J. Software Engineering & Applications, 2010, 3, 1032-1039
doi:10.4236/jsea.2010.311121 Published Online November 2010 (http://www.SciRP.org/journal/jsea)
Challenging the Evolutionary Strategy for
Synthesis of Analogue Computational Circuits
Yerbol A. Sapargaliyev, Tatiana G. Kalganova
Brunel University, London, UK.
Received July 21st, 2010; revised August 27th, 2010; accepted September 2nd, 2010.
ABSTRACT
There are very few reports in the past on applications of Evolutionary Strategy (ES) towards the synthesis of analogue
circuits. Moreover, even fewer reports are on the synthesis of computational circu its. Last fact is mainly due to the dif-
ficulty in designing of the comp lex nonlinear fun ctions that these circuits perform. In this paper, the evolving power of
the ES is challenged to design four computational circuits: cube root, cubing, square root and squaring functions. The
synthesis succeeded due to the usage of oscillating length genotype strategy and the substructure reuse. The approach
is characterized by its simplicity and represents one of the first attempts of application of ES towards the synthesis of
“QR” circuits. The obtained experimental results significantly exceed the results published before in terms of the circuit
quality, economy in components and computing resources utilized, revealing the great potential of the technique pro-
posed to design large scale analog circuits.
Keywords: Analog Circuit Synthesis, Evolutionary Electronics, Computational Circuits, SPICE
1. Introduction
The Evolvable Hardware (EHW) is one of the most
promising areas of today’s electronics. The EHW where
the ultimate goal is an electronic circuit is also known as
Evolutionary Electronics (EE) [1]. Evolutionary Algo-
rithms (EA) together with a circuit simulation tool (or
real hardware) automatically designs the circuit for a
given problem. This approach uses very little knowledge
of conventional circuit design theory and is mainly based
on the exploitation of “search and test approach”.
In general, EA navigated by fitness values, provides
randomly created and mutated chromosomes. Each chro-
mosome encodes a structure for a circuit in a form of
genotype and has to be evaluated by a fitness function.
The fitness function assigns each chromosome with a
fitness value that defines how close the current hardware
structure is to the target by its functioning. The circuits
evolved may have unconventional designs and less of all
depend on the personal knowledge of a designer. Nowa-
days, the EA is represented by Genetic Algorithm (GA),
Genetic Programming (GP) and ES. While GA is defi-
nitely the most popular tool, GP is rapidly developing in
recent years and is notable by its outstanding results in
the area of EE. On the other hand, ES that first was in-
troduced in [2], can be named as the simplest EA due to
it does not use the crossover between chromosomes dur-
ing mutation stage; each mutation involves only one
chromosome. One of the main targets of this paper is to
discover the potentials of ES in evolving large analog
circuits. The second column of Table 1 carries the statis-
tics on types of EAs used in other works.
The analogue CCs play an important role in applica-
tions where it is required to have a limited number of
mathematical functions. They suggest an economy in
components eliminating the analogue-digital-analogue
conversion that conventional designs usually incorporate,
and provide considerably shorter delay in circuit re-
sponse. The vitality of CCs is well described in [3].
Today the open-ended methods of evolutionary analog
circuit synthesis are questioned (i.e., in [1,4]) with an
important issue, whether they are able to create solutions
that are valid and trustworthy enough being realized in
silicon? In [5] the set of experiments have proven that
the open-ended methods enabled to design low/high-pass
filters with topology-based robustness. In [6] the fre-
quency discriminator robust to wide temperature range
was evolved with an open-ended GA intrinsically in
FPGA. The [7] describes experiments that allowed adap-
tive in-situ circuit reconfiguration in extreme tempera-
ture and radiation environments. In [8], the uncon-
strained evolution successfully created the analog vari-

Challenging the Evolutionary Strategy to Synthesis Analogue Computational Circuits1033
ability-tolerant CMOS circuits performing XOR and
XNOR functions. The literature review on that subject
enables to distinguish two approaches. The first tradi-
tional one follows the paradigm wherein the evolution is,
firstly, set to discover the unconventional design and later
the circuit is tuned to improve the robustness [6,7,9-11].
