Abstract:
Provided is an adaptive filter that can reduce the computational complexity. The adaptive filter includes: a segmentation unit for segmenting N number of input signals into G number of signal groups; sub-filter unit having G number of sub-filters, which are corresponding to each of the signal groups, for filtering the corresponding signal group; an addition unit for summating the output signals of the sub-filter unit; an error computing unit for generating an error signal by comparing the output signals of the addition unit with a desired signal; filter coefficient updating unit having G number of filter coefficient updating units, each of which is corresponding to each of the sub-filters, for updating the filter coefficient of the corresponding sub-filter; and a switching unit for inputting the error signal to any one of the filter coefficient updaters optionally with respect to an iteration number k.

Description:
FIELD OF THE INVENTION 
   The present invention relates to an adaptive filter and filtering method thereof; and, more particularly to an adaptive filter having a reduced computational complexity. 
   DESCRIPTION OF RELATED ART 
   Adaptive filters are used in various areas, such as identification, inverse modeling, prediction, and interference cancellation. 
     FIG. 1  is a block diagram illustrating a configuration of a typical adaptive filter. In the drawing, a variable ‘k’ is an iteration number; x i (k)(i=0,1,Λ,N−1) denotes an i th  input signal; y(k) is an output signal of the adaptive filter; and d(k) is a desired signal. An error signal e(k) is computed as d(k)−y(k). Adaptive filters appropriately update a filter coefficient W i (k)(i=0,1,Λ,N−1), based on the computed error signal e(k). In other words, the filter coefficient W i (k)(i=0,1,Λ,N−1) is updated in a way for minimizing a given objective function. 
   In a least square algorithm, the objective function is given as shown in Equation  1 , which is disclosed in “Adaptive Filtering: Algorithms and Practical Implementation” (Paulo Sergio Ramirez Diniz, Kluwer Academic Publisher, 1997) that is incorporated herein by reference. 
   
     
       
         
           
             
               
                 
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   , where X(i)=[x 0 (i),Λ,x N−1 (i)] T  denotes an input vector at an instant value i; 
   θ(k)=[W 0 (k),Λ,W N−1 (k)] T  denotes an adaptive filter coefficient vector; and 
   λ denotes an exponential weighting factor selected within a predetermined range. 
   Following Equation 2 can be obtained by differentiating J(k) with respect to θ(k) and equalizing the differential equation to zero. 
   
     
       
         
           
             
               
                 
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   where R(k) denotes a deterministic correlation matrix of an input signal; and 
   P(k) denotes a deterministic cross-correlation vector between the input signal and a desired signal. 
   A reversed function of R(k) is computed by an algorithm having an computational complexity of O(N 3 ). Conventional recursive least squares (RLS) algorithm updates the coefficient of a filter by using a matrix inversion lemma as described below, instead of performing the reversed function directly. 
   According to the matrix inversion lemma, the reversed function of R(k) is computed as shown in Equation 3. 
   
     
       
         
           
             
               
                 
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   Equations used in the conventional RLS algorithm are described as below. 
   At an initialization, below equations 4 and 5 are computed.
 
 S (−1)=δ −1   I   N×N   Eq. 4
 
 P (−1)= X (−1)=[00Λ0] T   Eq. 5
 
   When K≧0 then below equations 6 to 8 are computed. 
                   S   ⁡     (   k   )       =       1   λ     ⁡     [       S   ⁡     (     k   -   1     )       -         S   ⁡     (     k   -   1     )       ⁢     X   ⁡     (   k   )       ⁢       X   T     ⁡     (   k   )       ⁢     S   ⁡     (     k   -   1     )           λ   +         X   T     ⁡     (   k   )       ⁢     S   ⁡     (     k   -   1     )       ⁢     X   ⁡     (   k   )               ]               Eq   .           ⁢   6                 P ( k )=λ P ( K −1)+ d ( k ) X ( k )  Eq. 7 θ(k)=S(k)P(k)  Eq. 8 
   Occasionally, below equations 9 and 10 are computed.
 
