Adaptive Infinite Impulse Response Notch Filter Research Articles

Single and Multiple Continuous-Wave Interference Suppression Using Adaptive IIR Notch Filters Based on Direct-Form Structure in a QPSK Communication System

The removal filter coefficients in this technique are dependent on the jammer’s power and its Instantaneous Frequency (IF) information, which can both be obtained in the time–frequency domain (adaptive filtering techniques). The dependence of the removing/reducing filter characteristics on the interference power is critical, as it allows an optimal trade-off between removal interference and the amount of self-noise generated by the filter. This trade-off is bounded by the two extreme cases of no notch filter (no self-noise) and full suppression (k1 = 1) for both low- and high-power jammer values. In this paper, a cascade second-order adaptive direct Infinite Impulse Response (IIR) Notch Filter (NF) with a gradient-based algorithm to suppress the Continuous-Wave (CW and MCW) interference is proposed for maximizing the receiver Signal-to-Noise Ratio (SNR) in a Quadrature Phase-Shift Keying (QPSK)-modulated signal. The suppression approach consists of two Adaptive IIR NFs (ANFs) based on a direct-form structure: the Hd1(z) and Hd1(z). The proposal in this work presents a low-complexity Time-Domain (TD) algorithm for controlling the update filter coefficient and notch depth. Simulation results demonstrate that the proposed approach represents an effective method for removing/reducing the impacts of CWI/MCWI, resulting in improved system performance for low- and high-power jammer values when compared with the case of full suppression (k1 = 1); furthermore, it also improves the notch filter’s output SNR for a given Jamming-to-Signal Ratio (JSR) value and Bit Error Ratio (BER) performance. For example, the SNR output of the proposed IIR NF was enhanced by 7 dB versus the case without a filter when Eb/No = 15 dB and JSR = −5 dB. The proposed method can detect and mitigate weak and strong jamming with JSR values ranging from −30 to 40 dB, and can track the hopping frequency interference. Moreover, an improved BER performance is seen as compared to the case without an IIR NF.

BER of OFDM System with Multiple NBI Rejection Cascade Complex Coefficient Adaptive IIR Notch Filter

In this paper, rejection of multiple narrowband interferers in a binary phase shift keying modulated orthogonal frequency division multiplexing (BPSK-OFDM) system is investigated. The BPSK-OFDM system in consideration operates in an additive white Gaussian noise (AWGN) channel. A cascade complex coefficient adaptive infinite impulse response (IIR) notch filter with gradient-based algorithm is used to reject the interferers. Bit error ratio (BER) performance of the system is studied and a general closed-form expression is derived assuming negligible steady-state leakage NBI and by estimating the decision variable as Gaussian distributed based on Central Limit Theorem (CLT). Dependence of the BER performance on the notch bandwidth coefficient is demonstrated by the analysis. Extensive simulation results are included to substantiate accuracy of the analysis.

Design of three-dimensional adaptive IIR notch filters

This paper investigates a realization of a three-dimensional (3-D) adaptive notch filter. The procedures are mainly divided into two parts: frequency-detecting and sinusoidal interference removal. The detections are based on adaptive line enhancer on infinite impulse response (IIR) lattice structure. In the interference removal part, a non-separable version of a 3-D notch filter is effectively applied. The magnitude response of a 3-D adaptive IIR notch filter is illustrated. At the end of the paper, the implementation of an IIR notch filter on a 3-D image is also conducted in order to show how to remove a sinusoidal interference superimposed on a 3-D image.

Novel adaptive IIR filter for frequency estimation and tracking [DSP Tips&Tricks

In many applications, a sinusoidal signal may be subjected to nonlinear effects in which possible harmonic frequency components are generated. In such an environment, we may want to estimate (track) the signal's fundamental frequency as well as any harmonic frequencies. Using a secondorder notch filter to estimate fundamental and harmonic frequencies is insufficient since it only accommodates one frequency component. On the other hand, applying a higher-order infinite impulse response (IIR) notch filter may not be efficient due to adopting multiple adaptive filter coefficients. In this article, we present a novel adaptive harmonic IIR notch filter with a single adaptive coefficient to efficiently perform frequency estimation and tracking in a harmonic frequency environment. Furthermore, we devise a simple scheme to select the initial filter coefficient to insure algorithm convergence to its global minimum error.

Tracking Analysis of Complex Adaptive IIR Notch Filter for a Linear Chirp Signal

This paper analyzes frequency tracking characteristics of a complex-coefficient adaptive infinite impulse response (IIR) notch filter with a simplified gradient-based algorithm. The input signal to the complex notch filter is a complex linear chirp embedded in a complex zero-mean white Gaussian noise. The analysis starts with derivation of a first-order real-coefficient difference equation with respect to steady-state instantaneous frequency tracking error. Closed-form expression for frequency tracking mean square error (MSE) is then derived from the difference equation. Lastly, closed-form expressions for optimum notch bandwidth coefficient and step size constant that minimize the frequency tracking MSE are derived. Computer simulations are presented to validate the analysis.

