Abstract:
A digital filter for filtering an input signal to form an output signal containing a coefficient multiplier and a moving-average filter. The coefficient multiplier is embodied to multiply values of the input signal by coefficients of the filter to form an intermediate signal. The moving-average filter is embodied to generate the output signal as a moving average of the intermediate signal.

Description:
TECHNICAL FIELD 
     The invention relates to a filter, a filter system, a noise generator, and a method for filtering. 
     BACKGROUND 
     In order to realize an FIR filter, a number of multipliers that approximately corresponds to the order of the filter is conventionally required. Such an FIR filter is shown, for example, in the published European patent application EP 0 909 027 A2. The disadvantage here is that the large number of multipliers represents a significant circuit effort. 
     SUMMARY 
     The invention is based upon the object of providing a filter, a filter system, a noise generator and a method which achieve a filtering with low effort. 
     This object is achieved according to the following description for the filter, the filter system, the noise generator, and for the method described herein. Advantageous further developments are discussed herein. 
     A digital filter according to the invention for filtering an input signal to form an output signal contains a coefficient multiplier and a moving-average filter. The coefficient multiplier is embodied to multiply values of the input signal by coefficients of the filter to form an intermediate signal. The moving-average filter is embodied to generate the output signal as a moving average of the intermediate signal. In this manner, the filter properties of an FIR filter can be achieved with low hardware effort. 
     By preference, the input signal corresponds to an output signal of a sample-and-hold element. Accordingly, the signal properties can be used in order to select a particularly simple realization. 
     The coefficient multiplier preferably contains precisely one multiplier, which multiplies every sampled value of the input signal by a coefficient. The coefficient multiplier then provides a multiplexer which, after every multiplication of the multiplier, selects a next coefficient for a subsequent multiplication. Accordingly, a particularly low hardware effort is required. 
     By preference, the coefficient multiplier contains a coefficient buffer which stores the coefficients of the filter. In this case, the coefficient multiplier contains a modulus counter, which implements a counting process after every multiplication of the multiplier. The modulus counter then controls the multiplexer in such a manner that the multiplexer supplies a given coefficient from the coefficient buffer to the multiplier with every value of the modulus counter. A particularly simple circuit construction is achieved in this manner. 
     In each case L successive sampled values in the input signal are preferably identical. The modulus counter is accordingly embodied in such a manner that the first value of the modulus counter is synchronized with the first of each of the L successive sampled values. In this manner, a correct coefficient sequence is reliably guaranteed. 
     By preference, the moving-average filter provides a first delay element for delaying the intermediate signal by L sampled values and a subtractor for subtracting the intermediate signal delayed by L sampled values from the intermediate signal to form a subtracted signal. The moving-average filter then provides a second delay element for delaying the subtracted signal by one sampled value, and an adder for adding the subtracted signal delayed by one sampled value and the subtracted signal to form the output signal. The moving-average filter can thus be realized with particularly few components. 
     Alternatively, the moving-average filter provides L-1 delay elements which are connected in series. The intermediate signal is then supplied to a first delay element of the series circuit of L-1 delay elements. In this case, the moving-average filter provides an adder for generating the output signal by adding the output signals of the L-1 delay elements and the intermediate signal. The function of the moving-average filter can be realized in this manner without recursive elements. 
     A filter system according to the invention contains at least one first and one second filter described above. Furthermore, it provides a delay element and an adder. An input signal of the filter system is accordingly supplied to the first filter as an input signal and to the delay element. In this context, the delay element is embodied to delay the input signal by L sampled values and supply it to the second filter as an input signal. The adder is accordingly embodied to add an output signal of the first filter and an output signal of the second filter to form an output signal of the filter system. The impulse response of the filter system thus corresponds to a filter of the order (2×L)−1. Accordingly, a high filter order can be achieved with the use of the low-effort filters. 
     By preference, the filter system provides Y filters described above, Y-1 delay elements and Y-1 adders. The Y-1 delay elements are then connected in series. The Y-1 adders are then also connected in series. The Y filters are each connected to an output, in each case of precisely one of the delay elements or to the input signal. The impulse response of the filter system thus corresponds to a filter of order (Y×L)−1. An arbitrary order of the filtering can be achieved in this manner. 
     A noise generator according to the invention provides at least one filter described above or one filter system described above. A noise generator can thus be realized in a particularly simple manner. 
     With a method according to the invention for filtering an input signal to form an output signal, values of the input signal are multiplied by coefficients to form an intermediate signal. The output signal is generated as a moving average of the intermediate signal. Accordingly, an FIR filtering can be realized at particularly low effort. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In the following, the invention is described by way of example on the basis of the drawings in which advantageous exemplary embodiments of the invention are shown. The figures depict as follows: 
         FIG. 1 a    illustrates a first exemplary signal according to the application; 
         FIG. 1 b    illustrates a second exemplary signal according to the application; 
         FIG. 1 c    illustrates a third exemplary signal according to the application; 
         FIG. 2  illustrates a first exemplary embodiment of the filter according to the application in a block-circuit diagram; 
         FIGS. 3 and 4   a  illustrate a detailed view of the first exemplary embodiment of the filter according to the application; 
         FIG. 4 b    illustrates a detailed view of a second exemplary embodiment of the filter according to the application; 
         FIG. 5  illustrates an exemplary embodiment of the filter system according to the application; 
         FIG. 6  illustrates an exemplary embodiment of the noise generator according to the application; 
         FIG. 7  illustrates a use of a third exemplary embodiment of the filter according to the application in an IIR filter; and 
         FIG. 8  illustrates a flow diagram of an exemplary embodiment of the method according to the application. 
     
