Abstract:
The present invention provides a low-latency arc-tangent calculation structure and a calculation method thereof. The arc-tangent calculation structure comprises two lookup tables, a subtractor, a sign comparator, a numerical comparator and a shift encoder. The present invention divides the coordinate system into a plurality of sectors for simplifying the lookup tables. The first lookup table is used to perform logarithmic transformation so as to replace a divider with a subtractor. The second lookup table integrates an exponentiation table and an angle table to translate the output of the subtractor into arc-tangent value θ. Then, θ is shifted to a correct angle according to the output of the shift encoder.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    1. Field of the Invention 
         [0002]    The present invention relates to an arc-tangent calculation method, particularly to a low-latency and divider-free arc-tangent calculation structure and a calculation method thereof. 
         [0003]    2. Description of the Related Art 
         [0004]    The arc-tangent function is expressed by 
         [0000]    
       
         
           
             
               
                 tan 
                 
                   - 
                   1 
                 
               
                
               
                 ( 
                 
                   y 
                   x 
                 
                 ) 
               
             
             . 
           
         
       
     
         [0000]    The calculation method thereof normally needs a divider, and the division result is used to find a corresponding angle from a lookup table. However, the divider is bulky, needs complicated hardware, and occupies considerable area. 
         [0005]    In 1959, Volder published in IEEE a related paper “The CORDIC Trigonometric Computing Technique” that calculates the angle of an arc-tangent function without using any divider, wherein the coordinate system is divided into two parts, and wherein the part where the angle exists is determined. Next, the determined part is further divided into sub-parts again, and the sub-part where the angle exists is determined. The abovementioned dividing process is repeated until the angle is precisely defined. However, the prior art consumes too much time and greatly reduces the system efficiency. 
         [0006]    Accordingly, the present invention proposes a low-latency arc-tangent calculation structure and a calculation method thereof to overcome the abovementioned problems. The architecture and embodiments thereof will be described in detail below. 
       SUMMARY OF THE INVENTION 
       [0007]    The primary objective of the present invention is to provide a low-latency arc-tangent calculation structure and a calculation method thereof, wherein the coordinate system is divided into a plurality of parts, and wherein the part where the angle exists is determined to exclude the other part irrespective of the angle, whereby the quantity of lookup calculations is greatly reduced. 
         [0008]    Another objective of the present invention is to provide a low-latency arc-tangent calculation structure and a calculation method thereof, wherein the divider is exponentially converted into a subtractor, whereby is reduced calculation time and hardware complexity. 
         [0009]    To achieve the abovementioned objectives, the present invention proposes a low-latency arc-tangent calculation structure, which is used to work out an arc-tangent value of the ratio of an X-axis value I to a Y-axis value Q, and which comprises two lookup tables; two multiplexers respectively determining the order that the X-axis value I and the Y-axis value Q enter into the lookup table, and determining the signs of the comparison results of the second lookup table; at least one comparator determining the signs and magnitudes of the X-axis value I and the Y-axis value Q; a control unit determining the sector where the point corresponding to the X-axis value I and the Y-axis value Q exists; a shift encoder determining a shift angle according to the output of the comparator; and an adder adding the shift angle output by the shift encoder to the output of the multiplexer. 
         [0010]    The present invention also proposes a low-latency arc-tangent calculation method, which comprises steps: using a comparator to determine the signs and magnitudes of an X-axis value I and a Y-axis value Q; using two absolute generators to generate the absolute values of the X-axis value I and the Y-axis value Q; using a first lookup table to obtain the logarithms of the X-axis value I and the Y-axis value Q, and using a second lookup table to obtain the result of exponentiation with the base being the Euler&#39;s number and the exponent being the difference of the two logarithms, and obtain an angle corresponding to the arc-tangent value of the exponentiation result; using a control unit to determining the sector where the point corresponding to the X-axis value I and the Y-axis value Q exists; using a multiplexer to determine the sign of the output of the second lookup table according to the output of the control unit; using a shift encoder to determine a shift angle according to the output of the comparator; and adding the output of the shift encoder to the output of the multiplexer. 
