Application Note, V1.0, Feb. 2007 AP16105 XC166 family Software implementation of Trigonometric functions using CORDIC Algorithm Microcontrollers Edition 2007-06-21 Published by Infineon Technologies AG 81726 München, Germany © Infineon Technologies AG 2007. All Rights Reserved. LEGAL DISCLAIMER THE INFORMATION GIVEN IN THIS APPLICATION NOTE IS GIVEN AS A HINT FOR THE IMPLEMENTATION OF THE INFINEON TECHNOLOGIES COMPONENT ONLY AND SHALL NOT BE REGARDED AS ANY DESCRIPTION OR WARRANTY OF A CERTAIN FUNCTIONALITY, CONDITION OR QUALITY OF THE INFINEON TECHNOLOGIES COMPONENT. THE RECIPIENT OF THIS APPLICATION NOTE MUST VERIFY ANY FUNCTION DESCRIBED HEREIN IN THE REAL APPLICATION. INFINEON TECHNOLOGIES HEREBY DISCLAIMS ANY AND ALL WARRANTIES AND LIABILITIES OF ANY KIND (INCLUDING WITHOUT LIMITATION WARRANTIES OF NON-INFRINGEMENT OF INTELLECTUAL PROPERTY RIGHTS OF ANY THIRD PARTY) WITH RESPECT TO ANY AND ALL INFORMATION GIVEN IN THIS APPLICATION NOTE. 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AP16105 CORDIC AP16105 Revision History: 2007-02 Previous Version: none Page V1.0 Subjects (major changes since last revision) We Listen to Your Comments Any information within this document that you feel is wrong, unclear or missing at all? Your feedback will help us to continuously improve the quality of this document. Please send your proposal (including a reference to this document) to: [email protected] Application Note 3 V1.0, 2007-02 AP16105 CORDIC Table of Contents Page 1 Introduction ...................................................................................................................................5 2 2.1 2.2 The CORDIC Arithmetic Technique.............................................................................................5 Rotation Mode .................................................................................................................................6 Vectoring Mode ...............................................................................................................................6 3 3.1 3.2 Computation of Complex magnitude ..........................................................................................7 Implementation................................................................................................................................7 Results and discussion ...................................................................................................................9 4 4.1 4.2 4.3 Computation of Sine and Cos for an input angle ....................................................................11 Implementation..............................................................................................................................11 Pseudo code .................................................................................................................................12 Results and Discussion.................................................................................................................14 5 Numerical Error in CORDIC........................................................................................................14 6 Conclusions.................................................................................................................................15 7 Acknowledgements ....................................................................................................................15 8 References ...................................................................................................................................15 Application Note 4 V1.0, 2007-02 AP16105 CORDIC The CORDIC Arithmetic Technique 1 Introduction Trigonometric functions are often used in embedded systems. Motor drive control applications such as park transform, Clarke transform, PWM generation uses trigonometric functions extensively. Various methods