Patent Publication Number: US-2005123035-A1

Title: Multipath elimination filter

Description:
BACKGROUND OF THE INVENTION  
      The present invention relates to a multipath elimination filter, and more particularly to a multipath elimination filter for eliminating multipath distortion which occurs during the reception of an FM modulation signal or a phase modulation signal.  
      The present application claims priority from Japanese Patent Application No. 2003-405022, the disclosure of which is incorporated herein by reference.  
      For example, in the field of mobile communication, what is problematic when receivers receive and demodulate FM modulation signals or phase modulation signals is poor reception which occurs from multiple propagation of the radio waves, i.e., multipath distortion.  
      Examples of this multipath distortion include ones ascribable to the superposition of reflected-wave components from buildings on desired direct-wave components in urban areas, and ones ascribable to the superposition of reflected-wave components from mountains on desired direct-wave components. Since it deteriorates the reception quality, the multipath distortion is a problem of extreme significance.  
      For receivers intended to receive FM broadcasts, there has thus been proposed a multipath elimination filter which utilizes a digital filter as shown in  FIG. 5 , for the sake of eliminating this multipath distortion.  
      In  FIG. 5 , this multipath elimination filter is formed through the use of an FIR digital filter. A digital signal Xin(t) which is generated by an A/D converter performing analog-to-digital conversion on an intermediate frequency signal is input to the FIR digital filter.  
      Then, the digital signal Xin(t) is delayed by m stages of delay elements D 0  to D m−1 , each having a delay time T which is equal to the reciprocal of the sampling frequency. In the meantime, (m+1) multipliers MP 0  to MP m  (the number of taps of m+1) multiply the most recent digital signal X 0 (t) and the digital signals X 1 (t) to X m (t) output from the respective delay elements D 0  to D m−1  by tap factors K 0 (t) to K m (t). Moreover, an adder ADD adds the (m+1) outputs of the multipliers MP 0  to MP m  to obtain a desired signal Y(t) from which multipath distortion is eliminated. The desired signal Y(t) is output to a demodulator and the like.  
      Besides, a calculator CL determines the square of the absolute value of the desired signal Y(t). A subtractor SUB subtracts a predetermined fixed reference value Vth from the squared absolute value |Y(t)| 2  to obtain an error component e(t). Furthermore, a tap factor updating unit CQ adjusts the tap factors K 0 (t) to K m (t) of the respective multipliers MP 0  to MP m  so that the error component e(t) converges to near zero. The multiple distortion varying with time is eliminated in this way.  
      As above, the conventional multipath elimination filter approximates the inverse characteristic of the multipath propagation channel by using the FIR digital filter of  FIG. 5 . To eliminate the varying multipath distortion, it also adjusts the individual tap factors K 0 (t) to K m (t) in synchronization with the foregoing delay time T by using the calculator CL, the subtractor SUB, and the tap factor updating unit CQ so that the error component e(t) converges to near zero, based on a tap factor updating algorithm expressed by the following equation (1): 
 
 K   j ( t )= K   j ( t− 1)−α· e ( t )·{ X   j ( t )· Y ( t )+ X   j ( t− 1)· Y ( t− 1)}  [Eq. 1]
 
 where: j=0, 1, 2, 3, . . . m; α&gt;0; and t is a natural number indicating the point of time in units of delay time T. 
 
