM or LARGE, and are preferably selected so that their sum is equal to a power of two and they are arranged at uniform distances. 
     According to this method, given the rules R 1 , R 2  and R 3 , to calculate the weights P(k) to use in equation (2), the p i  values are determined on the basis of the rule of the minimum (p iSMALL  is determined, for example, as minimum between the corresponding values of μ LARGE  and η LARGE ) and hence are multiplied by the values of the corresponding centroid (C 1 , for example, in the case of p 1 ), to determine the weights P(k) on the basis of the equation:                  P        (   k   )       =         ∑     i   =   1     3            p   i     *     C   i             ∑     i   =   1     3          C   i                              (   6   )                                
     The overall diagram of this method is shown in FIG.  6  and will now be described in detail. 
     The method provides a sequence of phases repeated for each sample of the signal s(t) to be reconstructed. After a series of initial phases comprising the initialization of filter parameters and variables (block  5 ) and the definition of the variable k (block  6 ), the method begins with an assignment of the parameter t (block  10 ). The considered samples window is defined (by means of storage, for example, in a first register) by a vector {overscore (s)} of 2N+1 samples [s(t−N), . . . , s(t−1), s(t), s(t+1), . . . , s(t+N)] centered on s(t) (block  11 ). The estimated noise window is defined by means of previously calculated samples (by means of storage, for example, in a second register) of a vector {overscore (err)} of 2W+1 samples [err(t−2W−1), . . . , err(t−1)] centered on err(t−W−1) (block  12 ). Preferably, N=W, so that an equal number of signal and noise samples is considered. The method continues with the calculation, according to equation (1), of the 2K values of the variable D, for example, stored as vector {overscore (D)} in a third register (block  13 ). Alternatively, the vectors {overscore (s)}, {overscore (err)} and {overscore (D)} of the signal samples, of the noise samples and of the values of the variable D, respectively, are stored in three different parts of one register. 
     The minimum and maximum values MIN_S, MAX_S, MIN_ERR, MAX_ERR assumed by the signal samples s(t) and by the estimated noise err(t) in the windows considered, or of the samples stored above, (block  14 ) are then calculated (in parallel or in succession). The values V x  and V n  are determined, as MAX_S-MIN_S and MAX_ERR-MIN_ERR (block  15 ). On the basis of the values V x  and V n  just calculated, the membership functions of the variable D are then defined (block  16 ). 
     Then, for every value of k from −N to +N, with k≠0, the respective value of the variable P(k) is determined, so as to obtain a vector {overscore (P)}. In detail, starting from k=−N (block  20 ), on the membership functions of the variable D which have just been defined (FIG.  3 ), the respective values μ s , μ L  are determined respectively in the case of D=SMALL and D=LARGE (block  21 ). On the membership functions of the variable k (FIG.  4 ), the respective values η s , η L  respectively in the case k=SMALL and k=LARGE are determined (block  22 ). Then, by applying the rules R 1 , R 2  and R 3  described above and the rule of the minimum, the three associated weight values p 1 , p 2  and p 3  are determined (block  23 ). Based on the membership functions of the weight values, the respective centroids C 1 , C 2  and C 3  are determined (block  24 ). For example, for rule R 1 , the corresponding weight value p 1  is equal to the minimum among the values μ s  and η s  determined previously (e.g. μ s ) and the corresponding centroid (given that p 1 =LARGE, according to Rule  1 ) is equal to C 1 . 
     Next, on the basis of equation (6), the weight variable P(k) is calculated and is stored in an appropriate location in a respective register (block  25 ). The variable k is incremented (block  26 ) and a check is made as to whether k is different from zero (block  27 ), in which case p(k) is set equal to 1, block  28 , k is incremented again and, as long as k is not greater than N (output NO from block  29 ), the above-mentioned phases relating to blocks  21 - 26  are repeated. At the end, all the 2N+1 weights P(k) necessary to weigh the signal s(t) are available. The output signal o(t) is now calculated according to equation (2) (block  30 ). Finally, the new estimate of the noise at time t (err(t)) is calculated as the difference between the local value of the corrupted signal s(t) and the local value of the reconstructed signal o(t) (block  31 ). The procedure then begins again for a successive sample, as shown in the flow diagram with the return to block  10 . 
