Abstract:
A communications system ( 10 ) which utilizes an 128-ary QAM signal constellation suitable for non-linear applications. The communications system includes a modulator ( 18 ) for utilizing the 128-ary constellation to implement the modulation. The 128-ary constellation is a circular constellation which provides a simplified amplitude predistortion by utilizing the subject 128-ary constellations, enabling more efficient communications can then be achieved through a peak-power-limited non-linear channel ( 16 ). Such non-linear channels ( 16 ) are more power efficient at creating RF energy from DC energy.

Description:
CROSS REFERENCE TO AWAITED APPLICATIONS  
       [0001]    This application is a related application to U.S. application Ser. No. 09/883,651, filed Jun. 18, 2001, the disclosure of which is incorporated by reference. 
     
    
     
       FIELD OF THE INVENTION  
         [0002]    The present invention relates generally to communications systems utilizing 128-ary modulation formats and, more particularly, to an apparatus and method for efficiently communicating through a peak-power-limited, non-linear channel.  
         BACKGROUND OF THE INVENTION  
         [0003]    In high data rate communications systems, such as selected satellite communications systems, data transmission typically employs high power amplifiers such as traveling wave tube amplifiers (TWTAs) or solid state power amplifiers (SSPAs). Such high speed communications systems typically require a relatively high output power so that the signal being transmitted can travel greater distances before being significantly attenuated. However, such power is limited by several considerations, including the limited energy generation and storage in the satellite vehicle. In these types of communications systems, low frequency digital baseband signals comprising the stream of digital data bits are transmitted after being modulated onto a high frequency carrier wave.  
           [0004]    Various modulation schemes exist and distinguish between the digital bits. Examples of digital modulation schemes include amplitude-shift keying (ASK), binary phase-shift keying (BPSK), quadrature-phase shift keying (QPSK), and quadrature amplitude modulation (QAM). Further, the digital baseband signal may be multi-level (M-ary) signals requiring multi level modulation methods.  
           [0005]    Quadrature modulation schemes provide both amplitude and phase modulation of the carrier because both complex and imaginary representations of the signals are used. In quadrature amplitude modulation schemes, such as QAM, each bit is converted through a bit symbol representing a complex value having an in-phase, real component and a quadrature-phase, imaginary component. Each bit is represented on a graph having an imaginary axis and a real axis to form a constellation pattern representing a group of signals positioned within a circle around the origin of the axes. The distance from the origin represents the amount of power being transmitted. For example, four bits transmitted at a particular time may be represented as 16 symbols. Each symbol of the pattern identifies a complex voltage value having an in-phase component and a quadrature-phase component and represents the complex voltage value for a particular symbol period which is the time during which each symbol is transmitted. The symbols of the constellation pattern are geometrically spread so that they are more equally spaced apart to more readily distinguish the symbols and reduce bit errors. The constellation patterns are processed through the transmitter without being distorted so that the bits are readily distinguishable from each other at the receiver end.  
           [0006]    High power amplifiers are desirable in high speed communications applications because they provide high gain over wide bandwidths. However, the input signal to a high power amplifier must be controlled because the high power amplifier exhibits non-linear transfer characteristics. At lower input powers, the output-input power relationship of the high power amplifier is approximately linear. At peak power output, the high power amplifier saturates, and further increases the input power beyond the saturation point actually decrease the output power of the amplifier.  
           [0007]    Non-linear amplifiers are inherently more power efficient at creating radio frequency (RF) energy from direct current (DC) energy but create distortions in the process. Such distortions significantly complicate utilizing traditional signal constellations, such as M-ary QAM. Non-linear channels cause the constellation to rotate and expand non-uniformly. Various methods are available to compensate for this expansion and rotation, but such methods are complex and may be difficult to implement.  
           [0008]    The non-linearity of the high power amplifier affects the position of the symbols in the constellation pattern by moving them away from the origin. It is known to provide amplifier predistortion techniques in the amplifier when the transmitter is being operated in its non-linear range near peak output power.  