exist to compute the trigonometric functions. These include Taylor series, Curve fitting algorithms, CORDIC algorithm. This article describes software implementation of the following fixed point trigonometric routines using CORDIC Algorithm for Infineon’s XC164CS Microcontroller with MAC unit. The implementation of the algorithm is examined concerning accuracy and efficiency. • Complex Magnitude • Sine • Cosine Routines are provided for signed two’s complement arithmetic. First a brief description of the theory behind the algorithm is presented. Then the theory is extended to the implementation of algorithm in XC164CS Processor after which the numerical errors that occur in the fixed point implementation is discussed. 2 The CORDIC Arithmetic Technique The Coordinate Rotation DIgital Computer (CORDIC) algorithm is an iterative technique proposed by Volder in 1956. This algorithm can be a very powerful tool in areas where arithmetic or trigonometric function evaluation is heavily utilized, such as digital signal processing, motor control. The general vector rotational transform rotates a plane vector [X, Y] by an angle θ to produce a new vector point [Xi+1, Yi+1] as in (1) and (2). The CORDIC Rotation is achieved by the same principle. It rotates the point [X, Y] in series of steps, which are smaller than θ . This rotation may be in anti clockwise direction (increase in θ ) or clock wise direction (decrease in θ ). Suppose if we wish to achieve a total rotation of 35 o , we may rotate our point 30 o anticlockwise, followed by 10 o anticlockwise, followed by 5 o clockwise. Xi+1=Xi cos θ -Yi sin θ Yi+1= Yi cos θ + Xi sin θ (1) (2) These can be expressed as Xi+1=cos θ *[Xi - Yi tan θ ] (3) Yi+1=cos θ *[Yi + Xi tan θ ] (4) The reason for this simplification is to break down the rotation ( θ ) into many steps, each of decreasing size and has each step such that tan θ is the power of 2, where θ is the rotational angle. The first seven steps of the set of rotations are shown below. Table 1 CORDIC Rotation i θ Tan 0 45 1 1 26.565 0.49999 2 14.036 0.24999 3 7.125 0.12499 4 3.516 0.06249 5 1.784 0.03114 6 0.895 0.01562 Application Note 5 θ -i (Decimal) = 2 V1.0, 2007-02 AP16105 CORDIC The CORDIC Arithmetic Technique This would allow us to implement the multiplication by tan θ as a simple bit shift operation (2 ). Hence (3) and (4) reduces to -i Xi+1=cos θ *[Xi - Yi 2 ] (5) Yi+1=cos θ *[Yi + Xi 2 ] (6) -i -i From Table (1) it is clear that the total rotation step is 99.88 o . Since the rotation can be of clockwise or o anticlockwise direction these steps are used to approximate angles between +99.88 to -99.88 o . For o mathematical simplicity the rotation angles are limited to –90 and +90 . For rotation angles greater than +/90 o additional rotation is required. The cos θ term (K) is a constant which approaches to 0.6073 after ‘n’ iterations. The angle θ is accumulated in Zi+1 Zi+1= Zi - -1 -i δi * tan (2 ) Where δi determines the direction of rotation. The CORDIC Rotator is operated in one of the two modes • Rotation Mode • Vector mode The rotation mode rotates the input vector to a specified angle. The vectoring mode rotates the input vector to x axis while recording the angle required to make the rotation (i.e.) the direction of rotation is opposite in both the modes. The complex magnitude is computed using vectoring mode, the sine and cosine of the input angle is computed using rotation mode. 2.1 Rotation Mode The CORDIC equations for rotation mode is Xi+1=cos θ *[Xi - Yi Yi+1=cos θ *[Yi + Xi Zi+1= Zi Where (7) -i δi 2 ] -i δi 2 ] -1 -i δi * tan (2 ) δi = -1 if Zi <0, +1 otherwise After ‘n’ Iterations, Xn=K*[X0 Cos Z0 - Y0 Sin Z0] Yn=K*[Y0 Cos Z0 + X0 Sin Z0] Zn = 0 2.2 Vectoring Mode The CORDIC equations for vectoring mode is Xi+1=Xi - Yi Yi+1=Yi + Xi Zi+1= Zi Where (8) -i δi 2 -i δi 2 -1 -i δi * tan (2 ) δi = -1 if Yi <0, +1 otherwise