         [0011]    Below, the embodiments are described in detail to make easily understood the objectives, technical contents, characteristics and accomplishments of the present invention. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0012]      FIG. 1  is a diagram schematically showing an arc-tangent calculation structure according to one embodiment of the present invention; 
           [0013]      FIG. 2  is a flowchart of a low-latency arc-tangent calculation method according to one embodiment of the present invention; and 
           [0014]      FIG. 3  is a diagram schematically showing that a coordinate system is divided into eight sectors to calculate the arc-tangent value of a point (I i , Q i ) according to one embodiment of the present invention. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0015]    The present invention proposes a low-latency arc-tangent calculation structure and a calculation method thereof, wherein the present invention divides a coordinate system into a plurality of sectors, based on the characteristic of the arc-tangent function, and wherein the present invention completely replaces division operations with subtraction operations via logarithmic and exponential transformations, and wherein the present invention finds out the sector where the angle exists, whereby the calculation time is greatly reduced. 
         [0016]    In one embodiment, the X-Y coordinate system is equally divided into eight sectors, and each sector has an angle of 45 degrees. In such a case, the arc-tangent function can be expressed by Equation (1): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
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         [0017]    Refer to  FIG. 1  a diagram schematically showing an arc-tangent calculation structure for working out an angle corresponding to the arc-tangent value of a ratio of an X-axis value I to a Y-axis value Q. The structure comprises two absolute-value generators  10  and  10 ′, a first multiplexer  12 , a first lookup table  14 , a register  16 , a subtractor  18 , a second lookup table  20 , a complementer  22 , a second multiplexer  24 , an adder  26 , a sign comparator  28 , a numerical comparator  30 , a control unit  32  and a shift encoder  34 . 
         [0018]    The two absolute-value generators  10  and  10 ′ respectively generate the absolute values of the X-axis value I and the Y-axis value Q, and send the absolute values to the first multiplexer  12  and the numerical comparator  30 . The first multiplexer  12  determines the order of absolutely X-axis value I and absolutely Y-axis value Q go to the first look up table  14  according which number is smaller. The first look up table  14  works out the logarithms of the X-axis value I and the Y-axis value Q. The worked out logarithms are stored in the register  16 . The subtractor  18  calculates the difference between the logarithms of the X-axis value I and the Y-axis value Q. The second lookup table  20  includes an exponentiation lookup table and an angle lookup table. From the second lookup table is obtain the exponentiation result of the abovementioned difference. Further from the second lookup table is obtained an arc-tangent value (an angle) corresponding to the exponentiation result. The complementer  22  adds a minus sign to the output of the second lookup table  20 . Suppose that the output of the second lookup table  20  is a positive value. After the complementer  22  adds a minus sign to the positive value, the positive value is converted into a negative value. Thereby, the second multiplexer  24  simultaneously receives a positive input and a negative input. The second multiplexer  24  calculates to determine whether to output a positive value or a negative value. The sign comparator  28  and the numerical comparator  30  respectively determine the signs and magnitudes of the X-axis value I and the Y-axis value Q. The X-axis value I and the Y-axis value Q are directly sent to the sign comparator  28  before being processed by the absolute-value generators  10  and  10 ′. The X-axis value I and the Y-axis value Q are sent to the numerical comparator  30  after being processed by the absolute-value generators  10  and  10 ′. The two comparators  28  and  30  may be regarded as a single comparator. The control unit  32  determines the sector where a point corresponding to the X-axis value I and the Y-axis value Q exists. In one embodiment, the coordinate system is divided into eight sectors each having an angle of 45 degrees. The shift encoder  34  determines the shift angle according to the outputs of the sign comparator  28  and the numerical comparator  30 . The adder  26  adds the shift angle output by the shift encoder  34  to the output of the second multiplexer  24 . 
