Patent Publication Number: US-7224725-B2

Title: Method for determining coefficients of an equalizer and apparatus for determining the same

Description:
BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates to a method for determining coefficients of an equalizer and a device for determining the same. 
   2. Discussion of the Related Art 
   Asymmetric high speed digital subscriber line (ADSL) and very high speed digital subscriber line (VDSL) are examples of modern communication systems that permit transmission of data over communication lines at very high rates (e.g., up to 52 Mbits/s). The transmission of high-speed data over band-limited channels may be accomplished by means of discrete multitone (DMT)-based digital communication systems. DMT modems are multi-carrier transmission systems for dividing transmission data into several interleaved bit streams and using these bit streams to modulate several carriers. 
   Significant limitations in high data rate communication systems are inter-symbol interference (ISI) and inter-channel interference (ICI). One way to compensate for ISI in a DMT system is to add a cyclic prefix to the beginning of each transmitted DMT symbol. Unfortunately, while increasing the length of prefixes reduces ISI, it also decreases the effective data rate. Another approach to combat ISI is to employ an equalizer at the receiver. However, many equalizers need considerable and ongoing computational “overhead”. 
   In practical communications, the frequency response of a communications channel is not known. Accordingly, equalizers are designed using numerous parameters that need to be adjusted on the basis of measurements of characteristics having an influence on signals of the channel. 
   A typical equalizer comprises a transversal filter having a delay line spaced by T-seconds, where “T” is the sampling interval and “fs=1/T” is the sampling rate at the receiver. The outputs of filter taps are multiplied by a filter coefficient, summed, and input to a coefficient decision device for selecting coefficients. The coefficient values are typically selected to minimize either peak distortion or mean-squared distortion. The tap coefficients correspond to the channel parameters. Depending on which coefficients are selected, the equalizer can substantially remove the interference from DMT symbols. 
   There are at least two general approaches to obtain coefficients of an equalizer. One approach is a minimum mean-squared error (MSE) technology to minimize an MSE. Another approach is to obtain an eigenvalue and an eigenvector using the singular value decomposition (SVD). While the SVD approach obtains improved results as compared to the MSE approach, the SVD approach is not widely used in practical communication modems. The SVD approach can be classified into a direct matrix inversion approach or an adaptive algorithm approach. While the adaptive algorithm approach is substantially more efficient than the direct matrix inversion approach, it is not suitable for real-time communications because it is difficult to determine a degree of coefficient convergence. The direct matrix inversion approach is also computationally expensive for the matrix inversion. But since a matrix for inversion is a covariance matrix of a given communication channel, the computational expense is reduced to readily realize the direct matrix inversion approach. 
   Unfortunately, these approaches are limited in terms of reducing the inter-symbol interference (ISI) and the inter-channel interference (ICI). This is because a receiver installed at the DMT modem knows the frequency response characteristic of a downstream area in a channel for receiving data from a central office, but does not know the frequency response characteristic of an upstream area in a channel for sending data from the receiver to a central office or node along a communications line. 
   Therefore, a need exists for an equalizer, which amplifies and attenuates a received signal so that the whole band of a channel has a uniform gain, amplifies an upstream area without response and attenuates a downstream area with response. As a result, a channel of a practically used frequency area is attenuated to reduce a signal to noise ratio (SNR). 
   SUMMARY OF THE INVENTION 
   To overcome the foregoing disadvantages, a method according to an embodiment of the present invention determines coefficients of an equalizer by estimating a frequency response of an upstream area in a channel. 
   Further, an apparatus according to an embodiment of the present invention determines coefficients of an equalizer to minimize an inter-symbol interference (ISI) and an inter-channel interference (ICI). 
   According to an embodiment of the present invention, a method determines coefficients of a time domain equalizer in a receiver for receiving a reception signal transmitted through a downstream area in a communication channel having an upstream area and the downstream area. The method comprises estimating a frequency response of the upstream area in the communication channel, and determining the coefficients of the time domain equalizer from the estimated frequency response of the upstream area and a frequency response of the downstream area. 
   In this embodiment, the method further comprises determining a cost function using the estimated frequency response of the upstream area and the frequency response of the downstream area, and determining the coefficients using the determined cost function. 
   In this embodiment, the cost function is the sum of the square of a difference between a channel impulse response of the communication channel and an equalized channel impulse response and the square of the coefficient of the time domain equalizer. 
   In this embodiment, the channel impulse response and the equalized channel impulse response are determined according to an initial training signal comprising a unit pulse transmitted to the receiver through the communication channel. 
   In this embodiment, the frequency response of the upstream area comprises a phase response and an amplitude response of the upstream area. 
   In this embodiment, the phase response of the upstream area is estimated based on gradients of subchannels in the downstream area. 
   In this embodiment, the amplitude response of the upstream area is estimated wherein amplitudes of adjacent subchannels in the upstream area have different values. 
   In this embodiment, the amplitude response of the upstream area is estimated wherein even-number subchannels in the upstream area have a predetermined value h1 and odd-number subchannels have a value h2 that is smaller than the predetermined value h1. 
   According to another embodiment of the present invention, the invention provides an apparatus determines coefficients of a time domain equalizer in a receiver for receiving a reception signal transmitted through a downstream area in a communication channel having an upstream area and the downstream area. The apparatus comprises an estimator for estimating a frequency response of the upstream area, and a calculator for determining the coefficients of the time domain equalizer from the estimated frequency response of the upstream area and a frequency of the downstream area. 
   In this embodiment, the estimator has a phase estimator for estimating a phase response of the upstream area and an amplitude estimator for estimating an amplitude response of the upstream area. 
   In this embodiment, the phase estimator estimates the phase response of the upstream area based on gradients of any subchannels in the downstream area. 
   In this embodiment, the amplitude estimator estimates amplitudes of adjacent subchannels in the upstream area. 
   In this embodiment, the calculator determines a cost function using the estimated frequency response of the upstream area and the frequency response of the downstream area, and determines the coefficients using the determined cost function. 
   In this embodiment, the cost function is the sum of the square of a difference between a channel impulse response of the communication channel and an equalized channel impulse response and the square of the coefficient of the time domain equalizer. 
   In this embodiment, the channel impulse response and the equalized channel impulse response are determined according to an initial training signal comprising a unit pulse transmitted to the receiver through the communication channel. 
   According to another embodiment of the present invention, a method sets coefficients of a time domain equalizer in a communication system for transmitting a signal through a communication channel having an upstream area and a downstream area so that an error between an output of the time domain equalizer and an output of a channel target circuit becomes zero. The method comprises estimating a frequency response of the upstream area in the communication channel, estimating an output of the time domain equalizer from the estimated frequency response of the upstream area and the frequency response of the downstream area, determining a cost function J from the estimated output of the time domain equalizer and the output of the channel target circuit, and determining coefficients where the cost function J has a minimum value. The cost function J is given by the following equation:
 
