A coherent optical spatial integration approach to ambiguity function generation is described. It uses one dimensional acousto-optic Bragg cells as input tranducers in conjunction with a space variant linear phase shifter, a passive optical element, to generate the two dimensional ambiguity function in one exposure. Results of a real time implementation of this system are shown

Daehlin, O. T.

Lee, T. C.

Rebholz, J. J.

Tamura, P. N.

English

NASA Technical Reports Server

OPTICAL SPATIAL INTEGRATION METHODS FOR 
AMBIGUITY FUNCTION GENERATION 
P.N. Tamura, 5.5. Rebholz, O.T. Daehl in ,  and T .C .  L e e  
Honeywell Corporate  Technology Center 
Bloomington, MN. 55420 
SUMMARY 
An o p t i c a l  s p a t i a l  i n t e g r a t i o n  approach t o  ambiguity f u n c t i o n  g e n e r a t i o n  has  
been developed. It u s e s  acous to-opt ic  Bragg ce l l s  as i n p u t  t ransducers  and a s p e c i a l  
p a s s i v e  o p t i c a l  element c a l l e d  a space  v a r i a n t  l i n e a r  phase s h i f t e r  t o  create t i m e -  
delayed v e r s i o n s  of one of t h e  s i g n a l s .  R e a l  t i m e  p rocess ing  h a s  been achieved.  
INTRODUCTION 
The ambiguity f u n c t i o n  X ( V , T )  f o r  two given s i g n a l s  f ( t )  and f ( t )  1 2 i s  d e f i n e d  by: 
j ,  
X ( V 3 - r )  = 1; f I ( t )  f 2 ( t - r )  e x p ( - j 2 ~ v t )  d t .  (1) 
The ambiguity f u n c t i o n  i s  widely used i n  t h e  process ing  of r a d a r  and sonar  
s i g n a l s .  
ponent of a t a r g e t .  When t h e  v and T v a r i a b l e s  i n  (1) t a k e  on t h e  a p p r o p r i a t e  v a l u e s  
t o  compensate f o r  t h e  d e l a y  and Doppler d i f f e r e n c e s  between f l ( t )  and f 2 ( t ) ,  t h e  
ambiguity f u n c t i o n  w i l l  y i e l d  a peak v a l u e  a t  t h e  s p e c i f i c  p o s i t i o n  ( v , ~ )  i n  t h e  
range-Doppler plane.  
A rece ived  s i g n a l  f 2 ( t )  w i l l  d i f f e r  from a r e f e r e n c e  s i g n a l  f ( t )  by a 
t i m e  d e l a y  and a Doppler s h i f t  corresponding t o  t h e  range and r a d i a l  ve l o c i t y  com- 
Equat ion (1) imposes a severe computat ional  l o a d  on d i g i t a l  s i g n a l  process ing  
hardware,  e s p e c i a l l y  i n  s u r v e i l l a n c e  a p p l i c a t i o n s  where t imely  computation of l a r g e  
numbers of ambiguity f u n c t i o n s  cal ls  f o r  near  rea l  t i m e  p rocess ing .  Consequently,  
many o p t i c a l  schemes t o  perform t h e  ambiguity c a l c u l a t i o n  have been ' n v e s t i g a t e d  
over  t h e  y e a r s .  
r e c e n t l y ,  t i m e  i n t e g r a t i n g  a r c h i t e c t u r e s  t h a t  u t i l i z e  acousto-opt ic  Bragg cel ls  as 
These i n c l u d e  both  s p a t i a l  i n t e g r a t i o n  approaches'-', and more 
i n p u t  t r a n s d u c e r s .  7 
I n  t h i s  paper ,  a coherent  s p a t i a l  i n t e g r a t i o n  o p t i c a l  p rocessor  a r c h i t e c t u r e  
t h a t  computes t h e  ambiguity f u n c t i o n  w i l l  b e  descr ibed .  The r e s u l t s  of r ea l  t i m e  
p r o c e s s i n g  w i l l  a l s o  be shown. 
PROCESSING ARCHITECTURE 
I n s p e c t i o n  of e q u a t i o n  (1) shows t h a t  t h e  o p e r a t i o n s  t o  be performed by t h e  
167 
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s i g n  i s  oppos i te .  
Therefore ,  t h e  space  v a r i a n t  l i n e a r  phase s h i f t e r  can be  a c c u r a t e l y  f a b r i c a t e d  
by cementing a c y l i n d r i c a l  l e n s  and a s p h e r i c a l  l e n s  of oppos i t e  power t o g e t h e r ,  and 
o r i e n t i n g  them a t  4 5 " .  The f o c a l  l e n g t h  of t h e  c y l i n d r i c a l  l e n s  should be  h a l f  t h a t  
