Adaptive Digital Filtering Based on a Continuous-time Performance Index

Chapter 9 - Discrete-Time Signals and Systems

In this paper, the problem of sampling a continuous-time generalized almost-cyclostationary (GACS) signal is addressed. The class of such nonstationary signals includes, as a special case, the almost-cyclostationary (ACS) signals. ACS signals filtered by some linear time-variant channels are further examples. It is shown that the discrete-time signal constituted by the samples of a GACS signal is a discrete-time ACS signal. Thus, the nonstationarity kind of a continuous-time GACS signal cannot be deducted from that of the discrete-time signal of its samples. However, in the paper it is shown how, starting from the sampled signal, the GACS or ACS nature of the continuous-time signal can be conjectured, provided that analysis parameters such as sampling period, padding factor, and data-record length are properly chosen.

Chapter 1 - Continuous-Time Signals

Continuous-time or analog signals are processed using analog devices such as amplifiers, filters, etc. It is impossible to process signals multiplexed from various sources using a single hardware system in the analog domain. On the other hand, digital signals can be processed using both special-purpose hardware and software systems. Worldwide use of Internet, mobile communications, etc. demands all kinds of data such as video, audio, graphics, etc. In order to receive this information on a single device, computer, for instance, it is impossible to use analog signals and techniques. In order to be able to design and implement digitally based systems, it is absolutely necessary to have an understanding of digital signals and systems. Digital signals are discrete in time and amplitude. However, we will assume discrete-time signals to have a continuum of amplitude in order to be able to analyze such signals and systems mathematically. In this chapter we will describe typical discrete-time signals mathematically and then use them to describe and analyze linear time-invariant discrete-time systems. To help the readers understand the mathematical details, we will work out examples followed by MATLAB-based examples. Since digital signals are obtained from analog sources, we will also discuss the conversion of continuous-time signals into digital signals using analog-to-digital converters (ADC).

Hybrid systems modelers like Simulink come with a rich collection of discrete-time and continuous-time blocks. Most blocks are not defined in terms of more elementary ones—and some cannot be—but are instead written in imperative code and explained informally in a reference manual. This raises the question of defining a minimal set of orthogonal programming constructs such that most blocks can be programmed directly and thereby given a specification that is mathematically precise, and whose compiled version performs comparably to handwritten code. In this paper, we show that a fairly large set of blocks of a standard library like the one provided by Simulink can be programmed in a precise, purely functional language using stream equations, hierarchical automata, Ordinary Differential Equations (ODEs), and deterministic synchronous parallel composition. Some blocks cannot be expressed in our setting as they mix discrete-time and continuous-time signals in unprincipled ways that are statically forbidden by the type checker. The experiment is conducted in Zélus, a synchronous language that conservatively extends L ustre with ODEs to program systems that mix discrete-time and continuous-time signals.

A discrete time signal is one that has a value only for a finite or infinite number of time instants whereas a continuous time signal has a value for every (real) time instant. For example, consider: The sound pressure wave of a speech signal The electrical that is the output of a transducer that is used to mesdsure the cound pressure wave of a speech signal The sequence of numbers that is obtained from connecting an analg to digital converter (ADC) to the electrical signal sound pressure wave of a speech signal KeywordsSpeech SignalAnalog SignalAnalog InputLinear Predictive CodeDiscrete Time SignalThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A good echo cancellation algorithm should have a fast convergence rate, small steady-state residual echo, and less implementation cost. The normalized least mean square (NLMS) adaptive filtering algorithm may not achieve this goal. We show that using the Gabor expansion is a way to achieve this goal. For direct digital signal processing compatibility the Gabor expansion introduced in this paper is for discrete-time signals, although the Gabor expansion also can be used for continuous-time signals. The Gabor expansion can be defined as a discrete-time signal representation in the joint time-frequency domain of a weighted sum of the collection of functions (known as the synthesis functions). There are several design issues in the echo canceller based on the Gabor expansion: the design of the analysis functions for the far-end speech, the design of the analysis functions for the near-end signal containing the echo plus the near-end speech, the design of the adaptive filters in the subsignal path, and the design of the synthesis functions. All the adaptive filters are designed using identical NLMS adaptive filtering algorithms.

Active noise control systems that have been studied previously require a replica of the cancellation path to obtain the updating weight values, because the signals for the operation cannot be observed. In this paper we propose a structure and an adaptive algorithm for the active noise control system using an adaptive exponential filter that does not require any replica. The adaptive algorithm is based on the steepest descent method that can be easily realized and the updating weight value of the adaptive exponential filter is the correlation of the observable output signal of the adaptive system and the estimation error. The other adaptive filters are composed of the linear transversal system. It is proven that the transfer function after converging completely corresponds to the transfer function of the noise and the cancellation path under a certain condition. Finally, the convergence performance of the estimation accuracy and the tracking performance with respect to rapid changes of the noise and the cancellation paths are verified by computer simulation. © 2004 Wiley Periodicals, Inc. Electr Eng Jpn, 147(1): 53–59, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.10256