 y ( k )=θ T ( k ) X ( k )  Eq. 9
 
 e ( k )= d ( k )− y ( k )  Eq. 10
 
   , where δ denotes an inverse of an input signal power estimate; and I N×N  denotes a unit matrix of an N×N dimension. 
   As seen from Eqs. 4 through 10, since there are matrix multiplications with an order of N, the computational complexity is proportional to O(N 2 )in the conventional RLS algorithm. 
   The main concern in practical implementation of the adaptation algorithms is computational complexity, which generally depends on the number of filter coefficients. 
   To alleviate the computational cost of the RLS, a family of fast RLS algorithms such as fast transversal filters, RLS lattice filters, and QR-decomposition based lattice filters (QRD-RLS), have been explained by Simon Haykin in an article entitled “Adaptive Filter Theory” Fourth edition, Prentice Hall, 2002, which is incorporated herein by reference. Exploiting the special structure of the input data matrix, they can perform RLS estimation with O(N) complexity. One of the major disadvantages of these fast RLS algorithms is that they work for data with shifting input (structured data) only. 
   By applying multiple decompositions of the signal space, Wu and Tai proposed a method of reduced complexity RLS algorithms for non-structured data in “Split Recursive Least Squares: Algorithms, Architectures, and Applications,” by An-Yeu Wu and K. J. Ray Liu, IEE Trans Circuits and Systems Part II: Analog and Digital Signal Processing, Vol. 43, pp. 645–658, September 1996) and “HRLS: A more Efficient RLS Algorithm for Adaptive FIR Filtering,” by Tai-Kuo Woo, IEEE Communications Letter, Vol. 5, No. 3, March 2001. The proposed method of reduced complexity RLS algorithms are shown in  FIGS. 2 and 3 . 
     FIG. 2  is a block diagram showing a configuration of an adaptive filter according to a prior art. As shown in the drawing, the input signal {x 0 (k),x 1 (k),Λ,x i (k),x i+1 (k),Λ,x N−1 (k)} is grouped into two sets: {x 0 (k),x 1 (k),Λ,x i (k)} and {x i+1 (k),Λ,x N−1 (k)}. Each of them is provided to a first filter  201  and a second filter  202 , respectively. The outputs of the first and second filters  201  and  202 , y 1 (k) and y 2 (k), are inputted to a third filter  203 , which outputs y(k). The output y 1 (k) of the first filter  201  is compared with the desired signal d(k) at a first error computing unit  207  and an error signal e 1 (k) is outputted. The error signal e 1 (k) is fed back to a first filter coefficient updating unit  204 . 
   The output y 2 (k) of the second filter  202  is compared with the desired signal d(k) at a second error computing unit  208  and an error signal e 2 (k) is outputted. The error signal e 2 (k) is fed back to a second filter coefficient updating unit  205 . 
   The output y(k) of the third filter  203  is compared with the desired signal d(k) at a third error computing unit  209  and an error signal e(k) is outputted. The error signal e(k) is fed back to a third filter coefficient updating unit  206 . The first, second and third filter coefficient updating unit  204 ,  205  and  206  are used to update the first filter coefficient {W 0 (k),W 1 (k),Λ,W i (k)}, the second filter coefficient {W i+1 (k),Λ,W N−1 (k)}, and the third filter coefficient {W′ 1 (k),W′ 2 (k)}, respectively. 
   In this adaptation algorithm, all of the filter coefficients are updated simultaneously at each iteration number k. A first filter  201  and a second filter  202  can be further decomposed into sub-filters of a lower order. Thus, the minimum computational complexity can be O(N), if N is power of 2. 
     FIG. 3  is a block diagram describing a configuration of an adaptive filter according to another prior art. As shown in  FIG. 3 , the configuration of the adaptive filter is almost the same as  FIG. 2 . When the third filter  203  and the third filter coefficient updating unit  206  are removed from  FIG. 2  and replaced with an adder  305 , it becomes a simplified configuration of the filter shown in  FIG. 2 . In short,  FIG. 3  is a tabloid edition of the filter illustrated in  FIG. 2 . The problem of the two types of filters shown in  FIGS. 2 and 3  is that the filters are acceptable only for input vectors with almost non-correlated components. 
   SUMMARY OF THE INVENTION 
   It is, therefore, an object of the present invention to provide an apparatus and method for reducing the computational complexity of adaptive filters. 
