Characterization of the peak value behavior of the Hilbert transform of bounded bandlimited signals

The peak value of a signal is a characteristic that has to be controlled in many applications. In this paper we analyze the peak value of the Hilbert transform for the space B[∞ over π] of bounded bandlimited signals. It is known that for this space the Hilbert transform cannot be calculated by the common principal value integral, because there are signals for which it diverges everywhere. Although the classical definition fails for B[∞ over π] , there is a more general definition of the Hilbert transform, which is based on the abstract H [superscript 1]-BMO(ℝ) duality. It was recently shown in [1] that, in addition to this abstract definition, there exists an explicit formula for calculating the Hilbert transform. Based on this formula we study properties of the Hilbert transform for the space B[∞ over π] of bounded bandlimited signals. We analyze its asymptotic growth behavior, and thereby solve the peak value problem of the Hilbert transform for this space. Further, we obtain results for the growth behavior of the Hilbert transform for the subspace B[∞ over π,0] of bounded bandlimited signals that vanish at infinity. By studying the properties of the Hilbert transform, we continue the work [2]

Correlation background of the concept of active spectrum of nonstationary processes

The need of developing the known concept of active spectrum of nonstationary processes has been shown using its correlation background. Expressions for the input and response signals in complex form of the so-called active filter were derived in terms of the second known definition of this concept. A two-dimensional correlation function similar to the known uncertainty function was obtained using the known autocorrelation function of complex signals. In conclusion, some directions of the beneficial use of the correlation relationships obtained were noted, especially in solving the topical synthesis problem of optimal active filters for the specified signals