     The above-mentioned method is implemented by means of a hardware architecture that will now be described. In particular, in designing the architecture, the problem of finding data structures capable of meeting two contrasting requirements of information representation and data processability was addressed. This problem related in particular to the membership functions of the variable D which is the most costly from the point of view of the calculations required. 
     Generally, the membership functions are represented in analytical or vectorial manner. However, a first solution requires a computational complexity (in particular because of the number of divisions required) that is such that the requirements in terms of hardware are unacceptable for the majority of mass-market applications. In contrast, a second solution proves impossible to achieve in that, in the presence of input signals with 20-24 bit dynamics, the vectorization of the membership functions would require values of N (and hence of the associated samples) of the order of 2 20 -2 24  which are incompatible with any practical implementation. 
     To solve this problem, according to the present invention, the membership functions of both the variable D and the variable k are discretized, and a desired level of accuracy is pre-determined. This is advantageous in that k assumes only discrete values, so that it is sufficient to associate with each value of k a corresponding truth value (see FIG.  7 ). In the example considered, where N=5, there are six different truth values to store. By selecting an accuracy of representation of 4 bits it is thus possible to construct a table of 11×2 elements, containing the binary code of the values va10 . . . va15 (equal to 0000; 0011; 0110; 1001; 1100 and 1111) for the eleven values of k and for the two situations k=SMALL and K=LARGE. 
     Given that the variable D is linked by AND conditions to the variable k, and once the accuracy of the variable k has been chosen, the deduction is that it is not necessary to define truth values relating to the variable D with an accuracy greater than that relating to the variable k. This causes a notable simplification of the hardware structure. In particular, the increasing and decreasing sections of the level of truth associated with the variable D in both cases, SMALL AND LARGE, can be divided into sixteen horizontal levels, from 0 to 15, and these may be made to correspond to sixteen sub-intervals of variability of the variable D, characterized in that they have a same value of μ. According to this solution, furthermore, for values of the variable D that are smaller than V n  (where D=SMALL), the associated level of truth μ is set equal to the maximum value (15) and the discrete transfer function is shifted along the axis of the abscissae until the point D=V n  is brought to the origin of the axes. A discretized virtual membership function is thus obtained, shown in FIG. 8, in which D T  is the shifted variable D, comprised between 0 and (V x −V n ) inclusive, and its variability interval, denoted overall by A, has been divided into 16 sub-intervals, each equal to A/16. In this way, and as described with reference to FIG. 9, the level of truth associated with each value of D may be produced by means of simple additions, comparisons and repeated divisions by 2 without calculating or storing the function. In the case of D=LARGE, the membership function is obtained solely by discretizing the levels of truth associated with sixteen sub-intervals of variability of the variable D in the increasing section (between 0 and V x −V n ) and setting μ=15 when D&gt;V x −V n . 
     With reference to FIG. 9, the method for calculating the degrees of membership of the variable D in the case D=SMALL will now be described in detail; the calculation in the case D=LARGE is similar, apart from a number of inversions relating to the comparisons and the direct comparison between the variable D and the variability intervals. 
