Abstract:
A OAM receiver includes a source of a received hierarchical OAM signal. The hierarchical OAM signal represents successive data points in the I-Q plane, each data point being in one of four quadrants. Circuitry, coupled to the hierarchical OAM signal source, calculates the location in the I-Q plane of the center-of-gravity of successive received data points in a quadrant. A level 1 decoder is responsive to a received data point and detects the quadrant in the I-Q plane of a received data point. Further circuitry, coupled to the hierarchical OAM signal source and responsive to the calculating circuitry, translates the received data point in the I-Q plane such that the center-of-gravity of the detected quadrant is translated to the origin of the I-Q plane. A level 2 decoder is then responsive to the translated data point for detecting the quadrant of the translated data point.

Description:
This application claims the benefit under 35 U.S.C. § 365 of International Application PCT/US00/32008, filed Nov. 22, 2000, which claims the benefit of U.S. Provisional Application 60/167,022, filed Nov. 23, 1999. 

   FIELD OF THE INVENTION 
   The present invention relates to hierarchical quadrature amplitude modulation transmission systems. 
   BACKGROUND OF THE INVENTION 
   Hierarchical quadrature amplitude modulation (QAM) transmission systems are well known. For example, U.S. Pat. No. 5,966,412, issued Oct. 12, 1999 to Ramaswamy, discloses a modulation system which can remain backward compatible with older quadrature phase shift keyed (QPSK) receivers, while simultaneously further allowing additional data streams, for providing higher data rates or higher precision data, to be receivable by more advanced receivers.  FIG. 1  is a block diagram illustrating a hierarchical QAM transmission system as disclosed in this patent.  FIG. 1  discloses a data transmitter  100  coupled to a data receiver  300  via a transmission channel  200 . 
   In  FIG. 1 , a first input terminal DATA  1  is coupled to source (not shown) of a first data signal, and a second input terminal DATA  2  is coupled to a source (not shown) of a second data signal. The first and second data signals may represent separate and independent data, or may represent related data signals, such as signals carrying respective portions of the same data signal (for increasing the throughput of the transmission system) or a elementary data portion and a supplemental data portion of the same data signal (for transmitting enhanced signals while maintaining backward compatibility with existing older receivers, as described in more detail below). The first input terminal DATA  1  is coupled to an input terminal of a first error detection/correction encoder  102 . An output terminal of the first encoder  102  is coupled to an input terminal of a level 1 QPSK modulator  104 . An output terminal of the level 1 QPSK modulator  104  is coupled to a first input terminal of a signal combiner  106 . 
   The second input terminal DATA  2  is coupled to an input terminal of a second error detection/correction encoder  108 . An output terminal of the second encoder  108  is coupled to an input terminal of a level 2 QPSK modulator  110 . The level 2 QPSK modulator  110  is coupled to an input terminal of a variable gain amplifier  111 , having a gain of G. An output terminal of the variable gain amplifier  111  is coupled to a second input terminal of the signal combiner  106 . An output terminal of the signal combiner  106  produces a combined modulated signal and is coupled to the transmission channel  200 . In the illustrated embodiment, this channel is a direct satellite television signal transmission system, and the transmission channel includes a ground transmitting station at the transmitter  100  (represented by a transmitting antenna in phantom), a communications satellite (not shown), for receiving the data from the ground station and rebroadcasting that data to a plurality of ground receiving stations, one of which ( 300 ) is illustrated in  FIG. 1 , which receives and processes the rebroadcast data signal, as illustrated by a receiving antenna in phantom. 
   The output of the transmission channel  200  is coupled to an input terminal of a level 1 QPSK demodulator  302 . An output terminal of the level 1 demodulator  302  is coupled to respective input terminals of a first error detection/correction decoder  304  and a delay circuit  306 . An output terminal of the first decoder  304  is coupled to an output terminal DATA  1 ′, and to an input terminal of a reencoder  308 . An output terminal of the reencoder  308  is coupled to an subtrahend input terminal of an subtractor  310 . An output terminal of the delay circuit  306  is coupled to a minuend input terminal of the subtractor  310 . A difference output terminal of the subtractor  310  is coupled to an input terminal of a second error detection/correction decoder  312 . An output terminal of the second decoder  312  is coupled to a second data output terminal DATA  2 ′. 
