Abstract:
A method for correcting signals received on a channel. Signals are received along the channel and it is determined how many of the signals are outside a predetermined range relative to a plurality of predetermined constellation points (i.e., erasures). Then, the noise power is estimated the noise power based on the number erasures, and the noise power is used to correct the signals. Specifically, the estimated noise power can be used to correct the signals which have been determined to be outside the predetermined range and which have been determined to contain a large error component (i.e., based on distance from the closest constellation point). A look up table can be used to determine the correction to be applied, and a separate look up table can be used for each tone.

Description:
BACKGROUND  
       [0001]     Transferring digital data over a medium is performed using a modulation/demodulation scheme. A discrete multitone modulation method is commonly used in DSL. In DMT modulation, the transferred bits are divided between each one of the discrete tones in such a way to achieve maximum transmit rate with minimum Bit Error Rate (BER). Estimation of the signal to noise ratio (SNR) for each discrete tone is essential to determine how many bits will be assigned to each tone to achieve the desired BER.  
         [0002]     U.S. patent application Ser. No. 10/739,388, assigned to the same assignee as the present application, describes a method to estimate the signal to noise ratio that is used in ADSL link. To determine the signal to noise ratio in the channel, the transmit side transmits a known signal—a reverb signal using a 4QAM (quadratic Amplitude Modulation) modulation. 4QAM constellation has four constellation points, each representing two bits of information. The reverb signal is a fixed pseudo random sequence with equal probability to each of the four constellation points—P 1 , P 2 , P 3  and P 4 . The power of the transmitted signal is a constant. The receiver determines the power of the noise process by analyzing the distance of the received data values from the constellation points.  
         [0003]      FIG. 1  illustrates the constellation domain in the receiver. In  FIG. 1 , the location of constellation point P 1  is (1,1), the location of constellation point P 2  is (−1,1), the location of constellation point P 3  is (−1, −1) and the location of constellation point P 4  is (1, −1). A typical method of estimating the noise power is as follows: First, an acceptance square  10  is defined by the points (2,2), (−2,2), (−2,−2) and (2,−2). If the received data is in the acceptance square  10 , the hardware calculates the distance between the data and the nearest constellation point (i.e., P 1 , P 2 , P 3  or P 4 ), the distance is squared, and the resulting value is accumulated (i.e., summed) for N symbols. If the data is outside of the acceptance square (i.e., in area  12  in  FIG. 1 ), the data is marked as an erasure and is ignored by the algorithrn. An average is taken over a large number of samples to get the average noise power and to determine the signal to noise ratio.  
         [0004]     The signal to noise ratio is typically calculated using the following method: First, it is observed that for 4QAM the power of the signal is exactly 2. The noise power is calculated as: 
 
 NP=ΣD ( n ) 2   /N  
 
 Where N is the number of samples that fall within the acceptance square and D(n) is the distance between the n&#39;th sample and the nearest constellation point. 
 
         [0005]     It is well observed that the above method is efficient in high signal to noise ratio, but is not accurate when the signal to noise ratio is low. The reason is two fold: First, the computation ignores the samples that fall outside of the acceptance square even though these samples carry the biggest errors; second, if the error is large enough such as a sample that corresponds to a constellation point which is farther than the closest constellation point, the implementation will underestimate the error.  
       OBJECTS AND SUMMARY  
       [0006]     An object of an embodiment of the present invention is to provide a correction calculation algorithm which better estimates the signal to noise ratio for the case of low signal to noise ratio.  
         [0007]     Briefly, an embodiment of the present invention provides a method for correcting signals received on a channel. The method includes receiving a plurality of signals along the channel, determining how many of the signals are outside a predetermined range relative to a plurality of predetermined constellation points, estimating the noise power based on the number of signals which have been received and determined to be outside the predetermined range, and using the noise power which has been calculated to correct at least some of the signals.  
         [0008]     Specifically, preferably an algorithm is used which does the following: 
        Uses the number of erasures to estimate the noise power in the channel.     Uses the noise power estimation to add correction for the samples that were marked as erasure.     Uses the noise power estimation to correct for large errors that result in underestimation of the error.     Modify the sum of square errors by adding the previous two correction to the sum of the squared errors and calculate the modified average noise power.        
 
         [0013]     Preferably, the algorithm can be used for many modulation types, such as for BPSK, as well as for 4QAM and other QAM values.  
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0014]     The organization and manner of the structure and operation of the invention, together with further objects and advantages thereof, may best be understood by reference to the following description, taken in connection with the accompanying drawings, wherein:  
         [0015]      FIG. 1  illustrates the constellation domain in a receiver;  
         [0016]      FIG. 2  illustrates a correction-calculation algorithm which is in accordance with an embodiment of the present invention.  
         [0017]      FIG. 3  illustrates an algorithm to better estimate SNR;  
         [0018]      FIG. 4  shows the erasure region divided into several sub-regions;  
         [0019]      FIG. 5  illustrates an algorithm to estimate noise power; and  
         [0020]      FIG. 6  illustrates the areas of error due to the wrong constellation point.  
     
