Patent Publication Number: US-5832041-A

Title: 64 QAM signal constellation which is robust in the presence of phase noise and has decoding complexity

Description:
I. BACKGROUND OF THE INVENTION 
     A. Field of the Invention 
     The invention relates to the field of signal constellations for 64 QAM (quadrature amplitude modulated) signals and to consumer grade receivers for receiving such signals. 
     B. Related Art 
     For digital data transmission over CATV channels to be cost-effective, a consumer grade receiver must be built. At the front-end of such a receiver, a tuner frequency translates the RF band of interest down to baseband. A commercial grade tuner inserts a significant amount of phase noise. This phase noise can cause a rectangular 64 QAM signal to possess an irreducible error rate. In other words, the system will not operate reliably even with an infinite signal to noise ratio (SNR). 
     Previous efforts have been made to reduce phase noise in 64 QAM receivers. These include 
     G. J. Foschini, R. D. Gitlin, and S. B. Weinstein, &#34;On the selection of a two-dimensional signal constellation in the presence of phase jitter and Gaussian noise,&#34; BSTJ, vol.52, no. 6, pp.927-967, July-August, 1973. 
     B. W. Kernighan and S. Lin, &#34;Heuristic solution of a signal design optimization problem,&#34;BSTJ, vol. 52, no. 7, pp. 1145-1159, September, 1973 
     K. Pahlavan, &#34;Nonlinear Quantization and the design of coded and uncoded signal constellations,&#34; IEEE Trans Comm., vol. 39, no. 8, pp. 1207-1215, August, 1991 U.S. Pat. No. 4,660,213 
     These efforts have provided some QAM constellation design which provide improved performance in phase noise. However, the resulting decoders are unacceptably complex, because of arbitrary decision regions necessary to implement the arbitrarily spaced constellation points. 
     Decoder complexity becomes an important issue in high speed data communication, since decision directed implementations require extremely fast decoding, necessitating the use of an extremely complex ROM for general maximum-likelihood (ML) decoding. 
     Further background material may be found in: 
     E. A. Lee and J. G. Messerschmitt, Digital Communication, Kluwer Academic Publishers, Boston, 1988, in which chapter 6 deals with decoders for 64 QAM signals. and 
     J. Spilker, Digital Communication by Satellite, Prentice Hall, N.J., 1977, in which Chapter 12 deals with phase noise in 4 QAM signals 
     II. SUMMARY OF THE INVENTION 
     Accordingly, it is an object of the invention to create a 64 QAM signal constellation which has low phase noise and also low decoding complexity. 
     It is also an object of the invention to create a decoder for decoding such a 64 QAM signal. 
     This object is achieved using a signal having a constellation including the following points, or a scaled version thereof, in the first quadrant of the Cartesian plane: (1.500,7.500),(1.125,5.250),(3.375,5.625), (5.625,5.625),(0.750,3.750),(2.250,3.750), (3.750,3.750),(5.625,3.375),(0.750,2.250), (2.250,2.250),(3.750,2.250), (0.750,0.750), (2.250,0.750), (3.750,0.750), (5.250,1.125), (7.500,1.500). 
     Advantageously, this constellation may be scaled by a factor of 22/3 to reduce the number of bits used in decoding. 
     The object may also be achieved with a constellation using the following points, or a scaled version thereof, in the first quadrant of the Cartesian plane: (4,20), (4,14), (8,16), (16,16), (2,10), (6,10), (10,10), (2, 6), (6,6,), (10,6), (16,8), (2,2), (6,2), (10,2), (14,4), (20,4); although this constellation has been shown to give d signal more susceptible to additive noise than the first constellation. 
     Boundaries for decision regions for these points can be found according to the conventional technique of bisecting lines between adjacent points of the constellations. However, approximated decision regions, with shapes which are more nearly rectangular are cheaper to implement. The approximated decision regions create a slightly higher error rate in recognizing decision points, but reduce complexity of the resulting decoder. 
    
