CN114816333A - A Method of Realizing Arctangent Calculation Based on CORDIC Algorithm - Google Patents

Abstract

The present application relates to a method for realizing arctangent calculation based on the CORDIC algorithm. The method firstly processes the input (x, y) through the quadrant preprocessing module and the absolute value preprocessing module in sequence, and then iterates through the CORDIC algorithm module. Calculate to realize arctangent calculation; the quadrant preprocessing module first judges the input (x, y) to see if it needs to be converted, and ensures that the input value after completing this step is located in the first quadrant; the absolute The value preprocessing module first makes (x, y) corresponding to the above-mentioned quadrant preprocessing module respectively corresponding to binary numbers, to ensure that the absolute values of x and y are both less than or equal to their maximum absolute values and greater than half of their maximum absolute values. The method of the invention can significantly improve the range and precision of arctangent calculation realized by FPGA, and is simple in operation and easy to transplant.

Description

A Method of Realizing Arctangent Calculation Based on CORDIC Algorithm

technical field

The present application relates to the field of arctangent calculation, and in particular, to a method for realizing arctangent calculation based on a CORDIC algorithm.

Background technique

The application fields of trigonometric functions and inverse trigonometric functions are very wide, including the field of robot kinematics, image processing, navigation digital processing and calculators. The traditional inverse trigonometric function solving methods include ROM look-up table method, polynomial approximation method and iterative method, etc., but all have the disadvantages of low speed, few ROM hardware resources, low precision and inconvenient implementation. In the prior art, the CORDIC algorithm theory basically solves the above problems. The algorithm is compatible with the advantages of high speed, high precision, and convenient implementation of hardware resources, and has been widely used and popularized. However, the use of CORDIC algorithm for arctangent calculation has shortcomings in angle measurement range and measurement accuracy.

SUMMARY OF THE INVENTION

In view of the deficiencies of the prior art, the present application proposes a method for realizing arctangent calculation based on the CORDIC algorithm.

A method for realizing arctangent calculation based on the CORDIC algorithm of the present application, the method firstly processes the input (x, y) through the quadrant preprocessing module and the absolute value preprocessing module in sequence, and then iterates through the CORDIC algorithm module Calculated to achieve arctangent calculation;

The quadrant preprocessing module first judges the input (x, y) to see if it needs to be converted to ensure that the input value after completing this step is located in the first quadrant; the quadrant preprocessing module can make the arctangent The range of calculations is increased.

The absolute value preprocessing module first makes the above-mentioned (x, y) through the quadrant preprocessing module correspond to binary numbers respectively, and judges whether the absolute values of x and y satisfy simultaneously less than or equal to its maximum absolute value according to its binary digits, and is greater than half of its maximum absolute value; if the above conditions are not met, double x and y at the same time, and continue to judge until the above conditions are met. The absolute value preprocessing module can greatly improve the accuracy of the calculation anyway.

Further, both x and y in (x, y) input by the method described in this application are represented by signed numbers. In this way, the quadrant where (x, y) is located can be easily determined based on the highest bits of x and y, and it is convenient for the quadrant preprocessing module to process.

Further, the specific processing process of the absolute value preprocessing module in the method described in this application is:

1) let the binary digits corresponding to described x and y be m and n respectively, then the absolute values of x and y are respectively 2 ^(m-1) and 2 ^(n-1) at maximum;

2) Determine whether the absolute value of x is greater than 2 ^(m-2) and less than or equal to 2 ^(m-1) , and whether the absolute value of y is greater than 2 ^(n-2) and less than or equal to 2 ^(n-1) ;

3) If step 2) is satisfied, then x, y remain unchanged; if step 2) is not satisfied, proceed to following step 4);

4) Double x and y at the same time, and obtain the absolute values of x and y after the magnification;

5) Repeat steps 2), 3), and 4) until step 2) is satisfied.