    
     DESCRIPTION  
       [0021]     While the invention may be susceptible to embodiment in different forms, there is shown in the drawings, and herein will be described in detail, a specific embodiment of the invention. The present disclosure is to be considered an example of the principles of the invention, and is not intended to limit the invention to that which is illustrated and described herein.  
         [0022]     An embodiment of the present invention provides a correction calculation algorithm, such as is shown in  FIG. 2 , which better estimates the signal to noise ratio for the case of low signal to noise ratio. The algorithm can be used for many modulation types, such as for BPSK, as well as for 4QAM and other QAM values.  
         [0023]     The 4QAM signal is generated in the transmitter side as a fixed two bits value for each discrete tone. Let X(n) be the signal value for the n&#39;th symbol:  
                     1.   ⁢           ⁢     X   ⁡     (   n   )         =       ⁢       (     1   ,   1     )     ⁢           ⁢   with   ⁢           ⁢   probability   ⁢           ⁢   0.25                     ⁢       (       -   1     ,   1     )     ⁢           ⁢   with   ⁢           ⁢   probability   ⁢           ⁢   0.25                     ⁢       (       -   1     ,     -   1       )     ⁢           ⁢   with   ⁢           ⁢   probability   ⁢           ⁢   0.25                     ⁢       (     1   ,     -   1       )     ⁢           ⁢   with   ⁢           ⁢   probability   ⁢           ⁢   0.25                                 
 
 X(n) is a two dimension complex process, that is, 
 
 X ( n )= c ( n )+ j*d ( n ).  (2) 
 
 The one-dimensional processes c(n) and d(n) are independent of each other. 
 
         [0024]     In the transmitter, the signal X(n) goes through IFFT to convert it from frequency domain to time domain, digital filtering, digital to analog conversion, and analog filtering before being transmitted over a pair of copper wires. In the receive side, the analog signal goes through analog filters, analog to digital conversion, digital filtering and fft to return back to the frequency domain. All these processes add errors to the receive signal. Let Y(n) be the receive signal after the ffM. From the demodulation point of view, Y(n) can be considered as two dimension received signal, that is: 
 
 Y ( n )= X ( n )+γ  (3) 
 
 Where γ is a two-dimension vector error that is the accumulation of all the errors that are described above and Y(n) is a two-dimension complex, that is 
 
 Y ( n )= a ( n )+ j*b ( n ).  (4) 
 
         [0025]     The algorithm to evaluate the noise power for the process that is defined in (2) uses the role of the large number. It estimates the noise power by accumulating the square distance between the X(n) process and the Y(n) for a large number of symbols N, and then divides the sum by N. The hardware implementation is the following:  
         [0026]     First, the hardware defines a new process Z(n) as a hard-limited version of Y(n) and a one dimension count process I(n) that counts the number of erasures, as described below:  
         5.   ⁢             Z   ⁡     (   n   )       =       ⁢   0               ⁢       if   ⁢           ⁢          a   ⁡     (   n   )              &gt;     2   ⁢           ⁢   or   ⁢           ⁢     (     b   ⁡     (   n   )              &gt;   2                     ⁢         (       a   ⁡     (   n   )       -     p   x       )     2     +       (       b   ⁡     (   n   )       -     p   y       )     2                   ⁢   Otherwise             ⁢               
 
 Where p x  and p y  are the x and y coordinates of the constellation points that is the nearest to Y(n). 
 
         [0027]     Next, the count process I(n) is defined  
         6.   ⁢             I   ⁡     (   n   )       =       ⁢   0               ⁢     if   ⁢           ⁢            a   ⁡     (   n   )       &gt;     2   ⁢           ⁢   or   ⁢           ⁢     (     b   ⁡     (   n   )              &gt;   2                         ⁢   1               ⁢   Otherwise             ⁢               
 
 Let S be the sum of Z(n) and I counts the number of non-zero values of Z(n), that is 
 
 S=ΣZ ( n )  (7) 
 
 I=ΣI ( n )  (8) 
 
 The current approximation for the noise power is given by Power Noise =S/I 
 
 Since the power of the signal is always 2, the signal to noise ratio, SNR is given by 
 
 SNR= 2 *I/S   (9) 
 
 Or SNR is given in decibel units, 
 
 SNR   decibel =10*LOG 10 (2 *I/S )  (10) 
 
         [0028]     Even though the noise process γ is the sum of many errors, and thus not necessarily a Gaussian process, using a Gaussian approximation is justified because the average is that of a large number of symbols. The following assumption will be made for the model of the Noise process γ: 
        γ is a two dimensional complex process γ=(γ x , γ y )     γ x  and γ y  are two independent identical Gaussian processes with mean 0 and σ variance. That is for any value of α: 
 
 P (γ x &lt;α)= P (γ y &lt;α)=1/( sqrt (2*π)*σ)         ( e   −x2/2σ2 ) dx   (11) 
 