    
    
     DETAILED DESCRIPTION 
     Initially, the properties of the underlying input signal will be described with reference to  Fig. 1 a   - Fig. 1 c   . Following this, different exemplary embodiments of the filter according to the invention will be described with reference to  FIG. 2 - Fig. 4 b   . With reference to  FIG. 5 , an exemplary embodiment of the filter system according to the invention will then be explained in greater detail. Different exemplary applications of the filter according to the invention will be visualized on the basis of  FIG. 6  and  FIG. 7 . Finally, the functioning of an exemplary embodiment of the method according to the invention will be described with reference to  FIG. 8 . The presentation and description of identical elements in similar drawings have not been repeated in some cases. 
     The invention is based upon the exploitation of the special properties of an output signal of a 0th order interpolator to simplify the construction of a filter for filtering this signal. A 0th order interpolator is a simple holding element. The input value of every input clock period T in =1/f in  is held at the output for L output clock periods T out= 1/f out . In this context, L designates the interpolation factor, and the following applies:
 
 f   out   =L*f   in  
 
     In system-theoretical terms, the non-recursive part of a time-discrete filter can be described by its impulse response h FIR (n): 
     
       
         
           
             
               
                 h 
                 FIR 
               
               ⁡ 
               
                 ( 
                 n 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       b 
                       n 
                     
                   
                   
                     
                       
                         n 
                         = 
                         0 
                       
                       , 
                       1 
                       , 
                       … 
                       ⁢ 
                       
                           
                       
                       , 
                       
                         M 
                         - 
                         1 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     otherwise 
                   
                 
               
             
           
         
       
     
     The output signal of the filter y(n) is obtained as the convolution of the input signal x(n) with the impulse response h FIR (n): 
     
       
         
           
             
               
                 
                   
                     y 
                     ⁡ 
                     
                       ( 
                       n 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         x 
                         ⁡ 
                         
                           ( 
                           n 
                           ) 
                         
                       
                       ⊗ 
                       
                         
                           h 
                           FIR 
                         
                         ⁡ 
                         
                           ( 
                           n 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             x 
                             ⁡ 
                             
                               ( 
                               
                                 n 
                                 - 
                                 k 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               h 
                               FIR 
                             
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             x 
                             ⁡ 
                             
                               ( 
                               
                                 n 
                                 - 
                                 k 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             b 
                             k 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
             
           
         
       
     
     Different forms are known for the realization of equation 1, for example, the direct normal form, the cascaded form, the poly-phase form, and the lattice structure. If the sampling rate of the input sequence corresponds to the available system clock, M multipliers are generally required in all of the above named structures. 
     However, if the system clock is higher than the sampling rate, M multipliers are required only if the following applies:
 
Sampling rate&gt;(( M− 1)/ M )*system clock.
 