Further, the binary digits corresponding to x and y in the method described in this application are both 16 bits, and it is necessary to judge whether the absolute values of x and y are both greater than 16384 and less than or equal to 32768; if the above conditions are met, then x and y remain unchanged. ; If not satisfied, double x and y at the same time, and continue to compare their absolute values with 16384 and 32768. If it is still not satisfied with greater than 16384 and less than or equal to 32768, continue to enlarge until the absolute values of x and y are greater than 16384 and less than or equal to 32768.

Further, the specific processing process of the quadrant preprocessing module in the method described in this application is as follows: if (x, y) is in the first quadrant, the data remains unchanged; if (x, y) is in the second quadrant, (x, y) is in the second quadrant. y) is preprocessed as (y,-x), that is, rotated 90° clockwise; if (x,y) is in the third quadrant, (x,y) is preprocessed as (-x,-y), that is, clockwise Rotate 180°; if (x,y) is in the fourth quadrant, preprocess (x,y) to (-y,x), that is, rotate 270° clockwise.

Further, the CORDIC algorithm module in the method described in this application specifically performs 15 iterative calculations to realize arctangent calculation based on FPGA.

Further, the specific processing process of the CORDIC algorithm module in the method described in this application is:

Rotate the vector (x, y) with a fixed angle θ _i each time, the rotation direction is d _i =sign(y(i)), the target of the rotation is the x-axis, and after N iterations, there is an arc tangent angle:

The rotation angle θ _i satisfies:

2 ^-i = tanθ _i (2)

The iterative equation of the simplified iterative shift-add algorithm is:

z after N iterations is the settlement of the arctangent calculation, namely:

z=arctan(y/x) (4).

Further, the calculation range in the method described in the present application is 0° to 360°.

Further, the calculation error in the method described in this application is ≤0.01°.

The beneficial effects of the present application include: providing a method for realizing arc tangent calculation based on CORDIC algorithm, by adding two preprocessing modules before CORDIC algorithm to realize the optimization of arc tangent calculation angle range and accuracy. The range of arc tangent calculation can be increased from -99.99°～99.99° to 0°～360°, and the arc tangent calculation error is reduced from 8.01° to 0.01°, which significantly improves the range and accuracy of FPGA to realize arc tangent calculation. In addition, the hardware cost required by the method described in the present application is low, the operation is simple, and the transplantation is easy.

Description of drawings

Fig. 1 is the system block diagram of the method of the present invention;

Fig. 2 is the vector rotation schematic diagram in the comparative example 1 of the present invention;

Fig. 3 is the angle output situation after inputting four quadrants x, y in the method of the present invention;

FIG. 4 is the variation of the output angle in the second embodiment of the present invention.

Detailed ways

In order to make the purposes, technical solutions and advantages of the specific embodiments of the present application more clear, the technical solutions in the specific embodiments of the present application will be described clearly and completely below. If the specific conditions are not indicated in the specific implementation manner, the conventional conditions or the conditions suggested by the manufacturer are used.

Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art. The terms used herein in the specification are for the purpose of describing particular embodiments only and are not intended to limit the application. As used herein, the term "and/or" includes any and all combinations of one or more of the associated listed items.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS As shown in FIG. 1 , a method for realizing arctangent calculation based on CORDIC algorithm described in this application, firstly, the input (x, y) is processed through the quadrant preprocessing module and the absolute value preprocessing module in turn. , and then perform iterative calculation through the CORDIC algorithm module to realize arctangent calculation; both x and y in the input (x, y) are represented by signed numbers. The method described in this application can increase the arctangent calculation range from -99.99° to 99.99° to 0° to 360°, and reduce the arctangent calculation error from 8.01° to 0.01°, which significantly improves the range and accuracy of FPGA arctangent calculation. .

