A New Mutated Quantum-Behaved Particle Swarm Optimizer for Digital IIR Filter Design

Adaptive inﬁnite impulse response (IIR) ﬁlters have shown their worth in a wide range of practical applications. Because the error surface of IIR ﬁlters is multimodal in most cases, global optimization techniques are required for avoiding local minima. In this paper, we employ a global optimization algorithm, Quantum-behaved particle swarm optimization (QPSO) that was proposed by us previously, and its mutated version in the design of digital IIR ﬁlter. The mechanism in QPSO is based on the quantum behaviour of particles in a potential well and particle swarm optimization (PSO) algorithm. QPSO is characterized by fast convergence, good search ability, and easy implementation. The mutated QPSO (MuQPSO) is proposed in this paper by using a random vector in QPSO to increase the randomness and to enhance the global search ability. Experimental results on three examples show that QPSO and MuQPSO are superior to genetic algorithm (GA), di ﬀ erential evolution (DE) algorithm, and PSO algorithm in quality, convergence speed, and robustness.


Introduction
Adaptive IIR filters have been proven to be useful in many fields such as channel equation, noise reduction, echo cancelling, and system identification [1,2].A major problem in adaptive IIR filters design is that their error surface may be usually nonquadratic and multimodal.If this problem is considered, global optimization technique is required to get the global minima in a multimodal error surface.In the recent years, population-based intelligent algorithms and heuristic algorithms, such as genetic algorithm (GA) [3][4][5][6], simulated annealing (SA) algorithm [3,7], differential evolutionary (DE) [8] algorithm, particle swarm optimization (PSO) algorithm [9,10], Tabu search (TS) algorithm [11], ant colony optimization (ACO) [12] algorithm, and artificial bee colony algorithm [13], have been proposed and used in the digital IIR filter design.GA has received considerable attention for the digital IIR filter design.However, its disadvantages are lack of good local search ability and premature convergence.The drawback of standard SA algorithm is that it can be very slow and often requires much more number of cost function evaluations to converge to the global minima.The problem with DE algorithm is that it is sensitivity to the choice of its control parameters.PSO is quite easy to be programmed and outperforms GA in many practical applications.However, as the particles in PSO only search in a finite sampling space, PSO can easily get trapped in local optima.TS algorithm and ACO algorithm are more suitable for the combinatorial type problems.As a variant of PSO, Quantum-behaved Particle Swarm Optimization (QPSO) was proposed [14,15] in 2004.It is inspired by quantum mechanics and fundamental theory of particle swarm.It is convenient for PSO and QPSO to apply in the digital IIR filter design as the design can be reduced to a minimization problem and solved by these algorithms.In addition, the particle is real coded and represents a set of filter's parameters and then the swarm represents all the candidate solutions.In this paper, a mutated version of QPSO (MuQPSO) is proposed by introducing a random vector in QPSO in order to enhance the randomness and global search ability.Then both QPSO and MuQPSO are applied to digital IIR filter design.Three examples for the purpose of system identification are tested and compared with the results of using GA, DE, and PSO.This paper is organized as follows.In Section 2, QPSO is described and MuQPSO is proposed.In Section 3, the problem for digital IIR filter design is formulated and the method of applying QPSO and MuQPSO to the design of IIR filters is presented, and the experimental results are given in this section.A conclusion is given in Section 4.

