Optimal design of L sub(1)-norm based IIR digital differentiators and integrators using the bat algorithm

In this study, new infinite impulse response (IIR) digital differentiators of second, third and fourth orders based on optimising the L sub(1)-error fitness function using the bat algorithm (BA) are proposed. The coefficients of numerator and denominator of the differentiators are computed by minimising the L sub(1)-norm of the error fitness function along with imposing the constraint for the location of poles and zeros within the unit circle to ensure minimum phase. The transfer function of the differentiators are inverted and transformed into the digital integrators of the same orders. The results obtained for the solutions by the proposed L sub(1)-based BA (L sub(1)-BA) are superior to the designs using other techniques such as particle swarm optimisation and real-coded genetic algorithm. The designed optimal differentiator and integrator are compared with the existing models and are found to be of high accuracy and flatness in a wide frequency range along with minimum absolute magnitude error. The mean relative error (dB) is obtained as low as -67 dB and -73 dB for the proposed differentiators and integrators, respectively.