Tracking of real and complex sinusoids using piloted adaptive notch filter
Chobey, Dinesh Kumar
Date of Issue2018
School of Electrical and Electronic Engineering
Centre for Signal Processing
In digital signal processing, frequency estimation and tracking of sinusoids in noise using adaptive notch filter is an important area of research to solve problems in radar, communications, biomedical and other related areas. This filter is popularly implemented using the least mean square (LMS) algorithm in which the notch frequency is updated iteratively using either a fixed or an adaptive step-size to eventually converge to the frequency of the input signal. A large step-size will increase the rate of convergence but will result in larger misadjustment, while a small step-size will decrease the rate of convergence but will yield smaller misadjustment. So, in view of this trade-off, a variable step-size LMS algorithm is preferred over a fixed step-size LMS algorithm. Conventional variable step-size algorithms determine the step-size based on time-domain averaging of the gradient estimate at each sampling instance. The piloted adaptive notch filter (PANF) is a new concept in adaptive filtering which determines the step-size value based on the estimated distance between the main notch frequency and the input frequency with the help of pilot notches. A steady-state performance comparison of the PANF with conventional adaptive notch filters is conducted to verify the excellent performance exhibited by the PANF over other algorithms. Hence, this thesis aims at investigating the concept of PANF, originally proposed to estimate and track real sinusoid, to further enhance its performance and to extend the concept of pilot notches for signals in complex domain. A new steering direction determination mechanism, using a time-domain averaging based gradient analysis of the piloted notch cost function at several frequency points at the same sampling instant, is proposed to determine the direction of the main notch with respect to the input sinusoid frequency. This frequency domain information is then combined with time domain information to develop an algorithm for the determination of variable step-sizes for improved speed of convergence with significant reduction in steady-state mean square error (MSE). An improved PANF with a variable pole radius mechanism is introduced. This scheme significantly reduces the transient effect and improves the steering direction determination mechanism of the notch filter. Computer simulations demonstrate the excellent performance of the variable pole radius PANF (VP-PANF) to significantly outperform the PANF with respect to the speed of convergence and steady-state MSE. Aforementioned work leads to the development of a generalized formulation for multiple pilot-pairs adaptive notch filter (MPPANF) structure with the introduction of more number of pilots, variable pole radius mechanism, and improved steering direction determination. The transition from a very large step-size value to a very small step-size value is achieved in one or more intermediate step-size values. This reduces the probability that the main notch overshoots the optimum with large step-size producing large output error when it is close to the input frequency. Simulation results are presented to verify the excellent performance exhibited by the MPPANF and variable pole radius MPPANF (VP-MPPANF) over single-pair PANF with respect to the speed of convergence and steady-state MSE. A detailed theoretical analysis of the proposed adaptive notch filter and mathematical formulation for the determination of the probability of obtaining a correct steering direction along with the effect of input SNR, notch bandwidth and notch frequency on probability of obtaining a correct steering direction is presented. We formulate a complex piloted adaptive notch filter (CPANF) to estimate and track the frequency of complex sinusoidal signal. A novel complex coefficient filter structure with a main notch and pilot notches to track the frequency of the input complex sinusoid with a variable step-size least mean squares (LMS) based algorithm is presented. Simulation results verify the excellent performance exhibited by the CPANF over conventional complex adaptive notch filter (CANF) with respect to the speed of convergence and steady-state MSE. Theoretical analysis is also presented which closely follows the simulation results. Finally, the filter is implemented to suppress complex sinusoidal interference in a QPSK spread spectrum communication systems which shows improvement over conventional CANF in overall bit error rate (BER). This research leads to a detailed understanding and development of piloted adaptive notch filter to estimate and track both real and complex sinusoid.
DRNTU::Engineering::Electrical and electronic engineering::Electronic systems::Signal processing