      Now, the conventional multipath elimination filter has had the problem that the effect of eliminating multipath distortion tends to depend on the characteristic of the reflected-wave components, or in other words, the actual characteristic of the multipath propagation channel.  
      For example, in urban areas and the like, reflected-wave components reflected from buildings have propagation delay times of 10 μs (micro seconds) to 20 μs or so with respect to direct-wave components. In mountainous areas and the like, on the other hand, the propagation delay times sometimes exceed 50 μs. If such reflected-wave components as exceed 50 μs in propagation delay time are superimposed on direct-wave components to cause multipath distortion, the multipath components of the digital signal Xin(t) might have delay times longer than the total delay time (m×T) of the multipath elimination filter, which is determined by the delay elements D 0  to Dm−1 arranged inside. In such cases, it is difficult to make the error component e(t) converge to near zero even if the individual tap factors K 0 (t) to K m (t) are adjusted based on the foregoing tap factor updating algorithm. Then, the convergence failure may later make the multipath elimination filter unstable in operation.  
     SUMMARY OF THE INVENTION  
      It is an object of the present invention to provide a multipath distortion elimination filter which is less susceptible to such a characteristic of the multipath propagation channel, can avoid instability, and can eliminate multipath distortion appropriately.  
      A multipath elimination filter according to a first aspect of the present invention comprises: a digital filter for receiving a digitalized FM modulation signal or phase modulation signal as its input signal, and eliminating multipath distortion from the input signal by adjusting tap factors for the input signal; envelope detecting means for detecting an envelope of the input signal to generate a reference signal; error detecting means for generating an error signal by performing a differential operation on a desired signal generated by the digital filter and the reference signal; error component limiting means for monitoring the error signal for a direct-current component and controlling the amplitude of the error signal to generate a corrected error signal; and tap factor updating means for adjusting the tap factors based on the corrected error signal so that the amplitude of the error signal approaches zero.  
      According to a second aspect of the invention, in the multipath elimination filter according to the first aspect, the tap factor updating means adjusts the individual tap factors according to an algorithm expressed by 
 
 K   j ( t )= K   j ( t− 1)−α· e   cp ( t )·{ X   j ( t )· Y ( t )+ X   j ( t− 1)· Y ( t− 1)}, 
 
 where X j (t) and X j (t−1) are the input signal, Y(t) and Y(t−1) are the desired signal, K j (t) and K j (t−1) are the tap factors, α is a predetermined constant, t is a delay time determined based on a sampling frequency, j is the order of the tap factors, and e cp (t) is the corrected error signal. 
 
      According to a third aspect of the invention, in the multipath elimination filter according to the first or second aspect, the tap factor updating means adjusts the individual tap factors according to an algorithm expressed by 
 
 K   j ( t )=γ· K   j ( t− 1)−α· e   cp ( t )·{ X   j ( t )· Y ( t )+ X   j ( t− 1)· Y ( t− 1)}
 
 where X j (t) and X j (t−1) are the input signal, Y(t) and Y(t−1) are the desired signal, K j (t) and K j (t−1) are the tap factors, α is a predetermined constant, t is a delay time determined based on a sampling frequency, j is the order of the tap factors, e cp (t) is the corrected error signal, and γ is any predetermined constant in the range of 0 and 1.
 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
      These and other objects and advantages of the present invention will become clear from the following description with reference to the accompanying drawings, wherein:  
       FIG. 1  is a block diagram showing the configuration of a multipath elimination filter according to an embodiment;  
       FIG. 2  is a block diagram showing the configuration of a multipath elimination filter according to a practical example;  
       FIG. 3  is a waveform chart for explaining the characteristic of the multipath filter shown in  FIG. 2 ;  
       FIG. 4  is a waveform chart showing the characteristic of a conventional multipath elimination filter for the sake of comparison and verification with the multipath elimination filter shown in  FIG. 2 ; and  
       FIG. 5  is a block diagram showing the configuration of the conventional multipath elimination filter. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
      Hereinafter, description will be given of a preferred embodiment of the present invention, or a multipath elimination filter to be incorporated in a radio receiver for receiving FM broadcasts and the like.  FIG. 1  is a block diagram showing the configuration of the multipath elimination filter according to the present embodiment.  
      In  FIG. 1 , the multipath elimination filter  1  comprises a digital filter  2 , an envelope detecting unit  3 , an error detecting unit  4 , an error component limiting unit  5 , and a tap factor updating unit  6 .  
      This multipath elimination filter  1  receives, as its input signal Xin(t), a series of digital signals which are generated by converting a reception signal of radio frequency received at a reception antenna into an FM modulation signal or phase modulation signal of intermediate frequency, followed by analog-to-digital conversion in an A/D converter. The multipath elimination filter  1  generates a desired signal Y(t) from which multipath distortion is eliminated, and outputs the same to a demodulation circuit and the like.  
      The digital filter  2  is made of an FIR digital filter which is approximated to the inverse characteristic of the multipath propagation channel through Taylor expansion and is capable of changing tap factors. The digital filter  2  eliminates multipath distortion from the input signal Xin(t) to generate the desired signal (in other words, predicated signal) Y(t) for output.  
      More specifically, for example, the sampling frequency fs is determined to be four times the frequency of the carrier. The input signal Xin(t) is delayed by m stages of delay elements D 0  to D m−1 , each having a delay time T which is equal to the reciprocal of the sampling frequency fs. In the meantime, (m+1) multipliers MP 0  to MP m  (the number of taps of m+1) multiply the most recent input signal X 0 (t) and the digital signals X 1 (t) to X m (t) output from the respective delay elements D 0  to D m−1 , by tap factors K 0 (t) to K m (t). Then, an adder ADD adds the (m+1) outputs of the multipliers MP 0  to MP m  to generate and output the desired signal Y(t) from which multipath distortion is eliminated.  
      The envelope detecting unit  3  detects the envelope component of the multipath-affected input signal Xin(t), and smoothens the detected envelope component to obtain the variation of the envelope component as a direct-current reference signal Vth(t).  
      The error detecting unit  4  calculates the difference between the desired signal Y(t) output from the digital filter  2  and the reference signal Vth(t), as an error component e(t).  
      The error component limiting unit  5  smoothens the error component e(t) to generate a direct-current error component. If the direct-current error component exceeds a predetermined value, the error component limiting unit  5  generates and outputs a signal into which the error component e(t) is suppressed, i.e., a corrected error component e cp (t).  
      The tap factor updating unit  6  adjusts the individual tap factors K 0 (t) to K m (t) of the digital filter  2  adaptively so that the corrected error component e cp  converges to near zero, based on a tap factor updating algorithm expressed by the following equation (2a) 
 
