2.1 Genetic Algorithms

GAs use an abstract model of natural evolution to solve optimization problems, originally introduced by John Holland in the 1960's and 1970's ⁷. GAs operate on the principle of "survival of the fittest", generating new designs or solutions from a population of existing solutions and discarding those designs that have an inferior performance or fitness. Each member of the population is a solution candidate, referred to as an individual, represented by a chromosome composed by several genes. The genes represent one part or parameter of the solution candidate. In standard GAs, the chromosome is a binary string with a fixed length. New solutions, referred to as offspring, are created by breeding existing solutions, referred to as parents. Parents are randomly selected from the present population, with a bias towards the fittest individuals. Offspring are generated either through mutation (using one parent), but primarily through crossover (using two parents). Mutation involves a random modification of a single gene in the chromosome, while crossover entails merging the chromosomes of two parents. Once the offspring are generated, a survival scheme determines which individuals from the set of parents and offspring will survive and the process is repeated once again, each iteration is referred to as generation. Over successive generations, it is expected that the population "evolves" toward the optimal solution ⁷.

An important aspect of the basic GA, just like any meta-heuristic, is the role of the objective function. It is mostly assumed that the objective function is equivalent to the fitness function, but a subtle difference should be pointed out. The objective function defines the performance measure from the point of view of the problem, while the fitness function defines performance from the point of view of the search process, in this case the GA. While most works do not differentiate between them, the difference is important because it is not a requirement that both measure be the same.

2.2 Electronic Design with GAs

GAs have been used to design electronic circuits in several works. For instance, an early work on this topic used a GA to generate the topology and the component values of electronic circuits in ⁴.

Another alternative was proposed by Koza et al. in ⁹, they use genetic programming (GP) with a variable-length representation containing topology modifying operators, component-creating operators and arithmetic operators. In Koza's approach the operators that modify the circuit topology and select component values are inseparable, and all are under the control of the evolutionary processes operating on the circuit representation.

Circuit synthesis involves both the selection of a suitable topology and the choice of component values, and as Koza has shown ⁹, these may be optimized simultaneously by the evolutionary process. However, there is no reason why these operations should not be performed separately, with different optimization methods being used, specifically tailored for each task.

The circuit topology can be chosen first, and an appropriate algorithm for this task can be a standard GA. For each circuit topology generated the component values can then be optimized, and the performance of the circuit used as the objective function for the GA. In such cases, circuit fitness can be evaluated using SPICE, a simulation program with IC emphasis. The code for which can be incorporated into the synthesis program.

The component values could also be optimized using a GA, but this is not the best choice for problems involving well-behaved objective functions dependent on a fixed number of variables. It is well established that numerical optimization methods converge much faster and involve fewer objective function evaluations ³. No optimization method guarantees to find the global optimum, but it has been found that numerical optimization of component values achieves results that are close to the global optimum.

This hybrid approach using a GA to select a suitable topology and numerical optimization to choose component values is likely to be more efficient than allowing evolution to perform both tasks concurrently. In this work, however, we will only focus on the first part of the problem, that of finding novel circuit topologies, leaving any further optimization as future work for real-world implementations.

In particular, this work focuses on the synthesis of current follower (CF) circuits. These circuits copy the value of a current to other parts of a circuit. In other words, the CF can maintain a current in less impedance loads, without altering the original source. CFs can be used in various analog circuits, such as filters, oscillators, data transmission, and Current Conveyors (CC) ^{3, 4, 28, 30}.

2.3 Objective Function

The objective function for a synthesized CF circuit K is adapted from ², and is given by

with θ, λ, and φ defined as

where p=BW-107107, and

In Equations 1 to 4, g is the circuit gain, BW is the circuit bandwith, Z_in is the input impedance, and Z_out is the circuit output impedance. These 4 values are obtained by SPICE.