           [0009]    Thus, it is desirable to provide an efficient communications system utilizing a peak-power-limited, non-linear channel which compensates for distortion.  
         SUMMARY OF THE INVENTION  
         [0010]    A communications system comprising a modulator for modulating a digital data stream onto a carrier wave to generate a modulated signal, the modulator converting data in the data stream into symbols for transmission by the communications system, the symbol being encoded into one of M possible symbols of an M-ary constellation, wherein each symbol is defined by one of a plurality of phases and one of a plurality of magnitudes and an amplifier for amplifying the modulated signal prior to transmission to generate an amplified signal, the amplifier having a non-linear characteristic that generates a non-linear distortion in the modulated signal, wherein the M-ary constellation is a 128 point constellation having varying magnitudes with a varying number of points located on each magnitude.  
           [0011]    For a more complete understanding of the invention, its objects and advantages, reference should be made to the following specification and to the accompanying drawings.  
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0012]    The drawings, which form an integral part of the specification, are to be read in conjunction therewith, and like reference numerals are employed to designate identical components in the various views:  
         [0013]    [0013]FIG. 1 is a schematic block diagram of a communications system arranged in accordance with the principles of the present invention;  
         [0014]    [0014]FIG. 2 is a constellation diagram for a first 128-ary modulation communications system;  
         [0015]    [0015]FIG. 3 is a constellation diagram for a second 128-ary, modulation communications system;  
         [0016]    [0016]FIG. 4 is a constellation diagram for a third 128-ary, modulation communications system;  
         [0017]    [0017]FIG. 5 is a constellation diagram for a fourth 128-ary, modulation communications system; and  
         [0018]    [0018]FIG. 6 is a constellation diagram for a fifth 128-ary, modulation communications system. 
     
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT  
       [0019]    [0019]FIG. 1 depicts a block diagram of communications system  10  for exchanging modulated data between a transmitter  12  and a receiver  14  via a communications link  16 . Communications link  16  may be an air link for satellite communications or hard-wired interconnection, such as an electrical connection or fiber optic connection. Transmitter  12  includes a modulator  18 . Modulator  18  receives a data stream at a baseband frequency and modulates the data stream utilizing a quadrature amplitude modulation (QAM) format. In particular, modulator  18  modulates the data utilizing a M-ary QAM modulation format, such as a 128-ary modulation communications system as will be described herein.  
         [0020]    Typically modulator  18  modulates data bits of the data stream onto an analog carrier wave using mixer  22 . During modulation, modulator  18  identifies for each bit pattern a symbol that includes an in-phase and quadrature-phase component, and maps the symbols into a 128-ary constellation pattern, as will be described in greater detail herein. Modulator  18  may be any quadrature amplitude modulator suitable for implementing the 128-ary constellations as described herein.  
         [0021]    Modulator  18  outputs a radio frequency (RF) signal at a baseband frequency. Typically for satellite communications, the RF signal is up-converted to a high frequency for transmission. A mixer  22  up-converts the baseband frequency with a high frequency signal, such as cos(ω o t). Mixer  22  up-converts the in-phase and quadrature-phase representation of the complex voltage from modulator  18  to a single high frequency RF signal. The up-converted RF signal is then applied to amplifier  24  to significantly increase the signal gain for transmission. Operation of the mixing step and amplification step for a transmitter of this type is well understood by those skilled in the art. The up-converted, amplified signal from amplifier  24  is applied to RF filter  26  for subsequent RF filtering, such as may be required by Federal Communications Commission (FCC) requirements. The filtered signal is output to an antenna  30  for transmission to receiver  14 .  
         [0022]    In the configuration of FIG. 1, amplifier  24  introduces a distortion into the signal output by modulator  18 . The output for amplifier  24 , which is applied to RF filter  26  has an inherent distortion. As will be described in greater detail herein with respect to FIGS. 7 and 8, modulator  18  operates so as to introduce a predistortion into the signal output by modulator  18  and applied to mixer  22 . Amplifier  24  thus adjusts the predistorted signal to output a distortion compensated signal input to RF filter  26 .  