   In operation, the first encoder  102  encodes the first data signal DATA  1  to provide error detection/correction capabilities in a known manner. Any of the known error detection/correction codes may be implemented by the encoder/decoder pairs  102 / 304 ,  108 / 312 , and those codes may be concatenated, as described in the above-mentioned patent. The first encoder  102  produces a stream of encoded bits representing the encoded first data signal DATA  1 . The level 1 modulator  104  processes successive sets of two encoded data bits, each set termed a symbol, to generate a QPSK signal which lies in one of four quadrants in a known manner. Similarly, the second encoder  108  encodes the second data signal DATA  2  to provide error detection/correction capabilities in a known manner. The level 2 modulator  110  processes sets of two encoded data bits to also generate a QPSK signal which lies in one of four quadrants. One skilled in the art will understand that additional data signals (DATA  3 , etc.) may be respectively error detection/correction encoded by additional encoders and additional QPSK modulators, (level 3, etc.) may be responsive to respective additional sets of two encoded data bits to generate additional QPSK signals. The QPSK signal from the level 1 modulator  104  is given a weight of 1; the QPSK signal from the level 2 modulator  110  is given a weight or gain of 0.5 by the variable gain amplifier  111 ; the third a weight of 0.25 and so forth. All the weighted QPSK signals are then combined into a single modulated signal by the signal combiner  106  and transmitted through a transmission channel  200 . 
   The level 1 QPSK modulator  104  causes the combined signal to lie within one of four quadrants in response to the set of two encoded data bits from the first encoder  102 . Each quadrant, in turn, may be thought of as divided into four sub-quadrants. The level 2 QPSK modulator  110  causes the combined signal to lie within one of the sub-quadrants within the quadrant selected by the level 1 QPSK modulator  104 , in response to the set of two input data bits from the second encoder  108 . The sub-quadrant may further be thought of as divided into four sub-sub-quadrants, and the combined signal caused to lie within one of those sub-sub quadrants in response to the set of two input data bits form a third encoder (not shown), and so forth. 
   An older receiver (illustrated in  FIG. 1  by a dashed line  300 ′) includes only a level 1 QPSK demodulator  302 , which can detect where in the I-Q plane the received signal lies. From that information, the error detection/correction decoder  304  can determine the corresponding two encoded bits in the received first data stream. The error detection/correction decoder  304  can further correct for any errors introduced by the transmission channel to generate a received data signal DATA  1 ′ representing the original first data signal DATA  1 . Thus, such a receiver can properly receive, decode, and process a first data signal DATA  1  in the presence of additionally modulated data signals DATA  2 , (DATA  3 ), etc. The signals included by the level 2 (and level 3, etc.) QPSK modulators look simply like noise to such a receiver. 
   A more advanced receiver  300 , on the other hand, can detect which quadrant the received modulated signal lies within, and, thus, can receive, decode, and process successive sets of two data bits representing the first data signal DATA  1 . The reencoder  308  in the advanced receiver then regenerates an ideal signal lying in the middle of the indicated quadrant, which is subtracted from the received modulated signal. This operation translates the center of the transmitted signal quadrant to the origin. What remains is a QPSK modulated signal, weighted by 0.5, representing the second data signal DATA  2 . This signal is then decoded by the second decoder  312  to determine which sub-quadrant the signal lies within, indicating the set of two bits corresponding to that signal. Successive sets of two received data bits representing the second data signal DATA  2  are, thus, received, decoded and processed, and so forth. Such a transmission system operates by modulating a carrier in quadrature with what is seen as a constellation of permissible symbols, and is a form of quadrature amplitude modulation (QAM). Such a system is termed a hierarchical QAM transmission system because it may be used to transmit other levels of data signals, or other levels of detail in a single signal, while maintaining backwards compatibility with older receivers. 
