EASTERN MEDITERRANEAN UNIVERSITY

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EASTERN MEDITERRANEAN UNIVERSITY DEPARTMENT OF
COMPUTER ENGINEERING
Evgeny
Dukhnich PRESENTATION
FAST HARDWARE-ORIENTED ALGORITHMS for DSP
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1. Introduction

• The very fast growth of modern VLSI complexity
  offers a hardware
• realization of an ever-growing share of
  mathematical means. It
• essentially raises the computer performance.
  However, familiar
• computational algorithms for signal processing
  are not hardware-
• oriented. The exceptions are the famous
  CORDIC-algorithms and
• some FFT-algorithms.

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MODERN SIGNAL PROCESSING

• Applications
• image processing, computer vision, speech, radar,
  and so on
• Algorithms
• Matrix transforms, convolution, filtering,
  positioning etc.
• Main Problems
• Linear systems eigenvalues, singular values,
  least squares and so on
• Candidate Solutions
• QR-decomposition by Givens rotations or
  Householder reflections

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AIMS OF THE RESERCH

• Increase the effectiveness of DSP hardware
• Design new hardware-oriented algorithms for DSP
• Implement algorithms by designing new VLSI- chips

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Software (a) and hardware (b) realization of the
algorithm f ab/c d
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VLSI-technology requirements

• - algorithms must have a guaranteed accuracy and
  convergence after a fixed number of steps
• - every step of the algorithm must have a limited
  set of simple operations (add, shift, etc) with
  the same short realization time
• - algorithms must have the possibility of
  decomposition on equal parts with a limited set
  of types
• - algorithms must realize the highest possible
  typical computing procedure which are frequently
  found in signal processing methods.

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What is the CORDIC Algorithm?

• CORDIC
• COordinate Rotation DIgital Computer
• Introduced by J. Volder in 1959 1.
• Aim to build a special-purpose digital computer
  for
• airborne navigation.
• performing rotations
• compute of sine, cosine and arctangent
• to multiply or divide numbers using only
  shift-and-add elementary steps.
• In 1971, J. Walther 2 generalized the algorithm
• computes logarithm, exponentials and square roots.

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What is the CORDIC Algorithm?

• Used in
• HP35 (pocket calculator), June 1972.
• Intel 8087 (arithmetic coprocessor), 1983.
• Solving Linear Systems, 1982.
• Signal Processing Applications, July 1995.
• Filtering, 1984.
• Single Value Decomposition (SVD), June 1993.
• Complex SVD, June 2000

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CORDIC AlgorithmKey Ideas
If we have a computationally efficient way of
rotating a vector, we can evaluate cos, sin, and
tan1 functions Rotation by an arbitrary angle is
difficult, so we perform psuedorotations Use
special angles to synthesize a desired angle z z
a(1) a(2) . . . a(m)
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2-D CORDIC Algorithm
Y
Vector R from its initial angle ? will be rotated
by an angle ?.
X cos? -sin? X Y sin? cos? Y
V(X,Y)
? Requires multiplication
R
R
V(X,Y)
?
?
X
Using a series of smaller rotation angles ?i
,we can avoid the multiplication
?S?i?i where i0 ?n, ?i-1,1 ,
?iatan(2-i)
An elementary rotation matrix (Pi)could be
derived where elements will be zeros ones or twos
with integer power.
1 -?i2-i Pi
?i2-i 1
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CORDIC AlgorithmKey Ideas
Rotate the vector OE (i) with end point at (x
(i), y (i)) by a (i) x (i1) x (i)cos a (i) y
(i) sin a (i) (x (i) y (i) tan a (i))/(1
tan2a (i))1/2 y (i1) y (i) cos a (i) x (i)
sin a (i) (y (i) x (i) tan a (i))/(1 tan2a
(i) ) 1/2 z (i1) z (i) a (i) Goal eliminate
the divisions by (1 tan2a (i)) 1/2 and choose a
(i) so that tan a(i) is a power of 2
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Basic CORDIC Iterations
Pick a (i) such that tan a (i) di 2 i, di Î
1, 1 x(i1) x(i) di y(i)2i y (i1) y
(i) di x(i)2iCORDIC iteration z (i1) z
(i) di tan1 2i If we always pseudorotate by
the same set of angles (with or signs), then
the expansion factor K is a constant that can be
precomputed Example pseudorotation for 30
degrees 30.0 _at_ 45.0 26.6 14.0 7.1 3.6
1.8 0.9 0.4 0.2 0.1 30.1
e (i) tan 1 2-i
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Basic CORDIC Iteration
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CORDIC Rotation Mode
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CORDIC Vectoring Mode
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Generalized CORDIC
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Rotation Modes
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CORDIC Rotation/Vector Modes

