Fundamentals of Recurrent Neural Network (RNN) and Long Short-Term Memory (LSTM) Network

The outline of this chapter is as follows. Section 2 reviews various types of existing finite impulse response (FIR) and infinite impulse response (IIR) two-channel filter banks. The basic operations of these filter banks are considered and the requirements are stated for alias-free, perfect-reconstruction (PR), and nearly perfect-reconstruction (NPR) filter banks. Also some efficient synthesis techniques are referred to. Furthermore, examples are included to compare various two-channel filter banks with each other. Section 3 concentrates on the design of multi-channel (M-channel) uniform filter banks. The main emphasis is laid on designing these banks using tree-structured filter banks with the aid of two-channel filter banks and on generating the overall bank with the aid of a single prototype filter and a proper cosine-modulation or MDFT technique. In Section 4, it is shown how octave filter banks can be generated using a single two-channel filter bank as the basic building block. Also, the relations between the frequency-selective octave filter banks and discrete-time wavelet banks are briefly discussed. Finally, concluding remarks are given in Section 5.

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Multirate Systems and Filter Banks

Unrestrained human arm trajectories between point targets have been investigated using a three-dimensional tracking apparatus, the Selspot system. Movements were executed between different points in a vertical plane under varying conditions of speed and hand-held load. In contrast to past results which emphasized the straightness of hand paths, movement regions were discovered in which the hand paths were curved. All movements, whether curved or straight, showed an invariant tangential velocity profile when normalized for speed and distance. The velocity profile invariance with speed and load is interpreted in terms of simplification of the underlying arm dynamics, extending the results of Hollerbach and Flash (Hollerbach, J. M., and T. Flash (1982) Biol. Cybern. 44: 67-77).

https://www.jneurosci.org/content/jneuro/5/9/2318.full.pdf

Kinematic features of unrestrained vertical arm movements

EEG alpha map series: brain micro-states by space-oriented adaptive segmentation

We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete fractional Fourier transform supports our confidence that it will be accepted as the definitive definition of this transform.

/pdf/the-discrete-fractional-fourier-transform-13dhl8vxtf.pdf

The discrete fractional Fourier transform

This paper presents a general-purpose computer program which is capable of designing a large Class of optimum (in the minimax sense) FIR linear phase digital filters. The program has options for designing such standard filters as low-pass, high-pass, bandpass, and bandstop filters, as well as multipassband-stopband filters, differentiators, and Hilbert transformers. The program can also be used to design filters which approximate arbitrary frequency specifications which are provided by the user. The program is written in Fortran, and is carefully documented both by comments and by detailed flowcharts. The filter design algorithm is shown to be exceedingly efficient, e.g., it is capable of designing a filter with a 100-point impulse response in about 20 s.

A computer program for designing optimum FIR linear phase digital filters

The principal results of this paper are listed as follows. 1) The eigenvalues of a suitably normalized version of the discrete Fourier transform (DFT) are {1, -1,j, -j} . 2) An eigenvector basis is constructed for the DFT. 3) The multiplicities of the eigenvalues are summarized for an N×N transform as follows.

Eigenvalue and eigenvector decomposition of the discrete Fourier transform

/pdf/fir-digital-filter-design-techniques-using-3fk0xpdtbr.pdf

J. McClellan

Papers

A computer program for designing optimum FIR linear phase digital filters

Eigenvalue and eigenvector decomposition of the discrete Fourier transform

FIR Digital Filter Design Techniques Using

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