Another approach suggests for use the evolutionary sys-
tem that originally purposed for the robust designs [4,5,8].
In a current work, we go along the first way focusing on
exploration of technique’s capabilities to create uncon-
ventional designs, leaving the evolution of robustness for
the next stage experiments.
The next section overviews the previous work in the
area. Section 3 introduces the whole evolutionary tech-
nique. Section 4 describes the experimental results to-
gether with comparison with the results obtained before.
And, finally, the last section concludes the paper.
2. Previous Work
In the past the low-pass filters [5,10-16], high-pass filters
[5,10,11,13,17,18] and amplifiers [1,9,11,12] were suc-
cessfully synthesized with the help of EA. In [1] the un-
constrained evolution, both spatially and temporally had
been applied towards a digital reconfigurable hardware -
FPGA. By releasing the full repertoire of behavior that
FPGA can be manifest, namely, allowing any connec-
tions among modules, letting to evolve the granularity
and synchronization, evolution had been able to find a
highly efficient electronic structure, which requires 1-2
orders less silicon area to achieve the same performance
as conventional design does. Natural behavior of ana-
logue components started to be exploited inside a digital
circuit.
In analogy to this approach, the unconstrained evolu-
tion in our previous endeavor to sharpen our technique
was applied in [16] towards the originally analogue cir-
cuits (low-pass filters) and excellent results were retained.
Most of the works in the area start from evolving a
low-pass filter (Table 1). The last one is a convenient tool
for the probation of evolutionary technique and tuning the
EA parameters towards the more sophisticated designs. In
this paper, we tried to evolve the computational circuits
(CCs) that perform the: cube root, cubing, square root [19]
and squiring functions. CCs, in contrast to filters, enable a
comparison of resulted designs with circuits published
before much easier. This is due to fewer numbers of
characteristics that describe the circuit. A filter, besides
main characteristics such as a stop-, pass- and transi-
tion-bands, has the attenuation values and ripples, which
are difficult to count during the comparison. In contrast,
a CC is characterized only by average error of the com-
puting function, which together with circuit size (com-
ponent amount) and evaluation efforts gives the
Table 1. Advances on the evolution of analogue circuits.
Researcher EA type GLVS CSCR
Koza et al. [9] GP ILG Partially
Mydlowec et al. [3] GP ILG Partially
Streeter et al. [20] GP ILG Partially
Lohn, Colombano [11] GA ILG Yes
Goh, Li [12] GA ILG Yes
Zebulum, et al. [10] GP,GA ILG,OLG,UDIP Yes
Grimbleby [14] GA ILG Data n/a
Dastidar, et al. [21] GA OLG Yes
Ando, Iba [13] GP,GA Data n/a Yes
Sripramong et al. [22] GP Fixed Yes
Walker et al. [8] GA Fixed Partially
Chang, Hou, Su [18] GP UDIP Yes
Mattiussi, Floreano [23]GA OLG Yes
Gan, Yang et al. [17] CS OLG Yes
McConaghy et al. [4] GP ILG Yes
Kim et al. [5] ES ILG Data n/a
Walker et al. [8] GP + ES OLG No
Das A., Vemuri [24] GA UDIP Yes
LCR Sapargaliy-
ev et al.
[16] OLG ES No
QR(current)OLG ES No
CSCR is for circuit-structure-checking rules; PO is for parameter optimiza-
tion; GLVS is for the genotype length varying strategies; CS is for Clonal
Selection.
sufficient picture to judge on capabilities of the method-
ology.