     The value D is initially compared with V n  (block  40 ). If D&lt;V n , μ=15 is set (block  41 ) and the calculation ends. Otherwise (output NO from block  40 ), the central point (V x −V n )/2 of the variability interval of the variable D T  is calculated and a variable DCOMP is initialized with this value (block  42 ). The value D−V n  is then compared with DCOMP (block  43 ) and, if D−V n &lt;DCOMP (or the variable D T  is to the left of the central point, output YES from block  43 ), μ=8 is set (block  44 ). Then in block  45 , the value DCOMP is updated by decrementing it by (V x −V n )/16, that is reducing the comparison value DCOMP by one sixteenth of the variability interval A of D T , corresponding to a different level of truth. The value D−V n  is then compared with the new value of DCOMP (block  46 ). If D−V n ≧DCOMP (that is D T  is to the right of the new DCOMP value) the value of μ is already the correct one and the procedure ends. Otherwise in the case of a YES output from block  46 , μ is incremented by one unit (block  47 ) and the method returns to block  45 , again decrementing DCOMP by a quantity equal to a sub-interval A/16, resuming the above-mentioned phases of updating DCOMP, incrementing μ and checking until D T  comes to be to the right of the current value of DCOMP, after which the procedure ends. 
     Similarly, where the comparison carried out in block  43  gives a negative outcome (the variable D T  is to the right of the central point), μ is set at equal to 7 (block  50 ). The DCOMP value is updated by incrementing it by (V x −V n )/16, that is increasing the comparison value by a sixteenth of the variability interval of D T  (block  51 ). The value D−V n  is then compared with the new value of DCOMP (block  52 ). If D−V n &gt;DCOMP (or D T  is to the right of the new DCOMP value), μ is decremented by one unit (block  54 ) and the method returns to block  51 , again incrementing DCOMP by a quantity equal to a sub-interval. Otherwise, in the case of a NO output from block  52 , the procedure ends. 
     FIG. 10 shows a hardware structure, denoted in its entirety by  60 , implementing the filtering method described above. In the particular example shown, N=W=5 has been set and the signals are coded by means of 24 bits. The structure  60  comprises a control unit  61 , such as a state machine, which determines the processing sequence and the loading and the passage of data between the various units. The structure  60  further comprises a first data store  62 , a minimum/maximum determination unit  63 , a second data store  64 , a difference unit  65 , a third data store  66 , a fuzzy calculation unit  67 , a fourth data store  68 , a weights calculation unit  69  and an output calculation unit  70 . 
     As well as control lines connecting the control unit  61  to the various units  62 - 70 , FIG. 10 shows lines representing the flow of data within the structure. In particular, for the first data store  62 , the inputs at which the signal samples from outside the structure and from the difference unit  65 , the noise samples and the samples of the variable D are supplied are shown. The output of the first data store  62  is connected to the minimum/maximum determination unit  63 , to the difference unit  65 , to the fuzzy calculation unit  67  and to the output calculation unit  70 . The output of the minimum/maximum determination unit  63  is connected to the second data store  64  whose output is connected to the difference unit  65 . The difference unit  65  has an input connected to the output of the output calculation unit  70 , and outputs connected to the first  62  and to the third data store  66 . The third data store  66  has an output connected to the fuzzy calculation unit  67 , whose output is connected to the fourth data store  68 . The fourth data store  68  has an output connected to the weights calculation unit  69 , which has an output connected to the output calculation unit  70 . 
