Microtrend Systems Inc. Fixed Point Two s Complement CORDIC Arithmetic on MSP430

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1 Fixed Point Two s Complement CORDIC Arithmetic on MSP430 Titi Trandafir, MSEE, PhD (abd),mba Slide 1

2 Agenda Part 1 General considerations Part 2 Lab Demo using Microtrend Sys development platforms and IAR IDE Slide 2

3 Summary A Cordic primer A little bit of history Iterative function generation Fractional two s complement fix point arithmetic Numbers and angle representation used in calculations Cordic modes (circular functions: sin(x), cos(x), atan(x), arcsin(x), arccos(x) in the first quadrant) 16(12) bit error tradeoffs, machine cycles, msp430 coding examples FPGA Cordic based approach to calculate functions Series expansion and other ways to calculate circular functions Vector mode, hyperbolic functions, exp(x), sqrt(x), ln(x) discussion Other functions calculations using a Cordic engine, multiplication and division coding example for multiplication on MSP430 16x16 fractional arithmetic Slide 3

4 Generalized Binary Cordic x k+1 = x k mδ k y k 2 -k y k+1 = y k + δ k x k 2 -k k=0,1, n z k+1 = z k δ k ε k δ K = ±1 for m = 0 ε K = 2 -K m = 1 ε K = arctg 2 -k 1/k = Π cos ε k m = -1 ε k = arcth 2 -k k 1 = Π cosh ε k Rotation mode (z k 0) 1, if z k 0 δ k = -1, if z k < 0 x 0 given, y 0 = 0, force z 0 = 0 = > y n+1 = x 0 z 0 x 0 = 1/k= ; y 0 = 0, z 0 = θ x n+1 = cos θ y n+1 = sin θ x 0 = 1/k 1 = ; y 0 =0, z 0 = θ x n+1 = cosh θ y n+1 = sinh θ θ Vector Mode (y k 0) 1, if y k < 0 δ k = -1, if y k 0 x 0 given, y 0 given, z 0 = 0 z n+1 = y 0 /x 0 x 0 given, y 0 given, z 0 = 0 z n+1 = arctg y 0 /x 0 x n+1 = k x y 0 2 x 0 given, y 0 given, z 0 = 0 z n+1 =tanh -1 y 0 /x 0 x n+1 = k 1 x 0 2-y 0 2 x 0 = w+1 ; y 0 = w-1 lnw= 2z n+1 x 0 =w+ 025; y 0 = w- 025 w= x n+1 /k 1 Inverse circular functions Arcsin(x), Arccos(x) x i+1 = x i δ i 2-1 y i y i+1 = y i + δ i 2-1 x i z+1 = z i + δ i atan2-1 δ i = +1 if x t -1 if x < t x i+1 = x i - δ i 2 -i y i y i+1 = y i + δ i 2-1 x i z i+1 = z i +δ i atan2-1 δ 1 = -1 if y t +1 if y < t k= (1+2-2i ) -1/2 i=0 y zn+1 = arccos(t) z n+1 = 1-t 2 x n+1 = t z n+1 = arcsin(t) y n+1 = t x n+1 = 1-t 2 Initial condition 1 x 0 =, y 0 = 0, z 0 = 0 k Slide 4

5 x 1 = α + β/2, y 1 = α β/2 X n+1 = 2αβ Other functions x 1 = α + β, y 1 = β - α x n+1 = 2 αβ α + β +1 α β -1 x 1 =, y 1 = x n+1 = α (1+ β) x 1 = α +1, y 1 = 2 α x n+1 = 1+ α 2 Slide 5

6 Table 1 defines the codes used in most signal processing applications Because many applications involve conversion of analog signals, the analog signal is given as a reference The table also helps users understand the basis for each of these formats The following definitions are offered as the basis for each code format Offset Binary: A binary code in which the code represents analog values between Full Scale and Full scale All zero corresponds to Full scale This code can be balanced by appending a 1 below the LSB 2 s Complement: A binary code in which positive and negative codes of the same magnitude sum to all zero s plus a carry The 2 s complement can be generated from the Offset Binary code by inverting the MSB A negative number is generated by inverting each bit of the positive number, then adding one Example: 011(+3) = 101 (-3) Fractional 2 s complement example: 5A80h= = 1/2+1/8+1/16+1/64+1/256= h= =05 1 s Complement: Bipolar binary code in which positive and negative codes of the same magnitude sum to all one s A negative number is generated by investing each bit of the positive number Example: 011 (+3) 100 (-3) Sign Magnitude: A binary code in which the MSB represents positive (1) and negative (0) polarities The code in the tables uses a offset binary code to represent the magnitude portion of the number Slide 6

