Identification of Unknown Filter in a Half-Strip

We consider the analogue of the classic Wiener filtering theory to a half-strip of complex domain. In this paper all test signal obtained that solving the filter identification problem for the Hardy spaces in a half-strip. This result can be used for the investigations of electrical, optical, acoustical signals.


Introduction
In Wiener's mathematical filtering theory [1] a signal is a function of the continuous (t ∈ γ , γ : R → C) time parameter t and a filter Φ is a device ("a box") transforming an input signal into a certain output signal, g → Φg. The energy of a signal g is proportional to γ g(z) 2 |dz|.
Without entering into the physical nature (optical, electrical) of a stationary filter Φ, we consider it as a translation invariant linear operator on the corresponding L 2 space. The function G = F −1 g, where g → F −1 g is the inverse Fourier transforms, represents what is called the amplitude spectrum of g. B K. Huk khrystyna.huk2711@gmail.com V. Dilnyi dilnyiv@gmail.com 1 The following are among the major problem of signal processing: determine an unknown filter Φ : g → ψ ("black box") from an analysis of g and ψ ; in particular, reconstruct, if possible, a filter knowing the energy densities |F −1 g| 2 , |F −1 f | 2 of an input-output pair.
The last problem is a variation of M. Kac' famous question "Can one hear the shape of a drum?" [2]. Above and similar problems are studied by N. Wiener, D. Newman, B. Nyman, A. Beurling, H. Reinhard, P. Masani [3].

The Half-Strip Case
The complex domain provides a natural processing framework for signals with intensity and direction components (see [4][5][6]).
We consider the above problem for the case of an unknown filter f on the half-strip D σ = {z : | z| < σ, z < 0}, σ > 0. The aim of this paper is to construct all detecting signals g on D * σ = C\D σ under some natural conditions. The circle of ideas surrounding signal processing and Fourier transform has a long history, and has found a number applications. J. Martinez, R. Heusdens and R. Hendriks [7] investigate the characteristics of the signal and the connections of the generalized Fourier transform to analyticity. E. Sejdic, I. Djurovic and L. Stankocic relate the Fractional Fourier transform to other mathematical transforms and discuss various approaches for practical realizations of this transform [8].
Let E p [D σ ] and E p [D * σ ], 1 ≤ p < +∞, σ > 0, be the (Hardy) spaces of holomorphic functions respectively in the domains D σ and D * σ , for which where supremum is taken over all segments μ, that are contained in D σ and D * σ respectively. The amplitude spectrum of a signal g ∈ E p [D * σ ] is defined by the formula Obviously, the above equality is a natural analogue of the Fourier transform for the half-strip.
The filter identification problem for the half-strip is to find, if possible, a test signal Let H p (C + ), 1 ≤ p < +∞, be the Hardy space of holomorphic in C + = {z : z > 0} functions f , for which Theorem S [9] The space H p (C + ), 1 ≤ p < +∞, coincides with the space of holomorphic in C + functions f , for which Let H p σ (C + ) be a space of holomorphic in C + functions, for which +∞ 0 g re iϕ p e −pσ r| sin ϕ| dr holds.

The Main Result
implies f ≡ 0 if and only if one of the following conditions holds: a) g admits a holomorphic continuation as an entire function and b) g does not admit an analytic continuation as an entire function.
Proof The integral boundary function h : R → R of a function G ∈ H p σ (C + ) is defined [10] up to an additive constant at points of continuity by the equality [10] Since G is continuous on C + and G(z) = 0 for all z ∈ C + , we have | ln |G(z)|| ≤ M R for z ∈ {z : z ≥ 0, |z| ≤ R}. Therefore by Fatou's lemma h ≡ const. Since G is zero-free in C + , by the criterion of solvability [11,12] equation (3) has only trivial (zero) solution iff We will show that (5) implies condition a) or b). Consider the two alternatives: From the last formula we obtain that the integral on the right hand side of (2) converges uniformly on any compact subset of C + , hence g is an entire function.
From a Phragmen-Lindelof type theorem for the half-plane [13], we obtain ψ ∈ H 2 (C + ). It means and This is a contradiction. The second alternative is Let g ∈ E 2 * [D σ ] be an entire function. Then g is an analytic function in each closed rectangle M k , k < 0, where M k = {z : z ∈ D σ , z > k}. By the Cauchy formula we obtain ∂M k g(w)e zw dw = 0, k <0, then by (7) we obtain for x > 0. Then, by Schwarz inequality analogously x .
Further  (6) is an arbitrary negative number, we can This is a contradiction. Conversely, b) suppose, condition (9) is not valid. Then by (6) the integral on the right hand side of (2) converges uniformly on any compact subset of C + , hence g is the entire function. This is a contradiction. Condition (5) is a simple consequence of (9).
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