     FIG. 2   a  is a diagram illustrating a constellation in the I-Q plane of permissible symbols for a hierarchical 16 QAM transmission system, as illustrated in the above mentioned patent. In  FIG. 2   a , a first set of two bits determine which quadrant the generated symbol lies within. If the first two bits are “00” then the symbol lies within the upper right hand quadrant, and the level 1 modulator  104  produces I-Q signals such that I=1 and Q=1; if the first two bits are “01” then the symbol lies within the upper left hand quadrant, and the level 1 modulator  104  produces I-Q signals such that I=−1 and Q=1; if the first two bits are “10” then the symbol lies within the lower right hand quadrant and the level 1 modulator  104  produces I-Q signals such that I=1 and Q=−1; and if the first two bits are “11” then the symbol lies within the lower left hand quadrant and the level 1 modulator  104  produces I-Q signals such that I=−1 and Q=−1. This is indicated in  FIG. 2   a  by the appropriate bit pair in the middle of the associated quadrant. 
   As described above, each quadrant may, itself, be considered to be divided into four sub-quadrants, as illustrated in the upper right hand quadrant in  FIG. 2   a . The second set of two bits determine which sub-quadrant the symbol lies within. The same mapping is used for determining the sub-quadrant as was described above for determining the quadrant. That is, if the second two bits are “00”, then the symbol lies within the upper right hand sub-quadrant and the level 2 modulator generates an I-Q signal such that I=1 and Q=1; if the second two bits are “01” then the symbol lies within the upper left hand sub-quadrant and the level 2 modulator generates an I-Q signal such that I=−1 and Q=1; if the second two bits are “10” then the symbol lies within the lower right hand sub-quadrant and the level 2 modulator generates an I-Q signal such that I=1 and Q=−1; and if the second two bits are “11” then the symbol lies within the lower left hand sub-quadrant and the level 2 modulator generates an I-Q signal such that I=−1 and Q=−1. The variable gain amplifier  111  (of  FIG. 1 ) weights the signal from the level 2 modulator  110  by a weight of 0.5, so the points in the sub-quadrants lie at ±0.5 around the center point of the quadrant. Each of these locations is shown as a solid circle in  FIG. 2   a , with a four bit binary number illustrating the combination of the first and second sets of two bits, with the first two bits being the right hand pair of bits and the second two bits being the left hand pair. 
   European Patent Publication 0 594 505 A1, published Apr. 27, 1994, illustrates a similar hierarchical QAM system. In this publication, a hierarchical QAM transmitter is illustrated in  FIG. 3   d . A corresponding receiver is illustrated in  FIG. 5   c . This receiver includes an input for receiving the transmitted hierarchical QAM signal. The receiver detects the quadrant in which the received symbol lies, and thus the level 1 bits, using a Viterbi decoder  8  (col. 15, lines 44–58). An ideal symbol corresponding to the detected level 1 bits is regenerated in the middle of the detected quadrant by a multiplexer  90  receiving at its inputs values corresponding to the locations in the I-Q plane of the ideal center of the detected quadrant (col. 16, lines 8–12). Because the locations of the points in the constellation are known in advance, a weighted mean, or center-of-gravity, of the constellation points is pre-calculated. This ideal symbol location is subtracted from the received symbol to translate the pre-calculated center of the quadrant to the origin of the I-Q plane by subtractor  91  (col. 16, lines 8–12). 
   European Patent Publication 0 366 159, published May 2, 1990, illustrates a non-hierarchical QAM receiver in which various characteristics of the communications channel are detected and monitored. For example, symbol frequency drift and jitter and amplitude jitter are all analyzed and made available for analysis. While signal processing loops are illustrated to compensate for frequency drift and jitter, only a common analog gain auto-ranging is disclosed for amplitude control (page 4, lines 39–40). 
   It is known that the bit error rate performance of the respective data streams through the different levels of a hierarchical QAM system such as described above are different. In general, the bit error rate of the level 1 data stream is better than the bit error rate of the level 2 (and higher) data streams. However, the overall performance of the hierarchical QAM transmission system is optimized when the bit error rate of the respective data streams through the different levels are the same. It is desirable, therefore, to optimize not only the overall bit error rate of the transmission system, but also to more closely match the respective bit error rates of the different levels in the transmission system. 
