Power-weighted estimators have recently been proposed for relative attenuation and delay estimation in blind source separation. Their provenance lies in the observation that speech is approximately windowed-disjoint orthogonal (WDO) in the time-frequency (TF) domain; it has been reported that using WDO, derived from TF representations of speech, improves mixing parameter estimation. We show that power-weighted relative attenuation and delay estimators can be derived from a particular case of a weighted Bregman divergence. We then propose a wider class of estimators, which we tune to give better parameter estimates for speech.

The triple autocorrelation of a sequence is shown to be a Lempel Costas array iff the sequence is maximal, assuming correlation is appropriately defined and the alphabet of the sequence is of prime size. A method that allows the construction of thumbtack autocorrelation sequences of arbitrary alphabet size is subsequently proposed.

Non-negative Matrix Factorization (NMF) has found use in single channel separation of audio signals, as it gives a parts-based decomposition of audio spectrograms where the parts typically correspond to individual notes or chords. However, a notable shortcoming of NMF is the need to cluster the basis functions to their sources after decomposition. Despite recent improvements in algorithms for clustering the basis functions to sources, much work still remains to further improve these algorithms. To this end we present a novel clustering algorithm which overcomes some of the limitations of previous clustering methods. This involves the use of Shifted Non negative Matrix Factorization (SNMF) as a means of clustering the frequency basis functions obtained from NMF. Results show that this gives improved clustering of pitched basis functions over previous methods.

In order to extrapolate a signal, Empirical Mode Decomposition is used to decompose it into simpler components. Each component is individually extrapolated linearly, and the final extrapolation value is produced as the sum of these individual values. This technique is applied on financial signals, with a view towards capturing the sign of the increment of the signal instead of the exact future value, and the results are compared to cubic spline extrapolation.

Families of Costas arrays with low pairwise cross-correlation are sought. The two families of all exponential Welch arrays and all Golomb arrays generated in a certain finite field are specifically studied, and the maximal cross-correlation is determined by exhaustive search. Mathematically rigorous explanations for some of the observed results are presented, a surprising link between Welch and Golomb arrays is revealed, and what remains to be proved is stated precisely. The results suggest that the families with uniformly low cross-correlation correspond to finite fields whose size is a safe prime power.

A Costas array is an arrangement of N dots on an N -by- N grid, one per row, one per column, such that no two dots share the same displacement vector with any other pair. Costas arrays have applications in SONAR/RADAR systems, communication systems, cryptography, and other areas. We present a standardization of notation and language which can be used to discuss Costas array generation techniques and array manipulations. Using this standardization we can concisely and clearly state various theorems about Costas arrays, including several new theorems about the symmetries of Costas arrays. We also define labels for each array (generated, emergent, and sporadic), which describe whether the array is generated using a known technique, generated using a semiempirical variation of a known technique, or of unexplained origin. A new method for obtaining emergent Costas arrays, the DRT expansion, is also given here for the first time. A MATLAB Costas array toolbox has also been developed which implements the proposed standardization. The toolbox contains a comprehensive set of functions covering Costas array generation, manipulation and classification.

We present the results of the enumeration of Costas arrays of order 28: all arrays found are accounted for by the Golomb and Welch construction methods, making 28 the first order (larger than 5) for which no sporadic Costas arrays exist. The enumeration was performed on several computer clusters and required the equivalent of 70 years of single CPU time.

In this paper, we show that in order to ascertain whether a permutation has the Costas property, only a restricted subset among the totality of pairs of entries in the same row of the difference triangle needs to be checked, and we explicitly describe such a subset. This represents a further refinement on the definition of a Costas permutation. This observation can be used to speed up algorithms that exhaustively search for Costas permutations. Asymptotically, the savings approaches 43% for large orders when compared with the previous standard efficient method.

Sparsity of representations of signals has been shown to be a key concept of fundamental importance in fields such as blind source separation, compression, sampling and signal analysis. The aim of this paper is to compare several commonly-used sparsity measures based on intuitive attributes. Intuitively, a sparse representation is one in which a small number of coefficients contain a large proportion of the energy. In this paper, six properties are discussed: (Robin Hood, Scaling, Rising Tide, Cloning, Bill Gates, and Babies), each of which a sparsity measure should have. The main contributions of this paper are the proofs and the associated summary table which classify commonly-used sparsity measures based on whether or not they satisfy these six propositions. Only two of these measures satisfy all six: the pq-mean with p les 1, q > 1 and the Gini index.

This correspondence presents the results of the enumeration of Costas arrays of order$27$: all arrays found, except for one, are accounted for by the Golomb and Welch construction methods.

Few source separation and independent component analysis approaches attempt to deal with noisy data. We consider an additive noise mixing model with an arbitrary number of sensors and possibly more sources than sensors (the "degenerate separation problem") when sources are disjointly orthogonal. We show how disjoint orthogonality can be viewed as a limit of a stochastic voice modeling assumption. This is the basis for our approach to noisy model estimation by maximum likelihood, under the direct-path far-field assumptions. The implementation of the derived criterion involves iterating two steps: a partitioning of the time-frequency plane for separation followed by an optimization of the mixing parameter estimates. The solution is applicable to an arbitrary number of microphones and sources. Experimentally, we show the capability of the technique to separate four voices from two, four, six and eight channel recordings in the presence of strong noise.

It is possible to blindly separate an arbitrary number of sources given just two anechoic mixtures provided the time-frequency representations of the sources do not overlap, a condition which we call W-disjoint orthogonality. We define a power weighted two-dimensional histogram constructed from the ratio of the time-frequency representations of the mixtures which is shown to have one peak for each source with: peak location corresponding to the relative amplitude and delay mixing parameters. All of the time-frequency points which yield estimates in a given peak are exactly all the non-zero magnitude components of one of the sources. We introduce the concept of approximate W-disjoint orthogonality, present experimental results demonstrating the level of approximate W-disjoint orthogonality of speech in mixtures of various order, and show that even with imperfect W-disjoint orthogonality the histogram can be used to determine the mixing parameters and separate sources. Example demixing results can be found online: http://www.princeton.edu/∼srickard/bss.html

We present a novel method for blind separation of any number of sources using only two mixtures. The method applies when sources are (W-)disjoint orthogonal, that is, when the supports of the (windowed) Fourier transform of any two signals in the mixture are disjoint sets. We show that, for anechoic mixtures of attenuated and delayed sources, the method allows one to estimate the mixing parameters by clustering ratios of the time-frequency representations of the mixtures. The estimates of the mixing parameters are then used to partition the time-frequency representation of one mixture to recover the original sources. The technique is valid even in the case when the number of sources is larger than the number of mixtures. The general results are verified on both speech and wireless signals.