Adaptive IIR notch filters based on new error criteria

Modified indirect and direct plain gradient (MIPG and MDPG) algorithms using new error criteria for second-order adaptive IIR (infinite-impulse response) notch filters (ANFs) with constrained poles and zeros are proposed in this paper. The proposed techniques are based on modifying error functions of the standard indirect and direct ANFs. In this work, the proposed error criteria are modified as the mean value of the product of the two signals which are, respectively, produced by the FIR (finite-impulse response) and IIR sections of the ANFs. Due to a non-zero slope straight line-like characteristic of the new cost functions, the gradient based adaptive algorithm rapidly converges to an optimum solution resulting in faster convergence speed than that of the conventional method. The closed form expressions for steady-state estimation bias and mean square error (MSE) of the proposed algorithms are also carried out. In addition, extensive simulations of the proposed algorithms compared to other methods are provided to support the theoretical analyses. Although, there are some limitations in the proposed algorithms, however, the demonstrated results obtained from tested algorithms at the same condition confirm that the proposed MIPG and MDPG algorithms are superior to others in convergent speed.

Adaptive IIR Filters for Single Interference Suppression in a BPSK DS CDMA System In Rayleigh Fading Channel

In this paper, effect of a single narrow-band interference (NBI) on bit error rate (BER) performance for a binary phase shift keying (BPSK) synchronous direct-sequence code-division multiple access (DS CDMA) communication system operating in a frequency nonselective Rayleigh fading channel is analyzed. Second-order adaptive infinite impulseresponse (IIR) notch filters with plain gradient algorithm (GA) for suppression of NBI in the DS CDMA system are proposed. A general closed-form BER expression for the DS CDMA system with NBI suppression second order adaptive IIR notch filters is derived based on the standard Gaussian approximation (SGA) method. BER expressions are then derived for the allpass filter-based adaptive IIR notch filter and adaptive IIR notch filter with constrained poles and zeros, the two structures that are commonly found in literature. It is observed that both adaptive IIR notch filter structures exhibit comparable BER performance. Extensive computer simulation results are presented to verify the accuracy and limitations of the analysis.

Recursive least squares-like algorithms for the adaptive second-order lattice notch filter

Two new adaptive second-order infinite impulse response (IIR) notch filters with fast convergence rate, accurate estimation of notch frequency, and modest realization complexity are proposed in this paper. It is proven that the optimum filter coefficient can be determined by using the minimum output energy (MOE) criterion. Based on it, a simplified equivalent adaptive system is derived, and two recursive least squares (RLS)-like algorithms are presented to adaptively estimate the filter coefficient. Extensive simulations show the effectiveness and robustness of the new adaptive IIR notch filters.

Performance Analyses of Adaptive IIR Notch Filters Using a PSD-Based Approach

In this letter we present steady-state analyses of a gradient algorithm (GA) for second-order adaptive infinite impulse response (IIR) notch filters. A method for deriving more accurate estimation mean square error (MSE) expressions than the recently proposed method is presented. The method is based on the estimation error power spectral density (PSD). Moreover, an expression for the estimation bias for the adaptive IIR notch filter with constrained poles and zeros is shown to be obtained from the estimation MSE expression. Simulations are presented to confirm the validity of the analyses.

Statistical performance of the memoryless nonlinear gradient algorithm for the constrained adaptive IIR notch filter

Gradient-type algorithms for the adaptive infinite-impulse response (IIR) notch filters are very attractive in terms of both performance and computational requirements for various real-life applications. This paper presents, in detail, a statistical analysis of the memoryless nonlinear gradient (MNG) algorithm applied to the well-known second-order adaptive IIR notch filter with constrained poles and zeros. This analysis is based on a proper use of Taylor series expansion and nonlinearization of output signals of the notch and gradient filters. Two difference equations are derived first for the convergence in the mean and mean square senses, respectively. Two closed-form expressions, one for the steady-state estimation bias and the other for the mean-square error, are then derived based on the difference equations, with the former valid for both fast and slow adaptations and the latter valid for slow adaptation only. A closed-form coarse stability bound for the step size parameter of the algorithm is also derived. Extensive simulations are performed to reveal the validity and limitations of the analytical findings. Comparisons between the MNG and the conventional plain gradient algorithm are also made.

