Abstract:
A preceding system to achieve rates higher than 33.6 kbps in the analog modem to digital modem direction. The preceding system modifies the standard THP algorithm to adapt it for use in PCM modems. The present invention overcomes the above-described difficulties by preceding an upstream signal before transmission. In one aspect of the invention, the present invention provides a precoder for generating a mapped constellation signal from an input signal. In one embodiment, the precoder includes a processor configured to generate the mapped constellation signal from the input signal by mapping input signals in a plurality of distinct ranges onto a basic level. The processor is configured to map the plurality of distinct ranges onto the basic level following different arithmetic rules for at least two of the plurality of distinct ranges.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
   This application is a continuation of file Ser. No. 09/540,475 filed on Mar. 31, 2000 now U.S. Pat. No. 7,292,648, entitled “PRECODING FOR A NON-LINEAR CODEC,” issued on Nov. 6, 2007 by Nuri R. Dagdeviren, and claims the benefit of U.S. Provisional Application No. 60/169,896 filed on Dec. 9, 1999. U.S. Pat. No. 7,292,648 is commonly assigned with the present invention and is incorporated herein by reference as if reproduced herein in its entirety. The present application is also related to U.S. Pat. No. 7,099,403 to Dagdeviren and U.S. Pat. No. 6,798,851 to Dagdeviren. 