 K   j ( t )= K   j ( t− 1)−α· e   cp ( t )·{ X   j ( t )· Y ( t )+ X   j ( t− 1)· Y ( t− 1)}  [Eq. 2a]
 
 where: j=0, 1, 2, 3, . . . m; α&gt;0; and t is a natural number indicating the point of time in units of delay time T. 
 
      According to the multipath elimination filter  1  having such a configuration, the tap factors K 0 (t) to K m (t) of the multipliers MP 0  to MP m  can be adjusted based on the tap factor updating algorithm of the foregoing equation (2a), whereby the digital filter  2  is operated as an adaptive digital filter capable of automatically following the input signal Xin(t) undergoing actual multipath effects. It is therefore possible to generate a desired signal Y(t) of high quality from which multipath distortion is eliminated.  
      In other words, the tap factor updating unit  6  adjusts the tap factors K 0 (t) to K m (t) of the multipliers MP 0  to MP m  based on the foregoing equation (2a) by utilizing the corrected error signal e cp (t) which is generated from the error component e(t), the difference between the direct-current reference voltage Vth(t) generated by the envelope detecting unit  3  and the desired signal Y(t) output from the digital filter  2 . This allows the digital filter  2  to generate the desired signal Y(t) having a constant envelope conforming to the characteristic that FM modulation signals or phase modulation signals originally have constant envelopes.  
      It is therefore possible to generate a desired signal Y(t) of high quality from which multipath distortion is eliminated.  
      Moreover, in cases where the foregoing error component e(t) makes no convergence toward zero because of actual multipath effects, the tap factor updating unit  6  adjusts the tap factors K 0 (t) to K m (t) based on the tap factor updating algorithm of the foregoing equation (2a), by utilizing the corrected error component e cp (t) which is generated by the error component limiting unit  5  through the suppression of the value (i.e., amplitude) of the error component e(t). Consequently, variations of the individual tap factors K 0 (t) to K m (t) are suppressed to inhibit the adaptive operation of the digital filter  2 .  
      Consequently, the multipath elimination filter  1  of the present embodiment can suppress the variations of the individual tap factors K 0 (t) to K m (t) for stable operation even in situations where the operation of the multipath elimination filter might become unstable due to the actual characteristic of the multipath propagation channel. Examples of the situations include the case illustrated as the conventional problem, in which the delay time of the multipath component exceeds the total delay time (m×T) determined by the delay elements D 0  to D m−1  in the digital filter  2  having the number of taps of m+1.  
      The tap factor updating unit  6  may also control the individual tap factors K 0 (t) to K m (t) according to the algorithm expressed by the following equation (2b). According to this algorithm, the operation of the multipath elimination filter can be stabilized more effectively. 
 
 K   j ( t )=γ· K   j ( t− 1)−α· e   cp ( t )·{ X   j ( t )· Y ( t )+ X   j ( t− 1)· Y ( t− 1)}  [Eq. 2b]
 
 where: j is the order of the tap factors, j=0, 1, 2, 3, . . . m; 
          α and γ are predetermined constants, α&gt;0 and 0&lt;γ&lt;1;     t is a natural number indicating the point of time in units of delay time T; and     e cp (t) is the corrected error signal.        