   SUMMARY OF THE INVENTION 
   The inventors have realized that when decoding a received signal, it is imperative that the gain be controlled properly so that the values of the received points in the constellation are in the proper range to be detected accurately. However, due to the non-linearity inherent in a direct satellite television transmission system, and to the practice of purposely distorting the location of the data points in the constellation to improve performance, the standard method of comparing the received constellation with an ideal constellation, and adjusting the gain to maximize the correspondence between the two will not result in optimum operation. 
   In accordance with principles of the present invention a QAM receiver includes an input for receiving a hierarchical QAM signal. The received QAM signal represents successive data points in the I-Q plane, each data point being in one of four quadrants. A level 1 decoder detects the quadrant in the I-Q plane of a received data point. Circuitry, coupled to the hierarchical QAM signal input, translates the received data point in the I-Q plane such that a center point of the detected quadrant is translated to the neighborhood of the origin of the I-Q plane. A level 2 decoder detects the quadrant of the translated data point. This system is characterized by circuitry, coupled to the hierarchical QAM signal input, for calculating the location in the I-Q plane of the center-of-gravity of successive received data points in a quadrant. In addition, the translating circuitry further comprises circuitry for translating the received data point in the I-Q plane such that the calculated center-of-gravity location is translated to the origin of the I-Q plane. 
   Such a system can adapt to the received constellation, regardless of any distortion introduced into the constellation data points by non-linearities in the transmission channel, or purposely introduced into the transmitted constellation. 

   
     BRIEF DESCRIPTION OF THE DRAWING 
       FIG. 1  is a block diagram of a transmission system in accordance with principles of the present invention; 
       FIG. 2  is a diagram illustrating a constellation of permissible symbols for a hierarchical 16 QAM transmission system; 
       FIGS. 3   a  and  c  are more detailed block diagrams of respective portions of the transmission system illustrated in  FIG. 1  and further including a gray code mapper, and  FIG. 3   b  is a table containing data controlling the operation of the gray code mapper; 
       FIG. 4  is a more detailed block diagram of a portion of the transmission system illustrated in  FIG. 1  illustrating the operation of differing error detection/correction codes for differing levels; 
       FIG. 5  is a diagram of a received constellation and  FIG. 6  is a diagram of one quadrant of a received constellation distorted by the transmission channel; 
       FIG. 7  is a block diagram of circuitry for determining the center of gravity of a quadrant of a received constellation of data points; and 
       FIG. 8  is a diagram of a constellation illustrating the use of grouping factors to vary the relative bit rate performance of the different level signals in a hierarchical QAM signal. 
   

   DETAILED DESCRIPTION 
     FIGS. 3   a  and  c  are more detailed block diagrams of respective portions of the transmission system illustrated in  FIG. 1  and further including a gray code mapper, and  FIG. 3   b  is a table illustrating the operation of the gray code mapper illustrated in  FIGS. 3   a  and  c . Referring first to  FIG. 2   b , a constellation in which adjacent points at all locations represent data values which differ in only one bit position is illustrated. To produce this constellation, the mapping of the set of two bits in the encoded level 2 data signal to locations in a sub-quadrant depends on which quadrant that sub-quadrant lies within. The upper right hand quadrant (00) in  FIG. 2   b  is identical to that in  FIG. 2   a . In the upper left hand quadrant, however, the left and right columns are switched. In the lower right hand quadrant, the top and bottom rows are switched, and in the lower left hand quadrant, the left and right hand columns and the top and bottom rows are switched. This may be performed by a simple mapping operation in the transmitter  100  prior to modulating the encoded second data signal DATA  2 , and then a simple demapping operation in the receiver  300  after the received encoded second data signal is demodulated. 