 J=E{e   2   }+λ|a|   2 
 
   where E is an error function, e is an error between an output of a time domain equalizer and an output of a channel target circuit, λ is an integer except 1, and a is coefficients of the time domain equalizer. 
   In this embodiment, the coefficients of the time domain equalizer are determined from the cost function J by means of a minimum mean square error (MSE) algorithm. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Preferred embodiments of the present invention will be described below in more detail, with reference to the accompanying drawings: 
       FIG. 1  is a structural view of a transmitter and a receiver, which are provided in a multi-carrier transmission system; 
       FIG. 2  is a structural view of a transversal filter, which is provided in a time domain equalizer (TEQ); 
       FIG. 3  is a view for explaining a method for determining coefficients of a time domain equalizer (TEQ) shown in  FIG. 1 ; 
       FIG. 4  is a graph showing the amplitude response of a conventional twisted-pair cable without a bridged tap in a discrete multitone (DMT) communication system; 
       FIG. 5  is a graph showing the phase response of a twisted-pair cable; 
       FIG. 6  is a graph showing the frequency response characteristic of a time domain equalizer (TEQ); 
       FIG. 7  is a block diagram of a digital data transmission apparatus comprising a coefficient determining unit according to an embodiment of the present invention; 
       FIG. 8  is a graph for explaining a method of estimating the phase response of an upstream area in a channel of a phase estimator shown in  FIG. 7 ; 
       FIG. 9  is a graph for explaining a method of estimating the amplitude response of an upstream area in a channel of a phase estimator shown in  FIG. 7 ; 
       FIG. 10  is a graph for explaining a method of estimating the amplitude response of an upstream area of a channel with the use of an amplitude estimator shown in  FIG. 7 ; 
       FIG. 11  is a graph showing the response characteristic of a time domain equalizer (TEQ) according to an embodiment of the present invention; and 
       FIG. 12  is a graph showing the characteristic of an equalizer when coefficients of the equalizer are set based on the coefficient determining method according to an embodiment of the present invention. 
   

   DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
   A transmitter  10  and a receiver  20  of a conventional multi-carrier transmission system are illustrated in  FIG. 1 . The transmitter  10  comprises a serial-to-parallel converting buffer (S-P buffer)  11 , an encoder  12 , an inverted discrete Fourier transformer (IDFT)  13 , a cyclic prefix adding unit (CPA)  14 , and an analog front end unit (AFE)  15  having a low pass filter and performing an digital-to-analog conversion. 
   The receiver  20  comprises an analog front end (AFE) unit  21  having a low pass filter and performing an analog-to-digital conversion, a time domain equalizer (TEQ)  22 , a cyclic prefix eliminating unit (CPE)  23 , a discrete Fourier transformer (DFT)  24 , a frequency domain equalizer (FEQ)  25 , and a parallel-to-serial converting buffer (P-S buffer)  27 . 
   As shown in  FIG. 1 , a channel  16  (i.e., a transmission path, for example, a telephone network) is provided between the transmitter  10  and the receiver  20 . A noise source  17  is disposed on the channel  16 . 
   Bit streams of transmission data (TD) are input to the S-P buffer  11  that converts serial bit streams into parallel bit streams. The S-P buffer  11  outputs parallel bit streams to the encoder  12 . 
   The encoder  12  divides parallel bit streams into a plurality of parallel bit string groups (N pieces), encodes each of the parallel bit string groups as coded information, and outputs the coded information to the IDFT  13 . In this case, coded N pieces (hereinafter referred to as “N-coded information”) are assigned to N carriers. Each carrier is transmitted as a “transmission symbol”. 
   The IDFT  13  performs the inverted discrete Fourier transform on the N-coded information and converts the N-coded information from a frequency base signal to a time base signal. In an actual determination of the inverted discrete Fourier-transform (IDFT), an inverted fast Fourier transform (IFFT) is used instead of the IDFT to improve the speed of the determination. 
   The time base signals converted by the IDFT  12  are transmitted to the CPA  14 . When a cyclic prefix adding process is performed, an inter-symbol interference (ISI) caused by a response characteristic of the channel  16  having a predetermined transmission delay can be substantially eliminated using the CPE and the TEQ  22 . 
   Signals, which are subjected to the cyclic prefix adding process, are transmitted to the AFE  15  that converts digital signals into analog signals. The analog signals are transmitted to the channel  16  through the low pass filter provided in the AFE  15 . 
   In a channel used for the data transmission, if an amplitude characteristic (i.e., gain) and a group delay characteristic of the channel are constant, there will be no channel distortion of the signals. However, because the frequency characteristic is not constant in the actual channel, the signals are influenced by the channel distortion. 
   The distorted signals are transmitted to the AFE  21  through the channel  16 . In the AFE  21 , noise components at a high frequency are substantially eliminated by the low pass filter and the signals are converted into digital signals. The digital signals are then output to the TEQ  22 . 
   If the channel distortion is large, the influence applied to the signals also becomes large. Accordingly, the large distortion results in inter-channel interference (ICI) and inter-symbol interference (ISI). As a result, the large distortion has an influence on the received signals. 
   The distorted signals are transmitted to the AFE  21  through the channel  16 . In the AFE  21 , noise components at high frequency are substantially eliminated by the low pass filter and the signals are converted into the digital signals. The digital signals are then output to the TEQ  22 . 
   A structure of a transversal filter used in a time domain equalizer (TEQ) is illustrated in  FIG. 2 . This transversal filter has tap length K. As shown in  FIG. 2 , (K−1) delay elements are connected in series. Each of the (K−1) delay elements has a delay time for each sampling period. An input signal (TEQ input) from the AFT  21  is input to a first delay element  31 . The input signal, delayed by the first delay element  31 , is input to a second delay element  32 . The signal continues through the TEQ  22  until the final delay element  3 (K−1). 