         [0023]    Antenna  30  receives the filtered signal and outputs over communications link  16  a communications signal which is received by antenna  32  of transmitter  12 . Antenna  32  is connected to an amplifier  34 , which is preferably a low-noise, linear amplifier. Note that although communication system  10  is shown as having a wireless communications link  16 , communications link  16  may be a hard-wired connection, as described above. In such a situation, antennas  30  and  32  are unnecessary.  
         [0024]    The signal received by antenna  32  at receiver  14  is input to a filter  36 . Filter  36  provides initial filtering of the received signal to filter channel noise and the like. Typically, filter  36  is closely matched to the transmitted signal frequency. The output of filter  36  is applied to a mixer  38  to down-convert the RF signal to an intermediate frequency signal by mixing the RF signal with a high frequency cos(ω o t) signal. The down-converted signal from mixer  38  includes baseband in-phase and quadrature-phase components. The down-converted signal is applied to low-pass filter  40  to provide filtering at baseband frequencies. Thus, in receiver  14 , filter  36  acts as a course filter.  
         [0025]    The filtered baseband signal from low-pass filter  40  is applied to a demodulator  42 . Demodulator  42  demodulates the received signal in accordance with the M-ary QAM format implemented in modulator  18 . Demodulator  42  thus outputs the data initially modulated by modulator  18 .  
         [0026]    In a particular feature of the subject invention, FIG. 2 depicts a 128-ary QAM constellation arranged on a Cartesian coordinate system defined by an in-phase axis  46  and quadrature-phase axis  48 . The 128-ary constellation of FIG. 2 includes 5 amplitude levels: a first amplitude level  50 , a second amplitude level  52 , a third amplitude level  54 , a fourth amplitude level  56 , and a fifth amplitude level  58 . First amplitude level  50  has an amplitude greater than second amplitude level  52 ; second amplitude level  52  has an amplitude greater than third amplitude level  54 ; third amplitude level  54  has an amplitude greater than fourth amplitude level  56 ; and fourth amplitude level  56  has an amplitude greater than fifth amplitude level  58 . The amplitude levels  50 ,  52 ,  54 ,  56 ,  58  define concentric circles. Each amplitude level represents differing power levels for driving amplifier  24  of FIG. 1. First amplitude level  50  represents the peak power of amplifier  24 , and the remaining amplitude levels  52 ,  54 ,  56 ,  58  represent a power level less than the peak power of amplifier  24 . First amplitude level  50  includes first amplitude symbols  60 , second amplitude level  52  includes second amplitude symbols  62 , third amplitude level  54  includes third amplitude symbols  64 , fourth amplitude level  56  includes fourth amplitude symbols  66 , and fifth amplitude level  58  includes fifth amplitude symbols  68 .  
         [0027]    In the 128-ary constellation of FIG. 2, first amplitude level  50  includes 48 first amplitude symbols  60 , second amplitude level  52  includes 32 second amplitude symbols  62 , third amplitude level  54  includes 24 third amplitude symbols  64 , fourth amplitude level  56  includes 16 fourth amplitude symbols  66 , and fifth amplitude level  58  includes 8 fifth amplitude symbols  68 . First amplitude level  50  has a unit radius of 1, second amplitude level  52  has a radius of 0.87, third amplitude level  54  has a radius of 0.71, fourth amplitude level  56  has a radius of 0.54, and fifth amplitude level  58  has a radius of 0.33. Upper amplitude symbols  60  are separated along first amplitude level  50  by 7.5° with one first amplitude symbols  54  located at Cartesian coordinates x=1 and y=0, (1,0). Second, amplitude symbols  62  are arranged along second amplitude level  52  and are separated by 11.25°, with one second amplitude symbol  62  located at Cartesian coordinate x=0.875 and y=0, (0.875, 0). Third amplitude symbols  64  are arranged along third amplitude level  54  and are separated by 15°, with one third amplitude symbol  64  located at Cartesian coordinate x=0.711 and y=0 (0.711, 0). Fourth amplitude symbols  64  are arranged along fourth amplitude level  54  and are separated by 22.5°, with one fourth amplitude symbol  64  located at Cartesian coordinate x=0.544 and y=0 (0.544, 0). Fifth amplitude symbols  68  are arranged along fifth amplitude level  58  and are separated by 45°, with one fifth amplitude symbol  68  located at Cartesian coordinate x=0.332 and y=0 (0.332, 0). The 128-ary constellation enables modulation of an 7 bit word or symbol. To implement a practical 128-ary system requires mapping of a large number of binary bits (M) to a number (M/7) of 128-ary symbols. The minimum distance between any pair of signal points is 0.125.  