   In  FIG. 3   a , a portion of the transmitter  100  is illustrated. A level 1 symbol (two bits from the first encoder  102  of  FIG. 1 ) is coupled to respective input terminals of the level 1 modulator  104  and a gray code mapper  112 . An in-phase (I) signal from the level 1 modulator  104  is coupled to a first input terminal of a first adder  106 (I) and a quadrature (Q) signal from the level 1 modulator  104  is coupled to a first input terminal of a second adder  106 (Q). The combination of the first adder  106 (I) and the second adder  106 (Q) form the signal combiner  106  of  FIG. 1 . A level 2 symbol (two bits from the second encoder  108 ) is coupled to an input terminal of the level 2 modulator  110 . An I output terminal of the level 2 modulator  110  is coupled to an I input terminal of the gray code mapper  112 , and a Q output terminal of the level 2 modulator  110  is coupled to a Q input terminal of the gray code mapper  112 . An I output terminal of the gray mapper  112  is coupled to a second input terminal of the first adder  106 (I) and a Q output terminal of the gray mapper  112  is coupled to a second input terminal of the second adder  106 (Q). The variable gain amplifier  111 , conditioned to have an attenuation factor of 0.5 and coupled between the gray code mapper  112  and the signal combiner  106 , is not shown to simplify the figure. 
   In operation, the level 1 symbol, represented by the set of two encoded data bits, is received from the level 1 encoder  102  (of  FIG. 1 ). The level 1 symbol is QPSK modulated by the level 1 modulator  104  to generate a set of I and Q component signals representing the quadrant of the modulated signal in a known manner. For example, if the symbol is 0, i.e. the two bits are 00, then the upper right hand quadrant is indicated (I=1, Q=1); if the symbol it 1, i.e. the two bits are 01, then the upper left hand quadrant is indicated (I=−1, Q=1); if the symbol is 2, i.e. the two bits are 10, then the lower right hand quadrant is indicated (I=1, Q=−1); and if the symbol is 3, i.e. the two bits are 11, then the lower left hand quadrant is indicated (I=−1, Q=−1). In a similar manner, level 2 symbol is QPSK modulated by the level 2 modulator  110  to generate a set of I and Q component signals representing the sub-quadrant of the modulated signal in a known manner. The level 2 modulator generates the modulated signal in exactly the same manner as the level 1 modulator  104 , i.e. if the two bits are 00 (0), then the upper right hand sub-quadrant is indicated (I=1, Q=1); if the two bits are 01 (1), then the upper left hand sub-quadrant is indicated (I=−1, Q=1); if the two bits are 10 (2) then the lower right hand sub-quadrant is indicated (I=−1, Q=1); and if the two bits are 11 (3) then the lower left hand sub-quadrant is indicated (I=−1, Q=−1). This modulated signal is then weighted by 0.5 (not shown). 
   The resulting constellation from the combination of these two modulated signals would be as illustrated in  FIG. 2   a . The gray code mapper  112  operates on the I and Q signals from the level 2 modulator  110  to produce the constellation illustrated in  FIG. 2   b .  FIG. 3   b  illustrates the mapping applied by the gray code mapper  112 . If the level 1 symbol is 0, indicating the upper right hand quadrant, then the sub quadrants are unchanged, that is the I and Q output signals from the level 2 modulator are left unchanged. Thus, the I output signal, Iout from the gray code mapper  112  is the same as the I input signal Iin (Iout=Iin), and the Q output signal, Qout from the gray code mapper  112  is the same as the Q input signal Qin (Qout=Qin). If, however the level 1 symbol is 1, indicating the upper left hand quadrant, then, referring to  FIG. 2 , the columns are switched. That is, positive I values become negative and vice versa. Thus when the level 1 symbol is 1, the I output signal is the negative of the I input signal (Iout=−Iin), while the Q output signal remains the same as the Q input signal (Qout=Qin). If the level 1 symbol is 2, indicating the lower right hand quadrant, then, the rows are switched. That is, positive Q values become negative and vice versa. Thus, when the level 1 symbol is 2, the I output signal is the same as the I input signal (Iout=Iin), while the Q output signal is the negative of the Q input signal (Qout=−Qin). If the level 1 signal is 3, indicating the lower left hand quadrant, then, both the columns and the rows are switched. That is, positive I values become negative, and positive Q values become negative, and vice versa. Thus, when the level 1 symbol is 3, the I output signal is the negative of the I input signal (Iout =−Iin), and the Q output signal is the negative of the Q input signal (Qout=−Qin). The gray code mapper  112  provides this function. The resulting I and Q values from the gray code mapper  112  are weighted with a weight of 0.5 as described above (not shown for simplicity) and combined by the signal combiner  106  with the I and Q values representing the level 1 symbol. The resulting constellation is that illustrated in  FIG. 2   b.    