 Where the integral goes from minus infinity to α. Using the notation for the cumulative distribution function of the normal distribution with mean 0 and variance 1, equation (11) becomes: 
 
 P (γ x &lt;α)= P (γ y &lt;α)=φ(α/σ)  (12) 
 
 Description of the New Algorithm 
       
 
         [0031]      FIG. 3  illustrates an algorithm to better estimate SNR.  
         [0032]     Step 1—Estimation of the Noise Power  
         [0033]     Because of the symmetry of the model, the independence of the X(n) process and the noise process, it is sufficient to analyze the results for one constellation point only. Assume X(n) has the value (1,1). The probability of erasure from equation (5) is the probability that either |a(n)| or |b(n)| is more than 2.  FIG. 4  divides the erasure region to several sub-regions.  
         [0034]     Using (12) and the independency between γ x  and γ y  it is easy to see that if X(n) is (1,1): 
 
 P ( Y ( n ) is in regions  A,C,D,E )= P (γ y &gt;1)=1−φ(1/σ)  (13) 
 
 P ( Y ( n ) is in regions  L,H,G,J )= P (γ y &lt;−3)=φ(−3/σ)  (14) 
 
 P ( Y ( n ) is in regions  C,B,K,J )= P (γ x &gt;1)=1−φ(1/σ)  (15) 
 
 P ( Y ( n ) is in regions  E,F,I,L )= P (γ x &lt;−3)=φ(−3/σ)  (16) 
 
 P ( Y ( n ) is in region  C )= P (γ y &gt;1)* P (γ x &gt;1)=(1−φ(1/σ))*(1−φ(1/σ))  (17) 
 
 P ( Y ( n ) is in region  E )= P (γ y &gt;1)* P (γ x &lt;−3)=(1−φ(1/σ))*φ(−3/σ)  (18) 
 
 P ( Y ( n ) is in region  L )= P (γ y &lt;−3)* P (γ x &lt;−3)=φ(−3/σ)*φ(−3/σ)  (19) 
 
 P ( Y ( n ) is in region  J )= P (γ y &lt;−3)* P (γ x &gt;1)=φ(−3/σ)*(1−φ(1/σ))  (20) 
 
 Using the identity 
 
1−φ( x )=φ(− x )  (21) 
 
         [0035]     it is easy to show that the probability that a signal that was originated from constellation point (1,1) is marked as an erasure is equal to  
       22.   ⁢             P   erasure     =         2   *     ϕ   ⁡     (       -   1     /   σ     )         +     2   *     ϕ   ⁡     (       -   3     /   ϕ     )         -       ϕ   ⁡     (       -   1     /   σ     )       *     ϕ   ⁡     (       -   1     /   σ     )         -     2   *     ϕ   ⁡     (       -   1     /   σ     )       *     ϕ   ⁡     (       -   3     /   σ     )         -       ϕ   ⁡     (       -   3     /   σ     )       *     ϕ   ⁡     (       -   3     /   σ     )           =       (       ϕ   ⁡     (       -   1     /   σ     )       +     ϕ   ⁡     (       -   3     /   σ     )         )     *     (       ϕ   ⁡     (     1   /   σ     )       +     ϕ   ⁡     (     3   /   σ     )         )                                 
 
         [0036]     Considering the symmetry of the model, and the equal probability of equation (1), equation (22) represents the probability of an erasure.  
         [0037]     Let N be the number of samples that are observed. If N&gt;&gt;1, and I is as defined in equation (8), then I/N converges to the probability of erasures.  
         [0038]     For practical reasons, when the number of erasures reaches 1/16 of the number of samples that are observed, the algorithm does not calculate the signal to noise ratio. In a typical case N=4096. Thus, the number of erasures can vary from 0 to 255. For each value of I, we have the following relationship:  
               23.   ⁢           ⁢     I   /   N       =         2   *     ϕ   ⁡     (       -   1     /   σ     )         +     2   *     ϕ   ⁡     (       -   3     /   σ     )         -       ϕ   ⁡     (       -   1     /   σ     )       *     ϕ   ⁡     (       -   1     /   σ     )         -     2   *     ϕ   ⁡     (       -   1     /   σ     )       *     ϕ   ⁡     (       -   3     /   σ     )         -       ϕ   ⁡     (       -   3     /   σ     )       *     ϕ   ⁡     (       -   3     /   σ     )           =       (       ϕ   ⁡     (       -   1     /   σ     )       +     ϕ   ⁡     (       -   3     /   σ     )         )     *     (       ϕ   ⁡     (     1   /   σ     )       +     ϕ   ⁡     (     3   /   σ     )         )                               
 
 Thus, the algorithm to estimate the noise power is as illustrated in  FIG. 5 . 
 