of t h e  s p h e r i c a l  l e n s .  
ACOUSTO-OPTIC BRAGG CELLS 
The key element i n  any o p t i c a l  p rocess ing  system i s  the  i n p u t  t r ansduce r  which 
conver t s  an e l e c t r o n i c  s i g n a l  i n  t i m e  t o  an o p t i c a l  s i g n a l  i n  space.  
such device  known i s  t h e  acous to-opt ic  Bragg c e l l ,  which r ece ives  a temporal  s i g n a l  
modulated on an RF carrier which propagates  through t h e  c r y s t a l  as a shea r  wave, and 
space-modulates an i n c i d e n t  o p t i c a l  beam. Speed i s  a c r i t i c a l  f a c t o r  i n  ambiguity 
f u n c t i o n  c a l c u l a t i o n ,  and s i n c e  d i g i t a l  s i g n a l  processor  throughput w i l l  soon 
approach t h e  v ideo  rates a t  which most two-dimensional s p a t i a l  l i g h t  modulators  
o p e r a t e ,  Bragg ce l l s ,  which have a frame t i m e  of 50 p s ,  are a ve ry  a p p r o p r i a t e  cho ice  
as inpu t  t r ansduce r s .  I n  a d d i t i o n  t o  having a f a s t  frame t i m e ,  Bragg ce l l s  a l s o  have 
t h e  advantages of e l e c t r o n i c  d a t a  composing and h igh  d i f f r a c t i o n  e f f i c i e n c y .  
The f a s t e s t  
The Bragg cel ls  used f o r  a l l  experiments  were TeO devices  t h a t  o p e r a t e  i n  t h e  
slow shea r  mode wi th  5 0  us frame t i m e s .  The c e n t e r  frequency w a s  56.5  MHz w i t h  
4 0  MHz bandwidth a t  t h e  3 dB p o i n t s ,  g iv ing  a time-bandwidth product  of 2000. They 
w e r e  designed t o  o p e r a t e  a t  514.5 nm. 
2 
REAL TIME PROCESSING 
The most impor tan t  des ign  c r i t e r i o n  i n  t h e  real t i m e  system i s  t h e  system t i m e -  
bandwidth product  (TBW). An a c o u s t i c  shea r  wave i n  a TeO Bragg c e l l  w i l l  p ropagate  
wi th  a speed of 617 me te r s / sec ,  wh i l e  t h e  p r a c t i c a l  p h y s i c a l  window s i z e  is  30 mm. 
It t akes  only  4 8  psec  f o r  t h e  a c o u s t i c  wave t o  propagate  through t h e  window. A 
s u f f i c i e n t  system TBW, t h e r e f o r e ,  r e q u i r e s  a f a s t  d a t a  rate and h igh  speed DACs.  
F a s t  8 b i t  DACs commercially a v a i l a b l e  o p e r a t e  a t  a 5 0  MHz d a t a  ra te ,  and a d i g i t a l  
t i m e  base  compression (TBC) b u f f e r  of comparable speed must b e  cons t ruc t ed  t o  o p e r a t e  
wi th  it. 
2 
The real t i m e  system i s  designed around t h e  o p t i c a l  ambiguity f u n c t i o n  genera- 
t o r  t h a t  u t i l i z e s  two Bragg c e l l s  and t h e  LIPS element.  
c o n s i s t i n g  of a TBC b u f f e r ,  two d ig i t a l - to -ana log  conve r t e r s  (DACS), and two RF 
d r ive r /modu la to r s ,  r e c e i v e s  t h e  i n p u t  s i g n a l s  from t h e  HP 9825 c a l c u l a t o r  i n  d i g i t a l  
form and conve r t s  them t o  analog s i g n a l s  which cause index  of r e f r a c t i o n  modulat ions 
i n  t h e  Bragg cel ls  which are p r o p o r t i o n a l  t o  t h e  i n p u t  s i g n a l s .  The r e a r  end 
e l e c t r o n i c s ,  c o n s i s t i n g  of a two-dimensional d e t e c t o r  a r r a y  camera and a v ideo  
memory, cap tu res  t h e  ambiguity f u n c t i o n ,  d i s p l a y s  i t  on a CRT, and d i g i t i z e s  and 
s t o r e s  i t  f o r  examinat ion.  A block diagram of t h e  rea l  t i m e  system i s  seen  i n  
F igure  3 .  
The f r o n t  end e l e c t r o n i c s  
The p rocess ing  ra te  of t h e  rea l  t i m e  system i s  l i m i t e d  by t h e  d e t e c t o r  a r r a y  
read  out  t i m e  t o  500 f rames /sec .  
enough t o  create 500 d i f f e r e n t  frames each second. I n s t e a d ,  t h e  s a m e  frame i s  
