Theory of the Design, and Operation of Digital Filters

Chapter 16 present the theoretical basis for digital filters, including issues related to their design. The basic characteristics and structures of finite impulse response and infinite impulse response filters are presented and discussed. In addition, the ideal and practical transfer characteristics of the digital filters are defined. The basic advantage of finite impulse response filters is that they can be designed to have an exact linear phase. However, infinite impulse response filters are generally more efficient computationally. The methods for filters design and related algorithms, which are based on the bilinear transformation method, windowed Fourier series, and algorithms based on iterative optimization, are also presented.

Multi-Rate Discrete-Time Signal Processing

Chapter 17 presents the multi-rate signal process, starting with explanations of the up-sampling and down-sampling procedures on a discrete signal in the time domain. The operations of a down-sampler (decimator) and an up-sampler (interpolator) are analysed in the frequency domain, emphasizing the problem of possible aliasing. Complex systems that include both up-sampling and down-sampling are analysed and the problem of complexity reduction is mentioned. The operation of systems that combine an interpolator and an interpolation filter, and a decimator and a decimation lowpass filter, is presented in the time and frequency domains. In particular, the problem of reducing the complexity of a multi-rate system is addressed, and a polyphase decomposition for both finite impulse response filters and infinite impulse response filters is offered as an efficient solution.

MEG-Based Detection of Voluntary Eye Fixations Used to Control a Computer

Gaze-based input is an efficient way of hand-free human-computer interaction. However, it suffers from the inability of gaze-based interfaces to discriminate voluntary and spontaneous gaze behaviors, which are overtly similar. Here, we demonstrate that voluntary eye fixations can be discriminated from spontaneous ones using short segments of magnetoencephalography (MEG) data measured immediately after the fixation onset. Recently proposed convolutional neural networks (CNNs), linear finite impulse response filters CNN (LF-CNN) and vector autoregressive CNN (VAR-CNN), were applied for binary classification of the MEG signals related to spontaneous and voluntary eye fixations collected in healthy participants (n = 25) who performed a game-like task by fixating on targets voluntarily for 500 ms or longer. Voluntary fixations were identified as those followed by a fixation in a special confirmatory area. Spontaneous vs. voluntary fixation-related single-trial 700 ms MEG segments were non-randomly classified in the majority of participants, with the group average cross-validated ROC AUC of 0.66 ± 0.07 for LF-CNN and 0.67 ± 0.07 for VAR-CNN (M ± SD). When the time interval, from which the MEG data were taken, was extended beyond the onset of the visual feedback, the group average classification performance increased up to 0.91. Analysis of spatial patterns contributing to classification did not reveal signs of significant eye movement impact on the classification results. We conclude that the classification of MEG signals has a certain potential to support gaze-based interfaces by avoiding false responses to spontaneous eye fixations on a single-trial basis. Current results for intention detection prior to gaze-based interface’s feedback, however, are not sufficient for online single-trial eye fixation classification using MEG data alone, and further work is needed to find out if it could be used in practical applications.

Finite Impulse Response Filters for Stagger-Period Signals, their Designs and Applications

Those theories of conventional filters for uniform-period signals do not apply to the analysis and design of the finite impulse response (FIR) filters for stagger-period signals. In this paper, we defined the fundamental concepts related to the stagger-period signals, derived the calculating equations, and described the time-variant property of the stagger-period filter; we proposed the Fourier transform pair between the frequency and impulse responses of this type filter, and proved the inverse of each other. Then, we discussed the design methods of stagger-period frequency-selective FIR filters, including lowpass, bandpass, and high-pass, presented the staggered windowing philosophies, illustrated different windows’ effectiveness, and described the principles and designs of optimized stagger-period high-pass filters with the match algorithm. As applications, we introduced three staggered optimization algorithms: eigenvalue, match, and linear prediction; and discussed performances of the filters designed for a moving target indication (MTI) radar. The stagger-period MTI filters not only extended the blind speed of flying targets, but also had an optimized improvement factor. Finally, we proposed a mathematical programming to search the best period code, which makes this type filter’s velocity response flattened. Meanwhile, we compared properties of the stagger-period to uniform-period filters, and provided with some examples to illustrate the theories and designs.

Finite Impulse Response Filters for Stagger-Period Signals, their Designs and Applications

Those theories of conventional filters for uniform-period signals do not apply to the analysis and design of the finite impulse response (FIR) filters for stagger-period signals. In this paper, we defined the fundamental concepts related to the stagger-period signals, derived the calculating equations, and described the time-variant property of the stagger-period filter; we proposed the Fourier transform pair between the frequency and impulse responses of this type filter, and proved the inverse of each other. Then, we discussed the design methods of stagger-period frequency-selective FIR filters, including lowpass, bandpass, and high-pass, presented the staggered windowing philosophies, illustrated different windows’ effectiveness, and described the principles and designs of optimized stagger-period high-pass filters with the match algorithm. As applications, we introduced three staggered optimization algorithms: eigenvalue, match, and linear prediction; and discussed performances of the filters designed for a moving target indication (MTI) radar. The stagger-period MTI filters not only extended the blind speed of flying targets, but also had an optimized improvement factor. Finally, we proposed a mathematical programming to search the best period code, which makes this type filter’s velocity response flattened. Meanwhile, we compared properties of the stagger-period to uniform-period filters, and provided with some examples to illustrate the theories and designs.

Epileptic Spike Detection by Using a Linear-Phase Convolutional Neural Network

To cope with the lack of highly skilled professionals, machine leaning with proper signal techniques is a key to establishing automated diagnostic-aid technologies to conduct epileptic electroencephalogram (EEG) testing. In particular, frequency filtering with appropriate passbands is essential to enhance biomarkers—such as epileptic spike waves—that are noted in the EEG. This paper introduces a novel class of convolutional neural networks (CNNs) having a bank of linear-phase finite impulse response filters at the first layer. These may behave as bandpass filters that extract biomarkers without destroying waveforms because of linear-phase condition. The proposed CNNs were trained with a large amount of clinical EEG data, including 15,899 epileptic spike waveforms recorded from 50 patients. These have been labeled by specialists. Experimental results show that the trained data-driven filter bank with supervised learning is dyadic like discrete wavelet transform. Moreover, the area under the curve achieved above 0.9 in most cases.