         [0028]    The arrangement of symbols on each amplitude level is particularly selected to maximize the number of points in which amplifier  24  can operate at saturation. In particular, by placing the maximum number of points on first amplitude level  50 , amplifier  24  operates in saturation mode for transmission of the maximum number of symbols. The symbols placed on the other amplitude levels represent operation of amplifier  24  in a backed-off mode. However, due to signal-to-noise-ratio (SNR) considerations, not all points can be placed on first amplitude level  50 . Arranging and placing symbols on each of first amplitude level  50  and other amplitude levels  52 ,  54 ,  56 ,  58  preferably maximizes the number of symbols for which amplifier  24  operates in saturation mode while pursuing good performance in the presence of noise.  
         [0029]    [0029]FIG. 3 depicts a constellation similar to FIG. 2, but shows six level 128-ary constellation for use by modulator  18  of FIG. 1. The 128-ary constellation of FIG. 3 also enables modulation of up to a 7 bit word or symbol. The 128-ary constellation of FIG. 3 includes six amplitude levels: a first amplitude level  70 , a second amplitude level  72 , a third amplitude level  74 , a fourth amplitude level  76 , a fifth amplitude level  78 , and a sixth amplitude level  80 . First amplitude level  70  has an amplitude greater than second amplitude level  72 ; second amplitude level  72  has an amplitude greater than third amplitude level  74 ; third amplitude level  74  has an amplitude greater than fourth amplitude level  76 ; fourth amplitude level  76  has an amplitude greater than fifth amplitude level  78 ; and fifth amplitude level  78  has an amplitude greater than sixth amplitude level  80 . The amplitude levels  70 ,  72 ,  74 ,  76 ,  78 ,  80  defines six concentric circles. First amplitude level  70  includes first amplitude symbols  82 , second amplitude level  72  includes second amplitude symbols  84 , third amplitude level  74  includes third amplitude symbols  86 , fourth amplitude level  76  includes fourth amplitude symbols  88 , fifth amplitude level  78  includes fifth amplitude symbols  90 , and sixth amplitude level  80  includes sixth amplitude symbols  92 . First amplitude level  70  has a unit radius of 1; second amplitude level  72  has a radius of 0.85; third amplitude level  74  has a radius of 0.70; fourth amplitude level  76  has a radius of 0.54; fifth amplitude level  78  has a radius of 0.39; and sixth amplitude level  80  has a radius of 0.24. First amplitude level  70  includes 32 first amplitude symbols  82 ; second amplitude level  72  includes 32 second amplitude symbols  84 ; third amplitude level  74  includes 24 third amplitude symbols  86 ; fourth amplitude level  76  includes 16 fourth amplitude symbols  88 ; fifth amplitude levels  78  includes 16 fifth amplitude symbols  90 ; and sixth amplitude level  80  includes 8 sixth amplitude symbols  92 .  
         [0030]    The following chart lists the position of each of the 128 points of FIG. 3 in polar coordinates and in Cartesian coordinates.  