   As illustrated in the figure, each of the K multipliers  41  is coupled to an output of a delay element T except the first multiplier, and (K−1) adders  51  are coupled to each output of the multipliers  41 . Each of the multipliers  41  multiplies an output of a delay element T by a corresponding coefficient, e.g., a0, a1, . . . , and ak. Outputs from the multipliers  41  are sequentially added by the (K−1) adders  51 , so that an output signal (TEQ output) from the last delay adder becomes an output of the TEQ  22 . 
   Returning to  FIG. 1 , the TEQ  22  reduces the length of the taps of a channel characteristic (i.e., impulse response characteristic) from an infinite length to a predetermined length L or less. According to this function, the influence of the inter-symbol interference (ISI) at a reception signal may be suppressed within a range of the cyclic prefix having the length L. 
   The DFT  24  converts an information symbol, having a length L, from a time base signal to symbol data. To enhance an operation speed, a fast Fourier transformer (FFT) may be used instead of the DFT. 
   The frequency domain equalizer (FEQ)  24  equalizes symbol data for each carrier on the frequency base. The decoder  26  decodes each symbol data to the data of parallel bit strings. The P-S buffer  27  converts data of parallel bit strings into data of serial bit strings and outputs the data of serial bits strings as reception data of the receiver  20 . 
   A training unit, which is provided in the TEQ  22  and reduces the number of taps of a channel characteristic into a predetermined length L, has been disclosed in U.S. Pat. No. 5,285,474 entitled “METHOD FOR EQUALIZING A MULTICARRIER SIGNAL IN A MULTICARRIER COMMUNICATION SYSTEM” issued to Jack Chow and John M. Chioffi. 
   A method for determining (training) coefficients of the TEQ  22  shown in  FIG. 1  is explained with reference to  FIG. 3 . 
   Referring to  FIG. 3 , a data generator  101  on a transmission side, a channel  102 , noise  103  superposed on the channel  102 , a time domain equalizer (TEQ)  104 , a data generator  105  on a reception side, a delayer  106  for compensating for delay in the channel  102 , a channel target circuit  107  for a channel target characteristic, and a subtracter  108  are illustrated. 
   For training the TEQ  104 , the data generator  101  on the transmission side generates a pseudo random signal x, and the data generator  105  on the reception side generates a pseudo random signal x′. The noise  103  is superposed on the pseudo random signal x through the channel  102 . A pseudo random signal y superposed with the noise is input to the TEQ  104  that outputs a signal u. 
   The pseudo random signal x′ from the data generator  105  is input to the channel target circuit  107  through the delayer  106 . The channel target circuit  107  outputs a signal u′. The TEQ  104  adjusts tap coefficients such that an error E between the signals u and u′ becomes zero. The channel target circuit  107  also adjusts the tap coefficients such that the error E therebetween becomes zero. The signals u and u′ are combined by the subtractor  108 . 
   A downstream amplitude response and a downstream phase response are shown in  FIG. 4  and  FIG. 5 , respectively. 
   Referring to  FIG. 4  and  FIG. 5 , in subchannels  0 -n 1 , neither an amplitude response nor a phase response substantially exists, and only noise exists. This is because the subchannels  0 -n 1  are in upstream areas of an FDM communication system. Since no signal is received through an upstream area, a receiver cannot know of the frequency and phase characteristics for an upstream area of a channel. Although the receiver receives a signal through the upstream area so as to know the characteristic of the upstream area, signals of the upstream area are all attenuated by a digital filter or an analog filter of the receiver. 
   As previously described, there is the minimum MSE approach and the SVD algorithm approach to obtaining coefficients of the TEQ  22  using channel information shown in  FIG. 4  and  FIG. 5 . Results obtained by these approaches are different from each other, but a frequency characteristic of a time domain equalizer is substantially identical as shown in  FIG. 6 . 
