cordic_for vsp ntuee

1

VLSI Signal Processing

台大電機吳安宇

CORDIC(Coordinate rotation digital computer)

For VLSI Signal Processing Course

Ref: Y. H. Hu, “CORDIC based VLSI architecture for digital signal processing,” IEEE Signal Processing Mag., pp.16-35, July 1992.

2001/4/30

2



Rotation Operation

)(

)(.

cossin

sincos

'

'

iy

ix

y

x

You need: 4 multipliers.

2 adders.

or ROM for Table Look-up

3



•What is “CORDIC” ?–COordinate Rotation DIgital Computer

•Why do we use “CORDIC” ?–MAC dominates the implementational cost in some DSP functions.–The DSP approach, CORDIC, helps to save the hardware cost.

4



Basic Concept of The CORDIC

•To decompose the desired rotation angle (θ)

into the weighted sum of a set of predefined

elementary rotation angles (am(i))

•Such that the rotation through each of them

can be accomplished with simple shift-and-

add operation.

5



Behavior of CORDIC

V(0)

V(1)V(3)

122 yx

6



)(

)(.

cossin

sincos

)1(

)1(

iy

ix

aa

aa

iy

ix

mm

mm

In General Case:

)(

)(.

1

1

)1(

)1(

22

iy

ixi

i

iy

ix

i

i

In CORDIC Algorithm:

)(

)(

1tan

tan1cos

)1(

)1(

iy

ix

a

aa

iy

ix

m

mm

7



CORDIC Algorithm

(i)ami

1-n

0

1-n

0

ii

i

)2(tana 1m

i

2atan )(1m

i

).........2(tan)2(tan)2(tan 211101

.........43211

8



Initiation:Given x(0),y(0),z(0)

For i=0 to n-1 ,Do

/*CORDIC iteration equation */

/*Angle updating equation*/

(i)a- miz(i)1)z(i

/*Scaling Operation (required for m=±1 only)*/

End i loop

)(

)(

)(

1

ny

nx

nKy

x

mf

f

)(

)(.

1

1

)1(

)1(

22

iy

ixi

i

iy

ix

i

i

m

n

i

m

anK

cos

1)( 1

0

9



X(i) Y(i)

X-Reg Y-Reg

+/- +/-

Barrel shifter

Barrel shifter

X(i+1) Y(i+1)

a(n-1)

a(1)

a(0) Z-reg

i

Z(i+1)

Basic processor for

CORDIC

10



Modes of Operations• Vector rotation mode (θ is given) :

determined by the set of

)(i

z(n)-z(n)-z(0)1

0

im

n

i

a

The objective is to compute the final vector (Usually, we set z(0)= θ.)

θ

= sign of z(i)

11



Modes of Operations (cont’d)• Angle accumulation mode (θ is

not given)

The objective is to rotate the given initial vector back to x-axis ,and the angle can be accrued.(Now, we let z(0)=0.)

= - sign of x(i)·y(i)θ

V(0)

V(1)

X-axis

12



Scaling Operation

bqp

Q

qq

iq

m

P

p

ip

m

n

i

imsim

kk

knK

Type

knK

Type

mK

q

p

2;1;1

)21()(

1:2.

2)(

1:1.

21

1

1

1

0

),(22

13



X(n) Y(n)

X(n) Y(n)

+/- +/-

Barrel shifter

Barrel shifter

X-Reg Y-Reg

)('2)(')1('

)('2)(')1('

:2

)(2)(')1('

)(2)(')1('

:1

nyiyiy

nxixix

Type

nyiyiy

nxixix

Type

q

q

p

p

i

i

i

i

ff y x

Scaling Stage

14



Advantages and disadvantagesSimple Shift-and-add Operation.(2 adders+2 shifters v.s. 4 mul.+2 adder)

-It needs n iterations to obtain n-bit precision.

-Slow carry-propagate addition.

-Low throughput rate

-Area consuming shifting operations.

15



How to improve CORDIC ?• Use Pipelined Architecture

• Improve the Performance of the Adders (redundant arithmetic, CSA)

• Reduce Iteration Number

– High radix CORDIC. (e.g., Radix-4, Radix-8)

– Find a optimized shift sequence (e.g., AR-CORDIC)

• Improve the Scaling Operation

– Canonical multiplier recoding

– Force Km to 2.

P

p

ip

m

pknK 1

2)(

1

1pk

16



Parallel and Pipelined Arrays

Basic

CORDIC

Processor1

Basic

CORDIC

Processor2

Basic

CORDIC

Processor n+s

x(0)

y(0) f

f

y

x

Basic

CORDIC

Processor

1

Basic

CORDIC

Processor

2

Basic

CORDIC

Processor

n+s

L

A

T

C

H

L

A

T

C

H

L

A

T

C

H

f

f

y

x

)0(

)0(

1

1

sn

sn

y

x

)1(snv )1(2 snv )(1 snv )2(1snv)0(1snv

17



)(

)(.

cossin

sincos

)1(

)1(

iy

ix

aa

aa

iy

ix

mm

mm

In General Case:

)(

)(.

1),(

),(1

)1(

)1(

22

iy

ixims

imsm

iy

ix

i

i

In CORDIC Algorithm:

)(

)(

1tan

tan1cos

)1(

)1(

iy

ix

a

aa

iy

ix

m

mm

18



Generalized CORDIC Algorithm

(i)ami

1-n

0

i

]2

[tanm

1a ),(1m imsm

12tanh1 2tan

0

),1(1

),1(1

)1,0(2

mmm

is

is

s

m0 , linear system ;

m=1 , circular system ;

m=-1 , hyperbolic system.

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Circular

Linear

V(2)

V(4)

V(0)

V(1)V(3)

122 yx

V(0)

V(2)

V(1)

V(3)

Hyperbolic

V(0)

V(1)

V(2)

V(3)

Different coordinates

20



Initiation: Given x(0),y(0),z(0)

For i=0 to n-1 ,Do

/*CORDIC iteration equation */

/*Angle updating equation*/

(i)a- miz(i)1)z(i

/*Scaling Operation (required for m=±1 only)*/

End i loop

)(

)(

)(

1

ny

nx

nKy

x

mf

f

)(

)(.

1),(

),(1

)1(

)1(

22

iy

ixims

imsm

iy

ix

i

i

21



Shift Sequence{s(m,i); 0in-1}

)1(

- )(1

0)(

na

a

m

im

n

ii

Determine the convergence of the CORDIC iteration, as well as the magnitude of the scaling factor Km(n).

m=0 or 1 , s=(m,i)=i

m=-1 , s(-1,i)=1,2,3,4,4,5,….,12,13,14,14,..

An angle approximation error:

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Application to DSP Algorithms

• Linear transformation:- DFT, Chirp-Z transform, DHT, and FFT.

• Digital filters:- Orthogonal digital filters, and adaptive lattice filters.

• Matrix based digital signal processing algorithms:- QR factorization, with applications to Kalman filtering - Linear system solvers, such as Toeplitz and covariance system solvers,……,etc.

23



FFT application

Nnkje 2

Nnkjebaa 2'

-1

'a

'b

a

b

Nnkjebab 2'

24



Butterfly unit

+

+

-

-

CORDIC processor

Ra

Ia

Rb

Ib

Ra'

Ia'

Rb'

Ib'

Nnkje 2

25



Conclusions1. In some cases, CORDIC evaluates rotational

functions more efficiently than MAC units.

2. CORDIC saves more hardware cost.

3. By the regularity, the CORDIC based architecture is very suitable for implementation with pipelined VLSI array processors.

4. The utility of the CORDIC based architecture lies in its generality and flexibility.

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