                                                                     Symbol   Radius   Angle   X   Y                                1   1   0   1   0       2   1   11.25   0.98   0.2       3   1   22.5   0.92   0.38       4   1   33.75   0.83   0.56       5   1   45   0.71   0.71       6   1   56.25   0.56   0.83       7   1   67.5   0.38   0.92       8   1   78.75   0.2   0.98       9   1   90   0   1       10   1   101.25   −0.2   0.98       11   1   112.5   −0.38   0.92       12   1   123.75   −0.56   0.83       13   1   135   −0.71   0.71       14   1   146.25   −0.83   0.56       15   1   157.5   −0.92   0.38       16   1   168.75   −0.98   0.2       17   1   180   −1   0       18   1   191.25   −0.98   −0.2       19   1   202.5   −0.92   −0.4       20   1   213.75   −0.83   −0.6       21   1   225   −0.71   −0.7       22   1   236.25   −0.56   −0.8       23   1   247.5   −0.38   −0.9       24   1   258.75   −0.2   −1       25   1   270   0   −1       26   1   281.25   0.2   −1       27   1   292.5   0.38   −0.9       28   1   303.75   0.56   −0.8       29   1   315   0.71   −0.7       30   1   326.25   0.83   −0.6       31   1   337.5   0.92   −0.4       32   1   348.75   0.98   −0.2       33   0.9   0   0.85   0       34   0.9   11.25   0.83   0.17       35   0.9   22.5   0.79   0.33       36   0.9   33.75   0.71   0.47       37   0.9   45   0.6   0.6       38   0.9   56.25   0.47   0.71       39   0.9   67.5   0.33   0.79       40   0.9   78.75   0.17   0.83       41   0.9   90   0   0.85       42   0.9   101.25   −0.17   0.83       43   0.9   112.5   −0.33   0.79       44   0.9   123.75   −0.47   0.71       45   0.9   135   −0.6   0.6       46   0.9   146.25   −0.71   0.47       47   0.9   157.5   −0.79   0.33       48   0.9   168.75   −0.83   0.17       49   0.9   180   −0.85   0       50   0.9   191.25   −0.83   −0.2       51   0.9   202.5   −0.79   −0.3       52   0.9   213.75   −0.71   −0.5       53   0.9   225   −0.6   −0.6       54   0.9   236.25   −0.47   −0.7       55   0.9   247.5   −0.33   −0.8       56   0.9   258.75   −0.17   −0.8       57   0.9   270   −0   −0.9       58   0.9   281.25   0.17   −0.8       59   0.9   292.5   0.33   −0.8       60   0.9   303.75   0.47   −0.7       61   0.9   315   0.6   −0.6       62   0.9   326.25   0.71   −0.5       63   0.9   337.5   0.79   −0.3       64   0.9   348.75   0.83   −0.2       65   0.7   0   0.7   .0       66   0.7   15   0.68   0.18       67   0.7   30   0.61   0.35       68   0.7   45   0.49   0.49       69   0.7   60   0.35   0.61       70   0.7   75   0.18   0.68       71   0.7   90   0   0.7       72   0.7   105   −0.18   0.68       73   0.7   120   −0.35   0.61       74   0.7   135   −0.49   0.49       75   0.7   150   −0.61   0.35       76   0.7   165   −0.68   0.18       77   0.7   180   −0.7   0       78   0.7   195   −0.68   −0.2       79   0.7   210   −0.61   −0.4       80   0.7   225   −0.49   −0.5       81   0.7   240   −0.35   −0.6       82   0.7   255   −0.18   −0.7       83   0.7   270   −0   −0.7       84   0.7   285   0.18   −0.7       85   0.7   300   0.35   −0.6       86   0.7   315   0.49   −0.5       87   0.7   330   0.61   −0.4       88   0.7   345   0.68   −0.2       89   0.5   0   0.54   0       90   0.5   22.5   0.5   0.21       91   0.5   45   0.38   0.38       92   0.5   67.5   0.21   0.5       93   0.5   90   0   0.54       94   0.5   112.5   −0.21   0.5       95   0.5   135   −0.38   0.38       96   0.5   157.5   −0.5   0.21       97   0.5   180   −0.54   0       98   0.5   202.5   −0.5   −0.2       99   0.5   225   −0.38   −0.4       100   05   247.5   −0.21   −0.5       101   0.5   270   −0   −0.5       102   0.5   292.5   0.21   −0.5       103   0.5   315   0.38   −0.4       104   0.5   337.5   0.5   −0.2       105   0.4   0   0.39   0       106   0.4   22.5   0.36   0.15       107   0.4   45   0.28   0.28       108   0.4   67.5   0.15   0.36       109   0.4   90   0   0.39       110   0.4   112.5   −0.15   0.36       111   0.4   135   −0.28   0.28       112   0.4   157.5   −0.36   0.15       113   0.4   180   −0.39   0       114   0.4   202.5   −0.36   −0.1       115   0.4   225   −0.28   −0.3       116   0.4   247.5   −0.15   −0.4       117   0.4   270   −0   −0.4       118   0.4   292.5   0.15   −0.4       119   0.4   315   0.28   −0.3       120   0.4   337.5   0.36   −0.1       121   0.2   0   0.24   0       122   0.2   45   0.17   0.17       123   0.2   90   0   0.24       124   0.2   135   −0.17   0.17       125   0.2   180   −0.24   0       126   0.2   225   −0.17   −0.2       127   0.2   270   −0   −0.2       128   0.2   315   0.17   −0.2                  
 
         [0031]    Symbols  1 - 32  define first amplitude symbols  82 ; symbols  33 - 64  define second amplitude symbols  84 ; symbols  65 - 88  define third amplitude symbols  86 ; symbols  89 - 103  define fourth amplitude symbols  88 ; symbols  105 - 120  define fifth amplitude symbols  90 ; and symbols  121 - 128  define sixth amplitude symbols  92 . As can be seen in the chart, each first amplitude symbol  82  is separated by 11.25°; each second amplitude symbol  84  is separated by 11.25°; each third amplitude symbol  86  is separated by 15°; each fourth amplitude symbols  88  and each fifth amplitude symbol  90  are separated by 22.5°; and each fifth amplitude symbol is separated by 45°. The minimum distance between any pair of signal points is 0.150.  
         [0032]    Similarly to FIG. 2, amplitude levels  70 ,  72 ,  74 ,  76 ,  78 ,  80  are selected to maximize the number of symbols for which amplifier  24  operates in saturation. Further, lower amplitude levels  72 ,  74 ,  76 ,  78 ,  80  are selected so that amplifier  24  operates as efficiently as possible when amplifying the symbols placed on these levels. Further yet, the symbols are selected in order to provide suitable signal-to-noise ratios for the symbols placed on each respective amplitude level.  
         [0033]    [0033]FIG. 4 depicts a 7 level 128-ary constellation utilized for QAM by modulator  18 . The 128-ary constellation is depicted as a seven level constellation on a Cartesian coordinate system having an in-phase axis  46  and a quadrature-phase axis  48 . The 128-ary constellation includes a first amplitude level  100 , a second amplitude level  102 , a third amplitude level  104 , a fourth amplitude level  106 , a fifth amplitude level  108 , a sixth amplitude level  110 , and a seventh amplitude level  112 . Similarly as previously described, each respective amplitude level has a plurality of first amplitude symbols  114 , second amplitude symbols  116 , third amplitude symbols  118 , fourth amplitude symbols  120 , fifth amplitude symbols  122 , sixth amplitude symbols  124 , and seventh amplitude symbols  126 .  