    
     III. BRIEF DESCRIPTION OF THE DRAWING 
     The invention will now be described by way of non-limiting example with reference to the following drawings. 
     FIG. 1 shows a prior art QAM receiver. 
     FIG. 2 shows phase noise characteristics of a prior art voltage-controlled oscillator/tuner. 
     FIG. 3a shows a signal constellation in accordance with the invention (Cons. A) with a first type of decision regions. 
     FIG. 3b shows Cons. B with a second type of decision regions 
     FIG. 4 shows a scaled version of FIG. 3. 
     FIG. 5a shows a second signal constellation in accordance with the invention (Cons. B) with the first type of decision regions. 
     FIG. 5b shows Cons. B with the second type of decision regions. 
     FIG. 6 comparative performance of signal decoding schemes. 
     FIG. 7 comparative performance of signal decoding schemes. 
     FIG. 8 shows a decoder for decoding Cons. A. 
     FIG. 9 shows an alternate embodiment for a decoder for decoding Cons. A. 
     FIG. 10 shows a decoder for decoding Cons. B. 
    
    
     IV. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     A baseband QAM transmitted signal can be written as ##EQU1## where T S  is the symbol time interval, α is a scaling factor, I is the space of integers and A n  is the QAM input symbol stream which can take on complex values corresponding to a given QAM constellation. Also g(t) is an arbitrary shaping function which is typically chosen to have a square-root raised-cosine (SQRC) spectrum to maximize the SNR. and minimize the inter-symbol interference (ISI). The transmitted signal is then given by 
     
         s(t)=Re{s(t)exp(jw.sub.c t},                               (2) 
    
     where w c  is the carrier frequency in radians. 
     At the receiver, the transmitted signal is first subjected to IF filtering. Then the signal spectrum is frequency shifted to close to DC, so that the entire two-sided QAM spectrum is in the passband. This frequency-shifting is accomplished using a tuner, The tuner introduces phase noise into the system. The output of the tuner, when subjected to programmable gain control used to restrict peak values, will be called r(t) herein, and is shown as the input to FIG. 1, which is a prior art QAM receiver, as shown in chapter 6 of the Lee et al book cited above. The signal r(t) is presented to the analog to digital converter 101, and then fed to a timing recovery circuit 102 which controls the clock phase of the analog to digital converter 101 with a feedback loop as shown. 
     An analytic filter 103, with a SQRC filter characteristic, operates on the output of the analog to digital converter 101. After a frequency shift to the baseband at 104, the signal, now called X k , is sent to an adaptive equalizer 105. Due to oscillator imperfections, there is a residual frequency offset between the transmitter and the receiver and also an associated phase jitter in the output Y k  of the adaptive equalizer 105 which must be corrected by the carrier recovery circuit 106 which follows the adaptive equalizer. This correction is implemented by using the complex multiplier 109 after the adaptive equalizer as shown. This multiplier is followed by the QAM decoder 107 which finally performs a mapping from the complex noisy input to the complex symbol output B k . A decision-feedback equalizer (DFE) 108 may also be used. The tap-adaptation of the adaptive equalizer uses the error between the decoded output and the noisy signal, along with the correction required due to the carrier recovery circuit, shown in the form of the complex multiplier 110. The DFE uses only the error signal out of the element 111. 
     For decision-directed implementations, an error signal e k  exp (jΔwt) is used as an input to fine-tune the equalizer tap coefficients. Also the QAM decoder acts as a the slicer. The output B k  is used in the carrier recovery loop as well as in the DFE. 
     The QAM decoding must be performed as quickly as possible to avoid delays in the decision-directed loops. 
     Assuming that the adaptive equalizers remove the inter-symbol interference (ISI) completely, it can then be seen that 
     
         q.sub.k =A.sub.k exp(jφ.sub.k)=n.sub.k,                (3) 
    
