The following is a summary review of “Design of minimum multiplier fractional order differentiator based on lattice wave digital filter” from Volume 66 of ISA Transactions (January 2017). If you would like to learn more about ISA Transactions, please visit this link.

See Samuel Ko Tak Shun's previous summary on "QRS detection using adaptive filters: A comparative study" here.

Fractional calculus is an emerging field of research and gaining popularity over the integer order system due to its flexibility and performance improvements. In recent years, the fractional order derivative has acquired the great attention of the research community in almost every field of engineering and science to model various problems of automatic control, image processing, signal processing, fluid dynamics, electromagnetic theory, physics, biology, hydrology, and electrical circuits. Fractional derivative is used to compute the time derivative of the applied signal. In order to obtain flexibility in design and better design accuracy, the integer order derivative is generalized to fractional order derivative. 

Several methods have been extensively explored in literature for the design and implementation of fractional order differentiator in discrete-time domain, such as the discretization method, which includes mapping of fractional operator from s-domain to z-domain. It can be classified as direct and indirect discretization. Each method has its unique feature, but there is always a scope for a new methodology to improve the design of fractional order differentiator (FOD) and reduce its hardware requirements. 

The realization of FOD using the concept of lattice wave digital filter (LWDF) is still an unexplored area and not mentioned in any literature to date. In this paper, LWDF is considered as a target filter for the realization of FOD. In fact, novelty lies in the FOD designed using LWDF systems, in that it will incorporate all the advantageous properties of LWDF with the utilization of minimum hardware. Also, proficient results are obtained by using optimization techniques. 

An overview of the LWDF structure is summarized and, like wave digital filters (WDF), LWDFs are also related to certain analog prototype networks (i.e., lattice network). LWDF is a more preferred class of WDF and one of the best structures for implementing infinite impulse response (IIR) digital filters. LWDF is represented by two parallel branches, which realize all-pass filters. These all-pass filters can be realized by using first- and/or second- order wave digital all-pass structures. These all-pass structures are implemented by using symmetric two-port adaptors and delay elements. An adaptor requires a single multiplication and three additions. The usage of adaptor produces an efficient realization in terms of the number of multipliers for a given order and have low sensitivity to coefficient quantization. Its related formulae are listed in page 406 of this article. Also in this paper, the main focus is on the design of FOD based on LWDF, so the generalized high-pass LWDF transfer function is considered, and its related formulae are listed in pages 406-407 in this article. 

The design of lattice wave digital fractional order differentiator (LWDFOD) is considered as an approximation problem, where the error function is to be minimized. The error function is the difference between the frequency response of an ideal FOD and the designed LWDFOD. The y coefficients of the LWDF are iteratively optimized until the minimum error function is obtained. 

Three different optimization algorithms are applied for the efficient design of LWDFOD and its performance evaluation: Genetic algorithm (GA), particle swarm optimization (PSO), and cuckoo search algorithm (CSA). 

In GA, first the fitness function is defined over the genetic representation and then GA proceeds to initialize a population. A set of coefficient chromosomes (population) in each iteration (generation) is randomly selected and the selection operator chooses the better individual chromosomes by computing their fitness values for finding a better solution. Then, the crossover operator flips some of the genes of one parent with the other to maintain diversity in genetics in search of better fitness. Finally, the mutation process allows the algorithm to escape from any local minima trapping by altering one or more genes from its initial values. The best fitted chromosomes, known as the elite chromosomes, survive, which are transmitted to the next generation. With each generation, better solutions are obtained and improved by the repetitive application of different steps like mutation, crossover, and selection. 

In PSO, each particle in the swarm is a potential solution of the problem under consideration in a search space. Each particle is pictured with its position and velocity vector. The position vector is the required solution, and the velocity vector determines the speed of a particle with which it can reach the optimal solution. Each particle is evaluated by its fitness value at every iteration. The velocities and the position of each particle is updated for the next iteration. 

CSA is inspired by the unique breeding behavior of cuckoo birds and the concept of the Lévy flights which are observed in some species of birds and animals. Cuckoo birds depend on some other bird’s nest for hatching their eggs. Cuckoos search a nest where the host bird has recently laid eggs. Cuckoos then use this nest to hide their own eggs. If the host bird identifies that the eggs are not their own, it may abandon the nest or choose to destroy the alien eggs. This leads to the evolution of the cuckoo eggs, which make attempts to mimic the eggs of the host bird. 

To apply the CSA, the following assumptions are considered: 

1. Cuckoo birds lay one egg at a time and hide it in the host’s nest, chosen randomly.
2. The nests with the best host environment and with the best eggs will survive and move forward to the next generation.
3. The number of host nests is fixed.
4. The probability of identifying alien eggs by the host bird is Pa.

The Lévy flights are the forward steps taken by living beings, such as birds, insects, and animals, in search of their food. Lévy flight refers to a series of straight-line flights followed by sudden 90 degree turns. The usage of Lévy flight for choosing a random new nest is the main factor for improving the performance of CSA. The implementation steps of CSA are explained with a flow chart presented in Figure 3 of page 407 of the article. 

For a fair comparison of employed optimization algorithms for designing of LWDFOD, the same values are chosen for a similar set of parameters. Absolute magnitude error, root mean square (RMS) magnitude error, computation time, convergence rate, and the number of arithmetic operations are the main parameters taken into consideration to evaluate the performance of the designed LWDFOD. A count of 50 independent trial runs are carried out for all algorithms and the optimized values of the y coefficients are procured. Here, the design of the 3rd and 5th order LWDFOD using GA, PSO, and CSA algorithms are constructed to demonstrate the effectiveness of the proposed design method. The fractional order v = 0.5 is considered for the design problem. The fitness function provided is evaluated iteratively to obtain the optimized y coefficient and scaling factor “a” of the LWDF. The RMS magnitude error of the proposed LWDFOD is computed for the sampling time of T = 0.01s. 

The results of LWDFOD using optimization algorithms are discussed from pages 410-411 in the article, and the comparison with the existing FODs are discussed from pages 411-412 in the article. 

In conclusion, a novel LWDF based FOD is designed here using nature-inspired algorithms and its structural realization with minimum multipliers presented. GA, PSO, and CSA are employed to determine the LWDF coefficients that approximates the response of the ideal 3rd and 5th order FOD. The computational complexity of the proposed LWDFOD is reduced significantly due to the reduction in the number of filter coefficients compared to that required in existing methods. Furthermore, the structural realization shows that the proposed LWDFOD requires 45% less multipliers as compared to the existing FOD designs. To prove the efficiency, the results are examined on the basis of magnitude response, absolute magnitude error, and number of multipliers where results of the proposed LWDFOD prevails. Simulation results affirm that a significantly reduced RMS magnitude error have been observed in LWDFOD with the percentage improvement of 29% in comparison with recently published works, which makes the proposed LWDF based method a good alternative for the efficient design of FODs as well as for other fractional order systems.

Reference 

Barsainya, R., Rawat, T. K., & Kumar, M. (2017). Design of minimum multiplier fractional order differentiator based on lattice wave digital filter. ISA transactions, 66, 404-413.