2.4 Objective-based Fitness Function

Traditionally, EA practitioners use the objective to define fitness, but this is not necessarily the case. A simple but important difference, that will be further discussed in Section 3. The objective function is used to evaluate the CF circuit is given as in Equation 1, posing a minimization problem for the search process. In particular, the parametrization used for Equation 1 is intended to steer the search to find circuits with g = 1, BW = 10MHz, Z_in = 100Ω and Z_out = 10KΩ, which are standard and good values for CF circuits ²⁴. Hereafter, we refer to the standard GA that uses this fitness function as GA-OS, for objective-based search.

2.5 CFs Representation for a GA

Using nullators and norators, it is possible to describe the behavior of current followers, where for synthesis purposes, a nullator (O) and a norator (P) always must form a joined-pair ^{2, 28, 29}.

The node in the joined terminals of the O-P pair is associated to the source (S) by a MOSFET, the other terminal of the O element is associated to the gate (G), and the other terminal of the P element to the drain (D). Figure 1(a) shows the nullor-based description of a CF, which consists of four P elements (P1-P4), each one joined with an O element.

Fig. 1 CF representation using (a) nullors and (b) MOSFETs 

In this work we use a binary genetic encoding introduced by ²⁹, proposed for the automatic synthesis of analog circuits, particularly to find CF topologies. The O-P pairs can be described by a small-signal gene called SS; the synthesis of each O-P pair are coded by the gene called SMos. Moreover, the current and voltage biases are codified by the gene Bias; and the synthesis of the current biases by the current mirror MOSFET by the gene CM.

The SS gene defines the joined terminals of O-P pairs, using two bits for each O-P pair. The SMos gene can define either a PMOS or NMOS, using one bit for each O-P pair. The Bias gene is three bits for each MOSFET, and the CM gene can define 4 CMs: simple, cascode, Wilson, and modified Wilson; using two bits for each MOSFET. As a result, the genetic representation of the CF consists of a chromosome Ch_CF of four ordered genes, given by

An example of the CF synthesis for the proposed chromosome is shown in Figure 1(a) is depicted in Figure 1(b) using MOSFETs, where the CF obtained is known. This description has been reported in ^{2, 24, 29}, where CFs were synthesized using a standard GA (Eq. 5). The lenght of the chromosome Ch_CF is computed by using n as the number of O-P pairs used to define each gene. For instance, gene SS needs 2n O-P pairs, SMos just n, while Bias needs 2n, and CM always requires just 2; therefore, the length is given by

In this work we consider three different topology sizes related with the number of MOSFETs used by the CFs, from 1 to 3. For the topology using just one MOSFET, the chromosome length is 8 bits, when using two MOSFETs 14 bits are used, and for three MOSFETs 20 bits.

3.1 Novelty-based Fitness

Instead of designing an objective function that summarizes the performance of each individual, to use NS successfully the concept of uniqueness must be grounded in some way. The uniqueness of a solution must be measured against the rest of the evolved solutions. For instance, solutions can be compared based on their genotype or phenotype, or in the case of many previous works on NS the concept of behavior ^{10, 16,18, 20}. Whichever the chosen characterization, let us assume it is stored in a vector β that describes an individual K in the particular application domain. Lehman and Stanley proposed a measure of sparseness ρ around each individual as

where α_i is the ith-nearest neighbor of β with respect to a distance or similarity measure d(· , ·), and the number of neighbors k is an algorithm parameter ¹⁰.

Given this definition, when the average distance is large then the individual is located within a sparse region of the search space, and it is located in a dense region if the measure is small, see Figure 2. The original NS proposal considers the current population and an archive of individuals to compute sparseness. An individual is added to the archive if its sparseness satisfies a certain threshold condition ρ_th, which is the second NS parameter. Several papers ^{10, 11, 12, 13, 34} have suggested implementing the archive as a FIFO queue of size q, this alleviates the cost of computing sparseness but adds another parameter.

Fig. 2 The original NS proposed in ¹⁰ uses a measure of local sparseness ρ around each individual described by β, considering the current population and novel solutions from previous generations based on a threshold ρ_th. The Figure shows three different scenarios from the most dense region (less novel) to the most sparse (the most novel) 

For a minimization problem the inverse of ρ(β) is used to assign fitness during the evolutionary process, since we want to maximize novelty (and thus increment sparseness). Hereafter, we will refer to a GA based on NS as novelty or novelty-based search (GA-NS).