     The first data store  62  is intended to store the signal samples s(t−5) . . . s(t+5), the noise samples err(t−11) . . . err(t−1) and the vector {overscore (D)} D(t−5) . . . D(t+5), and thus comprises 32 registers, 11 for the signal samples, 11 for the noise samples and 10 relating to the elements D(k), where k≠0, of the vector {overscore (D)}. The first data store  62  is updated at every cycle, and specifically at the start of each new cycle, by causing the preceding signal samples to shift so as to eliminate the “oldest” sample and inserting a new signal sample corresponding to the load phase indicated by block  11  in FIG.  6 . The value of the variables D calculated by the difference unit  65  in the first processing phases according to block  13  of the diagram of FIG. 6 are then stored. At the end of the procedure for calculating the reconstructed signal sample o(t), a new noise sample err(t) is inserted, after causing the preceding noise samples to shift and thus eliminating the oldest noise sample, corresponding to block  12  of FIG.  6 . 
     The structure of the minimum/maximum determination unit  63  is shown in FIG.  11 . The unit  63  comprises a multiplexer  73  having two data inputs  74  and  75 , a selection input  76  and an output  77  connected to an input of a demultiplexer  78  having a selection input  79  and two outputs  80  and  81 . The output  80  is connected to an input of a maximum register  82  having two outputs  83  and  84 .  25  The output  81  is connected to an input of a minimum register  86  having two outputs  87  and  88 . The outputs  83  and  87  of the registers  82  and  86  are connected to the inputs of a multiplexer  90  having a selection input  91  and an output  92  connected to a first input  93  of a comparison block  94 . Comparison block  94  has a second input  95  connected to the first data store  62  and an output connected to a first input  96  of an AND gate  97 , which has a second input  98  and an output connected to the selection input  76  of the multiplexer  73 . The input  74  of the multiplexer  73  is connected to the output  92  of the multiplexer  90  and the input  75  is connected to the first data store  62 , together with the second input  95  of the comparison block  94 . 
     According to the illustration in FIG. 11, the first data store  62  has a control input to which is supplied a pointer signal Pt which specifies successive registers of the first data store  62  relating to the signal samples s(t) and noise samples err(t) stored, whose contents are supplied, one at a time, simultaneously to input  75  of the multiplexer  73  and input  95  of the comparison block  94 . Furthermore, inputs  79 ,  91  and  98  all receive a 1-bit signal M which specifies each time whether a minimum or maximum comparison is being made and switches with every timing pulse so as to carry out, for example, firstly a minimum comparison and then a maximum comparison for one binary signal or noise sample, supplied by the first store  62 . 
     The minimum/maximum determination unit  63  operates as follows. Initially the registers  82  and  86  are initialized to arbitrary values, such as the minimum number, for the maximum register  82 , and the maximum number, for the minimum register  86 , permitted by the accuracy provided. Then, to the inputs  75  of the multiplexer  73  and  95  of the comparison block  94 , the first data store  62  supplies a first datum, such as the first signal sample s(t−5). The selection signal M is at the minimum comparison logic value. Then the multiplexer  90  supplies the input  93  of the comparison block  94  with the content of the minimum register  86  which is then compared with the sample present at the input  95 . The comparison block  94  then generates at its output a 1-bit signal C whose logic value indicates whether or not the signal at the input  93  is greater than the signal at the input  95 . The signal C is combined by the AND gate  97  with the selection signal M so as to supply to the input  76  a control signal SEL which specifies which of the two data input at the multiplexer  73  must be output, according to the table below. By way of example, it is supposed that the signal M is 1 in the case of seeking the maximum and 0 if it is not, the signal C is 1 in the case where the datum at the input  93  is greater than that present at the input  95  and 0 if it is not, and the signal SEL is equal to 1if the multiplexer has to output the datum present at its input  74  and 0 if it does not. 
     