     PRACTICAL EXAMPLE  
      Next, with reference to  FIGS. 2-4 , description will be given of a more specific practical example.  FIG. 2  is a block diagram showing the configuration of the multipath elimination filter of this practical example.  FIGS. 3 and 4  are waveform charts for explaining the characteristic of this multipath elimination filter.  
      In  FIG. 2 , parts identical or equivalent to those of  FIG. 1  are designated by the same reference numerals.  
      In  FIG. 2 , this multipath elimination filter  1  comprises the same digital filter  2  as that shown in  FIG. 1 . Note that the output of the adder AD of the digital filter  2  is connected with a delay element Db which is given a delay time T. The output of this delay element Db is output as the desired signal Y(t−1).  
      The circuit equivalent to the envelope detecting unit  3  of  FIG. 1  comprises a calculator  3   a , a delay element Da having a delay time T, an adder  3   b , and a digital low-pass filter  3   c.    
      The calculator  3   a  calculates the square of the absolute value of the input signal Xin(t), or |Xin(t)| 2 . The delay element Da delays the squared absolute value |Xin(t)| 2  by the delay time T, and outputs the squared absolute value before the delay time T, or |Xin(t−1)| 2 .  
      The adder  3   b  adds the foregoing values |Xin(t)| 2  and |Xin(t−1)| 2  to output an envelope signal Xe(t) which shows the envelope of the input signal Xin(t).  
      The digital low-pass filter  3   c  smoothens the envelope signal Xe(t) into a direct-current reference signal Vth(t) for output.  
      The circuit equivalent to the error detecting unit  4  of  FIG. 1  comprises a calculator  4   a , a delay element Dc having a delay time T, an adder  4   b , and a subtractor  4   c.    
      The calculator  4   a  calculates the square of the absolute value of the desired signal Yin(t−1), or |Yin(t−1)| 2 . The delay element Dc delays the squared absolute value |Yin(t−1)| 2  by the delay time T, and outputs the squared absolute value before the delay time T, or |Yin(t−2)| 2 .  
      The adder  4   b  adds the foregoing values |Y(t−1)| 2  and |Y(t−2)| 2  to output an envelope signal Ye(t−1) which shows the envelope remaining as an error in the desired signal Y(t−1).  
      The subtractor  4   c  subtracts the reference signal Vth(t) from the envelope signal Ye(t−1), thereby outputting an error signal e(t−1) which shows the difference therebetween. That is, as described above, the reference signal Vth(t) is generated by smoothening the envelope signal Xe(t) which shows the envelope of the input signal Xin(t). The reference signal Vth(t) thus makes a so-called evaluation criterion for adjusting the tap factors K 0 (t) to K m (t) of the digital filter  2  adaptively by utilizing the constant-amplitude characteristic of FM modulation signals or phase modulation signals. Then, the subtractor  4   c  subtracts the reference signal Vth(t), or evaluation criterion, from the envelope signal Ye(t−1) to generate the error signal e(t−1) for adjusting the tap factors K 0 (t) to K m (t) of the digital filter  2  adaptively.  
      Now, the circuit equivalent to the error component limiting unit  5  of  FIG. 1  comprises an absolute value detection circuit  5   a , a digital low-pass filter  5   b , an amplitude control circuit  5   c , and an amplitude limiting circuit  5   d  which is made of a digital attenuator or an amplifier circuit.  
      The absolute value detection circuit  5   a  determines the absolute value of the error signal e(t−1) to output an absolute value signal |e(t−1)|. The digital low-pass filter  5   c  smoothens the foregoing absolute value signal |e(t−1)| to output a smoothened error signal Dce(t−1).  
      The amplifier control circuit  5   c  monitors the amplitude of the error signal Dce(t−1) successively. When the amplitude of the error signal Dce(t−1) exceeds a predetermined value, the amplitude control circuit  5   c  controls the amplitude limiting circuit  5   d  to adjust its attenuation or gain, thereby suppressing the amplitude of the error signal e(t−1) and outputting the amplitude-suppressed, corrected error signal e cp (t−1).  
      More specifically, the amplitude limiting circuit  5   d , as described above, is made of a digital attenuator or an amplifier circuit.  
      If the amplitude limiting circuit  5   d  is made of a digital attenuator, it changes the attenuation under the control of the amplitude control circuit  5   c , whereby the amplitude of the error signal e(t−1) input thereto is suppressed to output the amplitude-suppressed, corrected error signal e cp (t−1).  