3.2 CF Synthesis with GA-NS

In this work, our main goal is to apply NS in a GA search for automated CF synthesis. In this scenario, we hypothesize that the very nature of circuit design requires an exploratory and unorthodox approach, that promotes uniqueness and creativity. It is precisely in these scenarios, Lehman and Stanley argue, that NS should be able to produce strong results relative to standard objective-based search ²⁷. To apply NS we choose to make β = Ch_CF , and thus sparseness is computed based on the Hamming distance in genotype space.

The hypothesis in this work is that NS will be able to find new circuit designs, designs that are left unexplored by a standard search process that relies on a fixed objective function. Therefore, in the following experimental section we will focus our work on comparing the performance of the GA-NS with the standard GA-OS approach, highlighting the unique solutions found by GA-NS. It must be stressed that Equation 7 is used to determine fitness in NS, and thus establish the selection pressure for the search. Nonetheless, the final solutions returned by the search are still chosen based on the objective function given in Equation 1. It must be understood that NS changes the manner in which the search progresses, but the underlying goal of the task is still to find the best possible solution based on a domain specific objective. The difference with a standard GA, however, is that the objective function is not used to directly guide the search, it is only used offline to choose the final solution returned by the algorithm.

4.1 CF Topologies

Tables 3 to 5 summarize the most notable topologies found by GA-OS and GA-NS, respectively for the three considered topology sizes (1, 2, and 3 MOSFETs). In these tables, the second column shows the generation where the best CF topology was found, and the third column shows the decimal conversion of the binary chromosome, while the last column shows its performance as given by the objective function. Moreover, Figures 3-5 present the corresponding circuit topologies; i.e., the phenotype of the solutions. For convenience, the legend of each topology in Figures 3-5 provides the corresponding decimal conversion of the genotype (chromosome) of the solution (we will use this number as a reference ID of each circuit).

Fig. 3 Three CF topologies size of one MOSFET 

Fig. 4 Five synthesis of a CF with the topology size of two MOSFETs found by GA-NS 

Fig. 5 Four CF topologies for the synthesis of a CF with the topology size of three MOSFETs found by GA-NS 

For the simplest case, considering a single O-P pair, Table 3 and Figure 3 show two topologies found by GA-NS and one by GA-OS. It is important to note that GA-OS consistently found the same topology in all runs. On the other hand, GA-NS found different topologies in several runs. All topologies found have been previously reported and studied in the literature ²⁴; this was expected given the small dimensions of the CF circuit to synthesize, and given that we use a previously proposed representation and fitness function. Nonetheless, both GAs synthesized non trivial circuits with proper behavior and operation. In the case of GA-NS this is important, since the fitness function does not promote any particular type of behavior, only novelty.

For the second series of experiments, considering two O-P pairs (two MOSFETs), Table 4 and Figure 4 present the best topologies found by the GA-NS, since GA-OS converged to previously published results. Topologies 4979 and 5059 can be considered trivial because you can find them manually quite easily. Moreover, topology 36387 shows a known circuit ²⁴; which confirms that the GA-NS is guiding the search to good solutions in the search space. Finally, topologies 4567 and 6147 are novel, and should be considered as new designs of CF circuits. In both topologies gain values are near to one, and even though Z_in and Z_out are not ideal, these impedance's can be useful for specialized filter designs.

The third set of experiments used three O-P pairs to build circuits with 3 MOSFETs, these results are summarized in Table 5 and Figure 5. Topology 152195 (Figure 5(a)) shows three serial CFs, this a clear example of a modular design where a synthesized CF is built from simpler circuits. Figures 5(b) and 5(c) show more elaborate constructions of CFs found by GA-NS. Finally, topology 252746 in Figure 5(d) shows a novel structure that has not been presented in any related literature. Such a design confirms the ability of the NS paradigm to explore the search space and find unorthodox solutions, even to long-standing and well-known problems.