     where n k  is the complex additive noise and φ k  is the residual phase noise that is left uncorrected by the carrier-recovery circuit. 
     FIG. 2 shows the phase noise characteristics of a voltage-controlled oscillator VCO in a tuner designed by ANADIGICS The phase noise level at a 100 kHz offset is about -100 dBc/Hz. At this bandwidth, phase noise level has essentially flattened out. As discussed in the Spilker book, chapter 12, mentioned above, this section of the phase noise will appear essentially unaltered by the phase-locked loop (PLL) in the carrier recovery circuit 106, even when the bandwidth is made considerably larger than 100 kHz. Typically, the PLL is used to track low-frequency jitter, while the high-frequency jitter is left uncorrected for optimal results. The output of the PLL has phase noise components because of the additive noise, n k , and also due to the low-frequency jitter. The phase noise components due to additive noise can be ignored by assuming that the SNR is large. Also, the low-frequency jitter is inversely proportional to the PLL bandwidth. Hence, with a proper selection of the PLL parameters, it is possible to reduce this jitter significantly. As a first-order approximation, then, the phase noise, φ k , can be assumed to be the uncorrected flat-portion of the tuner noise as shown in FIG. 2. In other words, for a symbol rate of 5 MHz, the flat portion is from about 100 kHz to about 3 MHz. This phase noise can then be conveniently modeled to have a Gaussian distribution with zero mean and the variance which is given by the area under the curve in FIG. 2 between 100 kHz and 3 MHz. This area is approximated by the level at 100 kHz times the frequency band. A Gaussian phase noise distribution can then be used to evaluate the performance of various QAM constellations in the presence of phase noise. 
     It is possible to design QAM constellations which are more robust to phase noise, as discussed in the background of the invention. However, a good QAM constellation must take decoding complexity into account. For small decoding complexity, the following constraints must be satisfied: 
     1. The decision regions should preferably be rectangular, such that the boundaries of these regions are representable by as small a number of bits as possible. Since arbitrary scaling can be assumed, this implies that all the decision boundaries should be representable by integer multiples of a fixed quantity, such that the largest such integer is a small number. 
     2. Quadrant decoding should be allowed. One way to ensure this is to make the constellation in a quadrant symmetric about the y=x line, or about an angle of 45 degrees. In general only the nearest points to the X and the Y axes need be symmetric, so that the decision regions include the X and the Y axes. 
     3. The constellation points must be representable by a small number of bits, using the same scaling assumed above. This is because the ROM size is directly proportional to this number. 
     In the above-mentioned article by Pahlavan a 64 QAM constellation is proposed which allows for quadrant decoding. The resulting decision boundaries, when determined by the conventional technique of bisecting lines connecting adjacent points of the constellation, are difficult to implement. 
     An improvement on the Pahlavan constellation according to the invention is shown in FIGS. 3a and 3b. The constellation points here are (1.500,7.500), (1.125,5.250), (3.375,5.625), (5.625,5.625), (0.750,3.750), (2.250,3.750), (3.750,3.750), (5.625,3.375), (0.750,2.250), (2.250,2.250),(3.750,2.250), (0.750,0.750), (2.250,0.750), (3.750,0.750), (5.250,1.125), (7.500,1.500). 
     The boundaries of the decision regions can be determined by bisecting lines between adjacent points as shown in FIG. 3a. For most of the points, this yields rectangular decision regions, which are easy to implement in a decoder. For some of the points, the resulting non-rectangular decision regions are more expensive to implement and are shown as jagged lines. 
     These jagged lines can be approximated in many cases by rectangular decision regions, shown in FIG. 3b. These approximated regions give slightly noisier performance in exchange for much decreased decoder complexity. The approximated decision regions are as indicated in the following table: 
     
