Download Block Linkage Learning Genetic Algorithm in the Design of Ternary

A Fast Converging Block Linkage Learning Genetic Algorithm
BRISHBHAN PANWAR AND PUSHKAR SAREEN
Centre for Applied Research in Electronics
Indian Institute of Technology Delhi
Hauz-Khas, New – Delhi, 110016, INDIA
Abstract: The representation of genes in a chromosome by (locus, value, block) has provided a richer
source of relations through representation, leading to a fast converging block linkage learning genetic
algorithm. This algorithm circumvents the limitations of linkage learning on the natural selection by
injecting the genetic material at high recombination centers, which are obtained by introducing the
fuzziness at the center of acceptance of the genetic material. The evolutionary advantage of block
linkage learning genetic algorithm is demonstrated in designing a withdrawal weighted surface
acoustic wave filter with discrete sample values {1,0,-1} in its impulse response. A powerful design
tool is developed to compensate the diffraction effects and eliminate the quantization problems of
smaller sample values in the impulse response. The block linkage genetic search, from 3 N possible
impulse responses for the ternary weighted impulse responses of length N, provides an excellent
selection process of the impulse response in approximating the desired frequency response.
Key-Words: - Block Linkage, Genetic Algorithm, Surface Acoustic Wave Filter
1. Introduction
The constraints relating to quantization
effects, and unrealizable smaller sample values in
the impulse response are best handled by
synthesizing the finite impulse response of a surface
acoustic wave filter using discrete {1,0, -1} samples
(withdrawal weighting). The existing methods of
simulated annealing, simple genetic algorithm and
its variants are quite slow and often yield
unsatisfactory results. Genetic algorithms are
successful and efficient when they can propagate
building blocks i.e. association of genes during the
evolution process. The undesirable bias in simple
genetic algorithm (sGA) [1,2] towards those
relations that have all of their equivalence defining
positions close together has been removed in messy
genetic algorithm (mGA) [3], where a single gene is
represented as ordered pair (locus and value). Unlike
the simple genetic algorithms, mGAs define the
chromosome as sequence-represented member of the
population as a collection of position-dependent
genes. In mGA, a string (0,1) (2,1) (1,0) is
analogous to the string 101 in fixed locus
representation scheme of sGA. The messy genetic
algorithm has been improvised in the linkage
learning genetic algorithms (LLGA) [4], where the
chromosome is connected end-to-end in a circle as a
necklace of genes. As the genetic material in the
chromosome approaches infinity, the building
blocks of genes in this chromosomal circle facilitate
“tight linkage”. This provides a faster and efficient
algorithm for converging to global minima.
However, the work of Betancourt et al. [5] predicts
that the linkage limits the power of natural selection
and suggests that the adaptive species differences
should
map
disproportionately
to
high
recombination regions.
A block linkage learning genetic algorithm
(BLLGA) is proposed to resolve the limitations of
LLGA on the natural selection process and its
drawback of not providing a better solution. In
BLLGA, the representation of gene by (locus, value,
and block) provides a richer source of relations
through representation. A two-dimensional block
and position crossover operator, in conjunction with
injection of genetic material at fuzzy center,
facilitates the process of combining low order
schemata to higher order schemata of increased
fitness. This provides a powerful algorithm for
getting better solution in comparison with LLGA
and its contemporary algorithms. This algorithm
suits well in finding the discrete impulse response
(sample values 1,0,-1) providing a close
approximation to the desired frequency response. In
case of narrow band filters the possible sequences
describing the impulse responses is very large (3N,
where N is the length of the impulse response). A
SAW filter realizing these specifications for CDMA
application might require the filter length exceeding
500 samples. This makes it virtually impossible to
find the impulse response providing the closest
approximation to the desired frequency response.
The BLLGA algorithm has been verified by
designing a finite impulse response (FIR) surface
acoustic wave filter operating at the center frequency
of 183.5 MHz, a fractional band width of 0.25%,
close-in side lobes better than 32 dB (within 0.9
MHz of band width), and out-of-band rejection
better than 40 dB. The more stringent specification
is to achieve the device specification within the
impulse response length of 9 mm as the active area
on the substrate. The device parameters on aperture
(transducer finger overlap defining the sample
value), harmonic operation, and aluminum thickness
are optimized using 3-port P-matrix parameters for
an insertion loss of 10 dB. The impulse response is
restricted to discrete values {1,0,-1} for reducing the
quantization effect, and problems relating to smaller
sample values. The adaptation of ternary samples in
the filter design, referred as withdrawal weighted
design methodology, reduces significantly the
distortions in the frequency response caused by
diffraction effects [6], which are attributed to the
anisotropy of the substrate and the aperture of the
samples getting comparable with acoustic
wavelength.