It bears mentioning that the current sources are ideal for the circuits presented in Figures 3, 4, and 5; the GAs generated a Wilson modified current mirror as the current sources on all the circuits. All topologies behave correctly as CFs, with the typical dimensions of technology, which are: Wn = 6μ, Wp = 4.4, and Ln = Lp = 1.2μ; with Vdd = 1.6V, Vss = 1.6V, and I = 20uA. A deeper analysis of these circuits and their properties is left as future work, as well as use these topologies in a real analog circuit.

4.2 Comparison between GA-OS and GA-NS

Let us now analyze the effect that the NS algorithm has on the search process for circuit synthesis, relative to objective-based search. Figure 6 shows convergence plots of the best solution found by each algorithm, showing the average behavior over all runs. The figure plots the objective function value of the best solution with respect to the number of generations. In particular, we focus on the experiments using two O-P pairs, for circuits with two MOSFETs. Based on this plot, we can see no significant difference between both algorithms. The quality of the solutions found is comparable and the convergence is similar for both algorithms, even though GA-NS reaches a better average performance. However, as stated before, the topologies found by GA-NS do not match those found by GA-OS. Therefore, in terms of the performance of the synthesized circuits both algorithms are more or less equivalent, the difference lies on the actual topologies found by each method. These results should be highlighted, since some have argued that the search carried out by the NS algorithm seems to be random, given its exclusion of the objective function. However, our results support findings in other domains ^{31, 32, 21}, that show that NS is not a random process and that it can at the very least achieve the same level of performance as objective-based search and in many cases outperforms it.

Fig. 6 Convergence plots for GA-NS and GA-OS for the two MOSFET CFs, showing the performance (objective function value) of the best solution found over the initial generations of the search. The lines represent the average over 10 runs of the algorithms 

Figure 6 only shows the convergence over the first generations based on the number of generations that GA-OS required to converge. Table 6 presents the average number of generations required by each algorithm to find the best solution for each experiment over 10 runs. It is evident from these results that the GA-OS converges much earlier than GA-NS, much earlier than the maximum number of generations allowed (after the fitness stagnates for 10 generations).

Since GA-NS fitness is based on novelty, it takes more generations for the search to converge. This behavior highlights the fact that when the search is driven directly by the objective it can become stagnated around local optima. GA-NS certainly finds similar local optima, based on the performance measured by the objective funtion.

However, the search does not stagnate given its ability to promote diversity and explore other regions of the search space, allowing it to find solutions that might become inaccessible to GA-OS. For simplicity, and given the quick convergence of GA-OS, we can take a snapshot of the type of solutions found by each algorithm after the first 10 generations. Figures 7-9 compare the composition of the best solutions found by both GA-OS and GA-NS.

Fig. 7 Histograms showing the average composition of the chromosome of the best solution found by each algorithm after 10 generations, for the single MOSFET circuits 

Fig. 8 Histograms showing the average composition of the chromosome of the best solution found by each algorithm after 10 generations, for the two MOSFET circuits 

Fig. 9 Histograms showing the average composition of the chromosome of the best solution found by each algorithm after 10 generations, for the three MOSFET circuits 

These figures present frequency histograms, where the height of each bar represents the percentage of runs for which a particular bit in the chromosome was set to a 1 value in the best solution found so far. For instance, if a bar reaches a value of 0.5 this means that 50% of the best solutions found after 10 generations have a 1 at that particular position within the chromosome. The plots are divided for each experimental configuration, with Figure 7 showing the results for the single MOSFET CFs, Figure 8 for two MOSFETs and Figure 9 for three MOSFETs.

These figures nicely illustrate our claim, that GA-NS finds solutions that are different from those found by GA-OS. Moreover, that the solutions found, while being different, achieve the same performance based on the objective function. This means that GA-NS explores other areas of the search space, some of which might contain local (or even global) optima that are not accessible to a standard GA-OS.