                       TABLE I                                                     
______________________________________                                    
         Intersection of the following regions being the decision         
Point    region for the corresponding point                               
______________________________________                                    
(1.500,7.500)                                                             
         x≧0;y≧6.375;y≧x+4.125                       
(1.125,5.250)                                                             
         y≦6.375;y≧4.5;x≧0;x≦2.250            
(3.375,5.625)                                                             
         x≧2.25;x≦4.5;y≧4.5;y≦x+4.125         
(5.625,5,625)                                                             
         y≧4.5;x≧4.5                                        
(0.750,3.750)                                                             
         x≧0;x≦1.5;y≧3;y≦4.5                  
(2.250,3.750)                                                             
         x≧1.5;x≦3;y≧3;y≦4.5                  
(3.750,3.750)                                                             
         x≧3;x≦4.5;y≧3;y≦4.5                  
(5.625,3.375)                                                             
         y≦2.25;y≦4.5;x≧4.5;y≧x-4.125         
(0.750,2.250)                                                             
         x≧0;x≦1.5;y≧1.5;y≦3                  
(2.250,2.250)                                                             
         x≧1.5,x≦3;y≧1.5;y≦3                  
(3.750,2.250)                                                             
         x≧3;x≦4.5;y≧1.5;y≦3                  
(0.750,0.750)                                                             
         x≧0;x≦1.5;y≧0;y≦1.5                  
(2.250,0.750)                                                             
         x≧1.5;x≦3;y≧0;y≦1.5                  
(3.750,0.750)                                                             
         x≧3;x≦4.5;y≧0;y≦1.5                  
(5.250,1.125)                                                             
         x≧4.5;x≦6.375;y≧0;y≦2.25             
(7.500,1.500)                                                             
         x≧6.375;y≧0;y≦x-4.125                       
______________________________________                                    
 
    
     Alternatively a scaled constellation, as shown in FIG. 4, may be used. Both types of decision region are shown in the figure, superimposed on one another. This constellation requires that incoming signal points be scaled by a factor of 22/3 prior to being presented to the decoder. The scaled points are (4,20), (3,14), (9,15), (15,15), (2,10), (6,10) (10,10), (15,9), (2,6), (6,6), (10,6), (2,2), (6,2), (10,2), (14,3), (20,4). The scaled decision regions, with approximated rectangular boundaries, are indicated in the following table: 
     
                       TABLE II                                                    
______________________________________                                    
      Intersection of the following regions being the decision region     
Point for the corresponding point                                         
______________________________________                                    
(4,20)                                                                    
      y≧17;x≧0;y≧x+11                                
(3,14)                                                                    
      x≧0;x≦6;y≧12;y≦17                       
(9,15)                                                                    
      x≧6;x≦12;y≧12;y≦x+11                    
(15,15)                                                                   
      x≧12;y≧12;                                            
(2,10)                                                                    
      x≧0;x≦4;y≧8;y≦12                        
(6,10)                                                                    
      x≦4;x≦8;y≧8;y≦12                        
(10,10)                                                                   
      x≧8;x≦12;y≧8;y≦12                       
(15,9)                                                                    
      x≧12;y≧6;y≦12;y≧x-11                    
(2,6) x≧0;x≦4;y≧4;y≦8                         
(6,6) x≧4;x≦8;y≧4;y≦8                         
(10,6)                                                                    
      x≧8;x≦12;y≧4;y≦8                        
(2,2) x≧0;x≦4;y≧0;y≦4                         
(6,2) x≧4;x≦8;y≧0;y≦4                         
(10,2)                                                                    
      x≧8;x≦12;y≧0;y≦4                        
(14,3)                                                                    
      x≧12;x≦17;y≧0;y≦6                       
(20,4)                                                                    
      x≧17;y≧0;y≦x-11                                
______________________________________                                    
 
    
     FIGS. 5a and 5b show a second constellation (Cons. B) in accordance with the invention. This constellation does not perform quite as well as the constellations of FIGS. 3a, 3b and 4. The constellation points of this constellation are: (4,20), (4,14), (8,16), (16,16), (2,10), (6,10), (10,10), (2,6), (6,6,), (10,6), (16,8), (2,2), (6,2), (10,2), (14,4), (20,4). 
     Again, decision regions can be determined by bisecting lines between adjacent points as shown in FIG. 5a. For many of the points, however, this yields non-rectangular decision regions, as indicated by the jagged lines in the figure. More nearly rectangular approximated decision regions can be chosen, as shown in FIG. 5b. These approximated decision regions give slightly suboptimal performance but are much cheaper to implement in the decoder. The more nearly rectangular approximated decision regions are as indicated in the following table: 
     