After ‘n’ Iterations ( X n = K X 02 + Y 02 ) (9) Yn = 0 Application Note 6 V1.0, 2007-02 AP16105 CORDIC Computation of Complex magnitude Z n = Z0 + tan 3 − 1 Y0 X0 Computation of Complex magnitude The Aim of the algorithm is to calculate the magnitude of a complex number C= X + jY. 2 2 Magnitude of this complex number is given by |C| = X + Y .The Cordic rotator rotates the input vector to angle Zi for aligning the result vector with the x axis (Figure 1). The result of the operation is a rotation angle and the scaled magnitude of the original vector Figure 1 The Complex Plane To extend the region of convergence greater than +/-90 o , the phase is rotated by -90 o if Y is positive and it is rotated by +90 o if Y is negative. The CORDIC Rotation is done with successively smaller values of Z, starting with Z = 45 o . The sign of Y decides to add or subtract the phase. In the rotation process C is multiplied with the Inverse CORDIC gain (K) of 0.6073. 3.1 Implementation The Library is C-callable, hand coded assembly routine written for Infineon’s XC164CS Microcontroller with MAC unit. Tasking tool is used for compilation. For the implementation we make the assumption that inputs are in 1Q15 format. The input data’s are scaled down by 2 to avoid the overflow. The rotational gain K needs to be compensated at some stage (i.e.) compensation can be done before or after the iteration. In order to scale down the input further, the input X is multiplied by the inverse of gain before the iteration. In Tasking tool chain the parameter transforms of the first four arguments of the function will be in R12 to R16. Using C166SV2 instruction set, the micro rotations according to (8) is given below as reference implementation. A fixed point number representation is used for the implementation. The registers R1, R2, R13 are assigned with X, Y, shift value respectively. Label3: MOV R12, #0h MOV R3, R1 MOV R5, R2 ASHR R5, #0fh CMP R5, #0 JMPA cc_NZ, Label1 Application Note 7 V1.0, 2007-02 AP16105 CORDIC Computation of Complex magnitude ; Micro rotation 1 ;I= I+ (Q>>K) MOV R7, R2 ASHR R7, R13 MOV MAH, R1 CoADD R12, R7 CoSTORE R1, MAS ; Q=Q-(I_tmp>>K) ASHR R3,R13 MOV MAH,R2 CoSUB R12,R3 CoSTORE R2,MAS JMPA cc_NZ,Label 2 ; Micro rotation 2 Label1: ;I=I-(Q>>K) MOV R7, R2 ASHR R7,R13 MOV MAH,R1 CoSUB R12,R7 CoSTORE R1,MAS ; Q=Q+ (I_tmp>>K) ASHR R3, R13 MOV MAH, R2 CoADD R12, R3 CoSTORE R2,MAS Label2: ADD R13, #1h CMPD1 R6,#0h JMPR cc_NZ, Label3 The number of iterations is fixed to 15 and the direction of rotation is depended on Y, therefore there is no need to record the degree of rotation (i.e.) the value of Z. This reduces the latency from n + 25 to n + 19 cycles, where ‘n’ is the number of iterations. In figure 2 the program flowchart of the full assembler program of the complex magnitude is shown. Application Note 8 V1.0, 2007-02 AP16105 CORDIC Computation of Complex magnitude 3.2 Results and discussion The result of complex magnitude is shown in Table 2. It is represented in 1Q15 format. For example complex input {30143, 21254} should be read as {0.91928, 0.64862}. This accuracy is obtained by keeping the number of iterations as 15. The error will be significantly reduced if this is implemented in a 32 bit processor with 31 iterations. Table 2 Results of complex magnitude in 1Q15 format Input {Real, imaginary} Output [Scaled down 0.5] Expected Output [Scaled down 0.5] Error [1Q15 Format] {12345,9728} 7856 7859 3 {-13254,-12543 9131 9124 7 {30123,21234} 18431 18427 4 {30143,21254} 18445 18442 3 {-30143,-21254} 18450 18442 8 Table 3 Cycle count and Code size Cycle count Code Size (Bytes) Store State 1 2 Read Input values and Initialization Extension of region of convergence CORDIC Rotation 11 22 8 16 n+19, where ‘n’ Number of Iterations 1 n+40 Cycles 38 Restore state Total 2 80 This Algorithm should be better in code size compared to the traditional Taylor