   Referring to  FIG. 6 , a frequency response of the TEQ  22  is high in an upstream area where a signal is not received and is low in a downstream area where a signal is received. That is, a channel gain in the upstream area is amplified and a channel gain in the downstream area is attenuated. This is because the TEQ  22  is a finite impulse response (FIR) type filter. Thus, a gain of a low-response area is amplified and a gain of a high-response area is attenuated so that a general characteristic of a high-response area may be uniform. 
   Further, a channel response of the downstream area is abrupt and a channel response cannot be obtained for subchannels antecedent to a subchannel n 1 . Under this state, a time domain equalizer having 8–32 equalizers cannot divide a communication channel response into an upstream area response and a downstream area response. That is, correct channel equalization cannot be performed for subchannels n 1  and n 2  where the downstream area begins. To overcome the above disadvantages, a characteristic of the upstream area in a channel is estimated to determine coefficients of a time domain equalizer. 
   A digital data transmission apparatus having a coefficient determining unit according to an embodiment of the present invention is schematically illustrated in  FIG. 7 . 
   Referring to  FIG. 7 , a data generator  201  on a transmission side, a channel  202 , a noise superposed on the channel  202 , and a digital data transmission apparatus  210  provided on a reception side are illustrated. The digital data transmission apparatus  210  comprises a time domain equalizer (TEQ)  211 , a delayer  213 , a channel target circuit  214  for a channel target characteristic, a subtracter  215 , and a coefficient calculating unit  216 . 
   During the signal processing in the TEQ  211 , the data generator  201  in the transmission side generates a pseudo random signal X and the data generator  212  on the reception side generates the same signal X′. Noise  203  is superposed on the pseudo random signal X through the channel  202 . A pseudo random signal Y, superposed with the noise  203 , is input to the TEQ  211  that outputs a signal U. 
   The pseudo random signal X′ from the data generator  212  is input to the channel target circuit  214  through the delayer  213 . The channel target circuit  214  outputs a signal U′. The signals U and U′ are combined by the subtractor  215  into a multi-dimensional signal and input to the coefficient calculating unit  216 . The coefficient calculating unit  216  adjusts tap coefficients of the TEQ  211  and tap coefficients of the channel target circuit  214  such that an error between the signals U and U′ becomes zero. The coefficient calculating unit  216  has a phase estimator  221 , an amplitude estimator  222 , and a coefficient calculator  223 . The coefficient calculating unit  216  estimates a phase response and an amplitude response of an upstream area in a channel, and determines coefficients of the TEQ  211  and the channel target circuit  214 . 
   A method for estimating a phase response of an upstream area in channels in the phase estimator  211  shown in  FIG. 7  is explained with reference to a graph of  FIG. 8 . 
   Referring to  FIG. 8 , phases of an upstream area in channels, i.e., a subchannel  0  to a subchannel antecedent to a subchannel n 1  are estimated by an extrapolation method. That is, assuming a phase characteristic of a given channel has a linearity, the phase characteristic thereof is expressed as a linear function to obtain phases of subchannels in the upstream area. In this case, a gradient of the linear function used in the extrapolation is determined by phases of the subchannels n 1  and n 2 . Sub Subchannel n 1  and n 2  are subchannels in a downstream area, and a frequency of subchannel n 2  is higher than that of subchannel n 1 . 
   If a phase response of subchannel n 1  is φ1 and a phase response of subchannel n 2  is φ2, a gradient of a linear function to be used for the phase extrapolation is 
               φ2   -   φ1           n   ⁢           ⁢   2     -     n   ⁢           ⁢   1       ⁢               .         
Thus, the phase extrapolation is given by the following:
 