   In accordance with an aspect of the present invention, there is provided an adaptive filter, including: a segmentation unit for segmenting N number of input signals into G number of signal groups wherein N and G are an integer not less than 1; sub-filter unit having G number of sub-filters, which are corresponding to each of the signal groups, for filtering the corresponding signal group; an addition unit for summating the output signals of the sub-filter unit; an error computing unit for generating an error signal by comparing the output signals of the addition unit with a desired signal; filter coefficient updating unit having G number of filter coefficient updating units, each of which is corresponding to each of the sub-filters in the sub-filter means, for updating the filter coefficient of the corresponding sub-filter; and a switching unit for inputting the error signal to any one of the filter coefficient updating means optionally with respect to an iteration number k, wherein the iteration number k is an integer number not less than 1 and represents the number of performing entire operations of the adaptive filter. 
   In accordance with another aspect of the present invention, there is provided a filtering method of an adaptive filter that includes G number of sub-filters and filter coefficient updating unit corresponding to each sub-filters, including the steps of: a) segmenting N number of input signals into G number of signal groups, wherein N and G are an integer not less than 1; b) performing sub-filtering with respect to each of the G number of signal groups corresponding to the G number of sub-filters; c) summating the output signals of the sub-filtering; d) generating an error signal by comparing the output signals with a desired signal; e) each of the filter coefficient updaters updating the filter coefficient of the corresponding sub-filters based on the error signal; and f) switching the error signal to any one of the filter coefficient updaters optionally with respect to an iteration number k to update the filter coefficient of the corresponding sub-filters. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The above and other objects and features of the present invention will become apparent from the following description of the preferred embodiments given in conjunction with the accompanying drawings, in which: 
       FIG. 1  is a block diagram illustrating a configuration of a typical adaptive filter; 
       FIG. 2  is a block diagram showing a configuration of an adaptive filter according to a prior art; 
       FIG. 3  is a block diagram describing a configuration of an adaptive filter according to another prior art; 
       FIG. 4  is a block diagram illustrating a configuration of an interactive adaptive filter in accordance with an embodiment of the present invention; 
       FIG. 5  is a flow chart describing an interactive adaptive filtering process in accordance with an embodiment of the present invention; and 
       FIG. 6  is a graph depicting Mean Square Errors (MSEs) of 16-Quadrature Amplitude Modulation (QAM) Orthogonal Frequency Division Multiplexing (OFDM) systems, each adopting a conventional recursive least squares (RLS) adaptive filter and the interactive RLS filter of an embodiment of the present invention, respectively. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   The computational complexity of the adaptive filter algorithm is proportional to the length of input signal vector and the number of filter coefficients, which are simultaneously updated. 
   To reduce the computational complexity of an adaptive filter algorithm, the filter is implemented by composing a plurality of sub-filters with a smaller number of filter coefficients to be updated, and at each iteration step, only the filter coefficients of a single sub-filter is modified by the adaptation algorithm, while the filter coefficients of other sub-filters remain unchanged. 
   Following description only exemplifies the principle of the present invention. Although the description of the principle may not be clear or all possible embodiments of the present invention is not illustrated in the specification, those skilled in the art can embody the principle of the present invention and invent various apparatus within the scope and concept of the present invention from the description. Also, all the conditional terms and embodiments described in the specification are intended to make the concept of this invention understood, in principle, and the present invention should be understood not limited to the described embodiments or conditions only. 
   All the detailed description on a particular embodiment as well as the principle, view points and embodiments should be understood to include structural and functional equivalents. Those equivalents should be understood to include those currently known and to be developed in future. That is, they include all devices developed to perform the function of the element disclosed in the present invention, regardless of their structures. 