         [0039]     As described later herein, the look-up table for the σ as a function of I/N is replaced with a table that gives the correction to the sum S from equation (7) as a function of I/N. The above table and algorithm are not used.  
         [0040]     Step 2—Estimation of the Correction Due to Erasures  
         [0041]     Because of the symmetry of the model, only the correction for signals that were generated by the (1,1) constellation point will be analyzed. The results will be true for all other constellation points.  
         [0042]     The average square error that is ignored by the sum S of equation (7) depends on the erasure region. For a generic rectangular region R={(x,y) such that a+1&lt;x&lt;b+1 and c+1&lt;y&lt;d+1} and a signal that is originated from the constellation point (1,1), the contribution of the average square error C is given by 
 
 C   R =(1/((2*π)*σ 2 ))                   ( u   2   +v   2 )( e   −u2/2σ2 ) du ( e   −v2/2σ2 ) dv   (24) 
 
 Where the 2-D integral is taken over the region R.={(u,v) such that a&lt;u&lt;b and c&lt;v&lt;d}. Separation of the integral in (24) to two integrals, one with u 2  and one with v 2 , and using well known integration formula yields: 
 
 C   R =σ 2 ( A ( d )− A ( c ))*{ A ( b )− A ( a )+ T ( a )− T ( b )}+σ 2 ( A ( b )− A ( a ))*{ A ( d )− A ( c )+ T ( c )− T ( d )}  (25) 
 
Where 
 
 A ( x )=φ( x /σ))  (26) 
 
And 
 
 T ( x )=1/( sqrt (2*π)*( x/σ*e **(−( x   2 /2σ 2 ))  (27) 
 
 Similarly, the following definition will be used later: 
 
 B ( x )=1/( sqrt (2*π)*( e **(−( x   2 /2σ 2 ))  (28) 
 
         [0043]     Region (C and A and D and E) a=minus infinity, b=infinity, c=1, d=infinity. The average error in this region is: 
 
 C   LHGJ =σ 2 {2(1 −A (1))+ T (1)}  (29) 
        1. Region (L and H and G and J) a=minus infinity, b=infinity, c=minus infinity, d=−3(d+1&lt;−2). The average error in this region is: 
 
 C   LHGJ =σ 2 {2( A (−3))− T (−3)}  (30) 
    2. Region (B and K) a=1, b=infinity, c=−3(c+1&gt;−2), d=1 (d+1&lt;2). The average error in this region is: 
 
 C   BK =σ 2 {( A (1)−( A (−3))*(1 −A (1)+ T (1))+(1 −A (1))*( A (1)− A (−3)+ T (−3)− T (1))  (31) 
    3. Region (F and I) a=minus infinity, b=−3, c=−3(c+1&gt;−2), d=1 (d+1&lt;2). The average error in this region is: 
 
 C   FI =σ 2 {( A (1)−( A (−3))*( A (−3)− T (−3))+( A (−3))*( A (1)− A (−3)+ T (−3)− T (1))  (32) 
       
 
         [0047]     So that the total contribution from the erasures regions is 
 
 C   erasure   =C   LHGJ   +C   LHGJ   +C   BK   +C   FI   (33) 
 
         [0048]     Step 3—Estimation of the Correction of Samples in the Wrong Constellation Point  
         [0049]     Because of the symmetry of the model, only the correction for signals that were generated by the (1,1) constellation point will be analyzed. The results will be true for all other constellation points.  
         [0050]      FIG. 6  illustrates the areas of error due to the wrong constellation point. First, we calculate the error in calculating the noise power for samples that was generated by the constellation point P 1  and is received in P 2 . In that case, −1&lt;γ y &lt;1 and −3&lt;γy x &lt;−1. The real square error is equal to: 
 Error —   P 2 real =γ y   2 +γ x   2   (34)  
         [0051]     The measured squared error is the distance between the sample point and Constellation point P 2  (−1,1) 
 
Error —   P 2 measured =γ y   2 +(2+γ x ) 2 =Error —   P 2 real +4+4*γ x   (35) 
 
 Thus, the correction to add to the sum of the error square due to samples in R 2  is: 
 
 D   —   P 2=(1/((2*π)*σ 2 ))                   (−4+−4 *u )( e   −u2/2σ2 ) du ( e   −v2/2σ2 ) dv   (36) 
 
 Where the u boundaries are between −3 to −1, and v boundaries are between −1 to 1. 
 
 D   —   P 2=( A (−1)− A (1))*4*{( A (−1)− A (−3))+σ*( B (1)− B (3))}  (37) 
 
 Next, we observed that because of the symmetry between γ x  and γ y  the contribution of P 4  is equal to the contribution of P 2 , that is 
 
D_P4=D_P2  (38) 
 
 Finally, we calculate the contribution from P 3 . Again the real error is γ y   2 +γ x   2  and the measured error is given by: 
 
Error —   P 3 measured =(2+γ y ) 2 +(2+γ x ) 2 =Error —   P 3 real +8+4*γ x +4*γ y   (39) 
 
 D   —   P 3=(1/((2*π)*σ 2 )                   −(8+4*γ x +4*γ y )( e   −u2/2σ2 ) du ( e   −v2/2σ2 ) dv   (40) 
 
 Where the u boundaries are between −3 to −1, and v boundaries are between −3 to −1. Again, using symmetry 
 
 D   —   P 3==( A (−3)− A (−1))*8*{( A (−1)− A (−3))+σ*( B (1)− B (3))}  (41) 
 
         [0052]     Step 4—Add All Corrections to the Sum S  
         [0000]     The total correction is: 
 
Total_correction= C   LHGJ   +C   LHGJ   +C   BK   +C   FI +2 *D   —   P 2 +D   —   P 3  (42) 
 
So 
 
 S=S +Total_correction  (43) 
 
And 
 
 SNR= 2 *N/S   (44) 
 
 When N is the total number of samples for which the statistics are taken. 
 
