Algorithms for matrix vector multiplication are implemented using acousto-optic cells for multiplication and input data transfer and using charge coupled devices detector arrays for accumulation and output of the results. No two dimensional matrix mask is required; matrix changes are implemented electronically. A system for multiplying a 50 component nonnegative real vector by a 50 by 50 nonnegative real matrix is described. Modifications for bipolar real and complex valued processing are possible, as are extensions to matrix-matrix multiplication and multiplication of a vector by multiple matrices

Caulfield, H. J.

Foster, M. J.

Horvitz, S.

Rhodes, W. T.

English

NASA Technical Reports Server

OPTICAL IMPLEMENTATION OF SYSTOLIC ARRAY PROCESSING 
H .  J .  Cau l f i e ld  , W. T. %odes+, M. J. F o s t e r  , and Sam Horvi tz-  t 9i 
* Innovat ive  Opt ics ,  Inc . ,  P.O. Box 1275, Concord, MA. 01742 
+ Georgia I n s t i t u t e  of Technology, School of E l e c t r i c a l  Engineer ing,  
-/- Carnegie-Mellon Un ive r s i ty ,  Computer Science Department, 
- Naval Underwater Systems Center ,  PA4, New London, CT. 06320 
A t l a n t a ,  GA. 30332 
P i t t s b u r g h ,  PA. 15213 
SUMMARY 
We show how certain a lgor i thms f o r  mat r ix-vec tor  m u l t i p l i c a t i o n  can b e  
implemented us ing  acous to-opt ic  c e l l s  f o r  m u l t i p l i c a t i o n  and i n p u t  d a t a  t r a n s f e r  and 
us ing  CCD d e t e c t o r  a r r a y s  f o r  accumulation and ou tpu t  of t h e  r e s u l t s .  No 2-D ma t r ix  
mask i s  requ i r ed ;  m a t r i x  changes are implemented e l e c t r o n i c a l l y .  
mu l t ip ly ing  a 50-component nonnegat ive- rea l  v e c t o r  by a 50 x 50 nonnegat ive- rea l  
matrix i s  desc r ibed .  Modif ica t ions  f o r  b i p o l a r - r e a l  and complex-valued p rocess ing  
are p o s s i b l e ,  as are ex tens ions  t o  matr ix-matr ix  m u l t i p l i c a t i o n  and m u l t i p l i c a t i o n  
of a v e c t o r  by m u l t i p l e  matrices. 
A system f o r  
INTRODUCTION 
During t h e  p a s t  several y e a r s .  Kung and Leiserson at Carnegie-Mellon 
Un ive r s i ty  ( r e f s .  1 ,2)  have developed a new type of computat ional  a r c h i t e c t u r e  which 
they ca l l  " s y s t o l i c  a r r a y  processing".  Although t h e r e  are numerous a r c h i t e c t u r e s  f o r  
s y s t o l i c  a r r a y  p rocess ing ,  a g e n e r a l  f e a t u r e  is  a f low of d a t a  through similar o r  
i d e n t i c a l  a r i t h m e t i c  o r  l o g i c  u n i t s  where f i x e d  o p e r a t i o n s ,  such as m u l t i p l e s  and 
adds,  are performed. The d a t a  tend  t o  f low i n  a p u l s a t i n g  manner, hence the name 
" sys to l i c " .  
advantages f o r  VLSI implementation over  prev ious  c a l c u l a t i o n a l  a lgor i thms f o r  such 
ope ra t ions  as mat r ix-vec tor  m u l t i p l i c a t i o n ,  matr ix-matr ix  m u l t i p l i c a t i o n ,  p a t t e r n  
r e c o g n i t i o n  i n  con tex t ,  and d i g i t a l  f i l t e r i n g .  This  paper grew out  of ou r  d e s i r e  t o  
exp lo re  t h e  p o s s i b i l i t y  of improving s y s t o l i c  array p rocesso r s  by us ing  o p t i c a l  i n -  
put and ou tpu t .  We w i l l  concen t r a t e  on d e s c r i b i n g  t h e  p a r t i c u l a r  ca se  of mat r ix-  
v e c t o r  m u l t i p l i c a t i o n ,  bu t  n o t e  t h a t  many o t h e r  o p e r a t i o n s  can be performed i n  an 
analogous manner. 