       
         
               
               
               
             
           
               
                   
               
               
                 M\C 
                 0 
                 1 
               
               
                   
               
             
             
               
                 0 
                 SEL = 1 
                 SEL = 0 
               
               
                 1 
                 SEL = 0 
                 SEL = 1 
               
               
                   
               
             
          
         
       
     
     If the datum at the inputs  75  and  95  is less than the datum at the inputs  74  and  93 , it is then supplied to the demultiplexer  78  which, on the basis of the low value of the signal M, supplies it to the output  81  for storage in the register  86  in the place of the preceding value. The signal M then switches to 1, the multiplexer supplies to the inputs  74  and  93  the content of the register  82  which is compared by the block  94  with the datum supplied by the first data store  62  and previously compared with the content of the register  86 . On the basis of the outcome of the comparison, and by analogy with the explanation of the minimum comparison, the maximum between the two data inputs is supplied at the output  77  and, via the demultiplexer  78 , to the maximum register  82 . In a successive cycle, the first data store  62  supplies a second signal sample s(t−4), on which the comparisons are made as described above, and so on, for all the signal samples as far as s(t+5). At the end of the comparisons, the maximum and minimum value of the signal are stored in the registers  82  and  86 , respectively, and these values, called MAX_S and MIN_S, are copied into two of the four registers of the second data store  64 . The process described above is then repeated for all the noise samples err(t), permitting the storage, in the other two registers of the second data store  64 , of the minimum and maximum value of the noise, denoted by MAX_ERR and MIN_ERR. 
     The second data store  64  comprises four registers for storing MAX_S, MIN_S, MAX_ERR AND MIN_ERR. The difference block  65  is implemented by a hardware adder of known type and is not therefore shown in greater detail. The third store  66  comprises two registers for storing the values of V x  and V n . 
     The structure of the fuzzy calculation unit  67  is shown in FIG.  12 . The unit  67  comprises two shift registers  100  and  101  and one simple register  102 , which are all 24-bit registers. The registers  100 - 102  each have a first input, respectively  103 ,  104  and  105 , a second input, respectively  106 ,  107  and  108 , and an output, respectively  109 ,  110 ,  111 . The shift registers  100  and  101  also each have a control input  113  and  114 , respectively, to which is supplied a control signal which specifies whether the content of the register should be shifted and by how many positions (to carry out a division by 2 or by 16, according to the provision of the flow diagram shown in FIG.  9 ). The outputs  109  and  111  of the registers  100  and  102  are connected to respective inputs of a multiplexer  117  having a selection input  118  and an output  119  connected to an input of an adder element  120 . Adder element  120  has a second input connected to the output  110  of the register  101 , a timing input  121  and an output  122 . The output  122  is connected to the input of a demultiplexer  123  having a two-bit selection input  124  and three outputs, one connected to each of the inputs  106 - 108  of the registers  100 - 102 . 
     The output  122  of the adder  120  is also connected to the control unit  61  (FIG. 10) which also supplies the selection and control signals to the multiplexer  117 , the demultiplexer  123  and to the shift registers  100  and  101  to control the timing of the flow of data between the various elements of the unit  67  and to determine the operations to be carried out. In particular, as explained above, on the basis of the output of the adder  120 , the control unit  61  determines the outcome of the comparisons described in the blocks  40 ,  46  and  52  of the flow diagram of FIG.  9 . It also determines whether the DCOMP value needs to be incremented or decremented, i.e., whether to move to the right or left of the mid-point of the discretized and moved membership function of FIG.  8 . Furthermore, the control unit  61  provides to update the value of μ. 
     Initially, the registers  100 - 102  are loaded with the values D, V n  and V x . Then the adder  120  receives at its inputs the values V x  from the register  101  and the value D from the multiplexer  117 , appropriately controlled via the selection signal present at its input  118 . In a successive phase, the adder  120  carries out the difference between D and V x , controlled via the control input  121 , and simultaneously receives the values V x  and V n  from the registers  102  (via the multiplexer  117 ) and  101 . The result of the difference (D−V n ) is supplied both to the demultiplexer  123  and to the control unit  61  which, according to whether it is positive or negative, checks the condition of the block  40  of FIG.  9 . If the difference is negative, the control unit sets μ=15 and ends the procedure, otherwise it generates a control signal for the demultiplexer  123  so that it supplies the output of the adder  120  to the register  102 , the content of which is not modified further. The control unit  61  then controls the adder  120  so that it carries out the difference between the inputted values V x  and V n . It also controls the demultiplexer  123  so that it supplies to the registers  100  and  101  the output of the adder  120  and the registers  100  and  101  so that they cause the value received to shift respectively by four bits and one bit to the right, determining a division by 16 and by 2. At the end of this phase the register  100  stores the value (V x −V n )/16, the register  102  stores the value D−V n  and the register  101  stores the value (V x −V n )/2. After that the content of the register  100  is not modified further. 