The locus string generated randomly is associated
with different discrete values in the solution string.
In this process each gene becomes an ordered pair
(locus, value), identical to the representation scheme
adopted in LLGA. Subsequently, the number of
blocks is selected in accordance with the likely
number of zero crossings in the impulse response of
the filter. The phenomenon is identical to different
organs (blocks) in the body being identified by
different protein structures. A 250x1000 matrix,
containing the block information, is generated
randomly for associating the block information with
each ordered pair of gene having (locus, value)
information. In this process each gene is represented
by (locus, value, and block), which not only
provides a richer source of relations through
representation, but also facilitates better ordering of
the genes during the crossover operation, thus
providing a better match to filter specifications.
1
Value
0
-1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Locus
(a)
Block
Type
c
b
a
1
Value
0
-1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Locus
(b)
2 Locus, value and block association
An impulse response/chromosome/ string of length
250 is selected on the basis of initial design
parameters on the filter specifications [7]. The
solution is evolved from an initial population
(number of impulse responses) of 1000, where each
gene is assigned discrete random value {1,0, -1}.
Fig. 1. Representation of genes in a chromosome /
string of length 20 with the values restricted to {1,0,1} (a) ordered pair (locus, value) in messy genetic
algorithm (b) (locus, value, block) in a block linkage
learning genetic algorithm.
The genes in a chromosome represented as ordered
pair (locus, value) in the messy genetic algorithm,
and (locus, value, block) format adopted in block
linkage learning genetic algorithm are shown in Fig
1 (a) and (b), respectively. In the present case the
randomly generated blocks are labeled a, b, c…z.
The block association of a gene is also changed
dynamically in case the higher order schemata starts
discovering certain block level associations in the
string. These associations are examined in top 5% of
the fittest chromosomes by searching for consistent
appearance of a gene in higher concentration region
of the genetic material corresponding to a different
type of block. A typical example of such an
occurrence is visible in a chromosome having the
(locus, value, block) string of (3,-1,a) (5,1,a) (6,*,b)
(2,1,a) (1,1,a) (4,-1,a) (8,0,c) (7,-1,c). In such cases
the block association of the gene (6, *, b) is changed
to (6, *, a).
3. Selection Process
The population variance information is used to
preprocess raw fitness values prior to scaling. This
technique, termed as sigma () truncation [8],
circumvents the problem of negative fitness
calculation encountered in the linear scaling when
most population members are highly fit, but a few
lethal have a very low value. The fitness level with
the desired frequency response of SAW filter is
obtained by subtracting the computed error function
 W (i) Hd (i) – Ho (i) from the predefined
threshold value of 60 dB [7]. In the present case the
error function defines the weighted (W (i)) sum of
the absolute value of the deviations between the
desired frequency response Hd(i), and the
frequency response Ho(i), obtained at the
predefined frequency grids. In the selection process
two strings are randomly selected and the optimal
member is chosen to be the recipient of the genetic
material from the deceptive member (weak string).
In this process, the linkage density and average
linkage for optimal building blocks shift towards
higher linkage, thereby evolving into a good
solution. The genetic material of a block is injected
within the heuristically derived closeness to the
fuzzy center, rather than the statistically derived
parameters on mean and variance. In case the
genetic material contains genes of more than one
block, the injection is carried out at the
corresponding fuzzy centers. This overcomes the
problem of statistical center being far off from the
region of concentration of the block identified for
injecting the genetic material.
4. The Exchange Crossover Operator
The linear string of chromosome is concatenated to
itself to provide a sense of circularity in the
chromosomal structure. The genes of this new string
are indexed linearly. The ascending index values,
corresponding to a particular block, are picked up
from this new string and function “fcm” of
MATLAB is applied to compute the fuzzy center of
this block. The length of the original chromosome is
subtracted from the fuzzy center in case its location
of fuzzy center exceeds this length. In this process
the recombination centers in the original
chromosomal circle are identified to carry out the cut
and splice operation. The genetic material in the
recipient chromosome is now injected at the fuzzy
centers of the block contained in the material to be
grafted. In this process, the blocks formed in the
previous generation are utilized to propagate in the
subsequent generation. It has been observed that
during the crossover operation, the population starts
discovering a certain type of block associations. This
phenomenon is analogous to the requirement of
different time side lobes (block association) in the
impulse response of a digital filter. This technique of
injecting the genetic material at high recombination
centers produces better-adapted genes [5] and
circumvents the problem of linkage in the genetic
algorithms and enhances the natural selection
process.