   
     
       
         
           
             
               
                 
                   
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   In the case of n 2 =n 1 +1, the above equation 1 is simplified as follows:
 
φ i ′=(φ2−φ1) i+φ 1+(φ2−φ1) n 1 , i= 0, 1 , . . . , n 1  [Equation 2]
 
   A graph of  FIG. 9  shows one method for estimating an amplitude response of an upstream area in a channel in the amplitude estimator shown in  FIG. 7 . 
   Referring to  FIG. 9 , an amplitude gradient used for the extrapolation can have a variety of estimated values. That is, the gradient can have a positive value, be zero, or have a negative value. If an amplitude response of the subchannel n 1  is h1 and an amplitude response of the subchannel n 2  is h2, an amplitude response by a linear extrapolation from the subchannel  0  to the subchannel n 1  is given by the following: 
                     h   i   ′     =             h   ⁢           ⁢   2     -     h   ⁢           ⁢   1           n   ⁢           ⁢   2     -     n   ⁢           ⁢   1         ⁢   i     +         h   ⁢           ⁢   1   ⁢   n   ⁢           ⁢   2     -     h   ⁢           ⁢   2   ⁢           ⁢   n   ⁢           ⁢   1           n   ⁢           ⁢   2     -     n   ⁢           ⁢   1             ,     i   =   0     ,   1   ⁢           ,           ⁢   …   ⁢           ,     n   ⁢           ⁢   1             [     Equation   ⁢           ⁢   3     ]                 h   i   ′=h 1 , i= 0, 1 , . . . , n 1  [Equation 4] 
   
     
       
         
           
             
               
                 
                   
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   For example, to perform the amplitude extrapolation using the same gradient as an amplitude response of a downstream area of the channel, the upstream amplitude of the channel is estimated from the gradient of the equation 3. On the other hand, to perform the amplitude extrapolation using a gradient in the reverse direction of the amplitude response of the downstream area in the channel, the upstream gradient of the channel is estimated from the gradient of the equation 5. Further, to perform the amplitude extrapolation irrespective of the amplitude response of the downstream area in the channel, the amplitude of the downstream area in the channel is set to any value. 
   According to the above-described method, the phase and amplitude characteristics of the upstream area in the channel are estimated to reduce the inter-symbol interference (ISI). However a channel gain is still distributed to the upstream area, therefore, the potential improvement in a signal to noise ratio (SNR) is limited. According to an embodiment of the present invention, a proposed solution to the aforementioned limitation is to apply virtual noise to the upstream area when the amplitude response of the upstream area is extrapolated. Therefore, it is possible to prevent a time domain equalizer from distributing a channel gain to the upstream area. 
   A graph of  FIG. 10  shows a method for estimating an amplitude response of an upstream area in a channel in the amplitude estimator  222  shown in  FIG. 7  according to an embodiment of the present invention. 
   Referring to  FIG. 10 , an amplitude response is extrapolated so that partial amplitude responses from a subchannel  0  to a subchannel n 1  can be h1 and amplitude responses of the other subchannels can be h1/256. Equations 6 and 7 exemplarily show that an amplitude response is extrapolated so that amplitude responses of even-number subchannels can be h1 and amplitude responses of odd-number subchannels can be h1/256.
 
 h   i   ′=h 1 , i= 0, 2, 4 , . . . , n 1′  [Equation 6]
 
   
     
       
         
           
             
               
                 
                   
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     FIG. 11  shows a response characteristic of a time domain equalizer when the amplitude response of the upstream area in the channel is distorted by strong noise. 
   Referring to  FIG. 11 , a gain from a subchannel  0  to a subchannel n 1  (i.e., an upstream area) is small, and a gain of subchannels precedent to the subchannel n 1  is relatively large. Therefore, the general performance of an FDM communication system is enhanced. 
   A frequency characteristic of an upstream area is estimated in the phase estimator  221  and the amplitude estimator  222 , the coefficient calculator  223  shown in  FIG. 7  determines coefficients of the time domain equalizer (TEQ)  211 . 
   If a reference symbol x denotes a pseudo random signal output from the data generator  201 , a reference symbol y denotes a pseudo random signal input to the TEQ  211 , reference symbols a 1 , a 2 , . . . , and ak denote coefficients of the TEQ  211 , and reference symbols b 1 , b 2 , . . . , and bk denote coefficients of the channel target circuit  214 , they have a relationship given by equation 8.
 
 y ( n )+ a 1 y ( n− 1)+ a 2 y ( n− 2)+ . . . + aky ( n−K )= b 0 x ( n −δ)+ b 1 x ( n −δ−1)+ . . . + bMx ( n−δ−M )  [Equation 8]
 
   The coefficients a1, a2, . . . , and ak of the TEQ  211  are defined as shown in equation 9, and the coefficients b1, b2, . . . , and bk of the channel target circuit  214  are defined as shown in equation 10. 
   
