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

Rotation Operation You need: 4 multipliers. 2 adders. or ROM for Table Look-up

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.

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.

Behavior of CORDIC V(0) V(1) V(3)

In General Case: In CORDIC Algorithm:

CORDIC Algorithm

Initiation:Given x(0),y(0),z(0) For i=0 to n-1 ,Do /*CORDIC iteration equation */ /*Angle updating equation*/ End i loop /*Scaling Operation (required for m=±1 only)*/

Basic processor for CORDIC X(i) Y(i) Basic processor for CORDIC a(n-1) a(1) a(0) X-Reg Y-Reg Barrel shifter Barrel shifter Z-reg +/- +/- Z(i+1) X(i+1) Y(i+1)

Modes of Operations Vector rotation mode (θ is given) : determined by the set of  The objective is to compute the final vector (Usually, we set z(0)= θ.) θ  = sign of z(i)

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.) V(0) θ X-axis  = - sign of x(i)·y(i) V(1)

Scaling Operation

Scaling Stage +/- +/- X(n) Y(n) X-Reg X(n) Y(n) Y-Reg Barrel shifter

Advantages and disadvantages Simple 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.

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.

Parallel and Pipelined Arrays Basic CORDIC Processor1 Processor2 Processor n+s x(0) y(0) Basic CORDIC Processor 1 2 n+s L A T C H

In General Case: In CORDIC Algorithm:

Generalized CORDIC Algorithm m0 , linear system ; m=1 , circular system ; m=-1 , hyperbolic system.

Different coordinates V(2) V(4) V(0) V(1) V(3) Different coordinates Circular V(0) V(2) V(1) V(3) Hyperbolic V(0) V(1) V(2) V(3) Linear

Initiation: Given x(0),y(0),z(0) For i=0 to n-1 ,Do /*CORDIC iteration equation */ /*Angle updating equation*/ End i loop /*Scaling Operation (required for m=±1 only)*/

Shift Sequence {s(m,i); 0in-1} 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:

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.

FFT application -1

Butterfly unit + + - CORDIC processor -

Conclusions In some cases, CORDIC evaluates rotational functions more efficiently than MAC units. CORDIC saves more hardware cost. By the regularity, the CORDIC based architecture is very suitable for implementation with pipelined VLSI array processors. The utility of the CORDIC based architecture lies in its generality and flexibility.