S y s t o l i c  a r r a y  processors  appear  t o  o f f e r  c e r t a i n  des ign  and speed 
SYSTOLIC MULTIPLICATION OF A VECTOR BY A MATRIX 
-f 
The problem w e  addres s  i s  t h a t  of eva lua t ing  a v e c t o r  y g iven  by 
-+ -+ 
where A i s  an n by n m a t r i x ,  and x and y are n-component v e c t o r s .  We assume t h a t  A 
has bazdwidth w, i . e . ,  a l l  of i t s  non-zero e n t r i e s  are c l u s t e r e d  i n  a band o f  width 
5 3  
https://ntrs.nasa.gov/search.jsp?R=19820008026 2020-03-21T09:54:13+00:00Z
w around t h e  major d i agona l .  
boundary v a l u e  problems f o r  o rd ina ry  d i f f e r e n t i a l  equat ions .  A s y s t o l i c  a r r a y  t h a t  
s o l v e s  t h i s  problem i s  in t roduced  by. Kung and Leiserson  ( r e f .  1 , 2 )  and w i l l  be 
reviewed b r i e f l y  h e r e .  
Such matrices arise f r e q u e n t l y  i n  t h e  s o l u t i o n  of 
A s y s t o l i c  a r r a y  f o r  mul t ip ly ing  a ma t r ix  of bandwidth w by a v e c t o r  of 
a r b i t r a r y  l e n g t h  has  inner-product  cel ls .  
f i g u r e  1. 
updat ing  t h e  v e c t o r  component yi accord ing  t o  t h e  replacement 
The a r r a y  f o r  bandwidth 4 i s  shown i n  
Each of t h e  f o u r  heavy boxes r e p r e s e n t s  an inner-product  cell ,  capable  of 
'.&e cells act t o g e t h e r  at d i s c r e t e  t i m e  i n t e r v a l s ,  o r  b e a t s ,  w i t h  h a l f  of t h e  ce l l s  
active on each  b e a t .  
v e c t o r  2 is  inpu t  from t h e  top.  
terms of t h e  v e c t o r  9 as they  move upward. 
The elements  of t h e  mat r ix  A are i n p u t  from t h e r i g h t ,  and t h e  
Zeroes are i n p u t  from t h e  bottom, and accumulate 
F igu re  2 traces t h e  a c t i o n  of t h e  a r r a y  f o r  several b e a t s ,  o r  p u l s a t i o n s ,  show- 
ing  t h e  t e r m s  of A and % and t h e  p a r t i a l  terms of ? t h a t  are in  each ce l l  on each 
p u l s a t i o n ,  Thus Zn p u l s a t i o n  1, y = 0 is en te red .  
I n  p u l s a t i o n  3, y l  becomes a l lx l .  
p u l s a t i o n  5, Y 1  e x i t s .  
ano the r  yk i s  i n s e r t e d  ( a t  an  i n i t i a l  va lue  of zero]. 
