CORDIC Hardware Implementations

Purpose edit

  • developing CORDIC architectures similar to DDR (Double Data Rate).


Background edit

General CORDIC Description edit

Software Simulations edit

Fixed Point Software Implementations edit

FPGA Hardware Implementations edit

Lookahead CORDIC edit

  • Lookahead CORDIC Literature Note (pdf)

Redundant CORDIC edit

  • Redundant Adders (pdf)
  • Redundant CORDIC Note 1 - Tagaki (pdf)
  • Redundant CORDIC Note 2 - Ercegovac (pdf)
  • Redundant CORDIC Note 3 - Timmermann - A.(pdf) - B.(pdf) - C.(pdf)
  • Redundant CORDIC Note 4 - Booth Encoding (pdf)
  • Redundant CORDIC Note 5 - Noll (pdf)

Angle Recoding CORDIC edit

  • AR CORDIC Note 1 - Hu (pdf)
  • AR CORDIC Note 2 - Wu (pdf)
0. Overview (pdf)
1. Conventional CORDIC (pdf)
2. AR (Angle Recoding) (pdf)
3. MVR (Modified Vector Rotation) (pdf)
4. EEAS (Extended Elementary Angle Set) (pdf)
5. Generalized EEAS (Extended Elementary Angle Set) (pdf)
  • AR CORDIC Note 3 - Swartzlander (pdf)

Hybrid CORDIC edit

  • Hybrid CORDIC Note 1 - Sine/Cosine Generator Algorithms (pdf)
  • Hybrid CORDIC Note 2 - Sine/Cosine Generator Architecture (pdf)
  • Hybrid CORDIC Note 3 - ROM free modified coarse fine approach (pdf)
  • Hybrid CORDIC Note 4 - Hybrid CORDIC Alogorithms (pdf)

Binary Angular Measurement edit

Let's identify the source of the prevalent statement of "50 % improvement".
An explanation using a uniform framework would be good.


Scaling-free CORDIC edit


Idea Sketch edit

CORDIC as a Search Algorithm edit

  • CORDIC as a Search Idea.3.A (pdf)
5. Search (pdf)

Quad Angle Tree Based CORDIC edit

  • CORDIC Quad Tree Angles (pdf)
6 Lookahead Idea (pdf)

Minimizing Latency edit

  • Latency Minimizing Idea.2.A (pdf)
7. Backtrack (pdf)

Maximizing Throughput edit

  • Throughput Maximizing Idea.1.A (pdf)
8. Serialize (pdf)

Bit-Serial & Bit-Parallel Trade-offs edit

  • Generalized Multi-Byte CORDIC Idea.4.A (pdf)
9. Precision (pdf)
Bit-serial Redundant CORDIC (pdf)


Implementation Issues edit

Implementation Technical Background edit

- [ Understanding VLSI Design ]
- [ Understanding Low Power Design ]
- [ Understanding Arithmetic Circuits ]
- [ Understanding FPGA Design ]

Fixed Point Simulation edit

-Octave
3. Octave Codes (pdf)
-SystemC
-VHDL Simulation
CORDIC Angle LUT ( scr.pdf, out.pdf)
CORDIC ROM Generation and Initialization ( c6.rom.file.vhdl )
CORDIC BEH Simulation ( cordic_beh.pdf )
CORDIC RTL Simulation ( cordic_rtl.pdf )
CORDIC testbench ( cordic_tb.vhdl )
-VHDL Testcases

Resource Sharing edit

- Area, Speed, Power Trade-offs between architectures


CORDIC Sensor Characteristics edit

  • Accuracy
  • Preceision
  • Sensitivity
  • Linearity
  • Resoltion

CORDIC Accuracy & Precision edit

CORDIC.AccPrec (pdf)

Binary Angle Tree Approaches (C++, gnuplot) edit

C++ Codes edit

CORDIC Source (pdf)
Makefile (pdf)
Core class (pdf)
Angles class (pdf, pdf)
GPData class (pdf)
Figures class (pdf)
Interfacing GHDL CORDIC simulation with C (pdf)
Calling C++ cordic function from C (pdf)
batch run bash file for Angles_tb (pdf)
fig_basic (pdf, note)
fig_tscale (pdf)
fig_uscale (pdf)

