Meir Zibulski

Speech reconstruction from mel frequency cepstral coefficients and pitch frequency

This paper presents a novel low complexity, frequency domain algorithm for reconstruction of speech from the mel-frequency cepstral coefficients (MFCC), commonly used by speech recognition systems, and the pitch frequency values. The reconstruction technique is based on the sinusoidal speech representation. A set of sine-wave frequencies is derived using the pitch frequency and voicing decisions, and synthetic phases are then assigned to each respective sine wave. The sine-wave amplitudes are generated by sampling a linear combination of frequency domain basis functions. The basis function gains are determined such that the mel-frequency binned spectrum of the reconstructed speech is similar to the mel-frequency binned spectrum, obtained from the original MFCC vector by IDCT and antilog operations. Natural sounding, good quality intelligible speech is obtained by this procedure.

Discrete multiwindow Gabor-type transforms

The discrete (finite) Gabor scheme is generalized by incorporating multiwindows. Two approaches are presented for the analysis of the multiwindow scheme: the signal domain approach and the Zak transform domain approach. Issues related to undersampling, critical sampling, and oversampling are considered. The analysis is based on the concept of frames and on generalized (Moore-Penrose) inverses. The approach based on representing the frame operator as a matrix-valued function is far less demanding from a computational complexity viewpoint than a straightforward matrix algebra in various operations such as the computation of the dual frame. DFT-based algorithms, including complexity analysis, are presented for the calculation of the expansion coefficients and for the reconstruction of the signal in both signal and transform domains. The scheme is further generalized and incorporates kernels other than the complex exponential. Representations other than those based on the dual frame and nonrectangular sampling of the combined space are considered as well. An example that illustrates the advantages of the multiwindow scheme over the single-window scheme is presented.

Multi-window Gabor schemes in signal and image representations

Motivated by biological vision, schemes of signal and image representation by localized Gabor-type functions are introduced and analyzed. These schemes, suitable for information representation in a combined frequency-position space are investigated through signal decomposition into a set of elementary functions. Utilizing the Piecewise Zak transform (PZT), the theory of the multi-window approach is given in detail based on the mathematical concept of frames. The advantages of using more than a single window are analyzed and discussed. Applications to image processing and computer vision are presented with regard to texture images, and considered in the context of two typical tasks: image representation by partial information and pattern recognition. In both cases the results indicate that the multi-window approach is efficient and superior in major aspects to previously available methods. It is concluded that the new multi-window Gabor approach could be integrated efficiently into practical techniques of signal and image representation.

Frame analysis of the discrete Gabor-scheme

The properties of the discrete Gabor scheme are considered in the context of oversampling. The approach is based on the concept of frames and utilizes the piecewise finite Zak transform (PFZT). The frame operator is represented as a matrix-valued function in the PFZT domain, and its properties are examined in relation to this function. The frame bounds are calculated by means of the eigenvalues of the matrix-valued function, and the dual frame, which is used in calculation of the expansion coefficients, is expressed by means of the inverse matrix. DFT-based algorithms for computation of the expansion coefficients, and for the reconstruction of signals from these coefficients, are generalized for the case of oversampling of the Gabor space. The algorithms are implemented in an example of representation of a nonstationary signal. >

Oversampling in the Gabor scheme

A method for calculating the coefficients of the Gabor expansion in the context of oversampling is presented. The method is based on the concept of frames and utilizes the Zak transform. The Zak transform further highlights the meaning and importance of frames in the context of oversampling and other aspects of signal representation by means of nonorthogonal bases.<<ETX>>

Gabor representation with oversampling

An approach for characterizing the properties of the basis functions of the Gabor representation in the context of oversampling is presented. The approach is based on the concept of frames and utilizes the Piecewise Zak Transform (PZT). The frame operator associated with the Gabor-type frame, the so-called Weyl-Heisenberg frame, is examined for a rational oversampling rate by representing the frame operator as a matrix-valued function in the PZT domain. Completeness and frame properties of the Gabor representation functions are examined in relation to the properties of the matrix-valued function. The frame bounds are calculated by means of the eigenvalues of the matrix-valued function, and the dual-frame, which is used in calculation of the expansion coefficients, is expressed by means of the inverse matrix.

Multiwindow Gabor-type transform for signal representation and analysis

The Gabor scheme is generalized to incorporate several window functions as well as kernels other than the exponential. The properties of the sequence of representation functions are characterized by an approach based on the concept of frames. the frame operator associated with the multi-window Gabor-type frame, is examined for a rational oversampling rate by representing the frame operator as a finite order matrix-valued function in the Zak Transform domain. Completeness and frame properties of the sequence of representation functions are examined in relation to the properties of the matrix-valued function. Calculation of the frame bounds and the dual frame, as well as the issue of tight frames are considered. It is shown that the properties of the sequence of representation functions are essentially not changed by replacing the widely-used exponential kernel with other kernels. The issue of a different sampling rate for each window is also considered. The so-called Balian-Low theorem is generalized to consideration of a scheme of multi-windows, which makes it possible to overcome the constraint imposed by the original theorem in the case of a single window.

The Generalized Gabor Scheme and Its Application in Signal and Image Representation

Abstract. . The single-window Gabor scheme is generalized by incorporating multi-windows as well as kernels other than the exponential. Two types of schemes are considered, a continuous-variable scheme and a discrete-variable scheme. The properties of the sequence of representation functions are characterized by an approach which combines the concept of frames and the Zak transform. The frame operator associated with the multi-window Gabor-type frame is examined by representing the frame operator as a finite order matrix-valued function. Completeness and frame properties of the sequence of representation functions are examined in relation to the properties of the matrix-valued function. Calculation of the frame bounds and the dual frame, as well as the issue of tight frames, are considered. It is shown that the properties of the sequence of representation functions are essentially unchanged by replacing the widely-used exponential kernel with other kernels. For the continuous scheme, the Balian-Low theorem is generalized to consideration of a scheme of multi-windows. For the discrete scheme, generalized inverses and representations other than those based on the dual frame are considered. Examples illustrate the advantages of the multi-window scheme over the single-window scheme.

On the role of biorthonormality in representation of random processes

The representation of a random process by a set of uncorrelated random variables is examined. The main result indicates that a basis decorrelates a random process if and only if it satisfies an integral equation similar to the type satisfied by the Karhunen-Loeve expansion, but by relaxing the requirement of orthogonality of the representation functions.

Signal- and image-component separation by a multi-window Gabor-type scheme

The discrete (finite) Gabor scheme is generalized by incorporating multi-windows. Two approaches are presented for the analysis of the generalized scheme: the signal domain approach and Zak transform domain approach. These approaches are based on representing the frame operator as a matrix-valued function, and are far less demanding from a computational complexity viewpoint than a straightforward matrix algebra in various operations such as the computation of the dual frame. Issues related to undersampling, critical sampling and oversampling are considered. Examples illustrating the advantages of the multi-window scheme over the single-window scheme are presented.