         [0034]    First amplitude level  100  has a radius of 1; second amplitude level  102  has a radius of 0.85; third amplitude level  104  has a radius of 0.70; fourth amplitude level  106  has a radius of 0.55; fifth amplitude level  108  has a radius of 0.4; sixth amplitude level  110  has a radius of 0.25; and sixth amplitude level  112  has a radius of 0.10. First amplitude level  100  includes 32 first amplitude symbols  114 ; second amplitude level  102  includes 24 second amplitude symbols  116 ; third amplitude level  104  includes 24 third amplitude symbols  118 ; fourth amplitude level  106  includes 16 fourth amplitude symbols  120 ; fifth amplitude level  108  includes 12 fifth amplitude symbols  122 ; sixth amplitude level  110  includes 8 sixth amplitude symbols  124 ; and seventh amplitude level  112  includes 4 seventh amplitude symbols  126 . First amplitude symbols  114  are separated by 11.25°, with one first amplitude symbols  114  falling at Cartesian coordinates x=0.1 and y=0 (1, 0). Second amplitude symbols  116  are separated by 11.25°, with one second amplitude symbols  116  being located at x=0.85 and y=0 (0.85, 0). Third amplitude symbols  116  are separated by 15°, with one third amplitude symbol  118  located at x=0.7 and y=0, (0.7, 0). Fourth amplitude symbols  120  are separated by 22.5°, with one fourth amplitude symbol  120  being located at x=0.55 and y=0, (0.55, 0). Fifth amplitude symbols  122  are separated by 30°, with one fifth amplitude symbol  122  being located at x=0.4 and y=0 (0.4, 0). Sixth amplitude symbols  124  are separated by 45°, with one fifth amplitude symbol  122  being located at x=0.25 and y=0 (0.25, 0). Seventh amplitude symbols  126  are separated by 90°, with one seventh amplitude symbol  126  being located at x=0.104 and y=0 (0.104, 0).  
         [0035]    [0035]FIG. 5 depicts a second implementation of a seven level 128-ary constellation. The 128-ary constellation of FIG. 5 includes a first amplitude level  130 , a second amplitude level  132 , a third amplitude level  134 , a fourth amplitude level  136 , a fifth amplitude level  138 , a sixth amplitude level  140 , and a seventh amplitude level  142 . The respective amplitude levels include respective first amplitude symbols  144 , second amplitude symbols  146 , third amplitude symbols  148 , fourth amplitude symbols  150 , fifth amplitude symbols  152 , sixth amplitude symbols  154 , and seventh amplitude symbols  156 . First amplitude level  130  has a unit radius of 1; second amplitude level  132  has a radius of 0.85; third amplitude level  134  has a radius of 0.72; fourth amplitude level  136  has a radius of 0.58; fifth amplitude level  138  has a radius of 0.45; sixth amplitude level  140  has a radius of 0.31; and seventh amplitude level  142  has a radius of 0.18. First amplitude level  130  has 32 first amplitude symbols  144 ; second amplitude level  132  has 24 second amplitude symbols  146 ; third amplitude level  134  has 24 third amplitude symbols  148 ; fourth amplitude level  136  has 16 fourth amplitude symbols  150 ; fifth amplitude level  138  has 16 fifth amplitude symbols  152 ; sixth amplitude level  140  has 8 sixth amplitude symbols  154 ; and seventh amplitude level  142  has 8 seventh amplitude symbols  156 .  
         [0036]    First amplitude symbols  144  are separated by 11.25°, with one first amplitude symbol  144  located at coordinates x=1, y=0, (1, 0). Second amplitude symbols  146  are separated by 15°, with a second amplitude symbol  146  located at coordinates x=0.85, y=0 (0.85, 0). Third amplitude symbols  148  are separated by 15°, with one third amplitude symbol  148  being located at coordinates x=0.716 and y=0 (0.716, 0). Fourth amplitude symbols  150  are separated by 22.5°, with one fourth amplitude symbol  150  located at coordinate x=0.581, y=0, (0.581, 0). Fifth amplitude symbols  152  are separated by 22.5°, with one fifth amplitude symbol  152  located at Cartesian coordinates x=0.446 and y=0, (0.446, 0). Sixth amplitude symbols  154  are separated by 45°, with one sixth amplitude symbol  154  located at Cartesian coordinates x=0.311 and y=0, (0.311, 0). Seventh amplitude symbols  156  are separated by 45°, with one sixth amplitude symbols  156  located at Cartesian coordinates x=0.177 and y=0 (0.177, 0). The minimum distance between each point is 0.135.  
