CORDIC Hardware Implementations

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Purpose [edit]

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


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]

3. Octave Codes (pdf)
-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


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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