Works [10,13] gave the comparison between GP and
GA. The first work was made as an analogy to the biology
concept with comparison of different types of variable
length genotypes strategies, whereas in the second one it
was intrinsic evolution of a real hardware for robustness
purposes. According to [10], the “genotype length varying
strategies” refer to the way in which the chromosome’s
lengths are sampled by the EA at each generation. It is
easy to follow this idea if one looks at sizes of the best
circuits throughout the generations. There were different
kinds of strategies introduced in [10], where two of them
have shown excellent results: Increasing Length Geno-
types (ILG) and Oscillating Length Genotypes (OLG). If
it occurred that the size of the best circuit at each fol-
lowing generation never decrease than it is ILG, otherwise
it is OLG. The OLG strategy is a kind of ILG in which the
genotypes are also allowed to decrease in size. The main
purpose of OLG is to create pathways from large to
smaller genotypes with improved fitness values. The third
column of Table 1 summarizes the information on OLG
vs. ILG.
Most of the works focus on such circuits like filters and
amplifiers which, we think, are not an adequate enough
challenge for the probation of the up-to-date evolutionary

Challenging the Evolutionary Strategy to Synthesis Analogue Computational Circuits 1034
techniques. Current research is devoted to CCs which are
the ones of most provoking issues for any automatic cir-
cuit synthesis system. It should be mentioned that the
largest analog circuit evolved in the area of EE is a squire
root circuit with 64 components in [9]. We found another
three papers on CCs [3,24,25] that regarded the same
circuits as in this paper. In [3,9,20] they used GP cir-
cuit-constructing program trees approach with four kinds
of functions. They also used automatically defined func-
tions and potentially enabled certain substructures to be
reused. The paper [9] suggests an attractive opportunity
among all others for judging on effectiveness of the evo-
lutionary tool. Targeting to the same arithmetic functions
and utilizing an identical evaluation procedure (fitness
function), one can directly compare the fitness values
(average error), circuit size (economy) and PC time spent.
In this paper, we took advantage of this opportunity.
In [3] two CCs were developed by the similar evolu-
tionary technique as in [9], however they used time-con-
tinues signals in time-domain simulations. The transient
analysis of a circuit in contrast to DC-analysis provided
more robust circuits despite the higher time-consumption
to complete the analysis. The patent in [25] presents the
conventionally designed cubing CC, that was improved in
[19] by iterative refinement method. Both are taken for
comparison in Section 4.
The work in this paper contributes to the following is-
sues of EE: a) discovers the potentials of ES towards the
design of nonlinear analog circuits, and b) reveals the
ability of unconstrained evolution to find more efficient
and unconventional designs.
3. Evolutionary Technique of QR Circuits
Reaching successful circuits most of all depends on an
evolutionary technique that worked out and applied. The
last one is a set of rules according to which, parameters of
EA (e.g., mutation rate, crossover, selection, etc.), geno-
type length varying strategies [10], mutation types and the
circuit representation technique are managed.
3.1. Encoding (Representation)
We use only three types of components as in [3,9]: Qn –
the n-p-n bipolar transistor, Qp – the p-n-p bipolar tran-
sistor and R – resistor. The linear circuit representation
proposed for use, i.e., every component of a circuit rep-
resented as a particular gene, and each gene consisted
Rx N1 N2 Pa Qx N1 N2 N3
(a) (b)
Figure 1. A gene coding (a) resistor; (b) bipolar transistor.
Rx-loci and Pa-loci are the resistor’s name and parameter;
Qx-loci is the transistor’s name; N1, N2, N3-loci are the
nodes for the first, the se c o nd and the thir d pins.
of 4 loci corresponding to component’s features: name,
node numbers to each pin and parameter (only for R). On
Figure 1 is a view of a gene coding a resistor (a) and a
bipolar transistor (b). The gene looks exactly the same as
a component line in the PSPICE netlist, so, there is no
necessity to convert a genotype into a netlist. The sim-
plicity of linear representation we utilized simplifies the
terminology, for example, we mean “a circuit” when we
mention “a chromosome”, we mean “a component” when
we mention “a gene”, we mean “a population” when we
mention “a netlist”, and vice versa.
