ECEn 621 Computer Arithmetic Project #3. CORDIC History

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1 ECEn 621 Computer Arithmetic Project #3 CORDIC Algorithms Slide #1 CORDIC History Coordinate Rotation Digital Computer Invented in late 1950 s Based on the observation that: if you rotate a unit-length vector (1,0) by an angle z its new end-point will be at (cos z, sin z) Can evaluate virtually all functions of interest k iterations required for k-bits accuracy Slide #2 1

2 1959 Slide # Slide #4 2

3 HP Journal 1977 HP-35 The first scientific calculator Slide #5 Perfect Rotations z R z R = e jθ z in x R y R = cosθ sinθ sinθ cosθ Rotate by angle θ x in y in z in Slide #6 3

4 Micro-Rotations Decompose the angle into a sum of elementary angles Decompose the rotation into a product of elementary rotations Slide #7 Simplifying the Rotations 1. Decompose the rotation into a scaling operation, and a pseudo-rotation Slide #8 4

5 Pseudo-Rotations Rotation by α j : Pseudo Rotation by α j : y (x PR, y PR ) (x R, y R ) Vector Scaling Constant α j (x in, y in ) x Slide #9 Pseudo-Rotations of a Vector Slide #10 5

6 Simplifying the Rotations 2. Carefully choose elementary angles ±α such that: Slide #11 Simplification by Choosing Elementary Angles Slide #12 6

7 Computing the Expansion Factor K By product of pseudo-rotations Depends on the rotation angles. However, if we always uses the same rotation angles (with positive and negative signs), then K is a constant Can be precomputed and stored Its reciprocal can also be computed and stored Slide #13 CORDIC Recurrence Equations (Pseudo-rotations) Vector Recurrences Scalar Recurrences Slide #14 7

8 Building the Basic CORDIC Iteration - Each CORDIC rotation requires: 2 shifts by j 1 table lookup 3 additions/subtractions By rotating by the same set of angles (with + or - signs), the expansion factor K can be precomputed - Slide #15 Inputs are X, Y, and θ Rotation Mode CORDIC Has Two Two Modes Vector Mode Slide #16 8

9 Running the CORDIC Recurrences in Rotation Mode Initialization CORDIC Recurrences After All Pseudo-Rotations Constant Scaling Factor Constant, and independent of the angle for σ = ±1 Slide #17 Example Slide #18 9

10 Initialize: Trig Function Computation z 0 = θ x 0 = 1/K = y 0 = 0 Iterate with σ i = sign( z i ) Finally (after n steps): z n 0 x n cos(θ ) y n sin(θ ) y n /x n tan(θ ) Slide #19 Precision in CORDIC For n bits of precision in trig functions, n iterations are needed. For large j > n/2, tan(2 j ) 2 j, (Angle in radians) and α j 2 j For j > n, change in z < ulp Convergence is guaranteed for angles in range 99.8 θ 99.8 (degrees) j= 0 tan 1 2 j radians degrees For angles outside this range, use standard trig identities to convert angle to one in the range (range reduction) j α j = tan 1 2 j ±45.0 ±26.6 ±14.0 ± 7.1 ± 3.6 tanα j = 2 j ±1.000 ±0.500 ±0.250 ±0.125 ± Slide #20 10

11 Compensation for Scale Factor K Slide #24 CORDIC Implementation Slide #25 11

12 Running the CORDIC Recurrences in Vector Mode Initialization CORDIC Recurrences After All Pseudo-Rotations Constant Scaling Factor Constant, and independent of the angle for σ = ±1 Slide #27 Example Slide #28 12

13 Initialize: z 0 = 0 x 0 = 1 y 0 = Y Trig Function Computation Iterate with d i = -sign( x i y i ) = -sign( y i ) Finally (after n steps): z n arctan(y) Use trig identity to limit range of fixed-point numbers: Slide #29 Unified CORDIC Algorithms Slide #30 13

14 Extension to Hyperbolic Functions Slide #31 Pseudo-Rotation in Hyperbolic Coordinate System Slide #32 14

15 Convergence Problem with Hyperbolic Rotations Slide #33 Final Values for Hyperbolic CORDIC Slide #34 15

16 Rotating in Linear Coordinates Slide #35 Unified CORDIC Description Slide #36 16

17 Unified CORDIC Summary Table Slide #37 Functions Computed Directly with CORDIC Slide #38 17

18 Functions Computed Indirectly with CORDIC 1 w 2 cos 1 w = tan 1 w w sin 1 w = tan 1 1 w 2 ( ) ( ) cosh 1 = ln w + 1 w 2 sinh 1 = ln w + 1+ w 2 w = (w +1 4) 2 (w 1 4) 2 Slide #39 Some Additional Functions Slide #40 18

19 Variations in CORDIC Implementations Slide #41 Bit-Serial CORDIC For low cost, low speed, low power applications (like hand held calculators) bit-serial implementations of CORDIC are possible. Slide #42 19

20 High Speed CORDIC Do first k/2 iterations as normal. z becomes very small for j > k/2 Combine remaining k/2 iterations into one step using multiplication. K doesn t change. Slide #43 CORDIC with Redundant (Carry-Save) Numbers Slide #44 20

21 Double Rotation Approach Slide #45 Recurrences for Double Rotation Slide #46 21

Trig/Math Anal Name No HW NO. SECTIONS ASSIGNMENT DUE TG 1. Practice Set J #1, 9*, 13, 17, 21, 22

Trig/Math Anal Name No HW NO. SECTIONS ASSIGNMENT DUE TG 1. Practice Set J #1, 9*, 13, 17, 21, 22 Trig/Math Anal Name No LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME OF THE CHAPTER TEST ON NO GRAPHING CALCULATORS ALLOWED ON THIS TEST HW NO. SECTIONS ASSIGNMENT DUE TG (per & amp) Practice Set