      If the amplitude limiting circuit  5   d  is made of an amplifier, it changes the gain under the control of the amplitude control circuit  5   c , whereby the amplitude of the error signal e(t−1) input thereto is suppressed to output the amplitude-suppressed, corrected error signal e cp (t−1).  
      In this practical example, the amplitude control circuit  5   c  determines the logarithm of the value of the error signal Dce(t−1) Based on a value proportional to the logarithmic value, the amplitude limiting circuit  5   d  increases its attenuation or decreases its gain to suppress the amplitude of the error signal e(t−1), thereby outputting the amplitude-suppressed, corrected error signal e cp (t−1).  
      The tap factor updating unit  6  adjusts the individual tap factors K 0 (t) to K m (t) of the digital filter  2  adaptively based on the tap factor updating algorithm expressed by the foregoing equation (2 a ) so that the corrected error component e cp (t−1) converges to near zero.  
      When the multipath elimination filter of this practical example having such a configuration receives an input signal Xin(t), it conducts processing in synchronization with the delay time T described above. Then, the digital filter  2  delays the input signal Xin(t) in steps of the delay time T by using the m stages of delay elements D 0  to D m−1  while the resultants are multiplied by the tap factors K 0 (t−1) to K m (t−1) of the multipliers MP 0  to MP m . Moreover, the adder ADD adds the (m+1) outputs of the multipliers MP 0  to MP m  to generate the desired signal Y(t) before the delay element Dc delays the desired signal Y(t) by time T to output the desired signal Y(t−1).  
      Here, since the delay element Db has the delay time T, it is possible to separate the processing of the digital filter  2  and that of the subsequent circuits including the calculator  4   a  temporally, with a reduced load on the digital filter  2 .  
      Furthermore, the calculator  3   a , the delay element Da, the adder  3   b , and the digital low-pass filter  3   c  described above generate the reference signal Vth(t), the evaluation criterion, while the calculator  4   a,  the delay element Dc, the adder  4   b , and the subtractor  4   c  calculate the error signal e(t−1) Subsequently, the error signal e(t−1) is processed by the absolute value detection circuit  5   a , the digital low-pass filter  5   b , the amplitude control circuit  5   c , and the amplitude limiting circuit  5   d , whereby the amplitude-suppressed error signal e(t−1) is generates as the corrected error signal e cp (t−1).  
      Then, the tap factor updating unit  6  adjusts the individual tap factors K 0 (t) to K m (t) of the digital filter  2  adaptively based on the tap factor updating algorithm expressed by the foregoing equation (2 a ) so that the corrected error component e cp (t−1) converges to near zero.  
      As above, according to the multipath elimination filter  1  of this practical example, when the amplitude of the error signal e(t−1) exceeds the predetermined value, the tap factors K 0 (t) to K m (t) are adjusted as shown by the foregoing equation (2a) based on the amplitude-suppressed error signal e(t−1), or the corrected error signal e cp (t−1). This can suppress variations of the tap factors K 0 (t) to K m (t) and stabilize the digital filter  2 . The stabilization of the digital filter  2  then makes it possible to achieve the multipath elimination filter  1  which is free from instability ascribable to the characteristic of the multipath propagation channel.  
      To be more specific, when the tap factor updating unit  6  adjusts the tap factors K 0 (t) to K m (t) according to the algorithm expressed by the foregoing equation (2a), the time necessary for the foregoing convergence is determined depending on the predetermined coefficient α.  
      Here, the error signal e(t−1) output from the subtractor  4   c  is input to the digital low-pass filter  5   b  through the absolute value detecting circuit  5   a . The error signal Dce(t−1) is thus settled gradually according to the time constant characteristic of the digital low-pass filter  5   b.    
      That is, in the period before the settled error signal Dce(t−1) is supplied to the amplitude control circuit  5   c , the digital filter  2  is capable of convergence. The error signal e(t−1) is thus output as the corrected error signal e cp (t−1), and the tap factor updating unit  6  adjusts the tap factors K 0 (t) to K m (t) based on this corrected error signal e cp (t−1) according to the algorithm expressed by the foregoing equation (2a). This allows the corrected error signal e cp (t−1) and the error signal e(t−1) to converge to near zero at a speed depending on the predetermined coefficient α, so that the digital filter  2  can eliminate the multipath noise (multipath distortion).  
