Nikhil Pre

7/29/2019 Nikhil Pre

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By

NIKHIL SURYANARAYANAN


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Outline

Motivation

Eigen Value Decomposition &Applications

Exact Jacobi Parallel Decomposition using Systolic

Array

Optimization of Systolic Array Interconnect optimized Systolic Array

Conclusion


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Motivation

Required in various fields

High Performance and Real time applicationsdemands Hardware implementation

SDMA Communication

Realizing optimal architectures with respect tospeed and power for respective applications


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Eigen Value Decomposition

Angle of Arrival Estimation

Face Detection

Image Compression

Eigen Beam-forming

Signal Subspace Estimation PCA

MUSIC & ESPRIT


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EVD Methods

Exact Jacobi

Systolic Array

Approximate Jacobi

Algebraic Method (only for 3x3 matrix)


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Eigen Value Decomposition

(EVD) Special case of Singular Value Decomposition(SVD) where the Matrix is Square-Symmetric

Consider a Matrix ARmxn

SVD: A = UDVT

EVD: A = UDUT

where,

DRmxn is diagonal matrix,URmxn & VRmxn are orthogonal


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CORDIC

COordinate Rotation DIgital Computer

Set of Shift Add Algorithms forcomputing Sine, Cosine, Arc,

Hyperbolic, Coordinate Rotation etc Eliminates complex computations

Single Shift-Add Multiplier, ROM/RAMfor lookup & Basic Logic gates

Hardware friendly

Iterative Algorithm


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Loop Unrolling


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CORDIC Modules

ArcTan Module

Used to compute the

tan-1

/ angle forconstructing the Jacobi

Rotation Matrix

Sine/Cosine Module

cos sin

-sin cos

2x2 matrix is constructedusing the angle from the

ArcTan module


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Exact Jacobi

Aims at annihilating the off diagonalelements using a series of orthogonaltransformations

A(k+1) = JTpqA(k) Jpq,

where A(0)=A

Jpq is called the Jacobi RotationDefined by the parameter (c s, -s c)


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Exact Jacobi

A=UDVT

UTAV=D After niterations,

Ai+1=JiTAiJi

Repeating for allpossible pairs, A canbe effectivelydiagonalized

1......0......0......0

. . . .

0......c......s......0 p. . . .

0......-s......c......0 q

. . . .

0......0......0......1

p q


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Limitations of Exact Jacobi

Implementation

Jacobi iterations are serial

Inability to derive parallelism as iterations have

large inter-loop Data Dependency

Inability to pipeline

Every iteration involves transfer of 4N-4 matrix

elements to the processor Even though it is MATRIX operation,

parallelism cannot be derived


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How to parallelize?

Systolic Array

Solve 2x2 EVD sub problems

For a matrix of size N we have N/2xN/2

EVD sub problems If N=6; possible sets are

{ (1,2), (3,4) }

{ (1,3), (2,4) }

{ (1,4), (2,3) }

Parallel Reordering


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Systolic Array for EVD

PE PE PE

PE PE PE

PE PE PE


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Structure of PE

CORDICATAN

CORDICROT

REG REG


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Data Exchange Sequence

in in

in

in

PEij


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Data Exchange PE11

in in

in in


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Data Exchange PE1j

in in

inin


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Data Exchange PEi1

in in

inin


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in in

inin

Data Exchange PEij


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Timing & Data Exchange


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Array Cycle = 1


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Array Cycle = 1


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Array Cycle = 1

DATA

EXCHANGE


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Array Cycle = 1DATA

EXCHANGE


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Array Cycle = 1DATA

EXCHANGE


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Array Cycle = 1

DATA

EXCHANGE


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Array Cycle = 1DATA

EXCHANGE


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Array Cycle = 1DATA

EXCHANGE


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Array Cycle = 1

DATA

EXCHANGE


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Staggered Processing?

Not realistic to broadcast row and column angles

in real time

ij is the distance of the processor Pij from the

diagonalAlso Pij needs data from neighbors Pi+-1,j+-1 (1< i,

j < n/2)

Can be made faster by allowing off-diagonalPE to allow execution as soon as thediagonal PE produce angles


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Optimizations

Improves the utilization time for each PE from 1/3 rd to 2/3 rd

CYCLE 2

CYCLE 1


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Comparisons. Matrix 8x8

EXACT JACOBI SYSTOLIC ARRAY

Iterations forConvergence 3

Additions 3500

Multiplications 7000

Swaps/Exchange 0

Slower

Iterations forConvergence 22-25

Additions 1500 (less

than half)

Multiplications 3000

Swaps/Exchange = 368

Faster


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Optimized Architecture

In the final Stages of Analyzing a simpler

Systolic Architecture

Matrix size=4x4

1 2

PE

PE

PE

PE


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GOALS

Achieved:

Pipelined Jacobi Architecture

S/W Implementation of Systolic Array

Simultaneous execution of off diagonal PE to

improve timing and reduce idle time

Optimized Systolic Array architecture forminimum swaps and angle transmission


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References

Andraka, Ray, Survey of CORDIC algorithms for FPGA based computers, ACM 1998

A Novel Implementation of CORDIC Algorithm Using Backward Angle Recoding (BAR), Yu Hen Hu & Homer H.M. Chern,IEEE Transactions on Computers, December 1996

Parallel Eigen Value Decomposition for Toeplitz and Related Matrices, Yu Hen Hu, IEEE Transactions-1989

Kim Y, Kim Y, Doyle James, A Low Power CMOS CORDIC Processor Design for Wireless Telecommunications, IEEE2007

Hemkumar N, Masters Thesis, Rice University

Yang Liu et al, Hardware Efficient Architectures for Eigen Value Computation;, EDA 2006

ASIC Implementation of Autocorrelation and CORDIC algorithm for OFDM based WLAN, Sudhakar Reddy & RamchandraReddy, European Journal of Scientific Research, 2009

Advanced Algorithmic Evaluation for Imaging, Communication and Audio Applications Eigenvalue Decomposition usingCATAPULT C Algorithmic Synthesis Methodology

Efficient Implementation of SVD on a Reconfigurable System, Christophe Bobda, Klaus Danne and Andre Linarth,Springer-Verlag Berlin Heidelberg 2003

Hardware Implementation of Smart Antenna Systems, H. Wang and M. Glesner, Adv in Radio Sciences 2006

Spectral Estimation using MUSIC Algorithm, Jawed Qumar, Nios II Embedded Processor Design Contest-2005

Hardware Efficient Architectures for Eigen Value Computation, Yang Liu, Christis-Savvas Bouganis, Peter Y.K. Cheung,Philip H.W. Leong, Stephen J. Motley, EDAA 2006

A Novel Fast Eigenvalue Decomposition based on Cyclic Jacobi Rotation and its application in eigen-beamforming, TechReport of IEICE-Japan

Efficient Hardware Architectures for Eigenvector and Signal Subspace Estimation, Fan Xu & Alan Wilson, IEEETransactions on Circuits & Systems-204

16 BIT CORDIC Rotator for High Sped Wireless LAN, Koushik Maharatna, Alfonso Troya, Swapna Banerjee, EckhardGrass, Milos Krstic, IEEE Transactions-2004

Survey of CORDIC Algorithms for FPGA Based computers, Ray Andraka, ACM-1998

Smart Antennas for Wireless Communications, Frank B Gross, Mc-Graw Hill,2005 ( Used forFacts & References forComparison purposes and Specifications of Different wireless standards)


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Ei V l d Ei


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Eigen Value and Eigen

Vector The non zero vector of any linear

transformation when applied to the vector

changes the magnitude but not the direction is

an Eigen Vector The scalar value associated with this vector is

called the Eigen Value

Ax=x

A is the transformation, x is the Eigen vector &

is the corresponding Eigen Value


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CORDIC contd

Convergence depends on number of iterations

Unrolled for Systolic and Pipeline

implementations

Iterative architecture unsuitable for FPGA Pipelined preferred as less complex H/W &

operates at data rate

Registers present on Logic cells in FPGAssupport pipelining better

Runs at 52 MHz on XC4013E-2 [1]

CORDIC It ti E ti


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CORDIC Iteration Equations

Givens rotation transformation

x = x cos y sin

y = y cos + x sin

The iteration equations are given as

xi+1 = xi yi. di. 2-i

yi+1

= yi

+ xi

. di

. 2-i

zi+1 = zi di. tan-1(2-i)


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CORDIC Algorithms

for i=1:n

x1= x y * d * (2^-(i-1)) ;

y1= y + x * d * (2^-(i-1)) ;

angle = angle d * (W(i)); // W(i) is the ith ROM reference

if (angle==0)d=0;

elseif (angle>0)

d=1;

else

d=-1;

endx=x1;

y=y1;

end


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Exact Jacobi Algorithm

for j=1:n-1for i=n:-1:j+1

J = jacobi_rot( A( i, i), A( j, j), A( i, j) );

A( [ i, j], :)=J'*A( [ i, j], : );A( : , [ i, j])=A( :, [ i, j] )*J;

end

end

REPEAT for n iterations for accuracy


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EVD contd

Also UTU=I and VTV=I

The Diagonal matrix contains Eigen

Values along its diagonals

U are the left Singular Vectors & V are

right singular vectors

U = {u1,u2,,un}

V = {v1,v2,,vn}


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Diagonal Processor


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Sub-Diagonal Processor


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Super-Diagonal Processor

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