CORDIC Hardware Implementations

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Purpose [edit | edit source]

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

Background[edit | edit source]

General CORDIC Description[edit | edit source]

Software Simulations[edit | edit source]

Fixed Point Software Implementations[edit | edit source]

FPGA Hardware Implementations[edit | edit source]

Lookahead CORDIC[edit | edit source]

  • Lookahead CORDIC Literature Note (pdf)

Redundant CORDIC[edit | edit source]

  • 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 | edit source]

  • 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 | edit source]

  • 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 | edit source]

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 | edit source]

Idea Sketch[edit | edit source]

CORDIC as a Search Algorithm[edit | edit source]

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

Quad Angle Tree Based CORDIC[edit | edit source]

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

Minimizing Latency[edit | edit source]

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

Maximizing Throughput[edit | edit source]

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

Bit-Serial & Bit-Parallel Trade-offs[edit | edit source]

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

Implementation Issues[edit | edit source]

Implementation Technical Background[edit | edit source]

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

Fixed Point Simulation[edit | edit source]

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 | edit source]

- Area, Speed, Power Trade-offs between architectures

CORDIC Sensor Characteristics[edit | edit source]

  • Accuracy
  • Preceision
  • Sensitivity
  • Linearity
  • Resoltion

CORDIC Accuracy & Precision[edit | edit source]

CORDIC.AccPrec (pdf)

Binary Angle Tree Approaches (C++, gnuplot)[edit | edit source]

C++ Codes[edit | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

Search Space Approaches using C[edit | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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

Test Cases[edit | edit source]

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 | edit source]

CORDIC Uncertainty Analysis[edit | edit source]

Haskell Implementation[edit | edit source]

Functional Programming Approach[edit | edit source]

Background (pdf)

Prolog Implementation[edit | edit source]

Binary Tree in Prolog[edit | edit source]

Background (pdf)

Recursion Implementation[edit | edit source]

Computing Gaussian Function[edit | edit source]

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

Related Links[edit | edit source]

See Burkardt's C++ Implementations

Exercise[edit | edit source]

Rotating Vector Plotting[edit | edit source]

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

Symbolic Computation of CORDIC Equations[edit | edit source]

Simulation in C, C++, Octave[edit | edit source]

Simulation using Multiple Precision Libraries (GMP, MPFR)[edit | edit source]

SystemC Model[edit | edit source]

VHDL Behavioral Model[edit | edit source]

VHDL Data Flow Model[edit | edit source]

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 | edit source]

FPGA CORDIC Design Examples[edit | edit source]

Old Versions[edit | edit source]

Old CORDIC Background[edit | edit source]

Papers and Reports[edit | edit source]
  • 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 | edit source]
  • Burkardt's MATLAB implementation (Matlab.1.A) (pdf)
  • Burkardt's C++ implementation (C++.1.A) (pdf)
Software Implementations[edit | edit source]
  • Dr. Dobbs Journal implementation (fixed.1.A) (pdf)
Hardware Implementations[edit | edit source]
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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