Comparative Example 1:

The CORDIC algorithm and its shortcomings: As shown in Figure 2, the vector (x ₀ , y ₀ ) is rotated counterclockwise by an angle θ in the coordinate plane to obtain the vector (x ₁ , y ₁ ), and the coordinate relationship between the two vectors is:

Divide both sides by cosθ to get the pseudo-rotation equation, namely:

After dividing by cosθ, the rotation angle does not change, but the modulo length of the vector changes, which has no effect on the angle we solve. The essence of the CORDIC algorithm to find the arc tangent is to rotate the vector (x, y), each time a fixed angle θ _i is rotated, the rotation direction is d _i =sign(y(i)), the target of the rotation is the x-axis, and N times Iterating, there is an arctangent:

In order to facilitate the implementation of the CORDIC algorithm on the hardware platform FPGA, the rotation angle θ _i should satisfy:

2 ^-i = tanθ _i (4)

Therefore, the multiplication operation can be completed by a simple shift, so the original algorithm can be simplified to an iterative shift-add algorithm, and the iterative equation is:

z after N iterations is the settlement of the arctangent calculation. which is:

z=arctan(y/x) (6)

But this algorithm has two shortcomings. On the one hand, because the rotation angle θi needs to satisfy the formula (4), the calculation range of the arc tangent angle can only be -99.99°～99.99°; During the shifting process of , xi and yi will directly become 0, which will greatly increase the error of arctangent calculation.

Example 1:

A method for realizing arctangent calculation based on the CORDIC algorithm described in this application, firstly, the input (x, y) is processed by the quadrant preprocessing module and the absolute value preprocessing module in turn, and then iterative calculation is performed by the CORDIC algorithm module To realize arctangent calculation; both x and y in the input (x, y) are represented by signed numbers.

The quadrant preprocessing module first judges the input (x, y) to see if it needs to be converted to ensure that the input value after completing this step is located in the first quadrant; the specific processing process is: if (x, y) ) in the first quadrant, the data remains unchanged; if (x,y) is in the second quadrant, preprocess (x,y) into (y,-x), that is, rotate 90° clockwise; if (x, y) y) In the third quadrant, preprocess (x,y) to (-x,-y), that is, rotate 180° clockwise; if (x,y) is in the fourth quadrant, preprocess (x,y) is (-y,x), which is a 270° clockwise rotation.

The absolute value preprocessing module first makes the above-mentioned (x, y) through the quadrant preprocessing module correspond to binary numbers respectively, and judges whether the absolute values of x and y satisfy simultaneously less than or equal to its maximum absolute value according to its binary digits, and is greater than half of its maximum absolute value; if the above conditions are not met, double x and y at the same time, and continue to judge until the above conditions are met.

The specific processing process of the absolute value preprocessing module is as follows:

1) Let the binary digits corresponding to the x and y be m and n respectively, in the present embodiment 1, m and n are both 16, then the absolute values of x and y are both 32768;

2) Determine whether the absolute value of x and y is greater than 16384 and less than or equal to 32768;

3) If step 2) is satisfied, then x, y remain unchanged; if step 2) is not satisfied, proceed to following step 4);

4) Double x and y at the same time, and obtain the absolute values of x and y after the magnification;

5) Repeat steps 2), 3), and 4) until step 2) is satisfied.

The specific processing process of the CORDIC algorithm module in the method is:

Rotate the vector (x, y) with a fixed angle θ _i each time, the rotation direction is d _i =sign(y(i)), the target of the rotation is the x-axis, and after N iterations, there is an arc tangent angle:

The rotation angle θ _i satisfies:

2 ^-i = tanθ _i (2)

The iterative equation of the simplified iterative shift-add algorithm is:

z after N iterations is the settlement of the arctangent calculation, namely:

z=arctan(y/x) (4).

In this embodiment, the CORDIC algorithm module specifically performs 15 iterative calculations to realize arctangent calculation based on FPGA. It can be seen from FIG. 3 that in the method of the present invention, after inputting (x, y) of the four quadrants, the angle output is correct.

Embodiment 2: In this embodiment, the only difference from Embodiment 1 is that the input (x, y) is specifically (8, 10). It can be seen from FIG. 4 that in the second embodiment, the angle error is reduced from 8.01° to 0.01°.

The above-described embodiments are some, but not all, embodiments of the present application. The detailed descriptions of the embodiments of the application are not intended to limit the scope of the application as claimed, but are merely representative of selected embodiments of the application. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present application.