 Implementation of the New Algorithm in Fixed Point Arithmetic 
 
         [0053]     The first part of the algorithm is building the correction tables. This is performed during initialization. The number of tables that are built is the number of various symbols that are integrated during the calculation of SNR. In a typical system, the integration is done either on 2048 symbols or 4096 symbols. Thus, two tables will be built, one with 127 entries and one for 255 respectively.  
         [0054]     The format of the tables must agree with the format of the noise power data as it is measured and manipulated by the hardware. The following are the considerations for the table&#39;s format.  
         [0055]     Let the (x,y) value of the constellation points have the format nQm, that is n binary digits with m digit on the right on the binary point. If the error in x or y is more than a unit, the error will be marked as erasure or the sample will be closer to another constellation point and thus the error will be under-estimated.  
         [0056]     The square error in x and y has 2n bits, where the binary point is after 2m bits. Depending on the multiplier, the result of the square might be shifted one place to avoid two sign bits.  
         [0057]     The square error is summed and stored in a register. The number of bits in the register is 2n+LOG 2 (M) where M is the number of symbols that are summed and n is at least m+1. A typical case is where M=4096 and m=12. In this case, the minimum number of bits in the summation register is 24+2+12=38.  
         [0058]     The tables are calculated using double precision floating point format. Since the table value is added to a sum that has a format of (2n+12)Q2m, the conversion between the floating point value and the fixed point value is performed by multiplying the floating point value by 2**12=4096.0, adding 0.5, and casting the value to long integer.  
         [0059]     A table for N=2048 is as follows: 
    15231     28609     42507     56793     101414     116772     132339     148101     164047     180169     196457     212905     229506     246256     263149     280182     297350     314650     332079     349633     367311     385109     403025     421058     439204     457463     475833     494311     512897     531589     550385     569285     588286     607389     626591     645892     665291     684797     704379     724066     743847     763722     783689     803749     823900     844142     864473     884895     905405     926004     946691     967464     988325     1009272     1030305     1051424     1072627     1093915     1115288     1136744     1158284     1179907     1201613     1333401     1245272     1267224     1289258     1311374     1333570     1355847     1378205     1400643     1423160     1445758     1468435     1491192     1514028     1536942     1559936     1583008     1606158     1629387     1652694     1676078     1699541     1723081     1746698     1770393     1794165     1818014     1841940     1865943     1890022     1914177     1938410     1962719     1987104     2011565     2036102     2060715     2085404     2110169     2135010     2159927     2184918     2209986     2235129     2260348     2285641     2311011     2336455     2361975     2387570     2413240     2438986     2464805     2480701     2516671     2542716     2568837     2595032     2621302     2647647     2674067     2400562    
 
         [0185]     A table for N=4096 is as follows: 
    8837     15231     21842     28609     35504     42507     49607     56793     64059     71399     78808     86283     93819     101414     109066     116772     124530     132339     140196     148101     156052     164047     172087     180169     188292     196457     204661     212905     221186     229506     237863     246256     254685     263149     271649     280182     288750     297350     305984     314650     323349     332079     340841     349633     358457     367311     376195     385109     394053     403025     412027     421058     430117     439204     448320     457463     466634     475833     485059     494311     503591     512897     522230     531589     540974     550385     559822     569285     578773     588286     597825     607389     606977     626591     636229     645892     655580     665291     675027     684787     694571     704379     714210     724066     733945     743847     753773     763722     773694     783689     793708     803749     813813     823900     834009     844142     854296     864473     874673     884895     895139     905405     915694     926004     936336     946691     957067     967464     977884     988325     98788     1009272     1019778     1030305     1040854     1051424     1062015     1072627     1083261     1093915     1104591     1115288     1126005     1136744     1147503     1158284     1169085     1179907     1190750     1201613     1212497     1223401     1234326     1245272     1256238     1267224     1278231     1289258     1300306     1311374     1322461     1333570     1344698     1355847     1367016     1378205     1389414     1400643     1411891     1423160     1434449     1445758     1457087     1468435     1479804     1491192     1502600     1514028     1525475     1536942     1548429     1559936     1571462     1583008     1594574     1606158     1617763     1629387     1641031     1652694     1664376     1676078     1687800     1699541     1711301     1723081     1734880     1746698     1758536     1770393     1782269     1794165     1806080     1818014     1729967     1841940     1853931     1865943     1877972     1890022     1902090     1914177     1926284     1938410     1950555     1962719     1974902     1987104     1999325     2011565     2023824     2036102     2048399     2060715     2073050     2085404     2097778     2110169     2122580     2135010     2147459     2159927     2172413     2184918     2197443     2209986     2222548     2235129     2247729     2260348     2272985     2285641     2298317     2311011     2323724     2336455     2349206     2361975     2374763     2387570     2400395     2413240     2426103     2438986     2451886     2464805     2477744     2490701     2503676     2516671     2529685     2542716     2555767     2568837     2581925     2595032     2608157     2621302     2634465     2647647     2660848     2674067     2687305     2700562     2713837    
 