   FIELD OF THE INVENTION 
   This invention relates to analog modem technology. Specifically, it proposes a new preceding scheme to achieve higher rates in the analog modem to digital modem direction. 
   BACKGROUND OF THE INVENTION 
     FIG. 5  shows the basic elements of an end-to-end transmission within the Public Switched Telephone Network (hereinafter “PSTN”). The PSTN shown includes first and second Users, first and second Central Offices, and a Switched Digital Network. An Analog Subscriber Loop connects the Users to their respective Central Offices, and the Switched Digital Network connects the Central Offices together. The Analog Subscriber Loops are conventional twisted pairs that transport analog signals from the User Equipment to the associated local Central Office. At the Central Office, the analog signals are converted to 64 kbps DSO digital data streams by a channel unit filter and codec, which together implement a bandlimiting filter followed by a nonlinear encoding rule and subsequent analog to digital conversion. The resulting DSO streams are transported to their respective destination Central Office via the Switched Digital Network. 
   At the Central Office  1 , User&#39;s  1  loop signal is first bandlimited. The bandlimited analog signal is then sampled at a rate of 8 k samples/second, and then converted into an 8-bit digital representation using a nonlinear mapping rule referred to as PCM encoding. This encoding is approximately logarithmic, and its purpose is to permit relatively large dynamic range voice signals to be represented with only 8 bits per sample. 
   Users  1  and  2  may use a conventional modem, as shown in  FIG. 6 , to transmit digital data over the configuration of  FIG. 5 . The conventional modem encodes the user&#39;s digital data into a symbol sequence. The symbol sequence is then represented as an appropriately bandlimited analog signal which can be transmitted over the approximately 3.5 kHz bandwidth available on the end-to-end connection. The exemplary modem of  FIG. 6  includes a Digital to Analog converter (i.e. D/A),) an Analog to Digital converter (i.e. A/D), and a hybrid. The A/D and the D/A perform PCM encoding. The non-linearity associated with the PCM coding is incorporated in the circuitry that converts the analog signals to digital signals, and vice-versa. That is, the analog voltage level of the received signal is mapped to the nearest PCM quantization level, and vice versa, so that the PCM quantization levels serve as the channel symbol alphabet. 
   PCM baseband modulation in the upstream direction, i.e. from User I to the Central Office, presents special equalization problems. For instance, one potential application for PCM baseband modulation in the upstream direction is in conjunction with “56 k” modems. However, “56 k” modems have a zero in the frequency band of interest. The zero at zero frequency comes from the transformer coupling of the analog subscriber loop to the central office equipment. Therefore, telephone lines do not pass DC signals. Low frequencies near DC are also attenuated significantly as to rule out linear equalization of this channel. Moreover, it is not possible to avoid the zero at DC for 56 k modems using pass-band modulation as in the case of earlier V.34 modems because the central site modem is limited to using the sampling rate and quantization levels of the PCM codec at the central office. 
   One possible way to equalize this channel is to use a linear equalizer to reduce the channel response to a simpler “partial” response that still possesses the zero in the channel but can be dealt with using a non-linear technique such as maximum likelihood sequence (MLSE) decoding or decision feedback equalization (DFE). This however is only possible in the direction of digital modem to analog modem, also referred as the downstream direction. The reason this approach or any linear equalization scheme does not work in the upstream direction is that only PCM codec levels themselves can pass through the PCM codec unscathed. Any filtered version of a sequence of PCM levels will be a linear combination of these levels and in general not be a PCM level itself. When such intermediate levels are quantized by the PCM codec, quantization noise is introduced into the signal erasing any advantage over V.34 techniques. 
   Accordingly, there exists a need for a system capable of equalizing transmissions from an analog modem. 
   SUMMARY OF THE INVENTION 
   The present invention overcomes the above-described difficulties by preceding an upstream signal before transmission. In one aspect of the invention, the present invention provides a precoder for generating a mapped constellation signal from an input signal. In one embodiment, the precoder includes a processor configured to generate the mapped constellation signal from the input signal by mapping input signals in a plurality of distinct ranges onto a basic level. The processor is configured to map the plurality of distinct ranges onto the basic level following different arithmetic rules for at least two of the plurality of distinct ranges. 
   In another aspect, the invention provides a method of preceding an input signal to generate a mapped constellation signal. The method includes mapping an input signal contained in a first distinct range onto a basic level according to a first arithmetic rule and mapping an input signal contained in a second distinct range onto the basic level according to a second arithmetic rule that differs from the first arithmetic rule. 