      Suppose, on the other hand, the case where the delay time of the multipath distortion is shorter than the total delay time, i.e., where multipath distortion possible for the digital filter  2  to eliminate is input. Here, the error signal Dce(t−1) exceeding the predetermined amplitude is settled after a lapse of the time determined by the time constant of the digital low-pass filter  5   b , and is supplied to the amplitude control circuit  5   c . Consequently, the amplitude control circuit  5   c  controls the attenuation or gain of the amplitude limiting circuit  5   d , so that the amplitude limiting circuit  5   d  suppresses the amplitude of the input error signal e(t−1) and outputs the amplitude-suppressed signal as the corrected error signal e cp (t−1) .  
      Based on this suppressed corrected error signal e cp (t−1), the tap factor updating unit  6  then adjusts the tap factors K 0 (t) to K m (t) according to the algorithm expressed by the foregoing equation (2 a ). Since the value of the coefficient α? substantially decreases as much as the amount of the foregoing suppression, the product of the corrected error signal e cp (t−1) and the coefficient α can suppress variations of the individual tap factors K 0 (t) to K m (t) This achieves the stabilization of the digital filter  2 .  
       FIGS. 3 and 4  show the characteristics of the multipath elimination filter according to this practical example described above.  FIG. 3  shows the waveform of an input signal Xin(t) which is difficult to converge, the waveform of the desired signal Y(t−1) which is generated in response to the input signal Xin(t), and the waveforms of the error signal e(t−1) and the smoothened error signal Dce(t−1).  
       FIG. 4  shows the waveform of an input signal Xin(t) for situations where the delay time of the multipath distortion (noise) is shorter than the total delay time, the waveform of the desired signal Y(t) generated in response to the input signal Xin(t), and the waveform of the error signal e(t).  
      As is evident from the characteristic of  FIG. 4 , according to the present embodiment, the amplitude of the error signal e(t) will not approach zero over a long time because of multipath effects. This precludes unstable operation. In contrast,  FIG. 3  shows that the error signal e(t−1) converges to near zero, and the digital filter  2  makes convergence. The digital filter  2  can thus generate the desired signal Y(t−1) which can eliminate multipath distortion appropriately for the sake of improved reception quality.  
      Incidentally, in the multipath elimination filter  1  described above, the tap factors K 0 (t) to K m (t) are adjusted while the signal generated by suppressing the amplitude of the error signal e(t−1), i.e., the corrected error signal ecp(t−1) is applied to the foregoing equation (2 a ) for stable operation. Nevertheless, the individual tap factors K 0 (t) to K m (t) may be adjusted according to an algorithm of multiplying the tap factor K j (t) by a variable coefficient γ, as shown by the following equation (3). Such a configuration allows further stabilization of operation.  
      According to the multipath elimination filter  1  having such a configuration, when the foregoing corrected error signal e cp (t−1) exceeds a predetermined value, the processing system including the digital filter  2  shown in  FIG. 2  operates to control the updating values of the individual tap factors K 0 (t) to K m (t) as well as make the individual tap factors K 0 (t) to K m (t) approach zero. Consequently, even if there is the possibility of unstable operation due to multipath effects, it is possible to achieve further stabilization of operation and generate the desired signal Y(t−1) which can eliminate the multipath distortion appropriately for the sake of improved reception quality. 
 
 K   j ( t )=γ· K   j ( t− 1)−α· e   cp ( t )·{ X   j ( t )· Y ( t )+ X   j ( t− 1)· Y ( t− 1)}  [Eq. 3]
 
 where: j is the order of the tap factors, j=0, 1, 2, 3, . . . m; 
          α and γ are predetermined constants, α&gt;0 and 0&lt;γ&lt;1;     t is a natural number indicating the point of time in units of delay time T; and     e cp (t) is the corrected error signal.        

      While there has been described what are at present considered to be preferred embodiments of the present invention, it will be understood that various modifications may be made thereto, and it is intended that the appended claims cover all such modifications as fall within the true spirit and scope of the invention.