In  p u l s a t i o n  2, x1 is  en te red .  1 I n  p u l s a t i o n  4 ,  y1 becomes allxl + a12x2. I n  
Every o t h e r  p u l s e  another  y i  exits and on t h a t  same p u l s e  
OPTICAL SYSTOLIC ARRAY PROCESSING 
Key f e a t u r e s  of t h e  s y s t o l i c  a r r a y  approach t o  mat r ix-vec tor  m u l t i p l i c a t i o n  are 
(1) a r e g u l a r ,  d i r e c t e d  f low of d a t a  streams, (2) m u l t i p l i c a t i o n ,  and ( 3 )  a d d i t i o n  o r  
accumulat ion.  These f e a t u r e s  are a l s o  c h a r a c t e r i s t i c  of many o p t i c a l  s i g n a l  process-  
i n g  systems, and it should come as no g r e a t  s u r p r i s e  t h a t  o p t i c a l  implementations of 
s y s t o l i c  a r c h i t e c t u r e s  are p o s s i b l e .  S ince  both bulk and s u r f a c e  a c o u s t i c  waves are 
r o u t i n e l y  used i n  o p t i c a l  s i g n a l  process ing  t o  produce a moving stream of d a t a  and 
f o r  m u l t i p l i c a t i o n  of d a t a ,  i t  seems n a t u r a l  to use t h e s e  components f o r  o p t i c a l  
s y s t o l i c  a r r a y  process ing .  
We choose as our  example t h e  s imple matr ix-vector  m u l t i p l i c a t i o n  
as sumin i n i t i a l l y  th t a l l  q u a n t i t i e s  i n  t h i s  e q u  t i o n  are r e a l  and onnegat ive.  
The b a s i c  concept is  i l l u s t r a t e d  wi th  t h e  he lp  of f i g u r e  3 .  The system shown con- 
sists of an acous toop t i c  modulator i l l umina ted  by t h e  co l l ima ted  l i g h t  from t h r e e  
LEDS, a Sch l i e ren  imaging system, and t h r e e  d e t e c t o r s  connected t o  a CCD analog 
s h i f t  r e g i s t e r .  A t  t h e  moment i l l u s t r a t e d  i n  t h e  f i g u r e ,  modulating s i g n a l s  pro- 
p o r t i o n a l  t o  XI and x2 have been i n p u t  t o  t h e  acous to-opt ic  modulator d r i v e r ,  
54 
producing s h o r t  g r a t i n g  segments i n  t h e  acous to-opt ic  c e l l .  A s  t h e  x g r a t i n g  
segment pas ses  i n  f r o n t  of LED number 2 ( t h e  s i t u a t i o n  shown in  t h e  f i g u r e ) ,  t h a t  LED 
i s  pulsed i n  p ropor t ion  t o  ma t r ix  c o e f f i c i e n t  a l l .  
i o n a l  i n  i n t e n s i t y  t o  a1 X I ,  i s  imaged onto  CCD d e t e c t o r  2 ,  which sends a p r o p o r t i o n a l  
charge t o  an a s s o c i a t e d  'bin" i n  t h e  s h i f t  r e g i s t e r .  
The t r ansmi t t ed  l i g h t ,  p ropor t -  
The x 1  and x2  g r a t i n g  segments now travel s o  as t o  b e  i n  f r o n t  of LED'S 1 and 3, 
r e s p e c t i v e l y .  A t  t h e  s a m e  t i m e ,  t h e  accumulated CCD charge  from d e t e c t o r  2 i s  s h i f t e d  
one b i n ,  i n  t h e  d i r e c t i o n  i n d i c a t e d  by t h e  arrow l a b e l e d  "output" i n  t h e  f i g u r e .  
LED'S 1 and 3 are now pulsed  i n  p r o p o r t i o n  t o  a21 and a12, r e s p e c t i v e l y .  S ince  t h e s e  
LED'S i l l u m i n a t e  d e t e c t o r s  3 and 1 via  g r a t i n g  segments x 1 and x2 ,  charge is  genera ted  
by t h e s e  d e t e c t o r s  i n  p ropor t ion  t o  a21x1 and a12x2, r e s p e c t i v e l y ,  and accumulated i n  
t h e  corresponding s h i f t  r e g i s t e r  b ins .  
