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
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