Testbench Codes and Results edit

cordic testbenches (pdf)
cordic testbench 01 (percent error)
cordic testbench 02 (path error)
cordic testbench 03 (varying tree levels)
fig_basic (pdf)
fig_tscale (pdf)
fig_uscale (pdf)
cordic testbench 04 (coarse-fine)

CORDIC Accuracy Notes edit

1. General (pdf)
2. Statistical Analysis (pdf)
3. Octave Fixed Point Simulation (pdf)
4. Scaling K (pdf)


batch run bash file for testbench 01 (pdf)
batch run bash file for testbench 02 (pdf)
Angles_wx using wxWidgets & wxGlade (pdf)

General Angle Tree Approaches (C, R) edit

Search Space Approaches using C edit

Tree type Memory bound Time bound
Binary Tree code, output pdf
Ternary Tree code, output pdf
Quaternary tree code, output pdf
Full Tree code, output pdf

Parallel Search Space Approaches using C + MPI edit

Tree type Memory bound Time bound
Binary Tree code, output pdf
Ternary Tree code, output pdf
Quaternary tree code, output pdf
Full Tree code, output pdf

Parallel Search Space Approaches using C + OpenMP edit

Tree type Memory bound Time bound
Binary Tree code, output pdf
Ternary Tree code, output pdf
Quaternary tree code, output pdf
Full Tree code, output pdf

Statistical Analysis using R edit

Tree Type R Script Output
Binary Tree pdf pdf
Ternary Tree pdf pdf
Quaternary pdf pdf

Test Cases edit

1. Powers of 2 angles testcase (code, summary)

Tree Type log file table plot
Binary Tree pdf pdf pdf
Ternary Tree pdf pdf pdf
Quaternary pdf pdf pdf


2. Leaf and internal node angles testcase

3. Uniformly distributed angles testcase

4. Subtree angles testcase

CORDIC Sensitivity Analysis edit


CORDIC Uncertainty Analysis edit


Haskell Implementation edit

Functional Programming Approach edit

Background (pdf)


Prolog Implementation edit

Binary Tree in Prolog edit

Background (pdf)


Recursion Implementation edit


Computing Gaussian Function edit

  • Scale Space Filtering (Gaussian Smoothing)
  • Parallel CORDIC
  • Neural Network (Computing Exponential Function)


Related Links edit

See Burkardt's C++ Implementations




Exercise edit

Rotating Vector Plotting edit

  • Using gnuplot_i package (pdf)
  • CORDIC Animation: Java Swing Based CORDIC Simulator

Symbolic Computation of CORDIC Equations edit

Simulation in C, C++, Octave edit

Simulation using Multiple Precision Libraries (GMP, MPFR) edit

SystemC Model edit

VHDL Behavioral Model edit

VHDL Data Flow Model edit

c1.adder.rtl.vhdl, c2.addsub.vhdl, c3.bshift.vhdl, c4.dffreg.vhdl, c5.counter.vhdl, c6.rom.vhdl, c7.mux.vhdl, m1.disp.vhdl,
cordic_rtl.vhdl, cordic_pkg.vhdl, cordic_tb.vhdl

CMOS CORDIC Design Examples edit



FPGA CORDIC Design Examples edit


Old Versions edit

Old CORDIC Background edit

Papers and Reports edit
  • Background.1.A - CORDIC FAQ (1.A.pdf)
  • Background.2.A,B - Andraka's paper (2.A.pdf, 2.B.pdf)
  • Background.3.A - Unified CORDIC (3.A.pdf)
  • Background.4.A - Synthesis of Arithmetic Circuits examples (4.A.pdf)
Software Simulations edit
  • Burkardt's MATLAB implementation (Matlab.1.A) (pdf)
  • Burkardt's C++ implementation (C++.1.A) (pdf)
Software Implementations edit
  • Dr. Dobbs Journal implementation (fixed.1.A) (pdf)
Hardware Implementations edit
Some Plots
Threshold=0.0 (pdf)
Threshold=0.001 (pdf)
Quantization Effects (pdf)
Percent Error (pdf)
Residue Statistics (tex file:pdf, pdf)


CORDIC.AccPrec (pdf)
Angles Class Source (pdf)
Angles_wx using wxWidgets & wxGlade (pdf)







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