I n  t h e  nex t  increment of t h e  system, charges  are a g a i n  s h i f t e d ,  wi th  accumulated 
charge i n  p ropor t ion  t o  a l l x l  + a12x2, o r  y1  being ou tpu t .  
a s s o c i a t e d  with d e t e c t o r  2 ( a l r eady  p r o p o r t i o n a l  t o  a x ) is  augmented by a f i n a l  
s t r o b e  of LED 2 by an amount p r o p o r t i o n a l  t o  a 2 2 ~ 2 .  k ' f i na l  two s h i f t s  of t h e  CCD 
charge  packets  b r i n g  charge p r o p o r t i o n a l  t o  a21X1 + a 22 x 2 ,  Or Y2' t o  t h e  ou tpu t ,  and 
t h e  o p e r a t i o n  is  complete. 
The charge packet  now 
The system i l l u s t r a t e d  i s  e a s i l y  expanded t o  accommodate matr ix-vector  o p e r a t i o n s  -+ -t of h i g h e r  d imens iona l i ty .  
t h e  maximum number of LED's  r e q u i r e d  i s  2N-1 ( t h e  number of d iagonals  of t h e  ma t r ix )  , 
and t h e  number can be smaller i f  .- A has  a smaller bandwidth. 
If y and x are N-component v e c t o r s  and A an N x N ma t r ix ,  
Numerous v a r i a t i o n s  of t h e  system of f i g u r e  3 are p o s s i b l e .  F igure  4 ,  f o r  
example, shows t h e  LED'S r ep laced  by a s i n g l e  l i g h t  source  and an a r r a y  of  modulators .  
The CCD s h i f t  r e g i s t e r  h a s  been r ep laced  by s t a t i o n a r y  d e t e c t o r s  and i n t e g r a t o r s  
combined wi th  a second acous to-opt ic  c e l l ,  which s e r v e s  t o  d e f l e c t  l i g h t  t o  t h e  c o r r e c t  
d e t e c t o r j i n t e g r a t o r .  
f a c i l i t a t e  g r e a t e r  system dynamic range than  i s  ach ievab le  wi th  CCD d e t e c t o r  a r r a y s .  
The acous to-opt ic  d e f l e c t o r  approach t o  s o r t i n g  ou tpu t  d a t a  may 
BIPOLAR AND COMPLEX-VALUED COMPUTATIONS 
It w a s  assumed i n  t h e  preceding s e c t i o n  t h a t  a l l  e lements  of t h e  matrix and i n -  
p u t  v e c t o r s  were nonnegat ive- rea l .  
o p e r a t i o n s  of importance involve  b i p o l a r - r e a l  o r  complex-valued v e c t o r s  and matrices, 
and some means must b e  employed f o r  handl ing  them. 
bu t  n o t  n e c e s s a r i l y  nonnegat ive,  a two-component decomposition scheme descr ibed  i n  
r e f .  3 can be  employed. For complex-valued process ing ,  several schemes have been 
desc r ibed  ( r e f .  4 ) .  One of  t h e s e  invo lves  a three-component decomposition of 
complex numbers according t o  r e f .  5, 
I n  p r a c t i c e ,  most matr ix-vector  m u l t i p l i c a t i o n  
I f  t h e  elements are real  va lued ,  
= z + z1 exp [ i 2 ~ / 3 ]  + z2 exp [ i4T/3)  0 ( 4 )  
where z z and z are nonnegat ive- rea l .  Another i nvo lves  b iased  real and imaginary 
components ( r e f .  6 f .  
and t o  a reduc t ion  i n  t h e  s i z e  of t h e  v e c t o r s  and matrices t h a t  can be  accommodated. 
0' A l l  such methods l e a d  t o  some a d d i t i o n a l  processor  complexity 
55 
OPERATING PARAMETERS OF A TYPICAL SYSTEM 
Matrix s i z e  l i m i t a t i o n s  are imposed by t h e  acous to-opt ic  modulator.  Consider a 
system us ing  f o r  i npu t  a bulk  acous to-opt ic  cell  w i t h  a 100 MKz bandwidth and a 10 
psec t i m e  window. W e  estimate t h a t  such a c e l l  should accommodate 100 LED/lenslet  
combinations o p e r a t i n g  s i d e  by s i d e ,  a l lowing  m u l t i p l i c a t i o n  of a 50-component 
nonnegat ive- rea l  v e c r o r  by a 50 x SO nonnegat ive- rea l  matrix. 
