Page 503 - Introduction to Information Optics

488 9. Computing with Optics

correlator (P3-P5) performs the substitution phase. In Fig. 9.11, plane PI is
Fourier-transformed by lens LI onto plane P2, where a matched spatial filter
that represents the Fourier transform of the encoded search pattern is placed.
Thus, at P2 the product of the Fourier transform of the input and that of the
search pattern is formed. This product is further Fourier-transformed by lens
L2 to produce the correlation intensities at P3. The peak intensities indicate
the presence of the search pattern. The correlation peaks can be thresholded
by using an optically addressed SLM. In the second correlator (P3-P5), the
Fourier transform of the substitution pattern is put at P4 and a substitution
pattern will appear at each location that corresponds to the occurrence of the
search pattern. This implements a single symbolic substitution rule. To perform
multiple substitution rules in parallel, multichannel correlator architectures can
be used.
Recently a novel approach which combines recognition and substitution into
a single step has been proposed [70]. In this technique, symbolic substitution is
formulated as a matrix-vector (M-V) multiplication for each pair of input
recognition digits (the input vector x) and the associated pair of output
substitution digits (the output vector y). The M-V multiplication is written as
y = MX, where y is the L x 1 output vector (substitution pattern), x is the K x 1
input vector (recognition pattern), and M is an unknown K x K recall matrix.
This M-V equation is solved for the matrix M that satisfies all the possible
input and output digit pairs of a truth table. The operation can be implemented
by a single stage correlator. (An example will be shown in Sec. 9.6.1.)

9.2.6. CONTENT-ADDRESSABLE MEMORY

All arithmetic algorithms can be implemented either by logic operations or
by truth-table look-up (shown in the following Section). For the truth-table
look-up implementation, parallel architectures such as location-addressable
memory (LAM) [71], content-addressable memory (CAM) [72,73,74], or
symbolic substitution can be used. It has been pointed out that a CAM is more
efficient than a LAM in terms of processing speed because a CAM implements
the truth-table look-up by directly addressing its content rather than its
location. In this scheme, the digit combinations of the input numbers are
compared with the predetermined reference digits for generating the corre-
sponding outputs. Therefore, the main objective in CAM implementation is to
minimize the number of minterms and computational steps while ensuring a
minimum number of variables in a minterm. For example, for different addition
algorithms, one can choose the optimum algorithm in terms of the speed
minterm product. Both coherent [74] and incoherent CAM architectures have
been proposed [72,73]. Examples of incoherent CAM implementations will be
shown in Sec. 9.6.2.