7 Slide 7

8 Fractional Arithmetic Two s Complement 16 bit number representation - Maximum positive number = 1 1/ 2 15 = 7FFFh - Minimum positive number = 1/ 2 15 = 0001h - Minimum negative number = -1 = 8000h - Maximum negative number = - 1/ 2 15 = FFFFh Angle representation 2 16 *G N= where G is the angle in degrees 180 N= decimal representation of the number Example: G = 45 ; N = 2 14 = 4000h G = 30 ; N = 2 16 /6 = 2AAAh Slide 8

9 Two quadrants angle coding N= 2 16 *a/180 where a is the angle in degrees N= decimal representation of the number 16 Bits atan table ε k = atan 2 -k ε0 = atan 2 0 = 45 N= 4000 ε1 = atan 2-1 = N= 25C8 ε2 = atan 2-2 = N= 13F6 ε3 = atan 2-3 = N= 0A22 ε4 = atan 2-4 = N= 0516 ε5 = atan 2-5 = N= 028b ε6 = atan 2-6 = N= 0145 ε7 = atan 2-7 = N= 00A2 ε8 = atan 2-8 = N= 0051 ε9 = atan 2-9 = N= 0029 ε10 = atan 2-10 = N= 0014 ε11 = atan 2-11 = N= 000A ε12 = atan 2-12 = N= 0005 ε13 = atan 2-13 = N= 0003 ε14 = atan 2-14 = N= 0002 ε15 = atan 2-15 = N= 0001 Slide 9

10 CORDIC Rotation Mode Y ) a X 0 =1/k= = i=15 (1+2-2i ) -1/2 X(i+1) = x(i) m δ(i) y(i) 2 -i Y(i+1) = Y(i) + δ(i) X(i) 2 -i S(i+1) = S(i) + δ(i) ε(i) Z(i) = α S(i) i=0,1,15 δ(i)=sgn Z(i) =1 if Z(i) 0, δ(i)= -1 otherwise, m=1 X Slide 10

11 CORDIC Rotation Mode (x7, y 7, z7) (x5, y 5, z 5) (x3, y3, z 3 ) (x 9, y 9, z 9 ) X(i+1) = x(i) m δ(i) y(i 2 -i Y(i+1) = Y(i) + δ(i) X(i) 2 -i (x11, y 11, z 11 ) (x1, y1, z1) (x 12, y 12, z 12 ) (x 10, y 10, z 10 ) (x8, y8, z8) (x6, y 6, z6) S(i+1) = S(i) + δ(i) ε(i) (x4, y4, z 4 ) Z(i) = α S(i) i=0,1 15 δ(i)=sgn Z(i) =1 if Z(i) 0, δ(i)= -1 otherwise,m=1 (x2, y2,z2) )a x 0 = y 0 = 0 Slide 11

12 Example In Cordic rotation mode,calculate sin15 and cos15 δ 0 =1,s 0 =0 y 0 =0, x 0 = z 0 = α s 0 = 15-0 > 0 δ 0 = 1 s 1 = s o + ε 0 = 0+45 i= 0 x 1 = x 0 - δ 0 y = y 1 = y 0 + δ 0 x = z1 = α s 1 = < 0 δ 1 = -1 s 2 = s 1 + δ 1 ε 1 = = i= 1 x 2 = x 1 δ 1 y = = y 2 = y 1 + δ 1 x = = i= 2 z 2 = α s 2 = 15 18, = -3, < 0 δ 2 = -1 s 3 = s 2 + δ 2 ε 2 = = x 3 = x 2 δ 2 y = = y 3 = y 2 + δ 2 x = = i= 12 z 12 α s >0 x 13 = x 12 y = ~ cos 15 y13= y 12 + x = ~ sin 15 s 12 = Slide 12