range depends on CCD d e t e c t o r  dynamic range  and on t h e  c o r r e c t i o n  of LED and 
acous to-opt ic  modulator n o n l i n e a r i t i e s ;  i t  i s  t o o  s p e c u l a t i v e  t o  sugges t  numbers a t  
t h i s  t i m e .  Operat ing speed i s  determined by t h e  amount of t i m e  i t  t akes  t o  s h i f t  
t h e  components of through t h e  acous to-opt ic  c e l l ,  p l u s  se tup  and f i n a l  readout  t i m e .  
For t h e  10 Usec window cel l  under c o n s i d e r a t i o n ,  it t a k e s  5 Psec t o  g e t  t h e  XI g r a t -  
ing  segment t o  t h e  middle of t h e  acous to-opt ic  c e l l ,  at which t i m e  t h e  f i r s t  LED 
p u l s e  occurs .  
midpoint of t h e  c e l l .  Following t h a t  p u l s e ,  an  a d d i t i o n a l  50 psec  are requ i r ed  t o  
read  y ou t  of t h e  s h i f t  r e g i s t e r .  The t i m e  r equ i r ed  f o r  t h e  50 x 50 matr ix-vector  
mul t ip3 ica t ion  is  t h u s  10 psec process ing  t i m e  and 10 vsec l a t e n c y ,  f o r  a t o t a l  of 
20 usec. During t h e  process ing  i n t e r v a l ,  a t o t a l  of 2500 m u l t i p l i c a t i o n s  are per-  
formed, a t  a rate of 2.5 x 108 m u l t i p l i c a t i o n s  p e r  second. With s u i t a b l e  encoding 
7 -  - of t h e  d a t a  ( r e f s .  3 , 4 ) ,  t h i s  corresponds t o  a p rocess ing  rate of 6.25 x 10 
real  m u l t i p l i c a t i o n s  p e r  second o r  2 .78  x lo7 complex m u l t i p l i c a t i o n s  pe r  second. 
Achievable dynamic 
The last  LED p u l s e  occurs  10 psec la ter ,  when x50 f i n a l l y  passes  t h e  
b i p o l a r -  
VARIATIONS 
The system desc r ibed  does n o t  e x p l o i t  t h e  two-dimensionality of t h e  o p t i c a l  
system. 
t h e  s i n g l e  l i n e a r  L E D / l e n s l e t  and d e t e c t o r  a r r a y s  are rep laced  wi th  a c o l l e c t i o n  of 
l i n e a r  a r r a y s ,  one above t h e  o t h e r .  
square  window formats ,  can accommodate perhaps 20 such linear a r r a y s ,  a l lowing  20 
s e p a r a t e  matrices t o  m u l t i p l y  t h e  same input  v e c t o r  a t  t h e  same t i m e .  
More than  one ma t r ix  can m u l t i p l y  t h e  s a m e  i npu t  v e c t o r  a t  t h e  same t i m e  i f  
Shear  wave acousto-opt ic  modulators ,  w i th  n e a r l y  
Matr ix-matr ix  m u l t i p l i c a t i o n  can be  performed wi th  r e l a t e d  systems us ing  
m u l t i p l e  acous to-opt ic  c e l l s ,  o r ,  a l t e r n a t i v e l y ,  s i n g l e  cel ls  wi th  m u l t i p l e  d r i v e r /  
t r ansduce r s .  F igu re  5 shows one p o s s i b l e  arrangement f o r  m u l t i p l i c a t i o n  of two 
2 x 2 nonnegat ive-real  matrices. I n  g e n e r a l ,  f o r  such a scheme , .mul t ip l i ca t ion  of two 
N x N matrices r e q u i r e s  two mul t i - t ransducer  acous to-opt ic  modulators w i th  2N-1 
t r ansduce r s  each.  A l t e r n a t i v e 1  one such mul t i - t ransducer  ce l l  could b e  used,  
i l l umina ted  by a 2-D a r r a y  of N 3 ,  -2 LED'S. 