The crossover operation suggested by Siddharth [9]
has been used in the present algorithm. The cut and
splice operation is performed by injecting the
genetic material from a deceptive chromosome to
the optimal chromosome. The basic nature of this
crossover operation enhances the genesis of a
general structure of the chromosome by gradually
grouping the genes with the same type of block.
After the grafting process is complete, a new
offspring is obtained by deleting the non-coding and
allele genes in the recipient chromosome. The
process of generating offspring by cut and splice
operation from the randomly selected chromosomes
of previous population is continued until fresh
population of initial size is generated. In this
process, a new population of offspring is generated
using the sigma () truncation, which replaces the
previous generation.
5. Surface Acoustic Wave Filter Design
The dependence of fitness on the number of
generations for LLGA, mGA, and BLLGA, for
achieving the desired frequency response of a
withdrawal weighted surface acoustic wave filter is
shown in Fig.2. The discrete {1,0, -1} sample values
have been considered in synthesizing the impulse
response of the filter. The adaptation of this
withdrawal weighted design philosophy minimizes
the influence of diffraction, quantization, and
smaller (comparable with acoustic wavelength)
sample value effects on the frequency response of
the filter. The simulation results in Fig. 2 show
saturating behavior in the fitness level of mGA and
LLGA at 92% after 100 generations. Whereas, better
fitness levels in much lesser number of generations
(50 generations) is obtained using the BLLGA
algorithm. The fitness level of approximately 98%,
achieved after 200 generations in BLLGA, provides
an excellent match with the desired frequency
response of the filter.
Exchange algorithm. The simulation and measured
response of the filter operating at 183.5 MHz, with
the fractional bandwidth of 0.25%, close-in side
lobes better than 40 dB, and out-of-band rejection
better than 50 dB are shown in Fig.3. The measured
frequency response in the pass band region is fairly
consistent with the simulated results. However, the
deviations between the simulated and measured
performance in the close-in and far-off side lobes
exceeds 20 dB. These results are true reflection of
dominance of diffraction effects caused by the
smaller sample values in the impulse response of the
filter.
These diffraction effects can be minimized by
magnitude and phase correction of the sample values
in the impulse response of the filter. However, the
anisotropic nature of quartz and lithium niobate
substrates, commonly used for SAW filter
realization, restricts the adaptation of such a design
philosophy. An alternate approach is to design the
withdrawal weighted SAW filters. In such a design
Fig.3. Comparison of the simulated and measured
responses of a SAW filter designed using the Remez
Exchange algorithm.
Fig. 2. Comparison of fitness level for MGA, LLGA
and BLGA with the number of generations in the
evolution process
The diffraction effects on the performance of a
surface acoustic wave filter have been studied by
implementing the design of the filter on ST-X
quartz. The filter with minimum length in the
impulse response is designed using Remez
the normalized sample values in the impulse
response are restricted to {1,0,-1}. This scheme is
quite effective in eliminating the diffraction effects.
However, the design complexity is quite high for the
design of narrow band (fractional bandwidth <
0.5%) low loss SAW filters. In such cases the order
of polynomial (length of the impulse response)
defining the impulse response of the filter in z
domain will exceed 200.
representation. The 3-D representation of gene,
linking of the functional blocks, and their
correspondence with the number of side lobes in the
impulse response has enhanced the achievable
fitness level in the frequency response in comparison
with mGA, LLGA. Further, the introduction of
blocks and injecting the genetic material at the fuzzy
center of the block provides the flexibility of
controlling the length of the impulse response. The
mutation at the block level and division of the string
into controlled blocks and tightly linked genes could
further improve the fitness level.
References
Fig. 4. Frequency response of a withdrawal weighted
SAW filter
These designs are very tedious, because we need to
find a sequence defining a closest approximation to
the desired frequency response from a possible
number of sequences of 3N, where N is the order of
the polynomial (length of the impulse response).
This problem fits in well in the domain of genetic
algorithm. However we need an efficient and fast
converging algorithm for the design of SAW filters
with discrete values {1,0,-1} in its impulse response.
A withdrawal weighted SAW filter operating at the
center frequency of 183.5 MHz, with a fractional
band width of 0.25%, and side lobe level less than
32 dB is designed using the block linkage learning
genetic algorithm. The measured response of this
filter in Fig.4 shows an excellent agreement with the
simulated results in the entire frequency band of
interest.
6. Results
An efficient fast converging block linkage learning
genetic algorithm (BLLGA) has been proposed for
the design of surface acoustic wave filter with
discrete {1,0,-1} samples in its impulse response.
The representation of a gene (sample in the impulse
response) in BLGA by (locus, value, and block)
provides a richer source of relations through
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