• Rotation Mode

• Vector Mode

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CORDIC processor
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Iterative CORDIC Structure
Taken from A Survey of CORDIC Algorithms for
FPGA Based Computers, R. Andraka, FPGA98
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Publications

• Doukhnitch E. Highly parallel multidimensional
  CORDIC-like algorithms, Artificial Intelligence,
  No3, pp.284-293, Ukraine, 2001.
• Doukhnitch E. One way to execute digital linear
  transform, Kibernetica, (Cybernetics and Systems
  Analysis, ISSN 1060-0396), No5, pp.96-98, Kiev,
  May 1982
• E. Doukhnitch, Multidimensional Cordic-like
  Algorithms for DSP, in Proc.of The Sixteenth
  Intern. Symp. on Computer and Information
  Sciences, ISCIS XVI, pp.368-375, Antalya, Turkey,
  Nov. 2001.
• 4. E. Doukhnitch, Octonion CORDIC Algorithms for
  DSP, in Proc. of the 6th Symp. on Signal
  Processing, DSPCS2002, pp. 158-163, Sydney,
  Australia, Jan. 2002.
• 5. E. Doukhnitch, Hardware-oriented Algorithms
  for Fast Householder Transform, in Proc. of the
  First Intern. Conference on Signal Processing
  and Applications DSPA-98, v.2e, pp.129-132,
  Moscow, July, 1998

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Performing any given matrix M into the triangular
form
Upper Triangular matrix
Orthogonal matrix x M
All element of the M
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WHAT IS GIVENS ROTATION?

• Givens Transformations is an orthogonal matrix
  used for zeroing a selected entry of the matrix.
• That is mean Givens Rotations introduce zeros one
  at a time.
• The Givens Rotation is realized by the well-known
  hardware oriented CORDIC algorithm.

.
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GIVENS ROTATION
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2-D CORDIC Algorithm

• kV PiV where i0?N
• N is number of iterations (approximately number
  of bits for result representation)
• To zero the element Y, operator of rotation
  direction can be taken. So, the computation of
  elements for matrix P is not required and a
  parallel implementation of rotation
  transformation on many vectors by the same angle
  is possible.

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CORDIC modules array for 4X4 matrix
triangularization
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CORDIC ALGORITHM of 2-D PLANE ROTATION
4-D QUTERNION CORDIC ALGORITHM

• Extension of the original 2-D CORDIC
• algorithms is being 4-D Euclidean algorithm

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Publications
1.E. Doukhnitch, Synthesis for the Discrete
Quaternion Transform Algorithms Class, in Proc.
of Intern. Conf. On Intelligent Multiprocessor
Systems, pp.44-48, Taganrog, Russia, 1999 2.
E. Doukhnitch, O. Strelnikov, A. Andreev,
Application of Kronecker Matrix Product for the
Synthesis of Hardware-oriented DSP Algorithms,
in Proc. of Intern. Conf. on Signal Proc.
DSPA-99, pp. 78-83, Moscow, Sept. 1999  
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CONTROL SIGNS are either 1 or -1

• The rotation parameters tk

f (k) is set of non-decreasing positive
integers
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Architecture of a 4-D quaternion processor
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HOUSEHOLDER TRANSFORMATION

• Householder matrix P is a symmetric and
  orthogonal matrix of the form PI-wwT with
  unit matrix I and real vector w (wTw2)
• If a1 first column of matrix A, then
• a11-s
• w ?u where u a21 and
  s?(a1Ta1)1/2, ?(s2 - a11s)-1/2
• am1
• Then result of Householder transformation is
• s a12 a1m
• PA 0 a22 a2m
• 0 am2 amm