QPSO and Its Mutated Version
2.1.QPSO Algorithm.PSO, proposed by Kennedy and Eberhart [16] and Shi and Eberhart [17], is a new global search technique.The underlying motivation for the development of PSO was social behaviour of animals such as bird flocking, fish schooling, and swarm theory.In the PSO algorithm, each particle is represented as a potential solution to a problem in D-dimensional space and is denoted as X i = (x i1 , . . ., x id . . ., x iD ).Each particle remembers its own previous best position and its velocity along each dimension as V i = (v i1 , . . ., v id , . . ., v iD ).The velocity and position of particle i at (t+1)th iteration are updated by the following equations: where c 1 and c 2 are two positive constants, known as the cognitive and social coefficients, which control the relative proportion of cognition and social interaction, respectively.Vector P i = (P i1 , . . ., P i j , . . ., P iD ) is the best previous position (the position giving the best fitness value) of particle i, which is called pbest.And vector P g = (P g1 , . . ., P g j , . . ., P gD ) is the best position discovered by the whole population, which is called gbest.Parameter w is known as inertia weight and the optimal strategy to control it is to initially set to 0.9 and reduce it linearly to 0.4 [17].QPSO is inspired by quantum mechanics and fundamental theory of particle swarm.In the QPSO algorithm with M particles in D-dimensional space, the position of particle i at (t+1)th iteration is updated by where parameter α is called contraction-expansion (CE) coefficient.P i and P g have the same meanings as those in PSO.GP is called Mean Best Position, which is defined as the mean of the pbest positions of all particles.The characteristics of QPSO algorithm are reflected mainly in two ways.First of all, the introduced exponential distribution of positions makes QPSO search in a wide space.Furthermore, the introduction of Mean Best Position into QPSO is another improvement.In the standard PSO, each particle converges to the global best position independently.In the QPSO with mean best position GP, each particle cannot converge to the global best position without considering its colleagues because that the distance between the current position and GP determines the position distribution of the particle for the next iteration.[18], a major problem also encountered by GA, PSO, and other evolutionary algorithms in multimodal optimization, which results in great performance loss and suboptimal solutions.In QPSO, although the search space of an individual particle is the whole feasible solution space of the problem throughout the iterations, diversity loss of the whole population is also inevitable due to the collectiveness.From (2), one can see that if |GP j − x i j | is small enough, the search space will be narrowed and x i j cannot obtain a new position in the upcoming iterations.The explorative power of particles is lost and the evolution process will stagnate.This case can even occur at an early stage if |GP j − x i j | is zero.In the latter stage of evolution process, the loss of diversity for |GP j − x i j | is often occurred.To prevent this undesirable trend, a random vector is constructed according to the difference between two positional coordinates that are rerandomized in the problem space, and the value of the random vector will replace |GP j − x i j | with a certain probability CR.Then the particle's position is updated by the following equation: where mu x k and mu x s are two random particles generated in the problem space.One can see that as the random vector is introduced, the particles may escape from the current position and locate in a new search area.
The variant of QPSO is called MuQPSO.The procedure of MuQPSO is listed as follows.
Step 1. Initialize particles with random position and set the control parameter CR.
Step 2. For t = 1 to maximum iteration, execute the following steps.
Step 3. Calculate the mean best position GP among the particles.
Step 4. For each particle, compute its fitness value f Step 5. Select gbest position P g (t) among particles.
Step 6. Generate a random number, denoted as RN, in the range of (0 1).

Application of QPSO and MuQPSO to the Design Problem
3.1.Problem Formulation.In general, the basic structure of an IIR filter is identical to that of the autoregressive movingaverage (ARMA) model, whose input-output relation is defined by the following difference equation [2]: where x(k) and y(k) are the filter's input and output, respectively, and M(>=L) is the filter order, a i and b i are the adjustable coefficients of the model.The transfer function of this IIR filter can be written in the following general form: Then an IIR filter design can be formulated as an optimization problem with the mean square error (MSE) as the cost function where d(k) is the filter's desired response, e(k is the filter's error signal, and the composite weight vector of the filter is defined by concatenating the two sets of coefficients {a i } L i=0 and {b i } M i=1 , according to the formula The goal is to minimize MSE (8) by adjusting .In practice, ensemble operation is difficult to realize, and the cost function ( 8) is usually substituted by the time-averaged cost function where N is the number of samples used for the calculation of cost function.In order to apply QPSO and MuQPSO to design the digital IIR filters, the filter coefficients defined in ( 9) are represented as a particle in QPSO and MuQPSO.The particle is real coded and is treated as a trial solution.The dimension of a particle is equal to the size of the parameter vector and each dimension of a particle is in correspondence to a filter coefficient.The stability of the filter is guaranteed by constraining the range of the particles' position [8].The fitness of a particle is evaluated by its position and the fitness value is calculated using the cost function (10).The length N of the cost function is selected according to the problem.