     Subsequently, the control unit  61  controls the multiplexer  117  so that it supplies to the first input of the adder the content of the register  102  (D−V n ). It also controls the adder  120  so that it carries out the difference between the datum present at its first input and the datum present at its second input (equal to the content of the register  101 , (V x −V n )/2) and checks whether the result is positive or negative, as specified in block  43  of FIG.  9 . This determines whether the value of D−V n  is to the right or left of the central point of the variability interval. On the basis of this result of the comparison, the content of the resister  101  is updated by adding, at every step, the content of the register  100 , according to block  51  of the diagram of FIG. 9, or by subtracting, at every step, the content of the register  100 , according to block  45  (updating of DCOMP). These updates are carried out by appropriately controlling the adder  120 , and storing the result of the addition or subtraction in the register  101 . Thus, in every checking cycle, the content of the register  101  is subtracted from the content of the register  102  (equal to D−V n ), to make the comparison provided for in block  45  or block  51 . On the basis of the result of this subtraction, the control unit  61  decides whether to proceed with the updating of the comparison value DCOMP stored in the register  101  and with the updating of the value of μ, or whether to interrupt the calculation cycle, according to the description in the diagram of FIG.  9 . 
     In this way, with seven iterations at the most, by carrying out simple additions and subtractions, it is possible to determine the value of μ for a pre-determined value of k and a specific condition (D=SMALL or D=LARGE). The repetition of the phases described for all the values of k (apart from k≠0) and both the conditions of D enables ten values of μS corresponding to D=SMALL and ten values of μL corresponding to D=LARGE to be calculated. 
     The fourth data store  68  comprises four groups of ten registers (40 registers in total) of 4 bits, for storing truth values μ and η for D=SMALL and D=LARGE, respectively, k=SMALL and k=LARGE for k=−5 . . . +5, with k≠0, as indicated in FIG. 1, from μS 1  . . . μS 10 , μL 1  . . . μL 10 , ηS 1  . . . ηS 10 , ηL 1  . . . ηL 10 . In particular, the values of μS and μL are supplied by the fuzzy calculation unit  67 , as described above, whilst the values of ηS and ηL can be stored permanently in the fourth data store  68 , given that they do not depend on the value of the signal samples s(t,k) or on other variables conditioned by local situations or events, but only on k, whose variability is pre-defined at the design stage, as explained above with reference to FIG.  7 . The phases of acquisition of the values k (block  6  of FIG. 6) and of the respective levels η s , η L  (block  22 ) may therefore be carried out once and for all, in the design phase or when the fuzzy filter  60  is used for the first time. 
     The weight calculation unit  69  provides to apply the rules to determine the weight values p i  and the associated centroids and to calculate the variables P(k), according to the description with reference to blocks  23 - 25  of FIG.  6 . This unit is produced by means of a dedicated ALU, produced in known manner and not therefore described, which takes advantage of the fact that the sum of the centroids is equal to a power of2, as described above, so as to reduce the final division provided in equation (6) to one shift in one register by a number of positions equal to that power. 
     The output calculation unit  70  determines the estimated value of the signal o(t) on the basis of equation (2) based on the weights P(k) and the signals s(t−k) supplied by the first data store  62  and by the weights calculation unit  69 . This unit  70  is also produced as a dedicated ALU, preferably separated from the previous one to allow pipeline processing of the data. The output o(t) thus calculated is output as a reconstructed signal sample and furnished to the difference unit  65  so that it calculates the difference between it and the signal sample s(t) to determine the noise err(t) required for the successive calculation cycle relating to the signal sample s(t+1). 
     The operation of the hardware structure  60  will be evident from the above and on the basis of the flow diagrams according to FIGS. 6 and 9. 
     The advantages that can be obtained with the method and the architecture described are as follows. First, they permit a good reconstruction of the signal affected by Gaussian white noise even when the signal has steep fronts (square-wave signal). Furthermore, the reconstruction of the signal can be carried out by means of a hardware structure that is relatively simple and inexpensive, and capable of calculating the reconstructed signal samples in real time. This structure may be implemented and integrated with ease, permitting an integrated circuit dedicated to the reconstruction of the signal to be obtained. The method and architecture described do not require the presence of ordered data, which would involve burdensome operations from the point of view of the calculations, and permit the calculation of variable membership functions (dependent on the values V x  and V n  determined from time to time), which could not be tabulated and the calculation of which by intersection would be burdensome with traditional methods. 
     Having thus described at least one illustrative embodiment of the invention, various alterations, modifications, and improvements will readily occur to those skilled in the art. Such alterations, modifications, and improvements are intended to be within the spirit and scope of the invention. Accordingly, the foregoing description is by way of example only and is not intended as limiting. The invention is limited only as defined in the following claims and the equivalents thereto.