Performance analysis of the sign algorithm for a constrained adaptive IIR notch filter

Adaptive infinite impulse response (IIR) notch filters are very attractive in terms of their reasonable performances and low computational requirements. Generally, it is very difficult to assess their performances analytically due to their IIR nature. This paper analyzes in detail the steady-state performance of the sign algorithm (SA) for a well-known adaptive IIR notch filter with constrained poles and zeros. Slow adaptation and Gaussianity of the notch filter output are assumed for the sake of analysis. Two difference equations are first established for the convergences in the mean and mean square in the vicinity of the steady state of the algorithm. Steady-state estimation error or bias and mean square error (MSE) of the SA are then derived in closed forms. A coarse stability bound is also derived for the algorithm. Theory-based comparison between the algorithm and the plain gradient (PG) algorithm is done in some detail. Extensive simulations are conducted to demonstrate the validity of the analytical results for both slow and relatively fast adaptations.

Steady‐state analysis of complex adaptive IIR notch filters for the detection of single sinusoid

AbstractIn this paper, we present a new structure for a complex adaptive infinite impulse response (IIR) notch filter which is useful for the detection of complex sinusoids in communication and radar systems. The proposed structure is based on the gradient algorithm. The adaptive detection of a complex sinusoid with additive complex white noise has been investigated. A quantitative analysis for the steady‐state coefficient fluctuation has been developed. Analytical expressions for the steady‐state variance of complex variable coefficient fluctuation have been obtained. It has been shown that the variance of the complex coefficient fluctuation does not depend on the angular frequency of the input sinusoid. Finally, the results of computer simulations which confirm the theoretical predictions are presented. © 2001 Scripta Technica, Electron Comm Jpn Pt 3, 85(1): 32–40, 2002

Steady-state analysis of a plain gradient algorithm for a second-order adaptive IIR notch filter with constrained poles and zeros

Gradient-type algorithms for adaptive infinite-impulse response (IIR) notch filters are very attractive in terms of performance and computation cost. However, it is generally quite difficult to assess their performances analytically. There are several trials to analyze adaptive algorithms, such as the sign and the plain gradient algorithms for some types of adaptive IIR notch filters, but the analysis techniques used cannot be directly applied to different types of adaptive IIR notch filters. This brief presents closed form expressions for steady-state estimation bias and mean square error (MSE) of a well known plain gradient (LMS-like) second-order adaptive IIR notch filter with constrained poles and zeros. First, theoretical expressions for output signals of the notch filter and its corresponding gradient filter at their steady states are developed based on the Taylor series expansions of transfer functions of these two filters in the vicinity of the sinusoidal signal frequency difference equations for convergences in the mean and mean square are then established by using these output signals, from which the steady-state bias and MSE of the algorithm are derived. Stability bound of the algorithm is also investigated based on the difference equations. Extensive simulations are provided to support the analytical findings.

A new memoryless nonlinear gradient algorithm for a second-order adaptive IIR notch filter and its performance analysis

Adaptive infinite impulse-response (IIR) notch filters have been widely studied for many years. However, further efforts are still to be made to pursue new algorithms which work better than the conventional plain gradient algorithm but have little increase in complexity. In this paper, we employ the gradient linearization, Taylor series expansion, and calculus of variations to derive a novel memoryless nonlinear gradient algorithm for a second-order adaptive IIR notch filter, which improves the estimation performance considerably. Approximate closed-form expressions for the stability bounds on the step size parameter and the steady-state coefficient variance are also derived. Extensive simulations indicate the significant improvement that may be achieved using the new algorithm, and verify the validity of the analytical results.

An improved lattice-based adaptive IIR notch filter

A novel lattice-based adaptive infinite impulse response (IIR) notch filter is developed which features independent tuning of the notch frequency and attenuation bandwidth. The internal structure is based on planar rotators, ensuring reliable numerical behaviour and high processing rates in CORDIC environments. A simple update law allows a simpler implementation than previously proposed designs. Rather than minimizing an output error cost function, the algorithm is designed to achieve a stable associated differential equation, resulting in a globally convergent unbiased frequency estimator in the single sinusoid case, independent of the notch filter bandwidth. Using a second-order structure in the multiple sinusoid case, unbiased estimation of one of the input frequencies is achieved by thinning the notch bandwidth. The tracking behavior is superior to conventional output error designs, and the estimation of extremal frequencies is less prone to overflow instability than previously reported structures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The isolation of undistorted sinusoids in real time

A single-input, multiple-output adaptive filter that takes a multisinusoidal input signal and produces the individual sinusoids at each of the outputs is described. The individual outputs have identical amplitudes and phases to the components of the original input signal. The filter can be adapted in real time using approximate Gauss-Newton updates without requiring any matrix operations or divisions. The filter is an extension of a cascade adaptive IIR (infinite impulse response) notch filter reported previously. The filter is very amenable to being realized using custom ICs or DSP (digital signal processing) ICs, as no divisions are required and most of the filter is composed of identical biquads (which are very insensitive to the errors caused by finite accuracy in fixed-point implementations). The structure is especially good at isolating small-amplitude sinusoids when in the proximity of much larger sinusoids of unknown frequencies.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>