Also,  t h e  HP 9825 cannot r e load  t h e  TBC b u f f e r  f a s t  
168 
o p t i c a l  s i g n a l  processor  a r e  a conjugat ion  o p e r a t i o n ,  a m u l t i p l i c a t i o n  of f ( t )  by 
f 2 ( t - T ) ,  and a one dimensional  F o u r i e r  t ransform a long  t h e  Doppler axis.  
o p e r a t i o n s  can e a s i l y  b e  performed i n  t h e  space  domain by F o u r i e r  o p t i c s  techniques ,  
and an o p t i c a l  a r c h i t e c t u r e  capable  of performing them i s  shown i n  F i g u r e  1. 
Tkese 
The f i r s t  o p e r a t i o n ,  conjugat ion ,  is  performed by p l a c i n g  each s i g n a l  on a 
carrier frequency and p a s s i n g  t h e  oppos i te  d i f f r a c t e d  o r d e r  from each Bragg c e l l .  
T h i s  he te rodyning  w i t h  a carrier i s  necessary  anyway s i n c e  t h e  s i g n a l  d a t a  i s  i n v a r i -  
a b l y  i n  complex form from a previous  basebanding o p e r a t i o n .  
m u l t i p l i c a t i o n ,  i s  performed by imaging Bragg ce l l  I t e l e c e n t r i c a l l y  on t o  Bragg 
c e l l  11. F i n a l l y ,  l e n s  S5 t a k e s  a one-dimensional F o u r i e r  t ransform a long  t h e  
frequency a x i s  t o  y i e l d  t h e  ambiguity f u n c t i o n .  T h i s  a r c h i t e c t u r e  has  been analyzed 
The second o p e r a t i o n ,  
i n  formal  F o u r i e r  mathematics i n  a previous  paper 8 . 
The t ime-sh i f ted  v e r s i o n  of f 2 ( t )  is  created by us ing  a p a s s i v e  o p t i c a l  element 
c a l l e d  a space  v a r i a n t  l i n e a r  phase s h i f t e r  (LPS). The a c t i o n  of t h i s  LPS element 
can b e  understood by cons ider ing  t h a t  i t s  h o r i z o n t a l  c r o s s  s e c t i o n  i s  t h a t  o f  a 
pr ism,  The r e s u l t  of imaging Bragg c e l l  I o n t o  Bragg c e l l  I1 through a pr ism i s  t o  
cause  a p o s i t i o n  s h i f t  of t h e  image. S ince  t h i s  i s  a s p a t i a l  i n t e g r a t i o n  approach, 
a m i s r e g i s t r a t i o n  i n  t h e  space  domain i s  e q u i v a l e n t  t o  a t i m e  s h i f t  i n  t h e  t i m e  
domain, and a delayed v e r s i o n  of f 2 ( t )  w i l l  be  c r e a t e d .  By c o n t i n u a l l y  v a r y i n g  
t h e  wedge a n g l e  of t h e  pr ism a long  t h e  ve r t i ca l  a x i s ,  a continuous range  of de lay  
v a l u e s  can b e  c r e a t e d .  
The f e a s i b i l i t y  of implementing t h e  o p t i c a l  a r c h i t e c t u r e  of f i g u r e  1 depends 
h e a v i l y  on t h e  m a n u f a c t u r a b i l i t y  of t h e  l i n e a r  phase s h i f t e r  element.  I t  is essen- 
t i a l l y  an o p t i c a l  wedge; i t s  wedge angle  changes l i n e a r l y  wi th  h e i g h t .  The complex 
t r a n s m i s s i v i t y  f u n c t i o n  of t h i s  component i s  given i n  r e c t a n g u l a r  c o o r d i n a t e s  by 
where a i s  a c o n s t a n t ,  
Conventional manufactur ing processes  such as g r i n d i n g  and p o l i s h i n g  a g l a s s  
p i e c e  would b e  d i f f i c u l t  i f  n o t  imposs ib le  t o  apply t o  t h e  f a b r i c a t i o n  of such an 
element .  W e  have invented  a method t o  f a b r i c a t e  t h i s  component out  of convent iona l  
o p t i c s ,  hence h igh  accuracy of t h e  t r a n s m i t t e d  wavefront i s  p o s s i b l e .  
By modifying e q u a t i o n  ( 2 ) ,  we  have 
where t h e  decomposition of equat ion  ( 3 )  i s  5 s s e n t i a l l y  t h e  same as t h a t  used i n  t h e  
chirp-z a lgor i thm.  
( x ' , y ' )  t h a t  i s  r o t a t e d  from (x,y> by 45" ( f i g u r e  2 ) ,  equa t ion  ( 3 )  can be  r e w r i t t e n  as 
I f  we  d e f i n e  r = x2 + y and i n t r o d u e e  a coord ina te  system 
a 2  
2 
12 -j- 