     The input signal x(n) of the 0th order interpolator is presented in  FIG. 1 a    as the signal  10 . 
     The signal to be filtered x interpol (n), which corresponds to the output signal of the 0th order interpolator, has the following structure: 
     
       
         
           
             
               x 
               interpol 
             
             = 
             
               { 
               
                 
                   
                     
                       
                         … 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               
                                 x 
                                 ⁢ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               , 
                               
                                 x 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               , 
                               … 
                               ⁢ 
                               
                                   
                               
                               , 
                               
                                 x 
                                 ⁢ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             ︸ 
                           
                           L 
                         
                       
                       , 
                       
                         
                           
                             
                               x 
                               ⁢ 
                               
                                 ( 
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             , 
                             
                               x 
                               ⁡ 
                               
                                 ( 
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             , 
                             … 
                             ⁢ 
                             
                                 
                             
                             , 
                             
                               x 
                               ⁢ 
                               
                                 ( 
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                           
                           ︸ 
                         
                         L 
                       
                       , 
                       … 
                       ⁢ 
                       
                           
                       
                       , 
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               
                                 x 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     k 
                                     + 
                                     N 
                                   
                                   ) 
                                 
                               
                               , 
                               
                                 x 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     k 
                                     + 
                                     N 
                                   
                                   ) 
                                 
                               
                               , 
                               … 
                               ⁢ 
                               
                                   
                               
                               , 
                               
                                 x 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     k 
                                     + 
                                     N 
                                   
                                   ) 
                                 
                               
                             
                             ︸ 
                           
                           ⁢ 
                           
                               
                           
                         
                         L 
                       
                       ⁢ 
                       … 
                     
                   
                 
               
               } 
             
           
         
       
     
     The signal x interpol (n) is illustrated in  FIG. 1 c    as signal  12 . To obtain a mathematical description x interpol (n), we use the auxiliary signal x e    
     
       
         
           
             
               
                 
                   x 
                   e 
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           
                             … 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     x 
                                     ⁢ 
                                     
                                       ( 
                                       k 
                                       ) 
                                     
                                   
                                   , 
                                   0 
                                   , 
                                   0 
                                   , 
                                   … 
                                   ⁢ 
                                   
                                       
                                   
                                   , 
                                   0 
                                 
                                 ︸ 
                               
                               L 
                             
                           
                           , 
                           
                             
                               
                                 
                                   x 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       k 
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                 
                                 , 
                                 0 
                                 , 
                                 0 
                                 , 
                                 … 
                                 , 
                                 0 
                               
                               ︸ 
                             
                             L 
                           
                           , 
                           … 
                           ⁢ 
                           
                               
                           
                           , 
                         
                       
                     
                     
                       
                         
                           
                             
                                 
                               
                                 
                                   
                                     x 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         k 
                                         + 
                                         N 
                                       
                                       ) 
                                     
                                   
                                   , 
                                   0 
                                   , 
                                   0 
                                   , 
                                   … 
                                   ⁢ 
                                   
                                       
                                   
                                   , 
                                   0 
                                 
                                 ︸ 
                               
                               ⁢ 
                               
                                   
                               
                             
                             L 
                           
                           ⁢ 
                           … 
                         
                       
                     
                   
                   } 
                 
               
               , 
             
             ⁢ 
             
               
 
             
           
         
       
     
     which is illustrated in  FIG. 1 b    as signal  11 . A mathematical formulation x e (n) for our auxiliary signal is given by: 
     
       
         
           
             
               
                 
                   x 
                   e 
                 
                 ⁡ 
                 
                   ( 
                   n 
                   ) 
                 
               
               = 
               
                 
                   ∑ 
                   
                     k 
                     = 
                     
                       - 
                       ∞ 
                     
                   
                   ∞ 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     x 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   ⁢ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         n 
                         - 
                         kL 
                       
                       ) 
                     
                   
                 
               
             
             , 
           
         
       
     
     where the delta function δ(n) is defined as 
     
       
         
           
             
               δ 
               ⁡ 
               
                 ( 
                 n 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       1 
                     
                     
                       
                         n 
                         = 
                         0 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       
                         n 
                         ≠ 
                         0 
                       
                     
                   
                 
                 . 
               
             
           
         
       
     
     If x e (n) is now filtered with h interpol (n), where 
     
       
         
           
             
               
                 h 
                 interpol 
               
               ⁡ 
               
                 ( 
                 n 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       1 
                     
                     
                       
                         0 
                         ≤ 
                         n 
                         ≤ 
                         
                           L 
                           - 
                           1 
                         
                       
                     
                   
                   
                     
                       0 
                     
                     
                       otherwise 
                     
                   
                 
                 , 
               
             
           
         
       
     
     the following is finally obtained: 
     
       
         
           
             
               
                 x 
                 interpol 
               
               ⁡ 
               
                 ( 
                 n 
                 ) 
               
             
             = 
             
               
                 
                   
                     x 
                     e 
                   
                   ⁡ 
                   
                     ( 
                     n 
                     ) 
                   
                 
                 ⊗ 
                 
                   
                     h 
                     interpol 
                   
                   ⁡ 
                   
                     ( 
                     n 
                     ) 
                   
                 
               
               = 
               
                 
                   
