3D CORDIC Algorithm Based Cartesian to Spherical Coordinate Converter - VLSI Design and Test - page 337

3D CORDIC Algorithm Based Cartesian
to Spherical Coordinate Converter

Anita Jain 1 and Kavita Khare 2

Dept. of Electronics and Communication Engineering, MANIT, Bhopal, India
[e-mail], [e-mail]

Abstract. The standard Cartesian coordinate set is suitable for most of 3D
applications but when dealing with spheres or spherical symmetry, it is easier to
use Spherical coordinates. This paper presents a new hardware efficient scaling
free 3D CORDIC algorithm based Cartesian to spherical coordinates Converter.
Two different architectures i.e. Fully Pipelined and Hybrid Recursive Pipelined
are proposed for implementation of above Converter on FPGA. Synthesis is
carried out on Xilinx ISE 9.2i, Virtex-5 device. Synthesis results show that the
two architectures successfully convert Cartesian coordinates to spherical with
maximum operating frequency of 95.786 MHz and 42.29 MHz respectively.
The minimum Bit error Position is 12 and region of convergence is entire 3D
coordinate space.

1 Introduction

Global positioning system (GPS) consists of three-dimensional Cartesian coordinate
system and an associated ellipsoid. It uses World Geodetic System WGS84 as datum
in which positions can be described as XYZ Cartesian coordinates or latitude,
longitude and ellipsoid height spherical coordinates. To make GPS complementary to
geographic information system (GIS), data-focused integration of the two
technologies is required. For transfer of data from GPS [1-2] and to conduct spatial
analysis directly in the field, Cartesian coordinates are to be converted to spherical
counter parts. The Multiple input multiple output (MIMO) systems as in [3] and
adaptive beam former in antenna array also require the same conversion.
Overall efficiency of systems that employ conversion definitely depends on the
process of conversion. Faster, accurate and area efficient conversion is the demand of
all practical applications. CORDIC algorithm has already proven its efficacy in
various fields of digital signal processing [4] and 2D coordinate conversions [5]. Here
in this paper the conversion through CORDIC has been extended from 2D to 3D.
CORDIC stands for COrdinate Rotation in Digital Computer and is a simple
hardware efficient algorithm for the implementation of various trigonometric
functions. In this paper Cartesian to Spherical Coordinate Converter based on novel
3D CORDIC algorithm is proposed. It covers entire 3D space as its Region of
Convergence (RoC). RoC is indicator of maximum angle of rotation which can be
realized using finite number of iterations. The proposed design is able to convert any
Cartesian coordinate in 3D space to its spherical counterpart.