g(x,y)  = ejax e ( 4 )  
The f i r s t  exponent i n  e q u a t i o n  ( 4 )  is  t h e  complex t r a n s m i s s i v i t y  f u n c t i o n  of a 
c y l i n d r i c a l  l e n s  o r i e n t e d  p a r a l l e l  t o  t h e  x' a x i s .  The second exponent i s  a spher-  
i c a l  l e n s .  The c y l i n d r i c a l  l e n s  i s  t w i c e  as powerful as t h e  s p h e r i c a l  l e n s ,  and t h e  
169 
r e p e t i t i v e l y  read t o  t h e  Bragg ce l l s  t o  demonstrate  h igh  speed process ing .  
The t e s t  s i g n a l s  used t o  demonstrate  t h e  rea l  t i m e  p rocess ing  c a p a b i l i t y  of 
t h i s  system were s y n t h e t i c a l l y  produced complex random s i g n a l s  w i t h  vary ing  amounts 
of n o i s e  added. The t i m e  bandwidth product  w a s  s e l e c t e d  t o  b e  256. The output  of 
t h e  100 x 100 d e t e c t o r  a r r a y  camera w a s  p l o t t e d  i n  an i s o m e t r i c  d i s p l a y  format.  One 
such p l o t  f o r  an i n p u t  s igna l - to-noise  r a t i o  of 0 dB ( s i g n a l  power e q u a l  t o  n o i s e  
power) i s  shown i n  f i g u r e  4. The o p t i c a l l y  produced r e s u l t s  compare favorably  w i t h  
d i g i t a l  c a l c u l a t i o n s ,  and t h e i r  r e s o l u t i o n s  along both  t h e  d e l a y  and Doppler axes  are 
i n  agreement w i t h  t h e o r e t i c a l  p r e d i c t i o n s .  
CONCLUSIONS 
A coherent  s p a t i a l  i n t e g r a t i o n  approach t o  ambiguity f u n c t i o n  g e n e r a t i o n  h a s  
been descr ibed .  It u s e s  one-dimensional acous to-opt ic  Bragg ce l l s  i n  conjunct ion  
w i t h  a space  variant l inear  phase s h i f t e r  o p t i c a l  element t o  g e n e r a t e  t h e  two- 
dimensional  ambiguity f u n c t i o n  i n  one exposure.  A rea l  t i m e  implementation of t h i s  
system h a s  been demonstrated.  
REFERENCES 
1. L . J .  Cutrona, E.N. L e i t h ,  C . J .  Palermo and L . J .  P o r c e l l o ,  "Opt ica l  d a t a  proces- 
s i n g  and f i l t e r i n g  systems", I R E  Trans.  IT-6, 386, (1960). 
2 .  R.A.K. Sa id  and D . C .  Cooper, "Crosspath real-time o p t i c a l  c o r r e l a t o r  and 
ambigui ty-funct ion p r o c e s s o r t t ,  Proc.  I E E ,  120,  423, (1973).  
3. Robert  J.  Marks 11, John F. Walkup and Thomas F. K r i l e  "Ambiguity f u n c t i o n  
d i s p l a y :  an improved coherent  processor" ,  Appl. Opt. - 16 ,  746 (1977). 
4 .  David Casasent and B.B.K. Ki jaya  Kumar, "Opt ica l  image p l a n e  c o r r e l a t o r  f o r  
ambiguity s u r f a c e  computation", Appl Opt. - 18, 1674 (1979). 
5 .  Poohsan N.  Tamura, Joseph J .  Rebholz and T.C. Lee,  "Ambiguity Map Generat ion wi th  
Coherent Optics",  J.  Opt. SOC. Am. - 69 1451, (1979). 
6 .  T.C. L e e ,  J .  Rebholz, P.N. Tamura and J .  Lindquis t  "Ambiguity process ing  by j o i n t  
F o u r i e r  t ransform holography" Applied Opt ics ,  19, 595, 1980. 
7.  P .  Kellman, " T i m e  I n t e g r a t i n g  O p t i c a l  P r o c e s s o r s t t ,  SPIE Proc.  - 185,  (1979). 
8.  P.N. Tamura, J.J. Rebholz, O.T. Daehl in ,  and T.C. L e e ,  " R e a l  T i m e  O p t i c a l  Compu- 
t a t i o n  of t h e  Ambiguity Function",  SPIE Proc.  _I 241, (1980).  
170 
", 
Y I 
Figure  1.- O p t i c a l  l ayout  f o r  ambiguity func t ion  genera t ion .  
Y 
Y 
Figure  2.- Cons t ruc t ion  of the  space  v a r i a n t  l i n e a r  phase s h i f t e r .  
171 
BRAGG 
DATA F L O W  
-- - CONTROL SIGNAL 
Figure  3 . -  Real t i m e  system block  diagram. 
f REOUENCY (NU) 
Figure  4.- Real t i m e  r e s u l t s ,  SNR = OdB. 
172 


Optical Spatial integration methods for ambiguity function generation

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