                     ∑ 
                     
                       l 
                       = 
                       0 
                     
                     
                       L 
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       
                         x 
                         e 
                       
                       ⁡ 
                       
                         ( 
                         
                           n 
                           - 
                           1 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         h 
                         interpol 
                       
                       ⁡ 
                       
                         ( 
                         l 
                         ) 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       ∑ 
                       
                         l 
                         = 
                         0 
                       
                       
                         L 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         x 
                         e 
                       
                       ⁡ 
                       
                         ( 
                         
                           n 
                           - 
                           1 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         l 
                         = 
                         0 
                       
                       
                         L 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           
                             - 
                             ∞ 
                           
                         
                         ∞ 
                       
                       ⁢ 
                       
                         
                           x 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         ⁢ 
                         
                           δ 
                           ⁡ 
                           
                             ( 
                             
                               n 
                               - 
                               1 
                               - 
                               kL 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     The filtering of x interpol (n) with h FIR (n) 
     
       
         
           
             
               
                 h 
                 FIR 
               
               ⁡ 
               
                 ( 
                 n 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       b 
                       n 
                     
                   
                   
                     
                       0 
                       ≤ 
                       n 
                       ≤ 
                       
                         L 
                         - 
                         1 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     otherwise 
                   
                 
               
             
           
         
       
     
     leads to: 
     
       
         
           
             
               
                 
                   
                     y 
                     ⁡ 
                     
                       ( 
                       n 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ∑ 
                       
                         r 
                         = 
                         0 
                       
                       
                         L 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         
                           x 
                           interpol 
                         
                         ⁡ 
                         
                           ( 
                           
                             n 
                             - 
                             r 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           h 
                           FIR 
                         
                         ⁡ 
                         
                           ( 
                           r 
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       ∑ 
                       
                         r 
                         = 
                         0 
                       
                       
                         L 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           0 
                         
                         
                           L 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             
                               - 
                               ∞ 
                             
                           
                           ∞ 
                         
                         ⁢ 
                         
                           
                             x 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                           ⁢ 
                           
                             δ 
                             ⁡ 
                             
                               ( 
                               
                                 n 
                                 - 
                                 r 
                                 - 
                                 1 
                                 - 
                                 
                                   k 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   L 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               h 
                               FIR 
                             
                             ⁡ 
                             
                               ( 
                               r 
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                     
                       
                         ( 
                         * 
                       
                       ) 
                     
                   
                   ⁢ 
                     
                   ⁢ 
                   
                     
                       ∑ 
                       
                         r 
                         = 
                         0 
                       
                       
                         L 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           0 
                         
                         
                           L 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             
                               - 
                               ∞ 
                             
                           
                           ∞ 
                         
                         ⁢ 
                         
                           
                             x 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                           ⁢ 
                           
                             δ 
                             ⁡ 
                             
                               ( 
                               
                                 n 
                                 - 
                                 r 
                                 - 
                                 1 
                                 - 
                                 
                                   k 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   L 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               h 
                               FIR 
                             
                             ⁡ 
                             
                               ( 
                               l 
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     The identity (*) applies for reasons of symmetry. 
     The following applies: 
     
       
         
           
             
               
                 δ 
                 ⁡ 
                 
                   ( 
                   
                     n 
                     - 
                     r 
                     - 
                     1 
                     - 
                     kL 
                   
                   ) 
                 
               
               ⁢ 
               
                 
                   h 
                   FIR 
                 
                 ⁡ 
                 
                   ( 
                   l 
                   ) 
                 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           h 
                           FIR 
                         
                         ⁡ 
                         
                           ( 
                           l 
                           ) 
                         
                       
                     
                     
                       
                         n 
                         = 
                         
                           r 
                           + 
                           1 
                           + 
                           kL 
                         
                       
                     
                   
                   
                     
                       0 
                     
                     
                       otherwise 
                     
                   
                 
                 . 
               