   For example, block diagrams of the specification should be understood to show a conceptual viewpoint of an exemplary circuit that embodies the principle of the present invention. Similarly, all the flow charts, diagrams and pseudo-code may can be implemented as software can be stored in a computer-readable medium, practically. Even though a computer or a process is not described evidently, they should be understood to show diverse processes performed by a computer or a processor. 
   The functions of elements illustrated in the drawing including a functional block marked as a processor or a similar concept of the processor can be provided not only by a dedicated hardware but also by using hardware capable of executing proper software that can perform the functions. 
   When the functions of elements are provided by a processor, the processor can be a single processor dedicated to the function only, a single shared processor or a plurality of individual processors, part of which can be shared. 
   The apparent use of terms such as processor, controller, or terms having similar concept should not be exclusively construed to be hardware capable of executing software, but include digital signal processor (DSP) hardware, ROM, RAM and non-volatile memory for recording software implicitly. Here, other known or commonly used hardware can be included. Similarly, switches illustrated in the drawings may be suggested conceptually only. Such switches should be understood being controlled by a program logic or a dedicated logic, or controlled manually. Particular technologies may be selected by a designer to make the present specification better understood. 
   In claims of the specification, an element expressed as a means for performing a function described in the detailed description part of this specification is intended to include a combination of circuits that performs the function, or all methods for performing the function including all formats of software including firmware and micro code. They are connected to a proper circuit for executing the software. The present invention defined by such claims is connected to other functions provided by various means in a method defined by the claims. Thus, any means that can provide the function should be understood to be equivalent to what is figured out from the specification. 
   Other objects and aspects of the invention will become apparent from the following description of the embodiments with reference to the accompanying drawings, which is set forth hereinafter. The same reference number is given to the same element, though it appears in different drawings. Also, any description that may unnecessarily blur the point of the present invention is omitted from the detailed description. Hereinafter, preferred embodiments of the present invention will be described with reference to the accompanying drawings. 
   As described before, the computational complexity of adaptive filter is a bottleneck to practical implementation of adaptive filter and there are a number of fast algorithms disclosed. 
   Although fast recursive least squares filters can reduce the computational complexity to O(N), compared to conventional methods with computational complexity of O(N 2 ), they have a shortcoming that they are applied to some data having a specific structure only. 
   Split Recursive Least Squares (SRLS) algorithm or High Dimensional Recursive Least Square (HRLS) algorithm are applicable to non-structured data and can reduce the computational complexity to O(N), when N is power of 2. However, there is a problem that the performance of these types of filters are acceptable to input vectors having almost non-correlated components. 
   Similarly to the SRLS or HRLS algorithm, the present invention also performs data processing on the non-structured input data, and the computational complexity can be reduced to O(1), regardless of the N value. 
     FIG. 4  is a block diagram illustrating a configuration of an interactive adaptive filter in accordance with an embodiment of the present invention. As shown in the drawing, an input signal {x 0 (k),x 1 (k),Λ,x i (k),x i+1 (k),Λ,x N−1 (k)} is grouped into two sets, {x 0 (k),x 1 (k),Λ,x i (k)} and {x i+1 (k),Λ,x N−1 (k)}. Each of them is inputted to a first filter  401  and a second filter  402 , respectively. Although a segmentation unit is not shown in the drawing, the input signal {x 0 (k),x 1 (k),Λ,x i (k),x i+1 (k),Λ,x N−1 (k)}, is divided into two sets, {x 0 (k),x 1 (k),Λ,x i (k)} and {x i+1 (k),Λ,x N−1 (k)}, by the segmentation unit. 