For a resistor’s Pa-loci, we set 64 possible values of
E-12 series, i.e., there were 5 decades from 10 Ω to 1E + 6
Ω available for evolution, plus four additional parameters.
3.2. Unconstrained Evolution of “QR” Circuit
In an analogue domain, the circuit-structure-checking
rules at the netlist composition stage that prohibit some
circuits to be tested/simulated were regarded as the main
constraints in evolution. The target of these rules is usu-
ally increasing the portion of rightly analyzed circuits as
well as the avoiding time consumptions during resolving
the implausible designs by simulation software. In [16]
we called “absolutely unconstrained evolution of an
analogue circuit” the process of circuit netlist generation
during which no circuit-structure-checking rules applied
and all the circuits are counted as valid graphs except
ones that have components with dangling nodes and with
isolated sub-circuits. We utilized special technique that
enabled us to avoid most of the errors inherent to circuits
built up of reactive components inductors and capacitors.
In a current work we use Qn, Qp and R components.
Here transistors can lead to unconvergences during anal-
ysis. Therefore, in most of the works in Table 1 (column
4) the rules banning some transistor connections, such
like emitter-to-collector and base-to-PS, were applied. In
this paper, we did not apply any prohibitions to any kind
of connections. We also did not prohibit the formation of
“loops” of components during circuit growth. “Loop” (or
sub-circuit) is a component or a group of components
aside from the main circuit that does not connect to the
main circuit or connects to it only via alone node.
“Loops” mainly do not influence on functionality of the
whole circuit, however, they participate in carrying the
neutral mutations [1]. The statistics on constraint/un-
constraint methods in the area of analog EE are shown in
the fourth column of Table 1.
The transient analysis applied to perform the evalua-
tion instead of DC-analysis. Due to the tolerance that
transient analysis expresses to the same circuits that under
the DC-analysis are treated as unconvergent [26], we
could maximize the portion of valid chromosomes in
each population. This hint allows the multitude of indi-

Challenging the Evolutionary Strategy to Synthesis Analogue Computational Circuits1035
viduals that could potentially carry the right structures, to
pass on to the next generation, making significant con-
tribution to the unconstraint evolution.
3.3. Experiment Settings
The embryo circuit is the component or a number of
components (including the voltage source), that can be
predetermined for the particular targeted circuit to ease
the further circuit growth. We defined the embryo circuit
for all four kinds of our targets the same: a pulse voltage
source, source resistance Rsource = 1 kΩ and the load
resistance Rload = 1 kΩ. These three components on Fig-
ure 2 compose the embryonic circuit and are absolutely
identical to that ones in most of the works in Table 1. The
embryo also has two sources of direct voltage suggesting
the evolution to choose between (or use both) +15V and
–15V, so that the initial node number at a start is five.
Figure 3 generally shows the algorithm of the ex-
periment. It consists of 4 main blocks. The PC program
written in C programming language described all four
parts and unites them in one code.
The Start-block provides population of chromosomes
in the form of PSPICE netlist. This block includes all the
data necessary for embryo circuit production. Being de-
livered to ES block, every chromosome at this stage is
grown up from the embryo to the individuals with the
same number of genes. At first generation to be analyzed
all chromosomes consisted of three components.
ES block performs the particular procedures of ES,
such as: cloning them the best chromosomes, mutating
and checking for termination criteria. It modifies the
genotype and produces the population of chromosomes in
-15V
Rs Rl
N
ode0
N
ode2
N
ode3
N
ode1
N
ode4
N
ode0
+15V
Figure 2. Embryo circuit.
Fitness
values
OrCAD
netlists
Gneration
cycle
2. ES
Analysis
results
4.Fitness
Evaluation
3.Orcad
Pspice
1. Start
Embryo chromosomes
Figure 3. Experimental system.
a form of cir-batch-file towards the PSPICE. Last one is
utilized in non-interactive batch simulation mode.
Block 3 runs PSPICE, downloads cir-file and receives
the result from PSPICE in a form of out-file, then passes it
for evaluation to Block 4. Block 4 contains the fitness
function that evaluates and assigns each chromosome
with a fitness value.