   In yet another aspect, the present invention provides a computer-readable medium having stored thereon a plurality of instructions, the plurality of instructions including instructions that when executed by a processor cause the processor to implement a method of preceding an input signal to generate a mapped constellation signal. The method includes mapping an input signal contained in a first distinct range onto a basic level according to a first arithmetic rule and mapping an input signal contained in a second distinct range onto the basic level according to a second arithmetic rule that differs from the first arithmetic rule. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The features and advantages of the invention will be apparent from the following description, as illustrated in the accompanying Figures in which like reference characters refer to the same elements throughout the different Figures: 
       FIG. 1  is a block diagram of a precoder in accordance the present invention; 
       FIG. 2  is a graphical representation of an exemplary table utilized by the precoder of  FIG. 1 ; 
       FIG. 3  is a block diagram illustrating the precoder of  FIG. 1  in a modem; 
       FIG. 4  is a flow chart illustrating the preceding method in accordance with the present invention; 
       FIG. 5  is a block diagram of a conventional Public Switched Telephone Network; and 
       FIG. 6  is a block diagram of a conventional modem. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   The inventor has recognized that one way to overcome the difficulties noted in the background of the invention is to use preceding in the transmitter, in place of MLSE or DFE in the receiver. In this way PCM levels can be used as the symbol constellation. The combination of the precoder and a linear equalizer will eliminate the inter-symbol interference (ISI) introduced by the channel. In this manner signals arriving at the PCM codec will be free of ISI and no quantization noise will be introduced. 
   The simplest manner of implementing preceding is to implement a feedback filter that equalizes the partial response. This however is not practical in the case where the channel and hence the partial response possesses a zero in the band of interest. The reason is that since the feedback filter equalizes the partial response, it has a very large gain at the frequency where the partial response has a zero. Components in the transmitted signal that correspond to this frequency will be greatly amplified leading to an unstable feedback loop. 
   Tomlinson Harashma Precoding (“THP”) has emerged as an attractive solution for equalization in the presence of severe channel attenuation in the frequency band of interest; See M. Tomlinson “New Automatic Equalizer Employing Modulo Arithmetic” Electronics Letters Vol. 7, pp. 138-139, March 1971, the contents of which are incorporated herein by reference; and See H. Harashima and H. Miyakawa “Matched-Transmission Technique for Channels with Intersymbol Interference” IEEE Trans. Commun. Vol. COM-20, pp. 774-80, August 1972, the contents of which are incorporated herein by reference. THP is equivalent to Decision Feedback Equalization (DFE) in the receiver without the potential problem of error propagation. 
   The clever solution to the problem of very large gain at frequencies where the partial response has a zero is provided in the THP as follows. Whenever the output of the feedback loop passes a present threshold, the transmitted signal is subjected to a modulo operation which brings it back within range. This removes the instability in the feedback loop of the transmitter. The receiver must also account for the modulo operation in the transmitter. The receiver, since the modulo operation can be expressed as the addition of a constant, will compensate by subtracting the constant from the received signal. The receiver knows when to perform this compensation because whenever the transmitter subtracts the constant to bring the transmitted value to within range, the received value in the receiver will be out of range. When the receiver compensates the received signal by adding the constant, the received signal is brought back within range. 
   However, the standard THP scheme is not effective for PCM encoding in the upstream direction because the receiver can not implement the modulo compensation without introducing quantization noise. If the transmitter implements the standard THP modulo operation, then the received signal will arrive at the PCM codec with a value that corresponds to a PCM value shifted by a constant. In general it is not possible to find a set of PCM values and a constant such that each PCM value, when shifted by a constant is another PCM value. Thus THP scheme as previously defined is not effective for PCM modems. 
   This invention modifies the standard THP algorithm to adapt it for use in PCM modems. Instead of an arithmetic modulo operation that is implemented in the transmitter, the invention defines a Discrete Modulo Operation as a function of the constellation of levels chosen. The Discrete Modulo Operation is defined as a mapping from a constellation level outside the basic constellation of levels for the precoder to a constellation level inside the basic constellation of levels for the precoder. The mapping is based upon an index associated with levels in the constellation of levels. This operation performs the function of limiting the amplitude of the transmitted signals, hence removing the instability of the feedback loop, while ensuring that received signals at the PCM codec are always within the PCM level set free of quantization noise. Similarly, a discrete modulo operation is defined for the receiver to map received PCM values correctly into the symbol constellation. 
   Accordingly,  FIG. 1  illustrates a block diagram of a precoder  10  in accordance with one aspect of the invention. The precoder  10  comprises a feedback filter  12  and a discrete modulo adder  14 . The feedback filter  12  generates a feedback signal  16  as a function of a mapped constellation signal  18 , and the discrete modulo adder  14  generates the mapped constellation signal  18  as a function of the feedback signal  16  and as a function of an input signal  20  to the precoder  10 . The discrete modulo adder  14  utilizes an index  31  (of  FIG. 2 ) to the constellation of levels chosen for the precoder  10 , such that the amplitude of the mapped constellation signal  18  is limited. 