13 Examples, sin(x) ***************************************************************************** ; This program is a Microtrend Sys Cordic implementation on MSP430F437 ; 12(16) steps to calculate sinx for a 16(12) bit alfa angle Copyright 2004 ; ( a=alfa angle in R12->R4) ; X(i+1) = x(i) d(i)* y(i)*2(^-i) d(i)=+/- 1 ; Y(i+1) = Y(i) + d(i)* X(i)* 2(^-i) e(i)= atan(2^-i) ; S(i+1) = S(i) + d(i)* e(i) a=angle ; z(i) = a - S(i) s(0)=0,d(0)=1 x0= , y=0 ;****************************************************************************** ;****************************************************************************** #include "msp430x43xh" NAME fixed_point_sine_as_function PUBLIC fixed_point_sine_as_function RSEG CODE fixed_point_sine_as_function; ;Program starts here mov R12, R4 mov #1h,R15 ;initial number of (K) shifts=1 mov R15,R10 ;save it in R10 also mov #000eh,R5 ;R5 steps (15) ; mov #000bh,R5 ;R5 steps (11)for 12 steps mov #0,R7 ;R7 is a pointer to atan(2^-k) values mov atan_table(r7), R6 ; atan(2^-k) table mov #4db9h,R8 ; =x1 the magic number mov #4db9h,R9 ; =y1 the magic number mov #4000h,R11 ;R11 holds 45deg angle initially Slide 13

14 ; for first comparison et2 cmp R11,R4 ;alfa-45 deg jn et1 mov R9,R14 ;y1-->r14 here1 rra R14 ;y1/2^k dec R15 jnz here1 ; do it k times for y1 mov R10,R15 mov R8,R12 ;x1-->r12 sub R14,R8 ;x1-y1/2^k-->r8 cos(x) here2 rra R12 ;x1/2^k dec R15 ;do it k times for x1 jnz here2 add R12,R9 ;y1+x1/2^k-->r9 sin(x) incd R7 ;R7<--R7+2 pointer to next value add atan_table(r7),r11 ;s(k+1)=s(k)+atan(2^-k) inc R10 ;increment number of shifts(k) in next iteration mov R10,R15 ;and move it in R15 dec R5 jnz et2 ;have we done 16(12) iterations? jmp et3 et1 mov R9,R14 ;y1-->r14 here3 rra R14 ;y1/2^k dec R15 ;do y1/2 k times for y1 jnz here3 mov R10,R15 mov R8,R12 ;x1-->r12 add R14,R8 ;x1+y1/2^k-->r8 cos(x) here4 rra R12 ;x1/2^k dec R15 ;do x1/2 k times for x1 jnz here4 sub R12,R9 ;y1-x1/2^k-->r9 sin(x) incd R7 sub atan_table(r7),r11 ;s(k+1)=s(k)-atan(2^-2) inc R10 ;increment number of shifts(k) in next iteration mov R10,R15 ;and reload it in R15 dec R5 ;have we done 11 or 15 iterations? jnz et2 Slide 14

15 et3 nop mov R9, R12 ret atan_table ;R9 holds sin(x) ;R8 holds cos(x) ; yes, it all ends here dw 0x4000 ; 45 deg dw 0x25c8 ; deg dw 0x13f6 ; deg dw 0x0a22 ; deg dw 0x0516 ; dw 0x028b ; dw 0x0145 dw 0x00a2 dw 0x0051 dw 0x0029 dw 0x0014 dw 0x000a dw 0x0005 dw 0x0003 dw 0x0002 dw 0x0001 end Slide 15

16 Some Issues Using step predictors to minimize machine cycle count Rescaling steps for hyperbolic functions Overflow in fixed point arithmetic 2 -i rotations (no instruction yet to accomplish this in one cycle) Guard bits, granularity Slide 16

17 CORDIC Vector Mode Y X(i+1) = x(i) m δ(i) y(i) 2 -i Y(i+1) = Y(i) + δ(i) X(i) 2 -i Y= t S(i+1) = S(i) + δ(i) ε(i) Z(i) = α S(i) i=0,1,,15 δ(i)=sgn y(i) =-1 if y(i) 0, δ(i)= +1 otherwise, m=-1 X Slide 17

18 Other ways to approximate trigonometric functions Series expansion : cos(x)=1+a 2 *x 2 +a 4 *x 4 sin(x)/x=1+a 2 *x 2 +a 4 x 4 a 2 = , a 4 = a 2 = , a 4 = Chebysheff recursive equation : y(i)=2cosβ*y(i-1)-y(i-2), with i=0, y(-1)= - sin β, y(-2)= - sin( 2*β) For n iterations the equation generates sin(n* β) Fractional aproximation : tg(θ)=[x(1-x+a)]/(1-x)(x+a) x=θ/90 and A= Slide 18

19 MSP430 machine cycles for 16 bits and 12 bits 2 s complement fixed point arithmetic operation Function No: of bits No: of machine cycles Sin16(x) Sin12(x) Arcsin16(x) Arcsin12(x) Mul16(axb) Mul12(axb) Note: For each of the above functions the machine cycle count includes the direction of rotation encoding and initialization overhead Slide 19