Repetitions of these macrooperations with vectors
wj (j2,3,,m) produces an upper triangular
matrix A
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HOUSEHOLDER TRANSFORMATION Example
From aT(1,4,7) construct the auxiliary vector
uT(-7.124,4,7) Normalize to
wT(-0.662,0.372,0.651)
Treat the lower 2x2 submatrix of P1A. From
aT(4.602,-0.696) construct a 2x2 Householder
matrix, add a trivial first line and column to
promote to a 3x3 matrix
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FAST HOUSEHOLDER TRANSFORM (1985)

• Method with simplified operations shifts and
  adds
• Suitable for VLSI-designing

Factorization of matrix P ? P
?Pi i0 Iterative process for PA
transformation Ai1 PiAi (1)
i0,1,2,,n A0A. The result is An?PA, if
n?? The process is represented as a sequence of
elemental reflections for m2 with vector wiT
(-2-ic1(i), c2(i)) c1(i) , c2(i) -
direction for every coordinate
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FAST HOUSEHOLDER TRANSFORM (Continue)

• In this case wiTwi ki ? 2, therefore
• Pi ki-1(kiI 2wiwiT) ki-1Ti
• where
• 1 2-2i -2-i1c1(i)c2(i)
• Ti
• -2-i1c1(i)c2(i) 2-2i 1
• FHT algorithm for m2
• Ai1 ki-1TiAi,
• c1(i) sgn a11(i),
• c2(i) sgn a21(i),
• i 0, 1, , n A0 A.

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ITERATIVE REFLACTIONS
i0
i1
a21
0
i2
w2
w0
w1
sign a21(i)
1
2
-2-i
0
a11
After n steps (in)
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FHT ALGORITHM FOR m3
wiT (-2-ic1(i), c2(i),c3(i)) ki
2-2i11. Pi ki-1(kiI 2wiwiT)
ki-1Ti  where   -2-2i 2
2-i1c1(i)c2(i) 2-i1c1(i)c3(i) Ti
2-i1c1(i)c2(i) 2-2i
-2c2(i)c3(i)   2-i1c1(i)c3(i)
-2c2(i)c3(i) 2-2i
Ai1 ki-1TiAi, c1(i) sign a11(i),
c2(i) sign a21(i), c3(i) sign a31(i), i
0, 1, , n A0 A.
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DESIGN OF CHIP
Let us represent matrix Ti as -2-(2i1) 1
2-ic1(i)c2(i) 2-ic1(i)c3(i) Ti 2
2-ic1(i)c2(i) 2-(2i1)
-c2(i)c3(i) 2-ic1(i)c3(i)
-c2(i)c3(i) 2-(2i1) ai1 Tiai For
n8, coefficient will be 7 K? (ki/2)
where ki/2 2-(2i1)1 i0
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DESIGN OF CHIP
a1 is the first column of matrix A and
a11(i)X(i), a21(i)Y(i), a31(i)Z(i)

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UNITS FOR STEP i
Each unit performs FHT algorithm for m3
c1 a1 c2 u1(i) a2 c3 a3
a11(i1)
a11(i)
a12(i)
a12(i1)
a13(i)
a13(i1)
c1 a1 c2 u2(i) a2 c3 a3

• s a12 a13
• PA 0 a22 a23
• 0 a32 a33

a21(i1)
a21(i)
a22(i)
a22(i1)
a23(i)
a23(i1)
c1 a1 c2 u3(i) a2 c3 a3
a31(i1)
a31(i)
a32(i)
a32(i1)
a33(i)
a33(i1)
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m-D HouseHolder CORDIC Algorithm

• rotational matrix Rm,i is
• Rm,i(1/(1(m-1)ti2))(1 Si)
• 1-(m-1)ti2 2ti 2ti
  2ti
• -2ti
  1(m-3)ti2 -2ti2 -2ti2
• -2ti -2ti2
  1(m-3)ti2 -2ti2
• . . .
  .
• . .
  . .
  (1 Si)
• . .
  . .
• - 2ti -2ti2 -2ti2
  1(m-3)ti2

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Octonion CORDIC Algorithm (2002)

• R8,i 1/cos?fi?8S?8 (E differs by e11-1 from a
  unit matrix)
• 1 ?iti ?iti ?iti ?iti
  ?iti ?iti ?iti
• -?iti 1 -?iti ?iti
  ?iti -?iti ?iti -?iti
• -?iti ?iti 1
  -?iti ?iti -?iti -?iti ?iti
• R8,i -?iti -?iti ?iti 1
  ?iti ?iti -?iti -?iti
  