                       TABLE III                                                   
______________________________________                                    
      Intersection of the following regions being the decision region     
Point for the corresponding point                                         
______________________________________                                    
(4,20)                                                                    
      x≧0;x≦12;y≧17;y≧x+12                    
(4,14)                                                                    
      x≧0;y≧12;y≦17;y≦-2x+27                  
(8,16)                                                                    
      y≧-2x+27;y≦x+12;y≧12;x≦12               
(16,16)                                                                   
      x≧12;y≧12                                             
(2,10)                                                                    
      x≧0;x≦4;y≧8;y≦12                        
(6,10)                                                                    
      x≧4;x≦8;y≧8;y≦12                        
(10,10)                                                                   
      x≧8;x≦12;y≧8;y≦12                       
(16,8)                                                                    
      y≦12;x≧12;x+2y≧27;y≧x-12                
(2,6) x≧0;x≦4;y≧4;y≦8                         
(6,6) x≧4;x≦8;y≧4;y≦8                         
(10,6)                                                                    
      x≧8;x≦12;y≧4;y≦8                        
(2,2) x≧0;x≦4;y≧0;y≦4                         
(6,2) x≧4;x≦8;y≧0;y≦4                         
(10,2)                                                                    
      x≧8;x≦12;y≧0;y≦4                        
(14,4)                                                                    
      x≧12;x≦17;y≧0;x+2y≦27                   
(20,4)                                                                    
      x≧17;y≦12;y≦x-12;y≧0                    
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     FIG. 6 shows the bit-error rate (BER) performance of the constellation in FIG. 3 (Cons. A) in the presence of Gaussian phase-noise as a function of SNR. The lines in this figure have the following significance: 
     Solid Line: Theoretical performance of 64 QAM in Additive White Gaussian Noise (AWGN) 
     Small Dashes: Differentially-Encoded (DE) Rectangular (Rect.) 64 QAM in AWGN 
     Dotted Line: DE Rect. 64 QAM with 1 degree rms phase noise 
     Asterisk only: DE Rect. 64 QAM with 2 degrees rms phase noise 
     Dash-Dot Line: DE modified (where &#34;modified&#34; means the use of approximated decision region boundaries, as shown in the non-jagged lines of FIGS. 3-5) 64 QAM Cons. A in AWGN 
     Dash-3Dots Line: DE modified 64 QAM Cons. A with 1 degree rms phase noise 
     Large Dashes: DE modified 64 QAM Cons. A with 2 degrees rms phase noise 
     Triangles Constellation of U.S. Pat. No. 4,660,213 using optimal decoding with 2 degrees rms noise 
     `⋄` Constellation of U.S. Pat. No. 4,660,213 with 1 degree rms noise 
     `+` Constellation of U.S. Pat. No. 4,660,213 with no phase noise 
     Surprisingly, the suboptimal decoding of Cons. A performs better than optimal maximum likelihood (ML) decoding using the constellation of U.S. Pat. No. 4,660,213 with two degrees rms Gaussian phase-noise. The rectangular QAM constellation cannot operate reliably with two degrees rms phase noise. Making the constellation robust to two degrees rms phase noise, when compared with one degree rms, yields an increased phase noise susceptibility in terms of the phase noise level of 6 dB. 
     FIG. 7 compares the performance of Cons. A and Cons. B in Gaussian phase noise. The symbols on the table can be interpreted as follows, the first three sets of symbols being the same as the corresponding symbols defined in FIG. 6: 
     Solid line: Theoretical performance of 64 QAM in Additive White Gaussian Noise (AWGN) 
     Dotted line: DE rect. 64 QAM in AWGN 
     Asterisk only: DE rect. 64 QAM with 2 degrees rms phase noise 
     Small dashes: DE modified 64 QAM Cons. A in AWGN 
     Dash-dot line: DE modified 64 QAM Cons. A with 2 degrees rms phase noise 
     Dash-3dots line: DE modified 64 QAM Cons. B in AWGN 
     Large dashes: DE modified 64 QAM Cons. B with 2 degrees rms phase noise 
     The performance of constellation B is slightly worse, on the order of 0.5 dB for two degrees rms phase noise. Table IV summarizes the performance obtained for the different constellation at the BER of 10 -5  using the simplified decision regions. 
     