         [0019]    Refer to  FIG. 2  for a flowchart of a low-latency arc-tangent calculation method. In Step S 10 , input an X-axis value I and a Y-axis value Q. In Step S 12 , use two absolute-value generators to respectively generate the absolute values of the X-axis value I and the Y-axis value Q. In Step S 14 , use a first look up table to convert the absolute values of the X-axis value I and the Y-axis value Q into the logarithms thereof (log I and log Q), store the logarithms in a register, and use a subtractor to calculate the difference of the logarithms (log Q−log I). According to the absolutely value of Q and I, it can be either log Q−log I or log I−log Q. In Step S 16 , find out an exponentiation result e log Q−log I  of the logarithm difference (log Q−log I) from a second lookup table, and find out an angle corresponding to the arc-tangent value of the exponentiation result from the second lookup table. Both the exponentiation result and angle are positive values. A complementer adds minus signs to the positive values to obtain two negative values. The positive and negative values are input to a second multiplexer. 
         [0020]    On the other hand, after the values of the X-axis value I and the Y-axis value Q are input in Step S 10 , use a comparator to determine the signs and magnitudes of the X-axis value I and the Y-axis value Q in Step S 18 . In the embodiment shown in  FIG. 1 , a sign comparator is used to determine the signs of the X-axis value I and the Y-axis value Q, and a numerical comparator is used to determine whether the magnitude of the X-axis value is smaller or greater than that of the Y-axis value. In the present invention, the coordinate system is divided into a plurality of sectors, for example, eight sectors each having an angle of 45 degrees. In Step S 20 , use a control unit to determine the sector where a point corresponding to the X-axis value I and the Y-axis value Q exists. In other words, determine the magnitudes of ∥I i ∥ and ∥Q i ∥, and {sign(I i ), sign(Q i )}. Thereby, the second multiplexer can determine the sign of the arc-tangent value and whether to add a minus sign to tan −1  shown in Equation (1) according to the magnitudes of ∥I i ∥ and ∥Q i ∥, and {sign(I i ), sign(Q i )}. Similarly, a shift encoder determines the shift angle according to the magnitudes of ∥I i ∥ and ∥Q i ∥, and {sign(I i ), sign(Q i )}. Then, in Step S 22 , add the shift angle output by the shift encoder to the output of the second multiplexer to obtain the arc-tangent value of (I i , Q i ). 
         [0021]    Refer to  FIG. 3  a diagram schematically showing a coordinate system divided into eight sectors according to one embodiment of the present invention. In this embodiment, each sector has an angle of 45 degrees. The eight sectors are respectively the first, second eighth, seventh, third, fourth, sixth and fifth sectors in counterclockwise sequence from the positive X axis. Below is demonstrated how to calculate the angle corresponding 
         [0000]    
       
         
           
             
               
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         [0000]    From the position where the point (I i , Q i ) is located in the coordinate system, it is known that ∥I i ∥&lt;∥Q i ∥ and that I i  has a positive value and Q i  has a negative value, i.e. {sign(I i ), sign(Q i )}={+, −}. Therefore, the point (I i , Q i ) is located in the sixth sector. Thus, the shift angle is 270 degrees, and the angle corresponding to 
         [0000]    
       
         
           
             
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         [0000]    has a value of tan −1 (e log|I     i     |−log|Q     i     | )+270°. 
         [0022]    In conclusion, the present invention proposes a low-latency arc-tangent calculation structure and a method thereof, which replaces a divider with a subtractor to reduce calculation time and decrease the area occupied by hardware. In the present invention, the coordinate system is divided into a plurality of sectors, whereby the corresponding angle can be worked out with only two lookup tables, and whereby the invention can further reduce calculation complexity and promote system efficiency. 
         [0023]    The embodiments described above are only to exemplify the present invention but not to limit the scope of the present invention. Any equivalent modification or variation according to the spirit of the present invention is to be also included within the scope of the present invention.