   The discrete modulo adder  14  can also include an adder  22  and a mapper  24 . The adder  22  sums together the feedback signal  16  and the input signal  20  to generate a partial result  26 . The mapper  24  generates the mapped constellation signal  18  by mapping a partial result  26  outside a basic constellation of levels onto the basic constellation of levels. 
   As further illustrated in  FIG. 1 , the mapper  24  can include a table  30  that identifies both the levels inside a basic constellation of levels and those levels outside a basic constellation of levels. The table  30  further identifies the mapping from levels outside the basic constellation to the levels inside the basic constellation as a function of the index  31  associated with the levels in the table  30 . Preferably, the levels outside the basic constellation are mapped onto only one level inside the basic constellation. 
     FIG. 2  illustrates a graphical representation of an exemplary table  30  utilized by the precoder  10  of  FIG. 1 . The exemplary table  30  has a total of 12 levels, each level being identified by a horizontal line. The table  30  also includes two columns, one labeled Amplitude and another labeled Index  31 . The Amplitude column has 12 entries, one for each level. The Index column also has 12 entries, one for each column. Thus, as shown in  FIG. 2 , amplitude  9  and index  6  are both associated with the first level; amplitude  7  and index  5  are both associated with the second level; amplitude  5  and index  4  are both associated with the third level; . . . and amplitude − 9  and index − 6  are both associated with the twelfth level. 
   The levels in the exemplary table  30  can also be subdivided into three separate constellations: a basic constellation  42 , a positive constellation  44 , and a negative constellation  46 . The basic constellation  42  extends into both the positive and negative directions from an amplitude level of zero. Typically, the basic constellation extends an equal distance from amplitude zero into both the positive and negative directions. The positive constellation  44  extends from the maximum level of the basic constellation upwards, and the negative constellation  46  extends from the minimum level of the basic constellation downwards. For example, as shown in  FIG. 2 , the basic constellation includes the amplitudes {2,1,−1,−2}, or alternatively the basic constellation includes the indexes{2,1,−1,−2}. The positive constellation includes the amplitudes {3,5,7,9} or the indexes {3,4,5,6}. The negative constellation includes the amplitudes {−3,−5,−7,−9} or the indexes {−3,−4,−5,−6}. In a preferred embodiment of the invention, the basic constellation includes a set of indexes extending from −k to k; the positive constellation includes a set of indexes extending from k+1 to 3 k; and the negative constellation includes a set of indexes extending from −k−1 to −3 k. 
   The amplitude entries show that the separation between levels in the table may vary, as is found in PCM codec levels. For instance, the separation between levels in the basic constellation  42  of  FIG. 2  equals one amplitude, while the separation between levels in the positive constellation  44  equals two amplitudes. In comparison, in a preferred embodiment of the invention, the separation between indexes is a constant, regardless of the constellation. As shown in  FIG. 2 , the index separation between successive levels always equals one. Accordingly, although the difference in amplitude between the successive levels shown in  FIG. 2  may vary, the difference in index between successive levels is a constant. 
   The exemplary table  30  of  FIG. 2  also uses a first set of arrows  48  to show a mapping from levels in the positive constellation  44  to levels in the basic constellation  42 . A second set of arrows  50  shows a mapping from levels in the negative constellation  46  to levels in the basic constellation  42 . The first set of arrows  48  identifies that the levels associated with indexes {3,4,5,6} in the positive constellation are mapped to the levels associated with indexes {−2,−1,1,2} in the basic constellation, respectively. The second set of arrows  50  identifies that the levels associated with indexes {−3,−4,−5,−6} in the negative constellation are mapped to the levels associated with indexes {2,1,−1,−2} in the basic constellation, respectively. Thus, there is a one-to-one mapping between levels in the positive constellation  44  and the basic constellation  42 , and there is another one-to-one mapping between levels in the negative constellation  46  and the basic constellation  42 . 
   In a preferred embodiment of the invention, each of the levels in the positive constellation are mapped onto levels in the basic constellation based on the indexing system chosen. This form of mapping between the basic constellation and those levels outside the basic constellation, based upon the indexes in the constellation, will be referred to as a discrete modulo operation. Preferably, the discrete modulo operation is defied as a shift operation between the indexes in the basic constellation and the indexes outside the basic constellation (i.e. the positive constellation  44  and the negative constellation  46 ). 
   An exemplary shift operation is as follows:
         if the indexes in the basic constellation are labeled, basic_const, where basic_const goes from −k to k, and   if the indexes in the positive constellation are labeled positive_const, where positive_const goes from k+1 to m,   then the levels in the positive constellation  44  are mapped onto levels in the basic constellation  42  according to the equations:   Index positive_const→positive_const−(2+k); while   positive_const&gt;m−k; and   Index positive_const→positive_const−(2*k); while   positive_const&lt;=m−k;    Wherein→identifies the mapping function.       