                       TABLE IV                                                    
______________________________________                                    
rms value    Cons. A (dB)                                                 
                        Cons. B (dB)                                      
______________________________________                                    
0 degree     0.38       0.64                                              
2 degrees    2.75       3.23                                              
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     FIG. 8 shows a decoder for decoding Cons. A. Real R and imaginary I components are input to the quadrant rotator 801, which outputs absolute value signals |R| and |I|. These address ROM 802. The contents of the ROM 802 correspond to the decision regions shown in FIG. 4 which are either rectangular or modified to be rectangular according to the sub-optimal decoding scheme. A five bit number corresponding to either the read or imaginary part can denote any integer from 0 to 31. Hence the two 5-bit numbers denote a point in the real-imaginary plane. For ROM 802, the two 5 bit numbers form a 10 bit address to a point where the constellation point closest to the received point is stored. As explained earlier, only 4 bits need to be stored for each axis to represent a decision region for a constellation point in ROM 802, because all the constellation points have integer values less than 16, as shown in FIG. 4. 
     If the absolute value signals correspond to rectangular decision regions, decision region indications are output at 803. If the absolute value signals correspond to non-rectangular signal regions a fail signal is output at 804. If there is no fail signal at 804, switch 805 selects the outputs of ROM 802. If there is a fail signal at 804, switch 805 selects the outputs of ROM 806. 
     Decision regions which are non-rectangular are implemented by this ROM 806. To address ROM 806, the values |R| and |I| are fed to a subtraction means 807 which outputs a sign bit (Sign(|R|-|I|)) and the 5-bit value |R|-|I|. The 5-bit value is in turn fed to comparator 808 which compares the input to the value 11. The ROM 806 is addressed by the sign bit in combination with the comparator output. This implements the decision region for points (4,20), (15,9), (9,15), and (20,4). ROM 806 outputs according to the following table: 
     
                       TABLE V                                                     
______________________________________                                    
Sign(|R|-|I|)                         
             ||R|-|I|.vertlin
             e. &gt; 11 or ≧ 11                                       
                            point output                                  
______________________________________                                    
+            true           (20,4)                                        
+            false          (15,9)                                        
-            true           (4,20)                                        
-            false          (9,15)                                        
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     As can be seen, even in this implementation, a large ROM table is required at 802, namely 2 10  ×4×2=2 13 . Further reductions could be made by implementing the rectangular region at the extreme bottom rights side and at the top left side, i.e. around points (3,14) and (4,13) as shown in FIG. 4, separately. FIG. 9 shows this implementation. In this figure, the unchanged elements retain the same reference numerals as in FIG. 8. A truncator 901 is added to truncate values |R| and |I|. The resulting 2 bit values are used to address ROM 902 which now has only 2 4  ×4×2=2 7  bits. ROM 902 now operates according to the following table: 
     
                       TABLE VI                                                    
______________________________________                                    
              imaginary input with two                                    
real input with two least                                                 
              least significant and one                                   
significant and one most                                                  
              most significant bit                                        
significant bits truncated                                                
              truncated       output point                                
______________________________________                                    
00            00              (2,2)                                       
01            00              (6,2)                                       
10            00              (10,2)                                      
00            01              (2,6)                                       
01            01              (6,6)                                       
10            01              (10,6)                                      
00            10              (2,10)                                      
01            10              (6,10)                                      
10            10              (10,10)                                     
11            11              (15,15)                                     
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     ROM 903 is now addressed by the sign bit (Sign(|R|-|I|)) and by the outputs of 3 comparators 808, 904, and 905, which compare their inputs with the values 11, 17, and 5 respectively. 5-bit switches choose between |R| and |I| based on the value of Sign(|R|-|I|), with the inputs of switch 907 being reversed in comparison with the inputs of switch 906. When Sign(|R|-|I|) is positive, switch 906 chooses |R| and switch 907 chooses |I|. When Sign(|R|-|I|) is negative, switch 906 chooses |I| and switch 907 chooses |R|. The comparators 808, 906, 907 each give a one bit output which indicates whether the input is greater than or less than the respective numerical values, 11, 17, and 6. These outputs, in conjunction with Sign(|R|-|I|) are used to address the ROM 903, which outputs the appropriate constellation points in response to its inputs. In embodiment the points (14,3), (15, 9), (20,4), (3,14), (9,15), and (4,20) are decoded by the ROM 903 according to the following table: 
     