   To describe the embodiment of the present invention,  FIG. 4  discloses the input signal {x 0 (k),x 1 (k),Λ,x i (k),x i+1 (k),Λ,x N−1 (k)} grouped into two sets, {x 0 (k),x 1 (k),Λ,x i (k)} and {x i+1 (k),Λ,x N−1 (k)}, and inputted to the first and second filter  401  and  402 , respectively. However, it is obvious to those skilled in the art that the number of groups segmented from the input signal {x 0 (k),x 1 (k),Λ,x i (k),x i+1 (k),Λ,x N−1 (k)}, and the number of filters for corresponding groups of segmented input signals could be changed. Therefore, the present invention should be understood not limited to a case where an input signal is grouped into two sets. 
   The first and second filters  401  and  402  corresponds to the segmented input signals {x 0 (k),x 1 (k),Λ,x i (k)} and {x i+1 (k),Λ,x N−1 (k)}, respectively, and they include a multiplication unit for multiplying each input signal by the first filter coefficient {W 0 (k),W 1 (k),Λ,W i (k)} and the second filter coefficient {W i+1 (k),Λ,W N−1 (k)}, respectively. 
   The first filter  401  outputs y 1 (k) which is a summation of the multiplications of {x 0 (k),x 1 (k),Λ,x i (k)} and {W 0 (k),W 1 (k),Λ,W i (k)}, and the second filter  402  outputs y 2 (k) which is a summation of the multiplications of {x i+1 (k),Λ,x N−1 (k)} and {W i+1 (k),Λ,W N−1 (k)}. 
   The outputs y 1 (k) and y 2 (k) of the first and second filters  401  and  402  are inputted to an adder  403  and the adder  403  outputs y(k). 
   The output y(k) of the adder  403  is compared with a desired signal d(k) in an error computing unit  404  and an error signal e(k) is outputted. The error signal e(k) is fed back to a first filter coefficient updating unit  405  or a second filter coefficient updating unit  406 . The first and second filter coefficient updating unit  405  and  406  are used to update the first filter coefficient {W 0 (k),W 1 (k),Λ,W i (k)} and the second filter coefficient {W i+1 (k),Λ,W N−1 (k)}, respectively. 
   All filter coefficients of sub-filters (i.e., a first filter  401  and a second filter  402 ) that are determined in the first and second filter coefficient updating unit  405  and  406  are modified based on an adaptation algorithm recursively. The adaptation algorithms used for updating the filter coefficients of the sub-filters need not be always the same. For example, the first filter coefficient updating unit  405  updates the first filter coefficient {W 0 (k),W 1 (k),Λ,W i (k)} based on the RLS algorithm, and the second filter coefficient updating unit  406  can update the second filter coefficient {W i+1 (k),Λ,W N−1 (k)} based on a least mean square (LMS) algorithm. 
   The difference between the embodiment of the present invention shown in  FIG. 4  and the prior art illustrated in  FIGS. 2 and 3  are as below. 
   1. From a viewpoint of error signals used in the adaptation algorithm, different error signals are applied to the first filter coefficient updating unit  204  and  307  and the second filter coefficient updating unit  205  and  308  for adaptation algorithm. However, the embodiment of the present invention use the same error signal for the adaptation algorithms of the first and second filter coefficient updating unit  405  and  406 . 
   2. From a viewpoint of the execution of the adaptation algorithm, the prior art simultaneously operates the adaptation algorithms at both of the first filter coefficient updating unit  204  and  307  and the second filter coefficient updating unit  205  and  308 . However, the embodiment of the present invention alternatively executes the adaptation algorithms of the first filter coefficient updating unit  405  and the second filter coefficient updating unit  406 . 
     FIG. 5  is a flow chart describing an interactive adaptive filtering process in accordance with an embodiment of the present invention. The drawing shows the operation processes performed in the interactive adaptive filter of  FIG. 4 . As shown in  FIG. 5 , at step S 501 , the interactive adaptive filter is initialized in accordance with the present invention, before the filtering process is performed. In this step, the filter coefficient and parameters of the adaptation algorithm is initialized to their initial values. 
   Subsequently, at step S 503 , the input signal {x 0 (k),x 1 (k),Λ,x i (k),x i+1 (k),Λ,x N−1 (k)} or a desired signal d(k) are received. At step S 505 , the input signal {x 0 (k),x 1 (k),Λ,x i (k),x i+1 (k),Λ,x N−1 (k)} is segmented into two sets, {x 0 (k),x 1 (k),Λ,x i (k)} and {x i+1 (k),Λ,x N−1 (k)}, for the first filter  401  and the second filter  402 . 
   At step S 507 , filtering is performed with respect to the two sets {x 0 (k),x 1 (k),Λ,x i (k)} and {x i+1 (k),Λ,x N−1 (k)} in the first and second filters  401  and  402  and the results are outputted to the adder  403 , which summates the signals y 1 (k) and y 2 (k) outputted from the first and second filters  401  and  402 , respectively, to output y(k). 