• -?iti -?iti -?iti -?iti
  1 ?iti ?iti ?iti
• -?iti ?iti ?iti
  -?iti -?iti 1 ?iti -?iti
• -?iti -?iti ?iti ?iti
  -?iti -?iti 1 ?iti
• -?iti ?iti -?iti ?iti
  -?iti ?iti -?iti 1
• Control Signs are
• ?i fi?sign(y2,i) ?i fi?sign(y3,i)
  f(i) 0, 1, 2, 3, 4, 5, 6, 7, 8, ,13,
  14, ,32,
• ?i fi?sign(y4,i) ?i
  fi?sign(y5,i)
• ?i fi?sign(y6,i) ?i
  fi?sign(y7,i) ?i fi?sign(y8,i) fi
  sign(y1,i) ti2-f(i) ? f(i) is the shift
  sequence

Scaling Factor
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Matrix Triangularization by Octonion CORDIC
Algorithm
8x8 Matrix OverTriangularization In 8 steps
matrix will be formed. Results with for each
step will be saved.
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Some guessed Questions

• Why not higher Dimensions like 16-D ?
• Because Cayley numbers are the end-point of a
  very interesting sequences of Algebras. Dimension
  greather than 8 is not possible.
• Since HouseHolder exists as m-D why will we need
  Octonion CORDIC?
• There is no shift sequence where the convergence
  is proved for 8-D HouseHolder.
• Octonion CORDIC generalizes the Quaternion CORDIC
  where the Original CORDIC algorithm is its
  particular case.
• Hardware complexity of Octonion will be less than
  the hardware complexity of HouseHolder and
  hopefully will work faster than HouseHolder.

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Some patents

• E. Doukhnitch , O. Strelnikov, Special-purpose
  processor  for eigenvalue decomposition, Patent
  2168760, published in Russian Patent Bulletin
  16, 2001.
• E. Doukhnitch, S. Derevenskov, Matrix
  processor, Patent 2079879, published in Russian
  Patent Bulletin 14, 1997.
• . E. Doukhnitch, Unit for m-D coordinate
  transformation, Patent 2029356, published in
  Russian Patent Bulletin 5, 1995.

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Fast Hardware-Oriented Algorithm for Cellular
Mobiles Positioning (2004)

• Finding the location of a cellular mobile phone
  is one of the important features of the 3G
  wireless communication systems. Many valuable
  location based services can be enabled by this
  new feature. All location determination
  techniques which are based on cellular system
  signals and global positioning system (GPS) use
  standard trigonometric complex computation
  methods that are usually implemented in software

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Publications
1. Doukhnitch E., Muhammed Salamah, Deniz
Devrim, A fast hardware-oriented algorithm for
cellular mobiles positioning, LNCS, Vol. 3280,
Spriger-Verlag (2004), pp.267-277, ISSN
0302-9743. 2.  Doukhnitch E., Muhammed Salamah,
Fast hardwareoriented algorithm for 2-D
Positioning, Artificial Intelligence, No4,
pp.69-78, Ukraine, 2004, ISSN 1561-5359.
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Time of arrival (TOA) position determination
method
MS
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. The traditional algorithm
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Positions of Base Stations
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The SDD algorithm
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Idea of Positioning
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Parallel rotations of vectors
While( xixi1 gt d) Rotate d1 While(yigtyi1) Rot
ate d2 End while End while
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FAST ROTATIONS

• The rotation matrix M is as follows
• We took the sin function as
• sinA2-k
• and we approximate the cos function as follows
• cosA1-2-(2k1)
• Therefore, BS or vector coordinates are
  recursively rotated as follows
• xi1xi-xiayib
• yi1yi-yia-xib
• Where, a sinA , and b cosA.

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Weights of the Operations
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Number of Operations Versus sin(s) for the
Traditional and Our Algorithms
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Conclusion
There is presented a fast hardware-oriented
algorithm for locating a mobile in a cellular
network with a high accuracy. The main benefit of
the algorithm is that it avoids the calculations
of trigonometric functions. The calculations are
based on simple add, shift, and compare logical
operations and therefore it can be implemented
easily in hardware. In addition, its shown that
the number of involved operations is lower than
that of the traditional algorithm which implies
less computation time.