         [0441]     A C code algorithm to build the table is as follows:  
                                   /***********************************************************************       ******       *       * build_table.c  A program to build a look-up table that       * will be used to improve the SNR calculations       * The table build program will be done in several stages -       *       * 1. Build PHI(x) table - cumulative distribution function of the normal       *  distribution and compare it to the given table, only to test the       *  accuracy of the integral method       * 2. Build a table for I/N of equation 23 of the paper and find the values for       *  I goes from 1 to 255, where N is 4096 - or any other number up to M/16       * 3. Build the tables for the various corrections only for the 255 values from above       * 4. Invert the tables and get the final look-up table of equation 42       *       * Ran Katzur starting day 11-18-03       *       ************************************************************************       **/       #include &lt;stdio.h&gt;       #include &lt;math.h&gt;       #define M   4096 // number of symbols       #define TABLE_SIZE 20000       #define END_POINT 4       #define TABLE_SIZE_M_DIVIDE_16 M/16       double delta_t ;       double t_vector[TABLE_SIZE+1] ;       double pfi_of_one_over_sigma[TABLE_SIZE+1] ;       double pfi_of_three_over_sigma[TABLE_SIZE+1] ;       double I_over_N[TABLE_SIZE+1] , I_N_M[TABLE_SIZE+1] ;       double integer_I_N_M[TABLE_SIZE_M_DIVIDE_16] ;       double       phi_minus_one_over_sigma_for_integer_I_N[TABLE_SIZE_M_DIVIDE_16] ;       double       phi_minus_three_over_sigma_for_integer_I_N[TABLE_SIZE_M_DIVIDE_16] ;       double one_over_sigma[TABLE_SIZE_M_DIVIDE_16] ;       double correction_table[TABLE_SIZE_M_DIVIDE_16] ;       double x_delta ;       double x_value1 ;       double x_value2 ;       double t_value ;       double integral_value ;       double *p_t1, *p_t2 , *p_pfi1, *p_pfi2 ;       double one_over_square_root_2_pi ;       double in ;       double integral_function(double t_start, double delta_t) ;       double interpolation_f(double xx,int loop9,double *vector) ;       FILE *fp1 ,*fp2 ;       #define ONE 1.01       #defineI_COMPARE_LIMIT (double) (M/16)       main( )       {        int loop1 , i_sigma3 ;       double i_compare ,diff ;       double xx ;        int loop9 ,index ;        double alfa, beta ;       double A_1, T_1, A_minus_3 , T_minus_3 ;        double A_minus_1, B_minus_1, B_minus_3;        double B_1, B_3 ;        double real_sigma ,sigma_square ;        double C_LHGJ, C_BK, C_FI, C_CADE ;        double D_P2, D_P3, D_P4 ;        double C_TOTAL, D_TOTAL , TOTAL ;        double h1,h2,h3,h4 ;        h1 = 8.0 * atan(1.0) ; // h1 is 2*pi       h2 = sqrt(h1) ;       one_over_square_root_2_pi = 1.0 / h2 ;        if (M == 2048)        {         fp1 = fopen(“correction_table_128.dat”, “w”) ;         fp2 = fopen(“integer_correction_128.dat”, “w”) ;       }        else if (M == 4096)        {        fp1 = fopen(“correction_table_256.dat”, “w”) ;         fp2 = fopen(“integer_correction_256.dat”, “w”) ;       }        else        {          printf(“ M is not supported YET ♭n”) ;          return −1 ;        }         t_vector[TABLE_SIZE/2 −1] = 0.0 ;         pfi_of_one_over_sigma[TABLE_SIZE/2 −1] = 0.5 ;         p_t1 = (double *) &amp;t_vector[TABLE_SIZE/2 −1] ;         p_t2 = p_t1 ;         p_pfi1 = (double *) &amp;pfi_of_one_over_sigma[TABLE_SIZE/2 − 1] ;         p_pfi2 = p_pfi1 ;         t_value = 0.0 ;         x_value1 = 0.5 ;         x_value2 = 0.5 ;        delta_t = (double)(END_POINT) / ( (double) (TABLE_SIZE/2) ) ;        p_t1++ ;         p_t2−− ;         p_pfi1++ ;         p_pfi2−− ;         for (loop1 = 0; loop1 &lt; TABLE_SIZE/2 ; loop1++)         {          x_delta = integral_function(t_value, delta_t);          t_value = t_value + delta_t ;         *p_t1++ = t_value ;          *p_t2−− = −t_value ;          x_value1 = x_value1+ x_delta ;          x_value2 = x_value2 − x_delta ;          *p_pfi1++ = x_value1 ;          *p_pfi2−− = x_value2 ;        }       /*   starting the second Table - find I/N  */       /*  How do we build the I/N tables?        