We obtain the approximate analytic time-frequency spectrum of the output of a dynamical system when the input is an arbitrary finite-energy nonstationary signal. Our method is based on three steps. First, we transform the dynamical system to the time-frequency domain. Second, we approximate the time-frequency spectrum of the input as a sum of short duration sinusoids through a Fourier series expansion. Finally, we combine the time-frequency outputs corresponding to each individual short duration sinusoid, which are known in exact analytic form. An example shows that the proposed method requires a few terms only to obtain an approximate time-frequency output which is indistinguishable from the exact one. Furthermore, our method can clarify how dynamical systems process nonstationary signals. This processing mechanism is of fundamental interest since dynamical systems are a common model for real-world signals.

Periodicities in a DNA sequence contain information about its function and structure. Frequency analysis can be used to reveal such periodicities. A definition of spectrum for DNA sequences is proposed, based on an entropy minimization criterion. It is shown that the properties of the proposed spectrum are those expected from the spectrum of a symbolic sequence. Applications to simulated and experimental DNA sequences are presented and compared against classical techniques to prove the effectiveness of the proposed method.

This paper presents the reply of the authors to the "Comments on dasiathe generalized Wiener process for colored noise" by P. Loughlin (see ibid., vol.14, no.11, 2007). The original paper was by the authors (L. Galleani and L. Cohen, see ibid., vol.13, no.10, p.608-11, 2006).

We define the generalized Wiener process as the output of a first-order differential equation when the input is an arbitrary stochastic input. This is in contrast to the standard Wiener process, where the input is white Gaussian noise. We obtain a simple explicit result for any input wide sense stationary random process, namely, that the Wigner spectrum of the output random process is the product of the power spectrum of the input process times a simple universal function of time and frequency. We also obtain the impulse response function wherein the output of the generalized process is expressed as the impulse response function integrated with the input random process. Various limiting values are derived. We apply the method to the case where the driving stochastic process has an exponentially decaying autocorrelation function

We present a new method to study linear systems that exhibit time-varying characteristics and are governed by ordinary differential equations. The method transforms the differential equation into the governing equation for the ambiguity function and to the time-frequency domain. The advantage of the method is that in the new domain an immense simplification occurs, both mathematically and physically. The method applies to both deterministic and stochastic differential equations. The approach is applied to the classical RC filter driven by white noise. We obtain a new characterization of the output spectra.

A new method is described to study dynamical systems characterized by linear ordinary differential equations. The method is aimed at studying the time-varying properties of the resulting solution of the differential equation. In contrast to the standard methods where one solves the differential equation and then uses a time-frequency distribution, for example the Wigner distribution, to ascertain the time-frequency properties of the solution we show that one can obtain a differential equation for the Wigner distribution of the solution. We discuss a number of advantages for doing so.