This paper presents a novel low-complexity method for extracting periodicity of signals based on their sine-wave representation. In this representation, the signal is modeled as a finite sum of sine-waves, with time-varying amplitudes, phases and frequencies. We describe how one can modify the familiar spectral-comb analysis method to obtain a guaranteed and effective procedure to find the fundamental-frequency which gives the best harmonic approximation of the signal spectrum. The search is efficiently carried out in the frequency domain. The procedure obtains a successive refinement of possible pitch values which are consistent with an increasing number of sine wave components. Other pitch intervals are pruned at an early stage of the search. The advantage of this algorithm is its high accuracy achieved at a relatively low complexity. We also briefly describe one possible application in the area of pitch determination of speech signals.

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.

The multi-window Gabor-type scheme is generalized to the multidimensional case with special emphasis on images. Sampling rate p/q along each dimension, where p and q are relatively prime integers, is considered. The problem of multidimensional matrix operations required in dual window computations gives rise to the stacking method that merges dimensions from hyperdimensional to a two-dimensional conventional matrix. Algorithms are generalized to Gabor-type representation of images and multidimensional signals that are sampled over different grids, such as the hexagonal grids.

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.

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.

Properties of basis functions which constitute a finite scheme of discrete Gabor representation are investigated. The approach is based on the concept of frames and utilizes the Piecewise Finite Zak Transform (PFZT). The frame operator associated with the Gabor-type frame is examined by representing it as a matrix-values function in the PFZT domain. The frame property 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 eignevalues 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. It is illustrated by an example that a better reconstruction is obtained in from the same number of coefficients in the case of oversampling.

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.>