     
       
         
           
             
               
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                   ⁢ 
                   9 
                 
                 ] 
               
             
           
           
             
               
                 b 
                 = 
                 
                   [ 
                   
                     
                       
                         b0 
                       
                     
                     
                       
                         b1 
                       
                     
                     
                       
                         ⋯ 
                       
                     
                     
                       
                         bM 
                       
                     
                   
                   ] 
                 
               
             
             
               
                 [ 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   10 
                 
                 ] 
               
             
           
         
       
     
   
   If an impulse response of the channel  202  is h(z), the coefficients a1, a2, . . . , and ak, the coefficients b1, b2, . . . , and bk, and the h(z) have a relationship given by equation 11.
 
 h ( z )= z   −δ   b ( z )/1+ a ( z )  [Equation 11]
 
   where the number of the coefficients of the TEQ  211  is K+1 and the number of coefficients of the channel target circuit  214  is M+1=CP+1 (CP being the length of a cyclic prefix). 
   The coefficients of the TEQ  211  are determined using the Minimum MSE algorithm. The Minimum MSE algorithm obtains the coefficients of the TEQ  211  and the coefficients of the channel target circuit  214 , which are used to minimize an error between an output signal u from the TEQ  211  and an output signal u′ from the channel target circuit  214 . A cost function of the Minimum MSE algorithm is given by equation 12.
 
 J=E{e   2 }  [Equation 12]
 
   While the Minimum MSE algorithm achieves desirable results in many cases, e.g., desirable coefficients of the TEQ  211  and desirable coefficients of the channel target circuit  214 , the coefficients of the TEQ  211  may diverge, or comprise noise  203  or distortion that decreases the stability of a general system. Therefore, according to an embodiment of the present invention, a square term of the coefficient of the TEQ  211  is added to a cost function.
 
 J=E{e   2   }+λ|a|   2 , λ≠1  [Equation 13]
 
   According to the equation 13, the cost function is obtained to minimize the MSE and restrict the sizes of the coefficients of the TEQ  211 . If the definition of  220   
               y   _     =         [           y   ⁡     (     n   -   1     )                 y   ⁡     (     n   -   2     )               ⋯             y   ⁡     (     n   -   K     )             ]     ⁢           ⁢   and   ⁢           ⁢     x   _       =     [           x   ⁡     (     n   -   δ     )                 x   ⁡     (     n   -   δ   -   1     )               ⋯             x   ⁡     (     n   -   δ   -   M     )             ]         ,         
y(n) is given by the following equation 14;
   y ( n )=− a   T     y +b   T     x     [Equation 14] 
an error between the output u of the TEQ  211  and the output u′ of the cannel target circuit  214  is expressed by equation 15.
   e ( n )= y ( n )+ a   T     y −b   T     x     [Equation 15] 
   Using the equation 15, the cost function of the equation 13 is rearranged as shown in equation 16.
 
 J=E{y   2 ( n )+2 a   T     y y ( n )−2 b   T     x y ( n )−2 b   T     x   y a   T   +a   T     y   y     T   a+b   T     x   x     T   b}+λ|a|   2   [Equation 16]
 
   In the cost function J of the equation 16, there are minimum values for coefficients a of the TEQ  211  and minimum values for coefficients b of the channel target circuit  214 . A minimum value of the cost function J is achieved where a gradient of J to a is zero and a gradient of J to b is zero, as shown in equations 17 and 18. 
   