   Such a mapping is reversible in the receiver  300  using a similar gray code mapper.  FIG. 3   c  illustrates a portion of a receiver  300  including such a gray code mapper  314 . In  FIG. 3   c , the output terminal of the reencoder  308  is coupled to an input terminal of the gray code mapper  314 . An I signal from the subtractor  310  (of  FIG. 1 ) is coupled to an I input terminal of the gray code mapper  314  and a Q signal from the subtractor  310  is coupled to a Q input terminal of the gray mapper  314 . An I output terminal of the gray code mapper  314  is coupled to an I input terminal of the second decoder  312  and a Q output terminal of the gray code mapper  314  is coupled to a Q input terminal of the second decoder  312 . 
   In operation, the reencoder  308  generates a signal which is an ideal representation of the received level 1 symbol. That is, if the received level 1 signal is determined to lie anywhere in the upper right hand quadrant, then the reencoder  308  produces a signal having the value 0; if anywhere in the upper left hand quadrant a value 1, if anywhere in the lower right hand quadrant a value 2 and if anywhere in the lower left hand quadrant a value 3. This symbol is supplied to a gray code mapper  314 . Respective I and Q signals from the subtractor  310  are processed by the gray code mapper  314  in the manner described above, and illustrated in  FIG. 3   b . One skilled in the art will appreciate that the gray code mapper  314  in the receiver  300  operates identically to the gray code mapper  112  in  FIG. 3   a , and will perform the inverse function performed in the transmitter  100 . 
   The use of gray code mappers ( 112  and  312 ) in the transmitter  100  and receiver  300  allow use of a constellation as illustrated in  FIG. 2   b , in the manner described above with respect to  FIG. 3   a . A transmission system using the gray code mapping function described above, to produce a constellation in which adjoining constellation points differ by no more than a single bit will increase the bit error rate of the system. Simulations have shown that using gray coding as described above will cut the number of level 2 bit errors in half. This provides an extra margin in the signal to noise ratio (SNR) of around ¼ dB. This improvement, while modest, will, along with other enhancements, provide improved performance of the transmission system as a whole. 
     FIG. 4  is a more detailed block diagram of a portion of the transmission system illustrated in  FIG. 1  illustrating the operation of differing error detection/correction codes for differing levels. As described above, different levels of QPSK modulation suffer from differing levels of degradation due to the compression of the distance between the constellation points in the higher levels of modulation by the non-linear high powered amplifiers employed in satellite broadcasting. More specifically, bit errors inherently occur more often at higher levels of the hierarchical modulation than lower levels. To more closely match the bit error rates of the level 1 and level 2 signals, error detection/correction codes having differing performance characteristics are used in the respective data streams. More specifically, more powerful error detection/correction coding will be used in higher level data streams while less powerful error detection/correction coding will be used on lower level data streams. This will optimize the overall performance and information transmission capacity of the transmission system. 
   In  FIG. 4 , those elements which are the same as those illustrated in  FIG. 1  are designated with the same reference number and are not described in detail below. In  FIG. 4 , the first error detection/correction encoder  102  in the transmitter  100  is partitioned into a serial connection of an outer encoder  102 (O) and an inner encoder  102 (I). Similarly, the second error detection/correction encoder  108  is partitioned into a serial connection of an outer encoder  108 (O) and an inner encoder  108 (I). In a corresponding manner, the first error detection/correction decoder  304  in the receiver  300  is partitioned into a serial connection of an inner decoder  304 (I) and an outer decoder  304 (O). Similarly, the second error detection/correction decoder  312  is partitioned into a serial connection of an inner decoder  312 (I) and an inner encoder  312 (O). As disclosed in the above mentioned patent, the outer encoder/decoder pairs implement a block coding technique, such as Hamming codes, Hadamard codes, Cyclic codes and Reed-Solomon (RS) codes, while the inner encoder/decoder pairs implement a convolutional code. 
