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15705024_g.zip

Implementation on FPGA for CORDIC-based Computation of

Arcsine and Arccosine

Xiaoning Liu

, Yizhuang Xie

, He Chen

, Bingyi Li

Beijing Key Laboratory of Embedded Real-time Information Processing Technology, Beijing Institute of

Technology, Beijing 100081, China

liuxiaoning@bit.edu.cn, xyz551_bit@bit.edu.cn

Keywords: CORDIC, Double Iteration Algorithm, arcsine,

arccosine

Abstract

This paper concentrates on the hardware implementation of

CORDIC algorithm for computing inverse trigonometric

functions like arcsine and arccosine. We improve the existing

algorithm by changing the initial rotating vector of the

iterations and modifying the judging condition of rotation

direction. Due to the improvement, two iterations are saved

and the drawback of the previous algorithm is corrected.

In contrasting with the previous implementation, the

improved algorithm consumes less hardware resources and its

computing results are more accuracy.

1 Introduction

The CORDIC algorithm was originally developed in 1959 by

Jack Volder as a digital solution for real-time navigation

problems

[1]

and then was extended to a broader class of

functions

[2]

. The algorithm has been used for calculating

various trigonometric functions such as sine, cosine, arctan

and it also can be used to compute hyperbolic functions,

exponentials and square roots. CORDIC has remained

popular for its ability to calculate the results using the simple

operations of add, subtract and shift while using small Look-

Up Tables (LUTs) instead of hardware multipliers on FPGA

[3-

6].

CORDIC is widely used in the implementation of real-time

SAR imaging processing since there are many trigonometric

functions in the SAR imaging algorithm. But there are no

available IP cores for computing arcsine and arccosine

functions in the IDE (Integrated Development Environment)

of FPGA, so users need to realize the functions by using

CORDIC algorithm.

In the computation of inverse trigonometric functions like

arcsine and arccosine, the original CORDIC algorithm

displays its drawback. The accuracy of the result will get

worse due to the gain of the rotator

[7]

. Mazenc, C corrected

the gain problem by using “double iteration algorithm” at the

cost of an increase in complexity

[8]

. T. Lang proposed a

method which requires a standard CORDIC module plus a

module to compute the direction of rotation, but this approach

need a hard ware multiplier

[9]

The present work concentrates on the improvement and

optimization of double iteration algorithm. For real-time

calculation, pipeline structure is needed instead of feedback

structure, so that the operation module can obtain the

maximum throughput rate. The pipeline structure consumes

much more hardware resources for it must increase iteration

numbers in order to improve the accuracy. This paper mainly

works on the reduction of hardware resources by saving the

first two iterations without affecting the result of the

algorithm and the correction of the algorithm drawback when

the input argument is approach to 1.

2 Computation of Arcsine and Arccosine

The standard CORDIC algorithm can hardly be used for

computing arcsine and arccosine functions since it involves a

square root operation which cannot be replaced by add,

subtract or shift operations. Mazenc. C proposed double

iteration algorithm by introducing double iterations to the

standard CORDIC algorithm

[8]

. The main advantage of

double iterations is that in step n, the iteration t

i+1

/cos(tan

-i

) becomes t

i+1

= t

/cos

(tan

-1

-i

) =t

(1+2

-2i

); now, a

multiplication by this term reduces to an add and a shift.

2.1 Double Iteration Algorithm

The double iteration algorithm in reference[8] for computing

arcsine function is:

0 0 0 0

1 0 0,

sign( ) sign( )

2 tan 2

i i i i i

i i i

x y z t t

d x if y t else x

z z d

t t t

















   





  





   











   



   





   



  



，，

(1)

The above formula gives

sin









and

cos t



can

be obtained by the relationship

cos sin= / 2tt







For double iterations, the convergence domain of the

algorithm becomes:

22tan 2 , tan 2 [ 3.486,3.486]





   













Therefore we can compute

sin t



and

cos t



for any

 

1,1t 

, since

 

1,1

is the domain of definition.

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