             
           
         
       
     
     If we now consider the expression: 
     
       
         
           
             
               
                 δ 
                 ⁡ 
                 
                   ( 
                   
                     n 
                     - 
                     r 
                     - 
                     1 
                     - 
                     kL 
                   
                   ) 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     m 
                     = 
                     
                       - 
                       ∞ 
                     
                   
                   ∞ 
                 
                 ⁢ 
                 
                   
                     h 
                     FIR 
                   
                   ⁡ 
                   
                     ( 
                     
                       n 
                       - 
                       r 
                       - 
                       mL 
                     
                     ) 
                   
                 
               
             
             , 
           
         
       
     
     this is equal to 0 for all n≠r+1+kL. For n=r+1+kL, we obtain: 
     
       
         
           
             
               
                 δ 
                 ⁡ 
                 
                   ( 
                   0 
                   ) 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     m 
                     = 
                     
                       - 
                       ∞ 
                     
                   
                   ∞ 
                 
                 ⁢ 
                 
                   ( 
                   
                     r 
                     + 
                     1 
                     + 
                     kL 
                     - 
                     r 
                     - 
                     mL 
                   
                   ) 
                 
               
             
             = 
             
               
                 
                   ∑ 
                   
                     m 
                     = 
                     
                       - 
                       ∞ 
                     
                   
                   ∞ 
                 
                 ⁢ 
                 
                   
                     h 
                     FIR 
                   
                   ⁡ 
                   
                     ( 
                     
                       1 
                       + 
                       
                         
                           ( 
                           
                             k 
                             - 
                             m 
                           
                           ) 
                         
                         ⁢ 
                         L 
                       
                     
                     ) 
                   
                 
               
               ⁢ 
               
                 = 
                 
                   
                     ( 
                     * 
                   
                   ) 
                 
               
               ⁢ 
               
                 
                   h 
                   FIR 
                 
                 ⁡ 
                 
                   ( 
                   l 
                   ) 
                 
               
             
           
         
       
     
     The identity (*) applies, since 0≦l≦L−1 and h FIR (I′)=0 if I′≧L or I′&lt;0. 
     Overall, therefore, the following applies: 
     
       
         
           
             
               
                 δ 
                 ⁡ 
                 
                   ( 
                   
                     n 
                     - 
                     r 
                     - 
                     1 
                     - 
                     kL 
                   
                   ) 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     m 
                     = 
                     
                       - 
                       ∞ 
                     
                   
                   ∞ 
                 
                 ⁢ 
                 
                   
                     h 
                     FIR 
                   
                   ⁡ 
                   
                     ( 
                     
                       n 
                       - 
                       r 
                       - 
                       mL 
                     
                     ) 
                   
                 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           h 
                           FIR 
                         
                         ⁡ 
                         
                           ( 
                           l 
                           ) 
                         
                       
                     
                     
                       
                         n 
                         = 
                         
                           r 
                           + 
                           1 
                           + 
                           kL 
                         
                       
                     
                   
                   
                     
                       0 
                     
                     
                       otherwise 
                     
                   
                 
                 . 
               
             
           
         
       
     
     Accordingly, we have shown that: 
     
       
         
           
             
               
                 δ 
                 ⁡ 
                 
                   ( 
                   
                     n 
                     - 
                     r 
                     - 
                     1 
                     - 
                     kL 
                   
                   ) 
                 
               
               ⁢ 
               
                 
                   h 
                   FIR 
                 
                 ⁡ 
                 
                   ( 
                   l 
                   ) 
                 
               
             
             = 
             
               
                 δ 
                 ⁡ 
                 
                   ( 
                   
                     n 
                     - 
                     r 
                     - 
                     1 
                     - 
                     kL 
                   
                   ) 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     m 
                     = 
                     
                       - 
                       ∞ 
                     
                   
                   ∞ 
                 
                 ⁢ 
                 
                   
                     
                       h 
                       FIR 
                     
                     ⁡ 
                     
                       ( 
                       
                         n 
                         - 
                         r 
                         - 
                         mL 
                       
                       ) 
                     
                   
                   . 
                 
               
             
           
         
       
     
     From this, the following is now obtained: 
     
       
         
           
             
               
                 
                   
                     y 
                     ⁡ 
                     
                       ( 
                       n 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         r 
                         = 
                         0 
                       
                       
                         L 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           0 
                         
                         
                           L 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             
                               - 
                               ∞ 
                             
                           
                           ∞ 
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               
                                 - 
                                 ∞ 
                               
                             
                             ∞ 
                           
                           ⁢ 
                           
                             
                               x 
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                             ⁢ 
                             
                               δ 
                               ⁡ 
                               
                                 ( 
                                 
                                   n 
                                   - 
                                   1 
                                   - 
                                   
                                     k 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     L 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   h 
                                   FIR 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     r 
                                     - 
                                     mL 
                                   
                                   ) 
                                 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     If we now define: 
     
       
         
           
             
               
                 b 
                 ⁡ 
                 
                   ( 
                   n 
                   ) 
                 
               
               = 
               
                 
                   ∑ 
                   
                     l 
                     = 
                     0 
                   
                   
                     L 
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       
                         - 
                         ∞ 
                       