   In  FIG. 4 , the coding used for the level 2 data stream is more powerful than the coding used for the level 1 data stream. More specifically, the convolutional code used in the inner encoder/decoder pair in the level 2 data stream is more powerful than the convolutional code used in the inner encoder/decoder pair in the level 1 data stream. For example, in a preferred embodiment, the first inner encoder/decoder pair, processing the level 1 data stream, implements a rate ½, constraint length 7 convolutional code punctured to a rate of            . The second inner encoder/decoder pair, processing the level 2 data stream, implements a rate ½ convolutional code without puncturing. The coding of the level 2 data stream is more powerful than that of the level 1 data stream. This more closely matches the bit error rate performance of the level 1 and level 2 data streams, and optimizes the performance of the transmission system as a whole.
   As described above, and illustrated in  FIG. 1 , the level 1 demodulator  302  and decoder  304  cooperate to detect the DATA  1  signal from the received constellation. Then a reconstructed ideal signal, from reencoder  308 , representing this detected DATA  1  signal is then subtracted from the received constellation, and ideally results in translation of the received constellation to form another constellation of the sub-quadrants within the detected quadrant. However, this translation operation is very sensitive to any mismatch between the actual “center point” of the quadrant as received, and the ideal center point (displaced by ±1 from the origin of the level 1 constellation) assumed by the reencoder  308 . Any mismatch in size between the received constellation and the ideal constellation results in the actual center point of the received quadrant being displaced from the assumed center point, and when the received constellation is translated by the reencoder  308  and subtractor  310 , results in the actual center point of the displaced sub-quadrant being displaced from the origin assumed by the second decoder  312 . Thus, the gain of the received channel must be accurately adapted to, in order to place the center point of the sub-quadrant in the proper location (origin) to be accurately decoded by the second decoder  312 . 
   In known transmission systems, the gain of the system is determined by comparing the received constellation of data points to a known ideal constellation of data points. There are several problems associated with accurate maintenance of the gain in this manner, however. First, in some transmission systems, the locations of the constellation points may be deliberately distorted from their ideal locations. The resulting constellation does not have the equi-spaced points illustrated in  FIG. 2 . Second, the transmission channel is not constant, and may be noisy with varying amounts of non-linearity. To determine the location of the center point of the quadrants, and thus the gain of the system, in such systems, the center-of-gravity of all the data points in the quadrants is determined. 
     FIG. 7  is a block diagram of circuitry for determining the center of gravity of a quadrant of a received constellation of data points. In  FIG. 7 , a rotator  321  receives I and Q values representing I and Q components of successive received data points from the level 1 demodulator  302  (of  FIG. 1 ). An I output terminal of the rotator  321  is coupled to an input terminal of an I low pass filter (LPF)  320 . A Q output terminal of the rotator  321  is coupled to an input terminal of a Q LPF  322 . Respective output terminals of the I and Q LPFs,  320  and  322 , are coupled to corresponding input terminals of a magnitude calculating circuit  324 . An output terminal of the magnitude calculating circuit  324  is coupled to the reencoder  308 . 
   In operation, the rotator  321  rotates all of the received values from whatever quadrant they were received in to the upper right hand quadrant in a known manner.  FIG. 5  is a diagram of a received constellation and shows the locations of a plurality of successive received modulated data points. The received data points form scatters in the respective neighborhoods of the assumed locations of the received constellation points in all four quadrants.  FIG. 6  is a diagram of the upper right hand quadrant of a received constellation all of whose data points have been rotated to this quadrant by the rotator  321 . The quadrant illustrated in  FIG. 6  represents a constellation which has been distorted by either deliberate pre-distortion of the transmitted constellation points and/or by the operation of the transmission channel  200 . 