Claims (9)

1. A method for realizing arc tangent calculation based on CORDIC algorithm is characterized in that the method comprises the steps of firstly, sequentially processing input (x, y) by a quadrant preprocessing module and an absolute value preprocessing module, and then performing iterative calculation by a CORDIC algorithm module to realize the arc tangent calculation;

the quadrant preprocessing module firstly judges the input (x, y) to see whether the input (x, y) needs to be converted or not, and ensures that the input value after the step is finished is positioned in a first quadrant;

the absolute value preprocessing module firstly makes the (x, y) passing through the quadrant preprocessing module respectively correspond to binary numbers, and judges whether the absolute values of the x and the y simultaneously meet the requirement that the absolute values are less than or equal to the maximum absolute value and are more than half of the maximum absolute value according to the binary numbers; if the conditions are not met, the x and the y are amplified by one time at the same time, and the judgment is continued until the conditions are met.

2. The method for performing arctan calculation based on CORDIC algorithm of claim 1 wherein x and y of (x, y) of the input are both expressed with signed numbers.

3. The method for realizing arctangent calculation based on the CORDIC algorithm as claimed in claim 2, wherein the specific processing procedure of the absolute value preprocessing module is as follows:

1) setting the binary digit numbers corresponding to x and y as m and n respectively, the absolute values of x and y are 2 respectively ^(m-1) And 2 ^{(n -1)} ；

2) Judging whether the absolute value of x is larger than 2 ^(m-2) And is less than or equal to 2 ^(m-1) While the absolute value of y is greater than 2 ^(n-2) And is not more than 2 ^(n-1) ；

3) If step 2) is satisfied, x and y are kept unchanged; if the step 2) is not satisfied, the following step 4) is carried out;

4) simultaneously amplifying x and y by one time, and obtaining absolute values of the amplified x and y;

5) repeating the steps 2), 3) and 4) until the step 2) is met.

4. The method for realizing arctangent calculation based on the CORDIC algorithm of claim 3, wherein the number of binary digits corresponding to x and y is 16, and it is determined whether the absolute values of x and y are both greater than 16384 and less than or equal to 32768; if the above conditions are satisfied, x and y remain unchanged; if not, simultaneously amplifying x and y by one time and continuously comparing the absolute values with 16384 and 32768, and if still not, continuously amplifying until the absolute values of x and y are both greater than 16384 and less than or equal to 32768.

5. The method for realizing arctangent calculation based on the CORDIC algorithm as claimed in claim 4, wherein the specific processing procedure of the quadrant preprocessing module is as follows: if (x, y) is in the first quadrant, the data remains unchanged; if (x, y) is in the second quadrant, (x, y) is preprocessed to (y, -x), i.e., rotated 90 ° clockwise; if (x, y) is in the third quadrant, (x, y) is preprocessed to (-x, -y), i.e., rotated 180 ° clockwise; if (x, y) is in the fourth quadrant, (x, y) is preprocessed to (-y, x), i.e., rotated 270 clockwise.

6. The method for realizing arctangent calculation based on the CORDIC algorithm of claim 5, wherein the CORDIC algorithm module performs 15 iterative calculations to realize the arctangent calculation based on the FPGA.

7. The method for realizing arctangent calculation based on the CORDIC algorithm of claim 6, wherein the specific processing procedure of the CORDIC algorithm module is as follows:

rotating the vector (x, y) by a fixed angle theta each time _i In the direction of rotation d _i Sign (y (i)), the target of rotation is the x-axis, N iterations are performed, and there is an arctangent:

angle of rotation theta _i Satisfies the following conditions:

2 ^-i ＝tanθ _i (2)

the iteration equation of the simplified iterative shift-add algorithm is as follows:

z after N iterations is the settlement of the arctan calculation, i.e.:

z＝arctan(y/x) (4)。

8. the method for performing arctan calculations based on CORDIC algorithm of claim 7 wherein the calculation range is 0-360 °.

9. The method for performing arctan calculations based on CORDIC algorithm of claim 8, wherein the error of the method is less than or equal to 0.01 °.

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