A Matrix Multiplication Implementation for Pre-Conditioning Back Propagated Errors on a Multi-Conjugate Adaptive Optics System Rigo Dicochea University of California at Santa Cruz Research Advisor: Dr. Donald Gavel Research Supervisor: Marc Reinig

Mission Statement The goal is to implement a matrix multiplication on a Field Programmable Gate Array (FPGA) to reduce the total number of iterations necessary to solve a system of equations with unknown variables. Without AO With AO

Background/Iterative Approach Light Rays from Excited Sodium Ions A We propagate our initial estimate of phase delay through each voxel. B Possibly take 100’s of iteration to converge! C However, since each of these rays passes through different voxels, the total effect of the atmosphere on each of them is different. D = E Photo credit: Marc Reinig

Evolution of Project A B State Machine C D Hardware = E Resources

State Machine for Matrix Multiplication mult1by11 alu_input = ^b0001; store1x11 alu_input = ^b0000; mult1by21 alu_input = ^b0001;

Pre-Conditioning Implementation

Resources Utilized 208 FPGA Slices 18 Registers 1 ALU Control Logic/Gates Multiplexor MANY Bus Lines(wires interconnecting different hardware) 6 Virtex 4 FPGA’s

Timing Initial Simulations yield a timing constraint of 200MHz. Must be able to converge in less than 1 milli-second.

RESULTS!! Previous iterative solutions took in excess of 90 iterations to converge. With Matrix Multiplication/Pre-Conditioning we NOW converge in approximately 25 to 50 iterations! A reduction of 50 to 75 iterations!

What's Next? Implement fast Fourier transform which will allow for more accurate convergent values. Multiplex existing hardware to reduce resource consumption and cost. Determine the total number of FPGA’s necessary to implement system on TMT size telescope.

Illustration of Matrix Benefit

Acknowledgments Lab for Adaptive Optics Center for Adaptive Optics Dr. Don Gavel Marc Reinig ATMAOS Project Leader Carlos Andres Cabrera Center for Adaptive Optics Xilinx This project is supported by the National Science Foundation Science and Technology Center for Adaptive Optics, managed by the University of California at Santa Cruz under cooperative agreement No. AST - 9876783.