The mutation process is applied to each chromosome
except the chromosome with the best fitness value. It
stays as reference for others during the generations. De-
pending on correlations between the best and current
chromosome’s lengths/fitness each individual is put under
one of the particular mutations.
Add_new_component_mutation (ANEM) and De-
lete_component_mutation (DEM) are the procedures,
during which one randomly generated gene is add-
ed/removed to/from each chromosome. ANEM or DEM
applies depending on whether is the genotype length of
current chromosome shorter/longer of that one of the best
chromosome’s. Due to DEM the circuits evolved are
supposed to be modest in components.
The Circuit_structure_mutation (CSM) performs mu-
tation over every loci of randomly chosen gene including
the component name, parameter or pin connections,
without changing the chromosome length. In CSM, de-
spite the total amount of components stays unchangeable,
the number of circuit nodes could be reduced or increased.
3.4. Substructure Reuse
The modification of chromosome by junction it with a
substructure is a kind of mutation routine that brings,
however, a radical modification. If an ordinary mutation
brings 3-5% of a new genotype, the substructure reuse
mutation (SUM) is applied regardless a mutation rate and
can exceed 50%. For instance, in the case of CCs in this
paper we, firstly, tried to reach the targets without SUM,
based only on three mutations mentioned in previous
subsection. The best fitness that seemed to be rapidly
improving at a start, later, when genotype length reached
12-17 genes, slowly and irrevocably decayed. And only
the introduction to a system the SUM, could make the
experiment progress. The system, automatically utilizing
the best chromosomes of miner sizes (limited from 2 to 7
genes) collected at earlier generations and joining them to
stagnated chromosomes, was able to bring up to 60% of
new genotypes in our experiments.
It should be noticed that the SUM procedure is pur-
posed to be used only toward chromosomes that showed
the signs in a fitness decline during the previous genera-
tions. This approach differs from that one where the sub-
structures accumulated in a permanent database are used
intensively along with single components [4,21]. The
wide use of substructures that already are silicon-proven

Challenging the Evolutionary Strategy to Synthesis Analogue Computational Circuits 1036
could fasten the road of evolved designs to a commercial
application, but in the scope of this paper, as mentioned
in Section 1, we focus first on the exploring power of the
evolutionary system.
Table 2. Statistics for evolution of the 4 targeted circuits.
Fitness
Com-
ponent
No
Gen-
eration
No
Fitness Compo-
nent No
Gen-
eration
No
No
Square Root Squaring
10.28343 119 0.0302 35 92
20.19423 123 0.0459 43 309
30.44350 208 0.0563 48 143
40.79838 97 0.0951 38 97
50.25550 200 0.0776 50 135
Cube Root Cubing
10.76444 115 0.0095 50 195
21.06049 179 0.0205 38 72
30.25139 152 0.0079 49 109
40.26850 201 0.0061 44 78
50.64340 294 0.0101 37 98
Altogether ANEM-CSM-DEM-SUM enables to hold a
population within restrained chromosome’s lengths and
for evolution to be more focused. In general, each chro-
mosome in a population is allowed gradually grow,
where the speed of growth is controllable with the help
of an automatic/manual tunable threshold coefficient.
3.5. Fitness Function
The target for evolutionary search is to evolve four CCs
which output voltages are: the cube root, cube, square
root and square of their input voltages. To enable our-
selves to make the estimation of the final results, we
have set the same fitness terms as in [9] for all four cases.
