Basics of CORDIC Goal Enhancement References

CORDIC - Basic Algorithm and Enhancements

K.Sridharan, IIT Madras

February 25, 2010

Basics of CORDIC Goal Enhancement References Example Conventional CORDIC architecture

The CORDIC algorithm provides an iterative method of performingvector rotations by arbitrary angles using only shifts and adds.

Vector rotation transform: For rotating in a Cartesian plane byangle φ.

x ′ = x cos φ− y sin φ

y ′ = y cos φ + x sin φ

OR

x ′ = cos φ[x − y tanφ]

y ′ = cos φ[y + x tanφ]

Basics of CORDIC Goal Enhancement References Example Conventional CORDIC architecture

The CORDIC algorithm provides an iterative method of performingvector rotations by arbitrary angles using only shifts and adds.

Vector rotation transform: For rotating in a Cartesian plane byangle φ.

x ′ = x cos φ− y sin φ

y ′ = y cos φ + x sin φ

OR

x ′ = cos φ[x − y tanφ]

y ′ = cos φ[y + x tanφ]

Basics of CORDIC Goal Enhancement References Example Conventional CORDIC architecture

If rotation angles are selected such that tan φ = ±2−i , then

Xi+1 = Ki (Xi − Yi · di · 2−i )

Yi+1 = Ki (Yi + Xi · di · 2−i )

Zi+1 = Zi − di · tan−1(2−i )

where

Ki = cos(tan−1(2−i )) =1√

1 + 2−2i(1)

The scale factor Ki can be accumulated and the vector is scaled by

An =∏n

√1 + 2−2i

Basics of CORDIC Goal Enhancement References Example Conventional CORDIC architecture

Polar (R , θ) to Rectangular (X,Y) transformation:

+X

+Y

θ = 65◦

Rotation by 65◦

Basics of CORDIC Goal Enhancement References Example Conventional CORDIC architecture

Polar (R , θ) to Rectangular (X,Y) transformation:

+X

+Y Initialize

X0 = R

Y0 = 0

Z0 = θ

Basics of CORDIC Goal Enhancement References Example Conventional CORDIC architecture

Polar (R , θ) to Rectangular (X,Y) transformation:

+X

+Y

45◦

Rotate by tan−1(20) = 45◦

X1 = X0 − Y0

Y1 = Y0 + X0

Z1 = Z0 − tan−1(20)

Basics of CORDIC Goal Enhancement References Example Conventional CORDIC architecture

Polar (R , θ) to Rectangular (X,Y) transformation:

+X

+Y

26.5◦

Rotate bytan−1(2−1) = 26.5◦

X2 = X1 −Y1

2

Y2 = Y1 +X1

2Z2 = Z1 − tan−1(2−1)

Basics of CORDIC Goal Enhancement References Example Conventional CORDIC architecture

Polar (R , θ) to Rectangular (X,Y) transformation:

+X

+Y

14◦

Rotate by tan−1(2−2) = 14◦

X3 = X2 +Y2

4

Y3 = Y2 −X2

4Z3 = Z2 + tan−1(2−2)

Basics of CORDIC Goal Enhancement References Example Conventional CORDIC architecture

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Basics of CORDIC Goal Enhancement References Example Conventional CORDIC architecture

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Basics of CORDIC Goal Enhancement References Bottlenecks for area

Reduce area consumption without affecting the performance interms of accuracy and number of iterations.

Basics of CORDIC Goal Enhancement References Bottlenecks for area

ROM : The size of the ROM is 2dlog2(no. of iterations)e.

Barrel shifters.

Range is limited to |Z | <= 99◦. Multiplexers (both at inputand output) are required to extend the range.

Basics of CORDIC Goal Enhancement References Features Iterative equations Convergence Architecture

1 Completely eliminates barrel-shifters.

2 Represents all the angles in [−180◦, 180◦] using combinationsof two signed elementary angles, tan−12−1 and tan−12−3.

OR

Z = k0 · tan−1(2−1) + k1 tan−1(2−3)

Basics of CORDIC Goal Enhancement References Features Iterative equations Convergence Architecture

Either

X = K1 · (X − (−1)sgn(k0) · Y .2−1)

Y = K1 · (Y + (−1)sgn(k0) · X .2−1)

Or

X = X − (−1)sgn(k1) · Y · 2−3

Y = Y + (−1)sgn(k1) · X · 2−3

Basics of CORDIC Goal Enhancement References Features Iterative equations Convergence Architecture

Either

X = K1 · (X − (−1)sgn(k0) · Y .2−1)

Y = K1 · (Y + (−1)sgn(k0) · X .2−1)

Or

X = X − (−1)sgn(k1) · Y · 2−3

Y = Y + (−1)sgn(k1) · X · 2−3

Basics of CORDIC Goal Enhancement References Features Iterative equations Convergence Architecture

max(|k0|+ |k1|) = 13

for170◦ = 4 · tan−1(2−1) + 9 · tan−1(2−3)

and177◦ = 8 · tan−1(2−1)− 5 · tan−1(2−3)

Found using C program.

Basics of CORDIC Goal Enhancement References Features Iterative equations Convergence Architecture

max(|k0|+ |k1|) = 13

for170◦ = 4 · tan−1(2−1) + 9 · tan−1(2−3)

and177◦ = 8 · tan−1(2−1)− 5 · tan−1(2−3)

Found using C program.

Basics of CORDIC Goal Enhancement References Features Iterative equations Convergence Architecture

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Basics of CORDIC Goal Enhancement References

1 J. Volder, The CORDIC trigonometric computing technique,IRE Transactions on Electronic Computers, Vol. EC-8, 1959,pp. 330-334.

2 J.S. Walther, A unified algorithm for elementary functions,Proceedings of 38th Spring Joint Computer Conference, 1971,pp. 379-385

3 R. Andraka, A survey of CORDIC algorithms for FPGA-basedcomputers, Proceedings of ACM/SIGDA Sixth InternationalSymposium on Field Programmable Gate Arrays, 1998, pp.191-200

4 Leena Vachhani, K. Sridharan, P.K. Meher, Efficient CORDICalgorithms and architectures for low area and high throughputimplementation, IEEE Transactions on Circuits and Systems -Part II:Express Briefs, Vol. 56, No. 1, January 2009, pp.61-65