20 X i+1 Y i+1 X i+1 = x i δ i y i 2 -i Y i+1 = Y i + δ i X i 2-1 S i+1 = S i + δ i ε i z i = α -S i (X 1,Y 1 ) (X 3,Y 3 ) (X n, Y n ) (X 2,Y 2 ) (X 0,Y 0 ) Stage i >> i >> i Z i+1 +/- +/- Sign i Z Stage i Atan(2 -i ) X Y +/- Sign i Slide 20

21 References [1] HM Ahmed, M Morf, DT Lee and PH Ang "A VLSI Speech Analysis Chip Set Based on Square-Root Normalized Ladder Forms" Proc Int Conf Acoustics, Speech and Signal Processing ICASSP-81 pp (1981) [2] E Antelo, JD Bruguera and EL Zapata "Unnormalized Fixed-Point Cordic Arithmetic for SVD Processors" Proc Int Conf Signal Processing, Applications and Technology ICSPAT-94 (1994) [3] H Dawid and H Myer "High Speed Bit-Level Pipelined Architectures for Redundant Cordic Implementations" Proc Int Conf Appliaction Specific Array Processors ASAP-92 pp (1992) [4] J Duprat and JM Muller "The Cordic Algorithm: New Results for Fast VLSI Implementation" IEEE Transactions on Computers Vol 42 No2 pp (1993) [5] MD Ercegovac and T Lang "Redundant and On-Line Cordic: Application to Matrix Triangularization and SVD" IEEE Transactions on Computers Vol 39 No6 pp (1990) [6] ND Hemkumar and JR Cavallaro "Redundant and On-Line Cordic for Unitary Transformations" IEEE Transactions on Computers Vol 43 No8 pp (1994) [7] YH Hu "Cordic-Based VLSI Architectures for Digital Signal Processing" IEEE Signal Processing Magazine No7 pp (1992) [8] YH Hu "The Quantization Effects of the Cordic Algorithm" IEEE Transactions on Signal Processing Vol 40 No4 pp (1992) [9] X Hu and SC Bass "A Neglected Error Source in the Cordic Algorithm" Proc Int Conf on Circuits and Systems ISCAS-93 pp (1993) [10] K Kota and JR Cavallaro "Numerical Accuracy and Hardware Tradeoffs for Cordic Arithmetic for Special-Purpose Processors" IEEE Transactions on Computers Vol 42 No7 pp (1993) [11] JA Lee and T Lang "Constant-Factor Redundant Cordic for Angle Calculation and Rotation" IEEE Transactions on Computers Vol 41 No8 pp (1992) [12] P Strobach "2-Fold Normalized Square-Root Schur RLS Adaptive Filter" IEEE Transactions on Signal Processing Vol 42 No5 pp (1994) [13] N Takagi, T Asada and S Yajima "Redundant Cordic Methods with a Constant Scale Factor for Sine and Cosine Computation" IEEE Transactions on Computers Vol 40 No9 pp (1991) [14] D Timmermann, H Hahn and BJ Hosticka "Low Latency Time Cordic Algorithms" IEEETransactions on Computers Vol 41 No8 pp (1992) [15] JE VoIder "The Cordic Trigonometric Computing Technique" IRE Transactions Electronic Computers Vol 8 pp (1959) [16] JS Walther "A Unified Algorithm for Elementary Functions" Proc Spring Joint Computer Conf pp (1971) [17] T Lang, Elisardo Antelo, Cordic vectoring with arbitrary target value, IEEE Transactions on Computers,vol47,no7,pp , July 1998 [18] T Lang, Elisardo Antelo, Cordic based computation of ArcCos and (1-t2), Internal Report Dept Electrical and Computer Eng University of California at Irvine, March 1997 [19] T C Mazenc,X Merrheim and JMMuller, Computing functions Cos-1 and Sin-1 using CORDIC, IEEE Transactions on Computers, vol 42,no1,pp ,1993 [20] C Krieger and BJ Hosticka, Inverse kinematics computations with modified CORDIC iterations, IEE Proc ComputDigitTech vol143,no1,pp87-92, January 1996 Slide 21

CORDIC, Divider, Square Root

4// EE6B: VLSI Signal Processing CORDIC, Divider, Square Root Prof. Dejan Marković ee6b@gmail.com Iterative algorithms CORDIC Division Square root Lecture Overview Topics covered include Algorithms and