That are, we made the PSPICE simulator to perform a
transient analysis of a source signal of length 0.2 second
at 21 equidistant time-points; the voltage source forms a
pulse signal arising from –250 mV to +250 mV for the
cube root, cubing and squaring, and from 0 mV to +500
mV for the square root [19]. A fitness value is set to the
sum, over these 21 fitness cases of the absolute weighted
deviation between the target value and the actual output
Qp 2
-15V
-15V
Qp 2 7
Qp 2 3
15V
15V
Qn13
Qp25
R31
1e+4
Qn1
R37
3.3e+6
V_IN
Rs 1K
V_OUT
Rl1K
Qn 1 5
R1
1.5e+4
R0
1.2e+4
R4
1.5e+6
R5
4.7e+3
Qp5
Qn 3
-15V
R13
1.34e+6
-15V
R364.3e+4
R6
5.6e+3
15V
R2
2.2e+4
Qp24
Qp10
Qp20
value: , where is the voltage



p
i
i
measured
i
ideal VVF
0
|| i
ideal
VFigure 4. The evolved square root circuit.
in i-th point for the ideal response and is the
voltagein the i-th point obtained for the evolved circuit;
p is a number of points evaluated equaling 21. The
smaller the fitness value, the closer the circuit is to the
target; the fitness penalizes the output voltage by 10 if it
is not within 1% of the target voltage value.
i
measured
V
R0
2.7e+4
Qp4 0
Qp 3
15V
Qp38
Qp2 0R42
6e+4Qp3 0
R21
2.27e+5
Qp10
Qp3 5
Qp3 3
Qp16
15V
Qp1 2
V_IN
Rs
1K
Qp1
R7
820
V_OUT
Rl
1K
Qp4 4
Qn1 7
Qn2 8
Qn4 1
Qn 1
Qn14
15V
R2 1.815e+1
Qn2 Qp0
R39
4.73e+6
Qn40
-15V
Qp39
Qp2 4
Qp 7
Qp21
Qp43
Qn3 5
Qn1 0
Qn7 15V
R10
2.82e+4
The circuits that were treated by PSPICE as error car-
rying were assigned to the worst fitness. We set as a ter-
mination criterion reaching either the fitness value did not
improve over 20 consecutive generations or the best cir-
cuit exceeded 70 components in size.
4. Experimental Results
The results presented are out of 5 runs for each case with
different seeds for the random number generator. The
aggregated data for all 20 runs are presented in Table 2,
where the best runs are marked in bold. We used 10 PCs
with Pentium-4/2.8 GHz processor running at the same
time independently from each other. The average time per
experiment is 43 hours. The total population was 30000
individuals, mutation rate 5% and selection 10%.
Figure 5. The evolved squaring circuit.
ponents with the fitness 0.0302.
The best-of-run circuit (Figure 6) for the problem of
designing a cube root circuit appeared at generation 152
and had 39 components with the fitness 0.2508. The
best-of-run circuit (Figure 7) for the problem of design-
ing a cubing circuit appeared at generation 78 and has 44
components with the fitness 0.00614.
The best-of-run circuit (Figure 4) for the problem of
designing a square root circuit has 23 components with
the fitness 0.194. The best-of-run circuit (Figure 5) for
the problem of designing a squiring circuit has 35 com-

Challenging the Evolutionary Strategy to Synthesis Analogue Computational Circuits1037
V_OUT
Rl1K
Qp 3
Qn10
Qp15
Qn 3
Qn 6
Qp17
R37 1.2e+4
Qp 16
Qp 9
Qp19
Qp26
R40
1.2e+4 R26
6.5e+3
Qn 9
Qn2
Qp 5
Qn5 Qn21
Qp8
Qn4
Qp32
R272.2e+4
Qp31
R3 4.7e+5
Qp1
Qn13
Qp 0
R1
1.8e+4
R391. 2e+4
R34
3.3
Qn19
Qn 15
Rs 1K
V_IN
Qp4
R0
1.111e+4
Qn 1
Qp30 Qn20
R38
2.2
Qn18
Figure 6. The evolved cube root circuit.