                       TABLE VII                                                   
______________________________________                                    
                             choice by                                    
                                    choice by                             
              Sign(|R|-                                 
                       ||R|-|I.vertlin
                       e.|                                       
                             906 and                                      
                                    907 and                               
                                           point                          
|R|                                                     
      |I|                                               
              |I|)                                      
                       &gt;11   &gt;17    &gt;6     output                         
______________________________________                                    
0-6   12-17   -        true and                                           
                             |I|                        
                                    |R|                 
                                           (3,14)                         
                       false false  false                                 
 0-12 &gt;17     -        false |I|                        
                                    |R|                 
                                           (4,20)                         
                             true   true or                               
                                    false                                 
 6-12 &gt;12     -        true  |I|                        
                                    |R|                 
                                           (9,15)                         
                             true or                                      
                                    true                                  
                             false                                        
&gt;12    6-12   +        false |R|                        
                                    |I|                 
                                           (15,9)                         
                             true or                                      
                                    true                                  
                             false                                        
12-17 0-6     +        true or                                            
                             |R|                        
                                    |I|                 
                                           (14,3)                         
                       false false  false                                 
&gt;17    0-12   +        true  |R|                        
                                    |I|                 
                                           (20,4)                         
                             true   true or                               
                                    false                                 
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     FIG. 10 shows a decoder for decoding Cons. B. Like elements have been given the same reference numerals as in previous figures. ROM 1001 has been reduced in size to 2 4  ×3×2. Rom 1001 uses the same decision table as ROM 902. However, the number of bits is decreased because the last row outputs point (16,16) rather than (15,15); and (16,16) can be expressed with three binary zeroes, while (15,15) requires four binary ones. Output 1002 now has only 6 bits rather than 8 as before. Switch 1003 has only 8 bits of output rather than 10 as before. Switch 1004 now has 5 bits of output to allow multiplication by two of its output. Comparator 1006 compares to 12, rather than 11 and comparator 1007 compares to 27 rather than to 5. Additional logic 1005 has been added, so that the input to comparator 1007 is the sum of the outputs of switches 906 and 1004. ROM 1008 decides according to the following table: 
     
                       TABLE VIII                                                  
______________________________________                                    
                  choice by 906                                           
                             choice by 1004                               
                                      point                               
Sign(|R|-|I|)                         
        ||R|-|I|| &gt; 
        12        and &gt;17    and &gt;27  output                              
______________________________________                                    
-       true and  |I|                                   
                             |R|                        
                                      (4,14)                              
        false     false      false                                        
-       false     |I|                                   
                             |R|                        
                                      (4,20)                              
                  true       true or false                                
-       true      |I|                                   
                             |R|                        
                                      (8,16)                              
                  true or false                                           
                             true                                         
+       false     |R|                                   
                             |I|                        
                                      (16,8)                              
                  true or false                                           
                             true                                         
+       true or false                                                     
                  |R|                                   
                             |I|                        
                                      (14,4)                              
                  false      false                                        
+       true      |R|                                   
                             |I|                        
                                      (20,4)                              
                  true       true or false                                
______________________________________                                    
 
    
     ROM 1008 now has 16×4×2 bits. Thus Cons. B has considerably reduced decoding complexity when compared to Cons. A.