In this paper, Recursive Least Square (RLS) and Affine Projection (AP) adaptive filters are designed using Xilinx System Generator and implemented on the Spartan6 xc6slx16- 2csg324 FPGA platform. FPGA platform utilizes the non-restoring division algorithm and the COordinate Rotation DIgital Computer (CORDIC) division algorithm to perform the division task of the RLS and AP adaptive filters. The Non-restoring division algorithm demonstrates efficient performance in terms of convergence speed and signal-to-noise ratio. In contrast, the CORDIC division algorithm requires 31 cycles for division initialization, whereas the non-restoring algorithm initializes division in just one cycle. To validate the effectiveness of the proposed filters, a set of ten ECG records from the BIT-MIT database is used to test their ability to remove Power Line Interference (PLI) noise from the ECG signal. The proposed adaptive filters are compared with various adaptive algorithms in terms of Signal-to-Noise Ratio (SNR), convergence speed, residual noise, steady-state Mean Square Error (MSE), and complexity.

The Least Mean Square (LMS) algorithm and its variants are currently the most frequently used adaptation algorithms; therefore, it is desirable to understand them thoroughly from both theoretical and practical points of view. One of the main aspects studied in the literature is the influence of the step size on stability or convergence of LMS-based algorithms. Different publications provide different stability upper bounds, but a lower bound is always set to zero. However, they are mostly based on statistical analysis. In this paper we show, by means of control theoretic analysis confirmed by simulations, that for the leaky LMS algorithm, a small negative step size is allowed. Moreover, the control theoretic approach alows to minimize the number of assumptions necessary to prove the new condition. Thus, although a positive step size is fully justified for practical applications since it reduces the mean-square error, knowledge about an allowed small negative step size is important from a cognitive point of view.

The non-contact current measurement method with magnetic sensors has become a subject of research. Unfortunately, magnetic sensors fail to distinguish the interested magnetic field from nearby interference and suffer from gauss white noise due to the intrinsic noise of the sensor and external disturbance. In this paper, a novel adaptive filtering-based current reconstruction method with a magnetic sensor array is proposed. Interference-rejection methods based on two classic algorithms, the least-mean-square (LMS) and recursive-least-square (RLS) algorithms, are compared when used in the parallel structure and regular triangle structure of three-phase system. Consequently, the measurement range of RLS-based algorithm is wider than that of LMS-based algorithm. The results of carried out simulations and experiments show that RLS-based algorithms can measure currents with an error of around 1%. Additionally, the RLS-based algorithm can filter the gauss white noise whose magnitude is within 10% of the linear magnetic field range of the sensor.