     
       
         
           
             
               
                 
                   
                     ∂ 
                     J 
                   
                   
                     ∂ 
                     a 
                   
                 
                 = 
                 0 
               
             
             
               
                 [ 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   17 
                 
                 ] 
               
             
           
           
             
               
                 
                   
                     ∂ 
                     J 
                   
                   
                     ∂ 
                     a 
                   
                 
                 = 
                 0 
               
             
             
               
                 [ 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   18 
                 
                 ] 
               
             
           
         
       
     
   
   If the definition is R yy =E{  y   y   T }, R xx =E{  x   x   T }, R yx =E{  y   x   T }, R xy =E{  x   y }, P y =E {  y  y(n)}, and r yy =E {y(n)y(n−k)}, the cost function J is to be arranged in equation 19.
 
 J=r   yy (0)+2 a   T   P   y −2 b   T   P   x −2 b   T R xy   a   T   +a   T   R   yy   a+b   T   R   xx   b+λ|a|   2   [Equation 19]
 
   Since the pseudo random signal x generated from the data generator  201  is an impulse signal, the result is R xx =I and R xy =R yx   T . 
   A J-a gradient and a J-b gradient are given by equations 20 and 21, respectively. 
   
     
       
         
           
             
               
                 
                   
                     ∂ 
                     J 
                   
                   
                     ∂ 
                     a 
                   
                 
                 = 
                 
                   
                     
                       2 
                       ⁢ 
                       
                         P 
                         y 
                       
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         R 
                         xy 
                         T 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         R 
                         yy 
                       
                       ⁢ 
                       a 
                     
                     + 
                     
                       2 
                       ⁢ 
                       λ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       a 
                     
                   
                   = 
                   0 
                 
               
             
             
               
                 [ 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   20 
                 
                 ] 
               
             
           
           
             
               
                 
                   
                     ∂ 
                     J 
                   
                   
                     ∂ 
                     b 
                   
                 
                 = 
                 
                   
                     
                       
                         - 
                         2 
                       
                       ⁢ 
                       
                         P 
                         x 
                       
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         R 
                         xy 
                       
                       ⁢ 
                       a 
                     
                     + 
                     
                       2 
                       ⁢ 
                       b 
                     
                   
                   = 
                   0 
                 
               
             
             
               
                 [ 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   21 
                 
                 ] 
               
             
           
         
       
     
   
   Equation 20 can be arranged for a and equation 21 can be arranged for b as shown in equations 22 and 23, respectively.
 
 a= ( R   yy   −λI−R   xy   T   R   xy ) −1 ( R   xy   T   P   x   −P   y )  [Equation 22]
 
 b=P   x   +P   xy   a   [Equation 23]
 
   The coefficients a and b, which are the determined results of the coefficient calculator  223  shown in  FIG. 7 , are output to the TEQ  211  and the channel target circuit  214 , respectively. 
   As described above, a phase response for an upstream area in a channel is estimated by means of an extrapolation. Following forcible distortion of an amplitude response, coefficients of a time domain equalizer and coefficients of a channel target circuit are determined using the Minimum MSE algorithm. When a cost function J of the Minimum MSE is determined, the square of the coefficients of the time domain equalizer is included for determining the cost function J. Therefore, a channel-shortening effect of the time domain equalizer is improved to reduce inter-symbol interference (ISI) and inter-channel interference (ICI). As a result, a signal to noise ratio (SNR) of a communication system is improved. 
   A graph of  FIG. 12  shows the characteristic of an equalizer when coefficients of the equalizer are set by the coefficient determining method according to an embodiment of the present invention, in which a transversal axis denotes time and a longitudinal axis denotes amplitude. A channel impulse response, (demarcated by a dotted line) transmitted from a transmission side has an influence on adjacent symbols. Meanwhile, a channel impulse response (demarcated by a solid line) equalized by an equalizer of this invention is shortened to a narrow amplitude. Accordingly, since the equalized channel impulse response becomes substantially similar to an original pulse signal transmitted from the transmission side, it does not overlap adjacent symbols. 
   Although particular embodiments of the present invention have been shown and described, it will be apparent to those skilled in the art that changes and modifications can be made without departing from the present invention in its broader aspects. Therefore, the invention is limited only by the following claims.