         [0037]    [0037]FIG. 6 depicts a third implementation of a seven level 128-ary constellation. The 128-ary constellation of FIG. 6 includes a first amplitude level  160 , a second amplitude level  162 , a third amplitude level  164 , a fourth amplitude level  166 , a fifth amplitude level  168 , a sixth amplitude level  170 , and a seventh amplitude level  172 . The respective amplitude levels include respective first amplitude symbols  174 , second amplitude symbols  176 , third amplitude symbols  178 , fourth amplitude symbols  180 , fifth amplitude symbols  182 , sixth amplitude symbols  184 , and seventh amplitude symbols  186 . First amplitude level  160  has a unit radius of 1; second amplitude level  162  has a radius of 0.85; third amplitude level  164  has a radius of 0.68; fourth amplitude level  166  has a radius of 0.54; fifth amplitude level  168  has a radius of 0.41; sixth amplitude level  170  has a radius of 0.27; and seventh amplitude level  172  has a radius of 0.11. First amplitude level  160  has 32 first amplitude symbols  174 ; second amplitude level  162  has 24 second amplitude symbols  176 ; third amplitude level  164  has 24 third amplitude symbols  178 ; fourth amplitude level  166  has 16 fourth amplitude symbols  180 ; fifth amplitude level  168  has 16 fifth amplitude symbols  182 ; sixth amplitude level  170  has 12 sixth amplitude symbols  184 ; and seventh amplitude level  172  has 4 seventh amplitude symbols  186 .  
         [0038]    First amplitude symbols  174  are separated by 11.25°, with one first amplitude symbol  174  located at coordinates x=1, y=0, (1, 0). Second amplitude symbols  176  are separated by 15°, with a second amplitude symbol  176  located at coordinates x=0.85, y=0 (0.85, 0). Third amplitude symbols  178  are separated by 15°, with one third amplitude symbol  178  being located at coordinates x=0.68 and y=0 (0.68, 0). Fourth amplitude symbols  180  are separated by 22.5°, with one fourth amplitude symbol  180  located at coordinate x=0.544, y=0, (0.544, 0). Fifth amplitude symbols  182  are separated by 22.5°, with one fifth amplitude symbol  182  located at Cartesian coordinates x=0.405 and y=0, (0.405, 0). Sixth amplitude symbols  184  are separated by 30°, with one sixth amplitude symbol  184  located at Cartesian coordinates x=0.268 and y=0, (0.268, 0). Seventh amplitude symbols  186  are separated by 90°, with one sixth amplitude symbols  186  located at Cartesian coordinates x=0.11 and y=0 (0.11, 0). The minimum distance between each point is 0.136.  
         [0039]    The number of symbol and position of each symbol placed on the respective amplitude levels for each constellation described above is selected so that amplifier  24  operates at peak efficiency for the greatest number of symbols. Thus, the particular number of amplitude levels and the particular number of symbols placed on each amplitude level and the relative position of each symbol is specifically selected to maximize operation of amplifier  24 .  
         [0040]    The above-described invention utilizes concentric constellations to provide simple compensation amplitude distortion. By utilizing concentric constellations, the expansion of inner constellations is controlled by five, six, or seven settings, depending upon the number of amplitude levels for a 128-ary constellation. The spacing between symbols in each 128-ary constellation is selected to arrive at a suitable tradeoff between resolution and power and enables best use of available power. Further, fewer amplitude levels may be used when employing the teachings described herein. Further, when compared to conventional square constellations, the circular constellations defined herein utilize peak-power more efficiently.  
         [0041]    While the invention has been described in its presently preferred form, it is to be understood that there are numerous applications and implementations for the present invention. Accordingly, the invention is capable of modification and changes without departing from the spirit of the invention as set forth in the appended claims.