CORDIC Hardware Implementations

Purpose

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

Background

General CORDIC Description

• General.1.A.pdf - Rotation as a complex number multiplication (in CORDIC FAQ)
• General.2.A.pdf - Rotation as a geometric interpretation (in Synthesis of Arithmetic Circuits examples)
• General.3.A.pdf - A set of CORDIC equations (in Andraka's paper )
• General.4.A.pdf - Unified CORDIC

Software Simulations

• Software.1.A.pdf - Burkardt's MATLAB implementation
• Software.2.A.pdf - Burkardt's C++ implementation

Fixed Point Software Implementations

• Fixed.1.A.pdf - Dr. Dobbs Journal implementation

FPGA Hardware Implementations

• FPGA.1.A.pdf - Monash Univ VHDL implementation (architecture and precision)
• FPGA.2.A.pdf - Monash Univ VHDL implementation (VHDL detail )
• FPGA.2.B.pdf - Andraka's paper (FPGA related)

Binary Angular Measurement

Let's identify the source of the prevalent statement of "50 % improvement".

An explanation using a uniform framework would be good.

• BAM.1.A.pdf - Angle Recording CORDIC- Y. H. Hu
• BAM.2.A.pdf - Trellis-based Searching CORDIC - C. S. Wu
• BAM.3.A.pdf - Look-ahead CORDIC - C. C. Kao
• BAM.4.A.pdf - Redundant CORDIC - T. Vladimirova
• BAM.5.A.pdf - Adpative CORDIC - T. K. Rodrigues
• BAM.6.A.pdf - MSE Variance CORDIC - S. Y. Park
• BAM.7.A.pdf - Angle Approx Error & Datapath Rounding Error - C. Dick

Idea Sketch

Maximizing Throughput

• Throughput Maximizing Idea.1.A (pdf)

Minimizing Latency

• Latency Minimizing Idea.2.A (pdf)

Bit-Serial & Bit-Parallel Trade-offs

• Generalized Multi-Byte CORDIC Idea.4.A (pdf)

CORDIC as a Search Algorithm

• CORDIC as a Search Idea.3.A (pdf)

Quad Angle Tree Based CORDIC

• CORDIC Quad Tree Angles (pdf)

Implementation Issues

Implementation Technical Background

- CMOS VLSI Circuits

- Low Power Techniques

- Arithmetic Circuits

- FPGA Design

Fixed Point Simulation

-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

CORDIC Sensor Characteristics

• Accuracy
• Preceision
• Sensitivity
• Linearity
• Resoltion

CORDIC Accuracy & Precision

CORDIC.AccPrec (pdf)

Makefile (pdf)

CORDIC Core class source (pdf)

Angles class source (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)

fig_tscale (pdf)

fig_uscale (pdf)

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)

batch run bash file for testbench 01 (pdf)

batch run bash file for testbench 02 (pdf)

Angles_wx using wxWidgets & wxGlade (pdf)

CORDIC Quad Tree Angles

QuadTree (pdf)

Resource Sharing

- Area, Speed, Power Trade-offs between architectures

Related Links

See Burkardt's C++ Implementations

Exercise

Rotating Vector Plotting

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

Symbolic Computation of CORDIC Equations

• Using Maxima / wxMaxima (pdf, matrix.pdf)

Simulation in C, C++, Octave

• CORDIC Octave Simulation ( CORDIC.m.pdf)
• CORDIC C++ Simulation ( CORDIC_cpp.pdf)
• CORDIC C Simulation ( CORDIC.c.pdf, CORDIC_tb.c.pdf)
• CORDIC Fixed Point C Simulation ( CORDIC.fixed.c.pdf)

Simulation using Multiple Precision Libraries (GMP, MPFR)

• CORDIC GMP Simulation ( cordic_gmp.pdf, normerror.pdf)

SystemC Model

• SystemC Data Types (my_cnt.systemc, dtype.systemc)
• Fixed point Simulation (cordic.scfix)
• Generating VHDL ROM model (cordic.rom)

VHDL Behavioral Model

• VHDL Behavioral Model and Testbench (cordic_beh in one file, cordic_beh.vhdl, cordic_tb.vhdl, cordic_pkg.vhdl)
• VHDL Data Conversion - some test pieces (test_conv.vhdl)

VHDL Data Flow Model

• VHDL Data Flow Model and Testbench ( cordic_rtl in one file, CORDIC.rtl)

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

Old Versions

Old CORDIC Background

Papers and Reports

• 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

• Burkardt's MATLAB implementation (Matlab.1.A) (pdf)
• Burkardt's C++ implementation (C++.1.A) (pdf)

Software Implementations

• Dr. Dobbs Journal implementation (fixed.1.A) (pdf)

Hardware Implementations

* Monash Univ VHDL implementation (VHDL.1.A.pdf, VHDL.2.A.pdf )

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)