Rs
1K
V_IN
Qp 0
Qn 0
Qn7
R1
2.2e+4
Qn20
Qn6 Qn 5
Qp12
Qn2
Qp 10
Qp9
Qp13
Qn1
R5
4.7e+2
R6
1.8e+3
Qn17
R12
8.16e+6
Qp6
R13
6.81e+4
Qp5 R3
97.6K
Qn16
Qp8
Qp4
Qp1
Qn3
Qp 2
Rl1K
V_OUT
Qn1 3
Qn9
R10
3.9e+6
Qn 19
Qn1 8
Qn14
R43.27e+4
Qn 15
R11
2.2e+4
R19
9.4e+5
Qn 8
Qn10
R14
4.7e+2
R15
2.2e+3
Qn4
Qn11
Qp3
Figure 7. The evolved cubing circuit.
The schematics published in [3,9,20,25] enabled us to
source-code them, analyze their netlists in PSPICE and
get the fitness values appropriate for comparison. Both
DC and transient analysis gave us identical results for
each schematic, what, together with other published data,
let us to aggregate all the data into Tables 3 and 4. For
some circuits from [9] we got exactly the same fitness
values, the last fact ensured us that we chose the right
transistor models (SPICE default models as well as in
[3,20]) and other simulation parameters. The most right
column of the tables suggests the relative comparison
between the value received in this paper and the best
corresponding values from the past. As it could be noticed,
by 15 from 16 comparable positions our results are

Challenging the Evolutionary Strategy to Synthesis Analogue Computational Circuits 1038
Table 3. Comparison with circuits published before.
Author
Feature
Koza
et al. [9]
Mydlowec
et al. [3] This work
Improve-
ment
(times)
Square root
Average error, mV 183.57 20.00 9.23 2.2
Fitness value 3.855 70.403 0.194 18.9
Component No 64 39 22 1.8
Evaluation No Data n/a 6,7E + 9 3,7E + 6 1800
Squiring
Average error, mV Data n/a 27.00 1.44 18.7
Fitness value Not converged 4.812 0.0302 159.3
Component No 39 37 35 1.1
Evaluation No Data n/a 1,1E + 9 2,7E + 6 407
Cube root
Average error, mV 80.00 - 11.90 6.7
Fitness value 1.68 - 0.2508 6.7
Component No 50 - 39 1.3
Evaluation No 3.8E + 7 - 4.5E + 6 8.4
Table 4. Comparison of evolved cubing circuit with ones
published previously.
Author
Feature
Koza
et al.[9]
Streeter
et al.[19]
Cipriani
et al. [25]
This
work
Improve-
ment
(times)
Cubing
Aver.error, mV 1.04 0.99 7.13 0.29 3.4
Fitness value 0.0219 Data n/aData n/a 0.0061 3.6
Component No 56 47 12 44 0.3
Evaluation No Data n/a 2.94E + 6- 2.34E + 61.3
considerably better. Notably, that the best by size (12
components) conventionally designed cubing circuit from
[25] has an average error (7.13 mV) 25 times larger than
that one (0.29 mV) of cubing circuit (44 components)
evolved by us.
5. Conclusions
In this paper, we applied the unconstrained evolution
towards the design of analogue computational circuits, on
the example of cube root, cubing, square root and squiring
functions. It was one of the first successful attempts of
application of Evolutionary Strategy (ES) towards the
large analog circuit synthesis. The technique presented is
based on ES, oscillating length genotype, three types of
mutation and substructure reuse. In general, the technique
is plain with a simple algorithm. In all four cases, we
successfully evolved circuits with fewer numbers of
components at much less computer efforts with signifi-
cantly better fitness.
The work discovered the capabilities of the technique
developed by authors to find unconventional, economic
and precise solutions. However, the designs presented are
just simulated models, and are not guaranteed to be robust
being in silicon. There are still “countervailing factors that
impede progress toward industrial-strength automated
design of analog circuits” [27] such as mismatching
properties of devices, variation in the circuit’s power
supply and different operating temperatures.
Reaching the powerful evolutionary technique that
capable of finding theoretical solutions was only the first
stage of the research, whereas the second round is pur-
posed to be strengthening the designs evolved towards the
robust circuits in silicon.
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