                     
                     ∞ 
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         
                           - 
                           ∞ 
                         
                       
                       ∞ 
                     
                     ⁢ 
                     
                       
                         x 
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       ⁢ 
                       
                         δ 
                         ⁡ 
                         
                           ( 
                           
                             n 
                             - 
                             1 
                             - 
                             
                               k 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               L 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           h 
                           FIR 
                         
                         ⁡ 
                         
                           ( 
                           
                             n 
                             - 
                             mL 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             , 
           
         
       
     
     the following applies: 
     
       
         
           
             
               
                 
                   
                     y 
                     ⁡ 
                     
                       ( 
                       n 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         r 
                         = 
                         0 
                       
                       
                         L 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         b 
                         ⁡ 
                         
                           ( 
                           
                             n 
                             - 
                             r 
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     Furthermore, if we define: 
     
       
         
           
             
               
                 
                   
                     
                       
                         h 
                         FIR 
                         ′ 
                       
                       ⁡ 
                       
                         ( 
                         n 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             
                               - 
                               ∞ 
                             
                           
                           ∞ 
                         
                         ⁢ 
                         
                           
                             h 
                             FIR 
                           
                           ⁡ 
                           
                             ( 
                             
                               n 
                               - 
                               mL 
                             
                             ) 
                           
                         
                       
                       = 
                       
                         
                           h 
                           FIR 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               ( 
                               n 
                               ) 
                             
                             L 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   3 
                 
               
             
           
         
       
     
     Where (n) L  designates the modulus L operator, we finally obtain:
 
 b ( n )=x interpol  ( n )*h′ FIR ( n ).
 