In this study, a new adaptive filter is proposed to eliminate salt and pepper noise (SPN). The basis of the proposed method consists of two-stages. (1) Changing the noisy pixel value with the closest pixel value or assigning their average to the noisy pixel in case there is more than one pixel with the same distance; (2) the updating of the calculated noisy pixel values with the average filter by correlating them with the noise ratio. The method developed was named as Nearest Value Based Mean Filter (NVBMF), because of using the pixel value which the closest distance in the first stage. Results obtained with the proposed method: it has been compared with the results obtained with the Adaptive Frequency Median Filter, Adaptive Riesz Mean Filter, Improved Adaptive Weighted Mean Filter, Adaptive Switching Weight Mean Filter, Adaptive Weighted Mean Filter, Different Applied Median Filter, Iterative Mean Filter, Two-Stage Filter, Multistage Selective Convolution Filter, Different Adaptive Modified Riesz Mean Filter, Stationary Framelet Transform Based Filter and A New Type Adaptive Median Filter methods. In the comparison phase, nine different noise levels were applied to the original images. Denoised images were compared using Peak Signal-to-Noise Ratio, Image Enhancement Factor, and Structural Similarity Index Map image quality metrics. Comparisons were made using three separate image datasets and Cameraman, Airplane images. NVBMF achieved the best result in 52 out of 84 comparisons for PSNR, best in 47 out of 84 comparisons for SSIM, and best in 36 out of 84 comparisons for IEF. In addition, values nearly to the best result were obtained in comparisons where the best result could not be reached. The results obtained show that the NVBMF can be used as an effective method in denoising SPN.

Basics of Multirate Digital Signal Processing

Gabor's expansion and the Zak transform for continuous-time and discrete-time signals

In this paper, an adaptive optimal predictive control method based on improved prescriptive performance is proposed for the path tracking control of an underactuated two‐wheeled mobile robot. The system is challenging to control due to its underactuated characteristics and nonlinear dynamics behavior. To improve the robustness of the system under uncertainty and reduce the jitter phenomenon, a new control strategy is designed by combining the improved prescribed performance control with adaptive optimal predictive control. The method ensures that the system output can satisfy the prescribed performance requirements within a specific time range and recover the stable operation faster under certain disturbances through the design of reasonable soft‐constraint performance indexes. In addition, a rigorous stability analysis of the control system is carried out to demonstrate the boundedness of the error function and the transient and steady‐state performances specified by the output tracking error. Simulation results show that the proposed control method can significantly improve the stability and control accuracy of the system under disturbances.

When discretizing continuous-time systems or signals, one is often interested in preserving a property termed covariance-invariance. In this paper a technique is outlined for synthesizing discrete-time systems and signals which are covariance-invariant with corresponding continuous-time systems and signals. Applications of the technique to process simulation, minimum mean-squared error estimation, and digital filter synthesis are outlined, with example designs presented for covariance-invariant Butterworth and Chebychev digital filters. Based on the frequency response of these designs it is argued that the method of covariance-invariance is superior to the methods of impulse-invariance and bilinear-z as a response matching design technique for the synthesis of digital filters. This superiority is especially apparent at sampling rates that are marginal with respect to filter critical frequencies.

In this paper, we report an interesting finding of the adaptive filter characteristic that aids in the accurate detection of the pulse event information. Accurate pulse detection is extremely critical as it provides sufficient cardia information. This will further enhance the chances of wearable cardiac activity monitors. The signal is acquired using the using MPVZ5004G6U piezo-resistive (PZR) sensors from the wrist of humans. The signal conditioning is performed and an adaptive FIR filter that uses the least mean squared (LMS) algorithm is designed and deployed for the system identification application. Upon careful examination of the experimental results, the error signal is found to provide the pulse event information. This is due to the sudden change in the pulse activity that causes the error signal to exhibit impulse train like characteristics. The baseline wander and low frequency motion artifacts are alleviated from the signal. The study is conduced using the LabVIEW Virtual Instruments(VI).

Variants of Least Mean Square algorithm, namely the Sign Error, Sign Regressor and Sign-Sign Least Mean Square algorithms are meant for reducing computational complexity in digital adaptive filters in denoising electrocardiogram signals. In adaptive filtering using Sign-Sign algorithm, the adders and multipliers in the existing method take the coefficients in canonic signed power of two form which assures that between two non-zero there should be one zero in signed power of two digits and they can process only one bit per clock cycle and so is the filter resulting in low speed. In this paper, digit serial adder and digit serial multiplier are proposed to design the Sign-Sign Least Mean Square filter, which increases the speed by processing two bits per clock cycle. The digit serial adder and digit serial multiplier are used to design the Sign-Sign based adaptive filter in which unfolding technique is used for digit serial realisation. The unfolding factor of K = 2 is chosen to increase the speed of the design. The coefficients and its inputs are represented in Canonic Signed Digit form. Adder, multiplier, adaptive filter of both existing and proposed method are simulated using Mentor Graphics and synthesised using Synopsys Design Compiler Tool with 90 nm technology. From the results it is observed that the speed of the proposed digit serial Sign-Sign Least Mean Square filter is increased by 54.5% and the area of the filter is decreased by 4.21% compared to the bit serial Sign-Sign Least Mean Square filter.