   For example, the basic constellation might include the indexes {−2,−1,1,2} and the positive constellation might includes the indexes (3,4,5,6}. Given this set of constellations, the mapping is calculated as follows:
         index  6  maps to 6−(2*k)=6−4=2;   index  5  maps to 5−(2*k)=5−4=1;   index  4  maps to 4−(2*k)−1=4−4−1; and   index  3  maps to 3−(2*k)−1=3−4−1=2.       

   In an analogous fashion, the indexes in the negative constellation can be mapped onto levels in the basic constellation:
         if the indexes in the basic constellation are labeled basic_const, where basic_const goes from −k to k, and   if the indexes in the negative constellation are labeled negative_const, where negative_const goes from −k−1 to −m,   then the levels in the negative constellation  46  are mapped onto levels in the basic constellation  42  according to the equations:   Index negative_const→negative_const+(2+k); while negative_const&lt;−(m−k); and   Index negative_const→negative_const+(2*k)+1; while negative_const&gt;=−(m−k);    Wherein→identifies the mapping function.       

   This discrete modulo operation performs the function of limiting the amplitude of signals by mapping signals in the table outside the basic constellation onto signals inside the basic constellation. This mapping function allows the precoder  10  to remove the potential instability caused by the feedback filter  12 . This completes the description of the basic elements of table  30 , as shown in  FIGS. 1 and 2 . 
   With further reference to  FIG. 1 , the mapper  24  can also include a comparator  32  and an output block  34 . The comparator  32  compares the partial result  26  with levels in the table  30 . For instance, the comparator  32  can identify the level in table  30  closest to the partial result  26 . 
   The output block  34  generates the mapped constellation signal  18 . The mapped constellation signal  18  is selected from the levels in the basic constellation even though the partial result may be a level outside the basic constellation. In particular, the mapped constellation signal  18  output by the block  34  is equal to the identified level closest to the partial result  26  if the identified level is inside the basic constellation. Alternately, if the identified level is outside the basic constellation, then the mapped constellation signal  18  is set equal to the sum of the partial result and a mapping distance signal. The mapping distance signal equals the distance between the index basic_const, associated with the basic constellation level of the input signal, and the index positive_const, associated with a level outside the basic constellation. Further details on determining the mapping distance are discussed under the description of  FIG. 4 . 
     FIG. 1  also illustrates details of the feedback filter  12 . The feedback filter can include one or more delay elements D 1 , D 2 , . . . , DN, and the feedback filter can include one or more weighting elements a 1 , a 2 , . . . , aN. The feedback filter  12  thus provides feedback connections whose weighting coefficients are a 1 , a 2 , . . . , aN. The feedback filter  12  can be used to model an impulse response of a communication channel over which the input signal  20  is transmitted. 
     FIG. 3  is a block diagram illustrating the precoder  10  of  FIG. 1  in a modem  70 . The modem  70  includes the precoder  10 , a digital to analog converter  60  (“D/A”), a hybrid  62 , and an analog to digital converter  64  (“A/D”). An analog subscriber loop  68  operably couples the modem  70  to a PSTN. 
   The hybrid  62  operably couples the modem  70  to the analog local loop  68 . A hybrid can generally be described as a passive device used for converting a dual analog signal that is carried on one pair of conductors (i.e. the analog local loop) to separate analog signals that are carried on two pairs of conductors. Those skilled in the art are familiar with the use and operation of hybrid devices and, thus, a detailed description thereof is not necessary to enable one of skill in the art to make and practice the present invention. 
   The D/A converts digital signals to analog signals for transmission over the analog local loop, and the A/D converts analog signal received from the analog local loop to digital signals. The A/D converter and the D/A converter can also be described as capable of implementing a CODEC (coder/decoder) function. In one embodiment of the invention, the A/D implements a mu-law CODEC. Those skilled in the art are familiar with the non-linear mu-law and A-law signal compression algorithms. The mu-law algorithm includes 255 discrete signal conversion values; A-law uses 256 values. The broad principles of the invention are not, however, limited to a specific quantization scheme. 