     A filter structure can now be derived from equation 2 -equation 4. 
     According to equation 4, b(n) is generated by the circuit from  FIG. 3 . Equation 2 describes a moving-average filter and can be realized alternatively by the circuits in  FIG. 4 a    or  FIG. 4 b   . Finally, the overall filter structure is obtained from a combination of the circuits from  FIG. 3  and  FIG. 4 a    or  FIG. 3  and  FIG. 4 b   . 
     Instead of L multipliers for an order L-1 filter, as is conventional, only a single multiplier is required. 
       FIG. 2  shows a first exemplary embodiment of the filter  1  according to the invention in a block-circuit diagram. The filter  1  contains a coefficient multiplier  3 , which is connected to a moving-average filter  5 . An input signal  2 , which corresponds, for example, to the signal  12  from  FIG. 1 c   , is supplied to the coefficient multiplier  3  and multiplied by filter coefficients by the latter. In this context, each sampled value of the input signal  2  is multiplied by precisely one filter coefficient. The sequence of sampled values of the input signal  2  multiplied by the filter coefficients is output from the coefficient multiplier  3  as an intermediate signal  4 . 
     In this context, the coefficient multiplier  3  accesses L coefficients. After every multiplication of a sampled value of the input signal  2  by one of the L coefficients, a next one of the L coefficients is used for the next multiplication. As soon as the last of the L coefficients has been used, the process switches back to the first coefficient again. The function of the coefficient multiplier  3  will be described in greater detail with reference to  FIG. 3 . 
     The intermediate signal  4  is supplied to the moving-average filter  5 , which forms a moving average of the intermediate signal  4  and outputs it as an output signal  6 . In this context, the moving-average filter  5  forms the moving average of a window length which corresponds to the number of filter coefficients L. Accordingly, a moving average of L sampled values of the intermediate signal  4  is formed by the moving-average filter  5  and output as the output signal  6 . 
       FIG. 3  shows a detailed view of the first exemplary embodiment of the filter  1 . Here, in particular, the coefficient multiplier  3  is shown in detail. The coefficient multiplier  3  comprises a multiplier  20 , a coefficient buffer  24 , a multiplexer  21 , and a modulus counter  22 . The input signal  2  is supplied to the multiplier  20  at a first input. An output signal of the multiplexer  21  is supplied to the multiplier  20  at a second input. The multiplexer  21  is further connected to the coefficient buffer  24 . Furthermore, the multiplexer  21  is connected to the modulus counter  22 . 
     The modulus counter  22  performs a counting process after every multiplication of the multiplier  20 , that is, after every sampled value of the input signal  2 . In this context, the modulus counter  22  counts from 0 to L-1. The output signal of the modulus counter  22  is supplied to the multiplexer  21  as a control signal. Through the output signal of the modulus counter  22 , a coefficient buffered in the coefficient buffer  24  is selected by the multiplexer  21  and supplied to the multiplier  20 . 
     Accordingly, L coefficients are stored in the coefficient buffer  24 . These L coefficients are supplied to the multiplier  20  in succession in a rigidly specified sequence. The sampled values of the input signal  2  are thus multiplied in succession by the L coefficients stored in the coefficient buffer  24 . 
     As already explained, identical values are repeated in the input signal  2 , in each case L times in direct succession. This is a direct consequence of the property that the input signal  2  corresponds to an output signal of a 0th order interpolator. 
     In this context, the first occurrence of a new value of the input signal  2  is synchronized with a first one of the L coefficients which is stored in the coefficient buffer  24 . This ensures that the L identical values of the input signal  2  following in succession are multiplied by all the filter coefficients in the correct sequence. The sequence of values of the input signal  2  multiplied by the filter coefficients is outputted from the coefficient multiplier  3  as the intermediate signal  4 . 
     Accordingly, the coefficient multiplier  3  contains only the precisely one multiplier  20 . However, in this context, the multiplier  20  is a logical multiplier. In one realization, for example, several multipliers can be connected together to form this multiplier, for example, on an FPGA, if the bit width of one single multiplier is not sufficient. Also, in the case of a complex signal, which contains an I-component and a Q-component, 4 physical multipliers, which perform the logical multiplication of the signals, are used in a real circuit instead of the precisely one logical multiplier  20 . Accordingly, in a real circuit implementation, several multipliers, which, however, always process at least parts of one sampled value of the input signal at the same time, can correspond to the logical multiplier  20 . 
       FIG. 4 a    shows a detailed view of the moving-average filter  5  from  FIG. 2 . Since a first alternative is presented here, the moving-average filter  5  is designated with the reference number  5   a  in this context. The moving-average filter  5   a  comprises a subtractor  31  which is connected at its first input to the intermediate signal  4 . Furthermore, the moving-average filter  5   a  comprises a first delay element  30 , which is also connected at its input to the intermediate signal  4 . At its output, the first delay element  30  is connected to a second input of the subtractor  31 . The subtractor  31  is connected at the output-end to a first input of an adder  32 . The moving-average filter  5   a  further comprises a second delay element  33 , which is connected at the input-end to an output of the adder  32 . At the output-end, the second delay element  33  is connected to a second input of the adder  32 . 
     The intermediate signal  4  is supplied to the subtractor  31  at its first input. Furthermore, the intermediate signal  4  is supplied to the first delay element  30 . The delay element  30  delays the intermediate signal  4  by L sampled values and supplies the intermediate signal  4  delayed by L sampled values to the second input of the subtractor  31 . The subtractor  31  subtracts the intermediate signal  4  delayed by L sampled values from the intermediate signal  4 . At the output of the subtractor  31 , a subtracted signal  34  is generated. This is supplied to the adder  32 . Furthermore, an output signal of the adder  32  delayed by one sampled value is supplied to the adder  32  via the second delay element  33 . The output signal of the adder  32  corresponds to the output signal  6  of the filter. 
     The embodiment of the moving-average filter presented above allows a very simple realization of a moving-average filter of window length L using a few components. 
     As an alternative, a further embodiment of a moving-average filter is presented in  FIG. 4 b   . The moving-average filter  5   b  contains a plurality of delay elements  40 - 43  which are connected in series. In this context, the intermediate signal  4  is supplied to the first delay element  40 . The output signal of the first delay element  40  is supplied as an input signal to the second delay element  41 . The output signals of the individual delay elements and the intermediate signal  4  are combined via an adder  44  to form the output signal of the filter  6 . 