   The I component of the rotated data points from the rotator  321  is low pass filtered in the LPF  320  with a sliding moving average of n points. In the illustrated embodiment, the sliding moving average is calculated using the preceding  500  data points. The Q component of the rotated data points from the rotator  321  is similarly low pass filtered with a sliding moving average. One skilled in the art will understand that the low pass filters  320 ,  322  may also be constructed using respective IIR digital filters. The low pass filtering operation produces the respective I and Q components of the center of gravity of the received data points in the quadrant. The estimate of the magnitude of the center of gravity is calculated in the magnitude calculating circuit  324 . For example if r i [n] is the filtered in-phase I component, and r q [n] is the filtered quadrature Q component, then the magnitude of the center of gravity is calculated as M=√{square root over (r i [n] 2 +r q [n] 2 )}. The magnitude of the center of gravity M should ideally be √{square root over (2)}=1.4. The magnitude of the ideal reconstructed signal from the reencoder  308  is adjusted in response to the magnitude of the calculated center of gravity M. By properly adjusting the magnitude of the reconstructed ideal signal from the reencoder  308 , the centers of the respective received quadrants will be properly translated to the origin by the subtractor  310 , and allow for accurate decoding of the level 2 and higher data signals. 
   The circuit illustrated in  FIG. 7  will operate independently of the method of transmission, whether linear or non-linear. It also operates properly in the presence of a pre-distorted transmission constellation, or with non-standard grouping factors (described in more detail below). It has been found that the circuit works well in practice with no measurable degradation when used on hierarchical 16 QAM transmission system over a linear channel when compared with exact knowledge of the locations of the centers of the quadrants. The circuit also operates well in the presence of noise and in particular in the presence of channel distortion caused by non-linear channels, such as found in direct satellite television signal transmission systems. Such a circuit improves the performance of the higher level data streams, and thus, improves the overall performance of the transmission system. 
   Referring again to  FIG. 1 , in known hierarchical QAM transmission systems, the constellation generated by the level 2 modulator  110  is combined in the signal combiner  106  with the constellation generated by the level 1 modulator  104  after being weighted in the variable gain amplifier  111  by a factor of 0.5. The weighting factor of 0.5 is termed the grouping factor and may be varied to change the relative performance of the level 1 and level 2 data streams, as described in more detail below. Referring to  FIG. 2   a , the resulting constellation consists of equi-spaced constellation points. As described above, such an arrangement results in a transmission system in which the performance of the level 1 data stream, in terms of bit error rate, is better than that of the level 2 data stream. By varying the grouping factor, the relative performance of the level 1 and level 2 data streams may be more closely matches. 
   Referring to  FIG. 8   a , the gain of the variable gain amplifier ( 111  of  FIG. 1 ) is conditioned to be 0.3. The resulting constellation points are spaced only 0.3 from the center point of the quadrant. One skilled in the art will recognize that in the constellation illustrated in  FIG. 8   a , the constellation points in a quadrant are further away from constellation points in other quadrants than in the constellation illustrated in  FIG. 2   a . Conversely, the constellation points within a quadrant are closer together than those illustrated in  FIG. 2   a . Such a system allows more accurate determination of which quadrant the level 1 data signal is in at the expense of less accurate determination of the constellation point of the level 2 data signal within the quadrant, thus, increasing the performance of the level 1 data stream and decreasing the performance of the level 2 data stream, when compared to the system of  FIG. 2   a.    
   Referring to  FIG. 8   b , the gain of the variable gain amplifier ( 111  of  FIG. 1 ) is conditioned to be 0.7. The resulting constellation points are spaced 0.7 from the center point of the quadrant. One skilled in the art will recognize that in the constellation illustrated in  FIG. 8   b , the constellation points in a quadrant are closer to constellation points in other quadrants than in the constellation illustrated in  FIG. 2   a . Conversely, the constellation points within a quadrant are further apart than those illustrated in  FIG. 2   a . Such a system allows more accurate determination of the constellation point of the level 2 data signal within the quadrant at the expense of less accurate determination of which quadrant the level 1 data signal is in, thus, increasing the performance of the level 2 data stream and decreasing the performance of the level 1 data stream, when compared to the system of  FIG. 2   a.    
   By proper setting of the gain of the variable gain amplifier  111  (of  FIG. 1 ), the grouping of the constellation points with each cluster may be placed optimally to more closely match the performance of the level 1 and level 2 data streams. It has been determined that for a 16 QAM transmission system transmitted through a non-linear direct satellite television channel, a grouping factor of around 0.6 to around 0.7 will more closely match the bit error rate performance of the level 1 and level 2 data streams. This will increase the overall performance of the transmission system as a whole.