approximation series, but not good in terms of speed [8]. The speed is less due to the ‘n’ Number of iterations. Therefore this algorithm fits into an application which demands less code size. Application Note 9 V1.0, 2007-02 AP16105 CORDIC Computation of Complex magnitude o o XN = K ( ) 2 2 x0 + y 0 YN = 0 Z N = Z0 + tan Figure 2 −1 Y0 X0 Flowchart for the implementation of complex magnitude using CORDIC Application Note 10 V1.0, 2007-02 AP16105 CORDIC Computation of Sine and Cos for an input angle 4 Computation of Sine and Cos for an input angle Sine and cosine of the input angles is calculated using CORDIC. If the initial Y component of rotation transform is set to zero the rotation mode reduces to Xi+1=K* Xi cos Zi (10) Yi+1= K* Xi sin Zi Where, K is the CORDIC Gain. By setting initial X component to 0.60725 the rotation process produces an unscaled version of sine and cosine term. Since the rotational angle is limited to –90 o and +90 o additional rotation is required. This is done by exploiting the symmetry property of the sine wave. The values in other Quadrants are computed by using the relations, Sine (-Z) = -Sine (Z) and Sine (180-Z) = Sine (Z). The absolute value of the input is calculated. If the input is negative (III/IV Quadrant), then sign=1. If absolute value of the input is greater than 1/2 (II/III Quadrant), it is subtracted from 1. If sign=1, the result is negated to give the final sine result. 4.1 Implementation The input vector Z contains the angle in radians between [- π , π ] which is normalized between (-1, 1) in 1Q15 format (Z=Z rad/ π ). For example, 45 o = π /4 is equivalent to Z = ¼ =0.25 (8192 in 1Q15 format. Denormalisation is done in the algorithm. The algorithm is presented in a ‘C’ like pseudo code. Note that the Cos θ constant for this algorithm is -1 i 0.60725. We also assume that the 12 values of tan (1/2 ) are stored as a look up table in 4Q12 format. Using C166SV2 instruction set, the micro rotations according to (7) is given below as reference implementation. A fixed point number representation is used for the implementation. The registers R1, R2, R11 and R13 are assigned with X, Y, Z, shift value respectively. Label3: MOV R12, #0h MOV R3, R1 MOV R5, R2 ASHR R5, #0fh CMP R5, #0 JMPA cc_NZ, Label1 ; Micro rotation 1 ; I=I+ (Q>>K) MOV R7, R2 ASHR R7, R13 MOV MAH, R1 CoADD R12,R7 CoSTORE R1,MAS ; Q=Q-(I_tmp>>K) ASHR R3, R13 MOV MAH, R2 CoSUB R12,R3 CoSTORE R2,MAS ; Z=Z+ atan(K[L]) MOV MAH, R11 CoADD R12,[R4+] Application Note 11 V1.0, 2007-02 AP16105 CORDIC Computation of Sine and Cos for an input angle CoSTORE R11,MAS JMPA cc_NZ, Label 2 ; Micro rotation 2 Label1: ;I=I-(Q>>K) MOV R7, R2 ASHR R7,R13 MOV MAH,R1 CoSUB R12,R7 CoSTORE R1,MAS ; Q=Q+ (I_tmp>>K) ASHR R3, R13 MOV MAH, R2 CoADD R12, R3 CoSTORE R2,MAS ;Z=Z-atan(K[L]) MOV MAH,R11 CoSUB R12,[R4+] CoSTORE R11,MAS Label2: ADD R13, #1h CMPD1 R6,#0h JMPR cc_NZ, Label3 We cannot neglect the angle information as in complex magnitude, since the direction of rotation is dependent on Z Parameter. To denormalise, the input is multiplied by 4DBAh (4Q12 format). Therefore the lookup table value (tan-1 (1/2i)) has to be in 4Q12 format. Due to this the number of iterations is reduced to 12. 4.2 Pseudo code Input vector Z is initialized to the desired angle, Y=0 and X=0.60725.The initialization of X specifies the constant 0.60725 which results from the Cos θ term. { short L; short I, Q; short Z; short tmp_I; short sign; Q=0; I=0.60725; Z=P*pi; //denormalization of input If (Z<0) { sign =1; Application Note 12 V1.0, 2007-02 AP16105 CORDIC Computation of Sine and Cos for an input angle } //If X is in III/IV Quadrant (Extension of region of convergence) If (abs (Z)>0.5) { Z=1-abs (Z); } //CORDIC Rotation For (L = 0; L < 15; L++) { tmp_I = I; If (Z < 0.0) { I += Q >>L; Q -= tmp_I >>L; Z=Z+ tan -1 -L (2 ); // value of tan -1 -L (2 ) is stored in lookup table } else { I -= Q >>L; Q += tmp_I >>L; Z=Z- tan -1 -L (2 ); } } if (sign==0) { I=I; Q=Q; } else { I=-I; Q=-Q; } } The Sine of the desired angle is now present in the variable I and the Cosine of the desired angle is in the variable Q. These outputs are within the integer range –32768 to +32767. Application Note 13 V1.0, 2007-02 AP16105 CORDIC Numerical Error in CORDIC 4.3 Results and Discussion The result of complex magnitude is shown in Table 2. It is represented in 1Q15 format. For example complex input 8192 should be read as 0.25. This accuracy is obtained by keeping the iterations as 12. So we can expect an accurate result in a 32 bit processor. Table 4 Results of Sine computation in 1Q15 format Input 0 8192 16384 24576 32767 -8192 -16384 -24576 Table 5 Output Expected Output Error 1 23169 32766 23172 1 -23169 -32766 -23172 0 23170 32767 23170 0 -23171 -32768 -23171 1 1 1 2 1 2 2 1 Cycle count and Code size Cycle count Code Size (Bytes) Store State 2 4 Read Input values and Initialization Extension of region of convergence CORDIC Rotation 4 8 10 20 n+25, where ‘n’ Number of Iterations 2 n+43 Cycles 50 Restore state Total 5 4 86 Numerical Error in CORDIC The error in CORDIC is split to different factors as approximation error and truncation error [1]. Theoretical realization of CORDIC has infinite iterations which produce the accurate result. But, the practical implementation of CORDIC has finite number of iterations. This is the cause for approximation error. In general CORDIC Algorithm produces one additional bit of accuracy for each iteration. The truncation error is due to the finite word length effect. For example, consider a fixed-point representation of 5 bits, with the lower order 3 bits after the binary point. If xi = 1.2345678, then the approximate representation of this l 0 -1 -2 -3 number is 01001. Hence Q[xi] = 0 * 2 + 1* 2 + 0 * 2 + 0*2 + 1 * 2 = 1 + 0.125 = 1.125. Hence the -3 quantization error due to finite word length is Ei = 1.2345678 - 1.125 = 0.1095678 < 2 (0.125). Due to these errors the precision of CORDIC is affected. A Sine wave of 21 samples with input frequency 50 Hz and sampling frequency of 4000 Hz is generated using Ideal CORDIC and the implemented CORDIC. The magnitude of the difference between the Ideal CORDIC and the implemented CORDIC is shown in figure 3. The X axis represents the input angle which is normalized and the Y axis represents the approximation and the truncation error. Application Note 14 V1.0, 2007-02 AP16105 CORDIC References 6 Conclusions In this paper, fixed point software implementation of CORDIC has been presented. The accuracy has been discussed using the error and the efficiency is calculated using the cycle count and code size. 7 Acknowledgements Thanks to Samuel Ginsberg, Richard for helping me to understand the concept. Thanks to Manoj palat, Raghunath lolur, Sonali nath for their valuable ideas and code review. 8 References [1] Ray Andraka, A Survey of CORDIC algorithms for FPGA based computers [2] Y. H. Hu, “The quantization effects of the CORDIC algorithm,” IEEE Trans. Signal Processing, pp. 834844, Apr. 1992. [3] Sang Yoon Park and Nam Ik Cho, “fixed point error analysis of CORDIC processor based on the variance propagation,” IEEE Transactions on Circuits and Systems I-Fundamental Theory and Applicat, vol. 51 no. 3 pp.573-584, Mar. 2004 [4] Samuel Ginsberg, “Compact and Efficient Generation of Trigonometric Functions using a CORDIC algorithm” [5] http://www.dspguru.com/info/faqs/cordic.htm [6] Infineon Technologies, C166S V2 User manual, 16-Bit Microcontroller V 1.7 Application Note 15 V1.0, 2007-02 AP16105 CORDIC References [7] Guangyu Wang, C166S V2 Lib A DSP Librarary for C166S V2 Core, User’s Manual, V1.1, Sep 2002 [8] TriLib, A DSP Library for Tricore, User’s Manual, V1.2, Jan 2001 Application Note 16 V1.0, 2007-02 http://www. inf ineon.com Published by Infineon Technologies AG