for each sigma (that is the t_value) we calculate the 3*sigma= sigma_3         if 3*sigma is less than the minimum, 3*sigma is the minimum         The phi sigma table is the x table,         we wll fill the pfi_of_three_over_sigma table         calculate the I/N value and the same value multiply by M         */          for (loop1 = 0; loop1 &lt; TABLE_SIZE + 1 ; loop1++)          {          i_sigma3 = (loop1 + 1 − (TABLE_SIZE/2) ) * 3 ;          if (i_sigma3 &lt; 0) i_sigma3 = 0 ;           if (i_sigma3 &gt; TABLE_SIZE ) i_sigma3 = TABLE_SIZE ;           x_value2 = pfi_of_one_over_sigma[i_sigma3] ;         pfi_of_three_over_sigma[loop1] = x_value2 ;         x_value1 = pfi_of_one_over_sigma[loop1] ;           in = 2.0 * x_value1 + 2 * x_value2 − x_value1 * x_value1            −2.0 * x_value1 * x_value2 − x_value2 * x_value2 ;         I_over_N[loop1] = in ;           I_N_M[loop1] = in * (double) M ;        }        i_compare = 1.0 ;         /* print the first value to a file and vector */        for (loop1 = 0; loop1 &lt; TABLE_SIZE ; loop1++)         {       /*   from now on, we print only the values that are the closest         for integer I. To do so, we first look for difference that goes           down. When it starts to go up again, we print       */         diff = i_compare − I_N_M[loop1] ;           if (i_compare &lt; I_COMPARE_LIMIT)           {           if (diff &lt; 0.0)           {       /*   In addition to print the results, we will build a new set of tables         where the index is the I of I/N           And we do linear interpolation for the exact location of I         Linear interpolation -         step 1. Find the value of x such that I_N_M is a exactly i_compare          the value is loop1−1 + (I_N_M[loop1−1]/(I_N_M[loop1] −I_N_M[loop1−1]         step 2: for any given function, the exact value of the function for x is fiven by            f(x) = f(loop1−1) + (x − loop1 +1) /(f(loop1) − f(loop1−1)          A small utility function - interpolate_f with parameters x, loop1 and a        pointer              to the table will calculate the interpolated value        */         loop9 = loop1 −1 ;         alfa = I_N_M[loop9] − i_compare ;           beta = I_N_M[loop1] − i_compare ;         xx = −alfa/(beta − alfa) ;           index = (int) i_compare − 1 ;           integer_I_N_M[index] = xx ;           phi_minus_one_over_sigma_for_integer_I_N[index] =       interpolation_f(xx,loop9,pfi_of_one_over_sigma) ;         phi_minus_three_over_sigma_for_integer_I_N[index] =       interpolation_f(xx,loop9,pfi_of_three_over_sigma) ;           one_over_sigma[index] = interpolation_f(xx,loop9,t_vector) ;           // fprintf(fp1, “%d %le %le %le %le \n”,index, xx,       phi_minus_one_over_sigma_for_integer_I_N[index],            //  phi_minus_three_over_sigma_for_integer_I_N[index],       one_over_sigma[index]) ;         //printf(“ %lf %lf \n”,t_vector[loop1], pfi_of_one_over_sigma[loop1]) ;           // fprintf(fp2,“ %d %le %le %le %le %f\n”,loop1,       t_vector[loop1],            // pfi_of_one_over_sigma[loop1],       pfi_of_three_over_sigma[loop1],I_over_N[loop1],I_N_M[loop1]) ;            i_compare = i_compare + ONE ;           }           }         }       /*   Equation (42) look-up table calculation ----&gt;       1.  Remember phi(−x) = 1 − phi(x)       2.  For each i build T(X) and A(X) and B(x)       3.  Build all the corrections C —  and D —         4.  