   For instance, the A/D converter can utilize 255 non-uniformly spaced quantization levels, which are closer together for small analog signal values and spread further apart for large signal values, to convert an analog signal received from the analog local loop to one of 255 unique “symbols” or “levels”. A DSP in the modem then uses a symbol table to convert the received symbol back to the data transmitted by the Central Office over the analog subscriber loop. 
     FIG. 4  is a flow chart illustrating the method of preceding an input signal to generate a mapped constellation signal, in accordance with the present invention. The method includes generating a feedback signal from the mapped constellation signal at step  82 , and performing a discrete modulo operation on the feedback signal and the input signal at steps  86 - 100 . The discrete modulo operation is based upon an index to the constellation of levels chosen for the precoder, such that the amplitude of the mapped constellation signal is limited. 
     FIG. 4 , also shows that the discrete modulo operation can include the steps of adding together the input signal and the feedback signal to generate a partial result at step  84 , determining whether the generated partial result is contained within a basic constellation of levels at step  86 , and generating the mapped constellation signal by mapping a partial result outside the basic constellation of levels onto a level inside the basic constellation of levels as a function of the index to the levels, at steps  89 - 100 . 
   In particular, at step  86  the method determines whether the partial result calculated in step  84  is in the basic constellation  42 . This can be implemented by comparing the partial result to entries in the table  30 . If the partial result is in the basic constellation  42 , then processing proceeds to step  88 , otherwise processing proceeds to step  89 . 
   At step  88 , the mapped constellation signal is set equal to the partial result  88 . For instance, if the partial result is in the basic constellation, then feedback has not caused the partial result to be out of range and accordingly no mapping is required. After step  88 , processing proceeds to step  98 . 
   At step  89 , the method determines whether the partial result is less than the minimum of the basic constellation or whether the partial result is greater than the maximum of the basic constellation. If the partial result is less than the minimum of the basic constellation, then the method branches to step  90 . If the partial result is greater than the maximum of the basic constellation, then the method branches to step  94 . 
   At step  90 , the method determines the mapping distance p j . The mapping distance p j =the distance between an index basic_const and the index positive_const. The index basic_const is the index associated with the basic constellation level of the input signal, and the index positive_const is an index associated with a level found in the positive constellation  44  of  FIG. 2 . In particular, the index positive_const is the index in the positive constellation  44  that maps onto the basic constellation level of the input signal. The index positive_const can be obtained from the table  30 . After step  90 , processing proceeds to step  92 . 
   At step  92 , the mapped constellation signal is set equal to the sum of the mapping distance p j  and the partial result. After step  92 , processing proceeds to step  98 . 
   At step  94 , which is reached from step  89 , the method determines the mapping distance n j . The mapping distance n j =the distance between an index basic_const and an index negative_const. The index basic_const is the index associated with the basic constellation level of the input signal, and the index negative_const is an index associated with a level found in the negative constellation  46  of  FIG. 2 . In particular, the index negative_const is the index in the negative constellation  46  that maps onto the base constellation level of the input signal. The index negative_const can be obtained from the table  30 . After step  94 , processing proceeds to step  96 . 
   At step  96 , the mapped constellation signal is set equal to the sum of the mapping distance n j  and the partial result. After step  96 , processing proceeds to step  98 . 
   At step  98 , the precoder  10  outputs the mapped constellation signal  18 . At step  100 , the method ends. 
   Exemplary Operation of the Precoding Method: 
   