   At step S 509 , the error computing unit  404  compares the signal y(k) with the desired signal d(k) to output an error signal e(k). At step S 511 , the switching unit  407  determines whether the iteration number k is an odd number or an even number. 
   At step S 513 , if the iteration number k is an odd number (or an even number), the switching is performed to input the error signal e(k) into the first filter coefficient updating unit  405  and thereby update the first filter coefficient {W 0 (k),W 1 (k),Λ,W i (k)}. Otherwise, if the iteration number k is an even number (or an odd number), at step S 515 , the switching is performed to input the error signal e(k) into the second filter coefficient updating unit  406  and thereby update the second filter coefficient {W i+1 (k),Λ,W N−1 (k)}. 
   To describe the embodiment of the present invention,  FIG. 4  shows the input signal {x 0 (k),x 1 (k),Λ,x i (k),x i+1 (k),Λ,x N−1 (k)}, grouped into two sets {x 0 (k),x 1 (k),Λ,x i (k)} and {x i+1 (k),Λ,x N−1 (k)}, each of which is inputted into the first and second filters  401  and  402 , respectively. Then, the error signal e(k) is switched to be inputted into the first filter coefficient updating unit  405  or the second filter coefficient updating unit  406  based on whether the iteration number k is an odd number or an even number, to thereby update the filter coefficients alternatively. However, as described before, it is obvious to those skilled in the art that the number G of sets the input signal {x 0 (k),x 1 (k),Λ,x i (k),x i+1 (k),Λ,x N−1 (k)} is grouped into and the number of filters based on the number of groups can be changed variously. 
   Therefore, it is obvious to those skilled in the art that the switching process of the error signal e(k) to the filter coefficient updating unit based on the modulo computation with respect to the iteration number k and the filter coefficient updating process, which is performed alternatively, can be changed variously. Thus, the present invention should be understood not limited to the structure where an error signal e(k) is switched to be inputted into the first filter coefficient updating unit  405  or the second filter coefficient updating unit  406  and the first and second filter coefficient updating units  405  and  406  update the filter coefficients alternatively based on whether the iteration number k is an odd number or an even number, as illustrated in the drawings. 
   After the step S 513  or S 515 , the iteration number k is increased, and the logic flow returns to the step S 503  to receive a new input signal {x 0 (k),x 1 (k),Λ,x i (k),x i+1 (k),Λ,x N−1 (k)} and a desired signal. 
     FIG. 6  is a graph depicting Mean Square Errors (MSEs) of 16-Quadrature Amplitude Modulation (QAM) Orthogonal Frequency Division Multiplexing (OFDM) systems, each adopting a conventional recursive least squares (RLS) adaptive filter and the interactive RLS filter of an embodiment of the present invention, respectively. The graph shows the MSE performances when the interactive adaptive filter of the present invention is applied to digital predistortion, which is disclosed by Mohammad Ghader in an article titled “Adaptive Linearization of Efficient High Power Amplifiers Using Polynomial Predistortion with Global Optimization”, thesis of University of Saskatchewan, November 1994 which is incorporated herein by reference 
   In the drawing, the reference numeral  601  indicates a case where the conventional RLS adaptive filter is used, and the reference numeral  602  indicates a case where the interactive adaptive filter of the present invention is used. 
   While the convention RLS adaptive filter has two filter coefficients, the interactive adaptive filter of the embodiment of the present invention has two sub-filters and each of the sub-filters has two filter coefficients, respectively. In this case, the conventional RLS adaptive filter and the interactive adaptive filter of the present invention have the same computational complexity, but the interactive adaptive filter of the present invention is superior in performance to the conventional one. 
   Not only the improvement in performance, the interactive adaptive filter of the present invention can reduce the computational load on the adaptation algorithm. 
   This is because that, in the embodiment of the present invention, a plurality of adaptation algorithms are alternatively executed during a repeated process, and thus they have the least computational complexity of O(1) than the least computational complexity of the conventional fast adaptation algorithm, which is O(N). 
   While the present invention has been described with respect to certain preferred embodiments, it will be apparent to those skilled in the art that various changes and modifications may be made without departing from the scope of the invention as defined in the following claims.