56 
REFERENCES 
e 
1, H .  T .  Kung and C. E.  L e i s e r s o n ,  " S y s t o l i c  Array Apparatuses For  Matrix 
Computations", U.S. P a t e n t  Appl ica t ion ,  F i l e d  December 11, 1978. 
2. H .  T .  Kung and C. E.  Le iserson ,  " S y s t o l i c  Arrays f o r  VLSI", i n  I n t r o d u c t i o n  To 
VLSI, by C. A. Mead and L. A. Conway (Addison-Wesley, Reading, Mass., 1980).  
3. H. J .  C a u l f i e l d ,  D. Dvore, J.  W. Goodman and W. T. Rhodes, "Eigenvector 
Determinat ion By Noncoherent O p t i c a l  Methods", Appl. Opt. 20, (1981) 2263-2265 
4.  A. R. Dias, "Incoherent O p t i c a l  Matrix-Vector M u l t i p l i c a t i o n  For High-speed 
Data Processing",  Ph.D D i s s e r t a t i o n ,  S tanford  U n i v e r s i t y ,  1980, (Univers i ty  
Microf i lm No. 8024641). 
5. J .  W. Goodman, A. R. Dias, and L. M. Woody, "Fully P a r a l l e l ,  High-speed 
Incoherent  O p t i c a l  Method For Performing Discrete F o u r i e r  Transforms", Opt. 
L e t t .  2, (1978) 1-3. 
6. J .  W .  Goodman, A. R. Dias, L. M. Woody, and J. Erickson,  "Some New Methods For 
Process ing  E l e c t r o n i c  Image Data Using Incoherent  Light",  I n  Optica Hoy y Manana 
(Proceedings Of ICO-11 Conference, Madrid, Spain,  19781, J. Bescos, A. Hidalgo,  
L. P l a z a ,  J. Santamaria,  e d s . ,  pp. 139-145. 
57 
Figure  
a l l  
a21 
a 31 
a23 0 
a33 a34 
a 4 3  a44 a45 
a 53 
e 
0 
( k +  
x1 
x 2  
x3 
X - A TIME STREAM 
- e  
- a l l  
- 0  
- e  
Y 
a12 
e 
a21 
0 
0 
a22 
e 
a3 1 
a 2 3  
e 
a32 
0 
-+ 
1.- S y s t o l i c  m u l t i p l i c a t i o n  of v e c t o r  x by banded m a t r i x  A. T r a d i t i o n a l  
r e p r e s e n t a t i o n  of t h i s  ope ra t ion  i s  shown i n  ( a ) .  Basic c e l l  f o r  t h i s  o p e r a t i o n  
i s  shown i n  ( b ) .  Flow of x, y and A d a t a  i s  shown i n  ( c ) .  - + - +  
58 
2 
3 
4 
6 
Figure  2.- The f irst  seven p u l s a t i o n s  of t h e  p rocesso r  of f i g u r e  l ( c )  are shown 
h e r e  and desc r ibed  i n  the  t e x t .  
DETECTORS 
a2 
CLOCK 
Figure  3 . -  O p t i c a l  implementation of s y s t o l i c  a r r a y  p rocesso r  of f i g u r e  
X 
A 0  CELL 
AC CU M U LATl NG 
DETECTORS 
A 
TRACKING 
A 0  CELL 
Figure  4.- An a l t e r n a t i v e  o p t i c a l  implementat ion of the p rocesso r  of f i g u r e  l ( c ) .  
F igure  5.- U s e  of c ros sed  acous to-opt ic  cel ls  t o  produce A X - -  B = C. I n p u t  
i n fo rma t ion  flow is  shown i n  ( a ) ,  and c a l c u l a t e d  - C valGes are produced 
as i n d i c a t e d  i n  ( b ) .  
60 


Optical implementation of systolic array processing

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