Positive time-frequency distributions via quadratic programming (Q1273792)

As is known, the Fourier Transform approach does not provide a good solution for nonstationary stochastic signals. A possible approach to such cases is provided by the so-called time-frequency analysis. There exist many ways to achieve it. One of them consists of a bilinear time-frequency representation, which needs an explicit description of a kernel in two variables (it is called distribution). The most widely known distribution is the Wigner distribution, which, however, has some drawbacks. To overcome them, a new object, the positive time-frequency distribution (TFD), is introduced. Simply speaking, a TFD is a time-varying spectrum \(P(t,\omega)\) which obeys the following: a) it is nonnegative; b) its integral with respect to \(\omega\) provides the square of the signal; c) its integral with respect to \(t\) provides the square of the modulus of the Fourier transform of the signal. The paper presents a procedure to obtain the TFD by the use of quadratic programming, i.e. a minimization with respect to an adequate template and two marginal constraints. The paper explains only the main ideas; no example is given.