Experimental Results
. Three examples are used in the simulation studies.GA, DE, and PSO algorithm are also used for the digital IIR filter design in order to make a performance comparison with QPSO and MuQPSO.For each simulation, 100 Monte Carlo simulations are performed.The parameter settings of each example for the competitor algorithms are shown in Table 1.
Example 1 (see [3,7,8,11,12]).The unknown plant and the filter have the following transfer functions: As the plant is a second-order system and the filter is a first-order IIR filter, local minima problems could be encountered.The system input, x(k), was chosen to be random Gaussian noise with zero mean and unit variance.The error surface has a global minimum at {a, b} = {−0.311,−0.906} and a local minimum at {a, b} = {0.114,0.519}.For all the five algorithms, the search space is (−1, 1).Randomly chosen initial positions and fixed initial positions are considered in the simulation.The fixed initial positions are {0.114,0.519}, {0.8, 0}, {0.9, −0.9}, and {0.9, 0.9} [7].
Table 2 shows the comparison of the number of global and local minimum hits by various algorithms.The results are given by 100 random simulations with randomly chosen initial positions.The table lists that GA is likely to converge to the local minimum.DE, PSO, and QPSO might jump to the global minimum valley with more opportunities and converge to the global minimum, but it also can jump to the local minimum valley and then converge to the local minimum.MuQPSO could converge to the global minimum in all the runs.Tables 3 and 4 demonstrate the mean values of filter coefficients along with the standard deviations and the CPU times of each algorithm.From Table 3, one can see that MuQPSO could find the global minimum with the least standard deviations among all the five algorithms.As seen from Table 4, MuQPSO can jump out of any of the settled fixed initial positions and find the global minimum while the other algorithms are all trapped in these fixed initial algorithms.Figure 1 presents the coefficients learning curves and convergence behaviours of the five algorithms applied to design Example 1.The results are averaged over 100 random runs with randomly chosen initial positions.
Example 2 (see [7,11]).The plant is a third-order system and filter is a second-order IIR filter with the following transfer functions: The input x(k), which takes values from (−0.5, 0.5), was a uniformly distributed white sequence, and the SNR = 30 dB.Since the reduced order filter is employed for the identification, the error surface of the cost function is multimodal.
Table 5 shows the experimental results by various algorithms in Example 2, which gives the mean best values and standard deviations of the filter coefficients.All the results are averaged over 100 random runs with randomly chosen initial positions.Figure 2(a) shows the comparison of convergence behaviours for Example 2. As seen from Table 5, mean best values produced by PSO, QPSO and MuQPSO are approximate, while QPSO has smaller standard deviation.In Figure 2(a), one can see that convergence speed of QPSO and MuQPSO is faster than the other three algorithms.
Example 3. The plant and the filter are both third-order system with the following transfer functions: The input x(k) was a white Gaussian noise with the mean of zero and unit variance.Since the filter order is equal to that of the system, the error surface is unimodal.The best solution should be located at {−0.2, 0.4, −0.5, −0.6, 0.25, −0.2}.
Table 6 shows the mean best values and standard deviations of filter coefficients in Example 3 averaged over 100 random runs with randomly initial positions.Figure 2(b) shows the comparison of convergence behaviours for Example 3 averaged over 100 random runs.As seen from Table 6, the filter coefficients found by QPSO are exactly located at the best solution and the standard deviation is smaller than that yielded by any other algorithms.QPSO and MuQPSO are the most and the second most robust algorithms among the five ones.From Figure 2(b), we can see that the convergence speeds of QPSO and MuQPSO are much faster than those of GA, DE, and PSO.
From the above three examples, QPSO and MuQPSO have shown their stronger search abilities both on the multimodal problem and on the unimodal one.QPSO and MuQPSO outperform GA, PSO, and DE in convergence speed, robustness and qualitatively of the final solutions.

Conclusions
In this paper, we have introduced the new global optimization technique, QPSO, and proposed its variation, MuQPSO.MuQPSO has enhanced the randomness by modifying the update equation of QPSO.The modified method replaces a part of the update equation with a random vector in a certain probability.QPSO and MuQPSO were both used in the design of digital IIR filters for the purpose of system identification.Experimental results have shown that the performance of QPSO and MuQPSO is superior to GA, DE, and PSO in the digital IIR filter design problem and they will be an efficient tool for this design problem.

Figure 1 :
Figure 1: Coefficients a (a) and b (b) learning curves for Example 1 and comparison of convergence behaviours for Example 1 (c).

Table 1 :
Parameter settings for the competitor algorithms.
*The sign of a → b represents that the parameter value is linearly decreased from a to b according to the iteration.

Table 2 :
Example 1 with randomly chosen initial positions for system identification.

Table 3 :
Mean values and standard deviations of the filter coefficients in Example 1 (mean of 100 random runs with randomly chosen initial positions).

Table 4 :
Mean values of the filter coefficients in Example 1 (mean of 100 random runs with four fixed positions).

Table 5 :
Mean values and standard deviations of the filter coefficients in Example 2 (mean of 100 random runs with randomly chosen initial positions).