Build the final table as a floating point and integer       */         for (loop1= 0; loop1 &lt; TABLE_SIZE_M_DIVIDE_16; loop1++)        {          real_sigma = 1.0/one_over_sigma[loop1] ;        //A_1 is phi(1/sigma) = 1.0 − phi(−1/sigma) = 1.0 −       phi_minus_one_over_sigma_for_integer_I_N[loop1]        A_1 = 1.0 − phi_minus_one_over_sigma_for_integer_I_N[loop1] ;          // A_minus_1 = phi_minus_one_over_sigma_for_integer_I_N[loop1]        A_minus_1 = phi_minus_one_over_sigma_for_integer_I_N[loop1] ;          //T_1 = 1/sqrt(2*pi) * 1/sigma * exp(−1/2*sigma*sigma)       // T_1 = −one_over_square_root_2_pi *one_over_sigma[loop1] *       //      exp(− one_over_sigma[loop1] * one_over_sigma[loop1] /2.0) ;          h1 = one_over_square_root_2_pi ;           h2 = −one_over_sigma[loop1] ;           h3 = −one_over_sigma[loop1] ;           h4 = h3 * h3 ;           h4 = − h4/2.0 ;           h3 = exp(h4) ;           T_1 = h1 * h2 * h3 ;          //A_minus_3 = phi_minus_three_over_sigma_for_integer_I_N[loop1]        A_minus_3 = phi_minus_three_over_sigma_for_integer_I_N[loop1] ;        //T_minus_3 = 1/sqrt(2*pi) * −3/sigma * exp(−9/2*sigma*sigma)        // T_minus_3 = −one_over_square_root_2_pi *3*one_over_sigma[loop1] *          //   exp(− 9.0*one_over_sigma[loop1] * one_over_sigma[loop1]       /2.0) ;        //  h1 = one_over_square_root_2_pi ;           h2 = 3*one_over_sigma[loop1] ;           //h3 = −one_over_sigma[loop1] ;           h4 = h2 * h2 ;           h4 = −h4/2.0 ;           h3 = exp(h4) ;           T_minus_3 = h1 * h2 * h3 ;        //B_1 and B_minus_1 are the same 1/sqrt(2*pi) * exp(−1/2*sigma*sigma)        B_1 = one_over_square_root_2_pi *            exp(− one_over_sigma[loop1] * one_over_sigma[loop1] /2.0) ;          B_minus_1 = B_1 ;        //B_3 and B_minus_3 are the same 1/sqrt(2*pi) * exp(−9/2*sigma*sigma)        B_3 = one_over_square_root_2_pi *            exp(− 9.0 * one_over_sigma[loop1] * one_over_sigma[loop1]       /2.0) ;          B_minus_3 = B_3 ;        h1 = 1.0/one_over_sigma[loop1] ;          sigma_square = h1 * h1 ;          C_CADE = ( sigma_square ) * ( 2.0 * (1.0 − A_1) +T_1) ;          C_LHGJ = ( sigma_square ) * ( 2.0 * ( A_minus_3) − T_minus_3) ;        C_BK = sigma_square*((A_1 − A_minus_3)*(1.0 −A_1+T_1) +             (1.0 −A_1)*(A_1−A_minus_3−T_1+T_minus_3) ) ;        C_FI = sigma_square*((A_1 − A_minus_3)*(A_minus_3 − T_minus_3) +             (A_minus_3)*(A_1−A_minus_3−T_1+T_minus_3) ) ;          C_TOTAL = C_CADE + C_LHGJ + C_BK + C_FI ;          D_P2 = ( A_minus_1 − A_1)* 4.0 * ( A_minus_1−A_minus_3 + h1       *(B_1 − B_3)) ;          D_P4 = D_P2 ;          D_P3 = 8.0 * (A_minus_3 − A_minus_1) * (A_minus_1−A_minus_3 +       h1 *(B_1 − B_3)) ;          D_TOTAL = D_P2 + D_P3 + D_P4 ;         TOTAL = C_TOTAL + D_TOTAL ;          correction_table[loop1] = TOTAL ;          fprintf(fp1,“ %d %le \n”,loop1, TOTAL) ;          fprintf(fp2,“%ld \n”,(long)(TOTAL * 16777216.0+ 0.5) ) ;         }       }       #define NUMBER_INTERVAL 2000       double integral_function(double t_start, double delta_t)       {         double x2 ;         double delta_x ;         double minus_x_square ;         double exponent_x ;        double sum ;        int loop ;         delta_x = delta_t /( (double) NUMBER_INTERVAL) ;        sum = 0 ;         x2 = t_start+ delta_x/2 ;         for (loop =0; loop &lt; NUMBER_INTERVAL; loop++)         {          minus_x_square = − x2 * x2 /2.0 ;          exponent_x = exp(minus_x_square) ;          sum = sum + exponent_x * delta_x ;          x2 = x2 + delta_x ;         }         return (sum * one_over_square_root_2_pi );       }       double interpolation_f(double xx,int loop9,double *vector)       {         double ff ;         double v1, v2, slop ;         v1 = vector[loop9] ;         v2 = vector[loop9+1] ;         slop = v2 − v1 ;       //  xx = xx − (double) (loop9) ;         ff = v1 + xx * slop ;         //ff = vector[loop9] + (xx ) * (vector[loop9+1] − vector[loop9]) ;         return (ff) ;       // return (v1) ;       }                  
 
         [0442]     While an embodiment of the present invention is shown and described, it is envisioned that those skilled in the art may devise various modifications of the present invention without departing from the spirit and scope of the appended claims.