       FIG. 5  shows an exemplary embodiment of a filter system according to the invention. The filter according to the invention provides the disadvantage that a maximal filter order of the filter of only L-1 can be achieved. That is, with an L-fold repetition of every value of the input signal, as presented in  FIG. 1 c   , a maximal L-1th order of the filter can be achieved. 
     In this context, the filter system shown here provides some assistance. The filter system  50  according to the invention comprises filters  60 - 62  according to the invention. A first filter  60  according to the invention is connected at the input end to an input signal  52  and at the output end to an adder  65 . The input signal  52  is further supplied to a delay element  63 , which implements a delay by L sampled values. The delay element  63  is connected at the output end to the input of a second filter  61  according to the invention. At the output end, this second filter  61  is connected to a second adder  66 . The output signal of the delay element  63  is further connected to a second delay element  64 , which again implements a delay by L sampled values. The output signal of this second delay element is connected to the input of the third filter  62  according to the invention. The third filter  62  according to the invention is again connected at the output end to the second adder  66 . The adders  65 ,  66  add the output signals of the filters  60 - 62  according to the invention to form the output signal  56 . 
     In this context, the filter system  50  is not restricted to precisely three filters according to the invention. A realization with only two filters  60 ,  61  according to the invention is also possible. Conversely, a realization with an arbitrary number of filters according to the invention is possible. The input signals of the individual filters in this context are each delayed by a further L sampled periods by comparison with the preceding filter. The output signals of the individual filters are then added by means of adders to form the output signal  56 . 
     With the filter system  50  according to the invention, an arbitrary order of the overall filter realized can be achieved. If Y is the number of the individual filters according to the invention, an order of the impulse response of the filter system of (Y×L)−1 can thus be achieved. 
       FIG. 6  shows an exemplary embodiment of the noise generator according to the invention. A random-number generator  80  is connected to a first filter  81 . The first filter  81  is connected to a 0th order interpolator  82 . The 0th order interpolator  82  is connected to a filter  83  according to the invention. 
     The random-number generator  80  generates random numbers which are filtered by the optional first filter  81 . In this context, the sampling frequency of the random-number generator  80  corresponds to the sampling frequency of the first filter  81 . The 0th order interpolator ensures a band limitation of the resulting signal. A sampling frequency of L times the sampling frequency of the random-number generator is accordingly present at the output of the 0th order interpolator  82 . As described with reference to  FIG. 1 c   , L sampled values are repeated here in each case. 
     The output signal of this 0th order interpolator  82  is supplied to a filter  83  according to the invention which performs an FIR filtering. Such a filter could serve for the spectral formation of the noise, for example, by increasing the stop-band attenuation. 
     A further application of the filter according to the invention is shown in  FIG. 7 . The use of the filter  90  according to the invention is shown here in an IIR filter  91 . An IIR filter  91  is made up from an FIR filter, which is realized here by the filter  90  according to the invention, and a recursive filter part  92 . This recursive filter part  92  contains a plurality of adders  93   a - 93   c  and a plurality of multipliers  94   a - 94   c . Furthermore, the recursive filter part  92  contains a plurality of delay elements  95   a - 95   c . The output signal of the filter  90  according to the invention is supplied to a first adder  93   a . The output signal of this adder  93   a  is supplied to a delay element  95   a . The delay element  95   a  delays the resulting signal by one sampled value and transmits the resulting signal to a multiplier  94   a , which multiplies it by a first coefficient a 1 . The resulting signal from the multiplier  94   a is supplied to an adder  93   b . 
     Furthermore, the output signal of the delay element  95   a  is supplied to a further delay element  95   b  which also performs a delay by one sampled value. The output signal of this delay element  95   b  is supplied in turn to a multiplier  94   b  which performs a multiplication by a coefficient a 2 . The output signal of this multiplier  94   b  is supplied to an adder  93   c . The recursive filter part  92  can contain an arbitrary number of these individual stages just presented. Here, only one further stage comprising a delay element  95   c  and a multiplier  94   c  is presented. The output signals of the multipliers  94   a - 94   c  are each supplied to the adders  93   a - 93   c  and added by the latter to form the output signal  96 . Accordingly, an IIR filter can be realized with low hardware effort. 
     Finally,  FIG. 8  presents an exemplary embodiment of the method according to the invention. In a first step  70 , a current value of an input signal which corresponds to the output signal of a 0th order interpolator is multiplied by a current coefficient. In a second step  71 , a next coefficient is determined by modulus calculation. In a third step  72 , a next value of the input signal is selected. In this context, the values of the input signal are each repeated L times, as described with reference to  FIG. 1 c   . Similarly, L different coefficients are used. In a fourth step  73 , a moving average of the current value and L-1 previous values is formed. The moving average corresponds to the present output signal. Steps  70 - 73  are repeated constantly. As soon as the last coefficient has been reached, the first coefficient is selected again on the basis of the modulus calculation. 
     The invention is not restricted to the exemplary embodiment presented. As already mentioned, different moving-average filters can be used. A use of the filter according to the invention in a plurality of different devices is also possible. For example, a use in the post-processing of a resolution enhancement of image signals is possible. In particular, in this context, the input signal need not have been generated by a sample-and-hold element. The exemplary embodiment of the coefficient multiplier shown in  FIG. 3  is therefore not a compulsory embodiment. Other designs are also conceivable. In particular, the coefficient buffer and the multiplexer can be replaced by a buffer module addressed by the modulus counter. All of the features described above or the features shown in the drawings can be combined with one another arbitrarily in an advantageous manner within the scope of the invention.