An exactly solvable self-convolutive recurrence

We consider a self-convolutive recurrence whose solution is the sequence of coefficients in the asymptotic expansion of the logarithmic derivative of the confluent hypergeometic function $U(a,b,z)$. By application of the Hilbert transform we convert this expression into an explicit, non-recursive solution in which the $n$th coefficient is expressed as the $(n-1)$th moment of a measure, and also as the trace of the $(n-1)$th iterate of a linear operator. Applications of these sequences, and hence of the explicit solution provided, are found in quantum field theory as the number of Feynman diagrams of a certain type and order, in Brownian motion theory, and in combinatorics

Determination of transmission patterns in multichannel data

The power of today's laboratory equipment allows more and more data channels to be easily recorded. However, some misunderstandings about processing such multivariate data may still be found in the literature. The typical mistake is to treat a multichannel system as comprising pairs of channels; this approach does not use all the collected information about the investigated system, and may lead to erroneous conclusions. In this paper, the differences between single- and multichannel approaches will be briefly summarized and some examples of problems will be described

2D-spectral estimation based on DCT and modified magnitude group delay

This paper proposes two new 2D-spectral estimation methods. The 2D-modified magnitude group delay (MMGD) is applied to 2D-discrete Fourier transform (2DDFT) for the first and to the analytic 2D-discrete Cosine transform for the second. The analytic 2D-DCT preserves the desirable properties of the DCT (like, improved frequency resolution, leakage and detectability) and is realized by a 2D-discrete cosine transform (2D-DCT) and its Hilbert transform. The 2D-MMGD is an extension from 1D to 2D, and it reduces the variance preserving the original frequency resolution of 2D-DFT or 2D-analytic DCT, depending upon to which is applied. The first and the second methods are referred to as DFT-MMGD and DCT-MMGD, respectively. The proposed methods are applied to 2D sinusoids and 2D AR process, associated with Gaussian white noise. The performance of the DCT-MMGD is found to be superior to that of DFT-MMGD in terms of variance, frequency resolution and detectability. The performance of DFT-MMGD and DCT-MMGD is better than that of 2D-LP method even when the signal to noise ratio is low