       
       A) Let&#39;s say the desired sequence to be transmitted
 
2,−2,−2,2,1,−1,−2,2
 
       B) Let&#39;s also assume that our feedback filter coefficients are 1 and −1 so what is to be transmitted is the current input signal minus the previously transmitted sample, then 
       C) With the first input signal=2, then the first output of the adder  22  (i.e. the partial result signal  26 ) is:
 
2−0=2
 
which is in range so it is transmitted as 2.
 
       D) With the second input signal=−2, then the second output of the adder  22  (i.e. the partial result signal  26 ) is:
 
−2−2=−4
 
     
  
   This value is out of range of the basic constellation  42 , so it must undergo the discrete modulo operation outlined in steps  89 - 96  of  FIG. 4 . Using the table shown  FIG. 2 , we can identify that when the input signal=−2 the corresponding index in the positive constellation  44  is 3. Thus: 
   p j =difference between the index of the input signal and the index associated with a level in the positive constellation that maps onto the basic constellation level of the input signal, and accordingly
 
 p   j =absolute value of [(−2)−(3)]=5.
 
   Then, in accordance with step  92 , the mapped constellation signal=p j +partial result=5+(−4)=1. 
   So the second transmitted value is 1.
     E) With the third input signal=−2, then the first output of the adder  22  (i.e. the partial result signal  26 ) is:
 
−2−1=−3.
   

   This value is out of range of the basic constellation  42 , so it must undergo the discrete modulo operation outlined in steps  89 - 96  of  FIG. 4 . Using the table shown in  FIG. 2 , we can identify that when the output signal=−2 the corresponding index in the positive constellation  44  is 3. Thus: 
   p j =difference between the index of the input signal and the index associated with a level in the positive constellation that maps onto the basic constellation level of the input signal, and accordingly
 
 p   j =absolute value of [(−2)−(3)]=5.
 
   Then, accordance with step  92 , the mapped constellation signal=p j +partial result=5+(−3)=2. 
   So the third transmitted value is 2.
     F) The fourth input signal is 2, and the output of the adder  22  (i.e. the partial result signal  26 ) is:
 
2−2=0
 
which is in range so it is transmitted as 0.
   G) The fifth input signal is 1, and the output of the adder  22  (i.e. the partial result signal  26 ) is:
 
1−0=1
 
which is in range so it is transmitted as 1.
   H) The sixth input signal is −1, and the output of the adder  22  is:
 
−1−1=−2
 
which is in range so it is transmitted as −2.
   

   Whenever a receiver in a digital modem receives a level in the positive or negative constellations, it maps the level to the corresponding level in the basic constellation as identified in the table  30 . This mapping in the receiver can be formulated as a shift operation that is dependent on the level being transmitted. If the difference between the level in the basic constellation and the corresponding level in the negative constellation is n; then the mapping in the receiver from the negative constellation can be thought of as an addition of offset n j  to the received value. Thus, if we completed the above example by showing the response in the receiver, we get the following: 
   
     
       
             
             
             
           
         
             
                 
                 
             
           
           
             
                 
               Transmitted Symbol Sequence 
               . . . 2, −2, −2, 2, 1, −1 . . . 
             
             
                 
               Partial Result 
               . . . 2, −4, −3, 0, 1, 2 . . . 
             
             
                 
               What is xmitted 
               . . . 2, 1, 2, 0, 1, −2 
             
             
                 
               Output of comm. Channel 
               . . . 2, 3, 3, 2, 1, −1 
             
             
                 
               (i.e. what is received) 
             
             
                 
               After Receiver mapping to 
               . . . 2, −2, −2, 2, 1, −1 
             
             
                 
               basic constellation 
             
             
                 
                 
             
           
        
       
     
   
   Having thus described a few particular embodiments of the invention, various alterations, modifications, and improvements will readily occur to those skilled in the art. Such alterations, modifications and improvements as are made obvious by this disclosure are intended to be part of this description though not expressly stated herein, and are intended to be within the spirit and scope of the invention. Accordingly, the foregoing description is by way of example only, and is not limiting.