A Gaussian regularization for derivative sampling interpolation of signals in the linear canonical transform representations

The linear canonical transform (LCT) plays an important role in signal and image processing from both theoretical and practical points of view. Various sampling representations for band-limited and non-band-limited signals in the LCT domain have been established. We focus in this paper on the derivative sampling reconstruction, where the reconstruction procedure utilizes samples of both the signal and its first derivative. Our major aim was to incorporate the reconstruction sampling operator with a Gaussian regularization kernel, which on the one hand is applicable for not necessarily band-limited signals and on the other hand hastens the convergence of the reconstruction procedure. The amplitude error is also considered with deriving rigorous estimates. The obtained theoretical results are tested through various simulated experiments.


Introduction
The linear canonical transform (LCT) is a three-parameter transform where a, b, d are fixed real constants and b = 0. While the case b = 0, cf. [7,11,13] can be also considered with a four-parameter transform, it is not of interest in this work as it is nothing but a chirp multiplication. Integration (1) converges for L p (R)-functions, p ≥ 1. For simplicity, we replace L a,b,d by L and use the subscripts only when it is necessary. The LCT turns out to be the fractional Fourier transform (FrFT) when a = d = cos α and b = sin α; to the Fourier transform when a = d = 0, b = 1; and to the Fresnel transform when a = 1, b ∈ R, b = 0, d = 0, see [16,26,30,31] for more details. For this reason, the LCT has become a basic tool in signal and image processing, cf., e.g., [7]. For the derivations of sampling theorems of Shannon type in the LCT domain, the space of band-limited signals in the LCT and FrFT domains is precisely investigated in [8,14,28,30]. Sampling theorem of Shannon type has been derived extensively, see, e.g., [1,3,5,6,10,11,15,19,23,24,29], using different approaches. The associated truncation, amplitude, and jitter errors are analytically considered in [2,3,9,22,25].
The space of band-limited signals in the LCT domain is defined as follows. Let > 0 be fixed. A signal f is said to be band-limited with band width if f ∈ L 2 (R) and (L f )(t) = 0, |t| > . This space is denoted by B 2 . Thus, f ∈ B 2 if and only if there is a bandlimited signal in the classical sense with band width /b, φ, for which f (t) = e i( a 2b )t 2 φ(t). The derivative sampling theorem for f ∈ B 2 is given in [11], see also [12], by where h ∈ 0, 2π b is fixed and The convergence of (2) is absolute on C and uniform on compact subsets of C. See also [21,27] for derivative sampling theorem in the FrFT domain.
The rate of convergence of (2) is slow, unless f decays fast. Our purpose in this work is to incorporate (2) with a Gaussian regularization kernel that hastens the reconstruction rate. In addition, we do not necessarily assume the reconstructed signals to be band-limited and/or to have a finite energy, i.e., square integrable. The regularized sampling theorem is derived in the next section. Section 3 is devoted to investigating the associated amplitude error, which results from using approximate samples in the reconstruction procedure. In Sect. 4, we carry out several numerical experiments.

Regularized Gaussian Sampling of Derivative Representation
For fixed x ∈ R, N ∈ N, define the integer-type interval where · is the floor function, see Fig. 1.
be also fixed. As we have indicated in the previous section, the reconstruction (convergence) rate of (2) is slow, unless f (t) decays fast. We incorporate (2) with the Gaussian function G(t) = e −t 2 , t ∈ C, which decays fast as |t| → ∞. Indeed, define the regularized Gaussian sampling of derivative representation operator for f : C → C to be The major result of this section is the following theorem, which gives an estimate of the error associated with (5). where locally uniformly on R.

Now, we estimate the integration
and c 1 , c 2 , c 3 , and c 4 are the line segments AB, BC, C D, and D A, respectively, cf. Fig. 2. Let us estimate these four integrals. For I 1 , let ζ = N 0 + m 0 + iη ∈ c 1 . Then, To estimate I 1 , we estimate the integrand over c 1 . Inequality (6) implies We also have Using (17)- (20), we obtain Using the fact t − 1 < t ≤ t, t ∈ R implies Substituting from (22) and (23) in (21) yields Using the estimate obtained by Schmeisser and Stenger for the last integral, cf. [20, p. 205], I 1 is estimated via In a similar manner, we estimate I 3 to have Then, I 2 turns out to Inequality (6) leads to Moreover, we have Hence, The inequalities and imply Likewise, Combining (25), (26), (34), and (35) yields where The proof is completed as λ N (·) has the asymptotic (8) locally uniformly on R.

Remark 1
The following special cases can be directly deduced from Theorem 1: (i) Let f be entire, and there be M, κ ≥ 0 such that For h ∈ 0, π b + 2κ , and |y| < N , estimate (7) becomes The proof is based on the fact that we can take then Theorem 1 implies (iii) If f ∈ B 2 , then we have, cf. [18, p. 319], f ∞ ≤ √ /πb f 2 and consequently

Numerical experiments
This section includes two examples illustrating the above method. In the first example, we compare the results obtained by Gaussian regularization of derivative sampling in the LCT domain, which is investigated in this paper, with the derivative sampling theorem in the LCT domain f D  N (t). Here, f D N (t) is the truncated reconstruction formula of (2), i.e., The other example is devoted to the comparison between the absolute error f (t) − H h,N f (t) and its associated bound. Let B(H h,N ; t) and A(ε, H h,N ; t) be the error bounds of (41) and (44), respectively. For t ∈ R, N ∈ Z + , we have Since H h,N f (t) duplicates f (t) at the points kh for k ∈ Z, it looks reasonable to study the absolute errors at the intermediate points where a = b = 1/ √ 2, h = √ 2, and = π/2. Table 1 and Figs. 3-4 exhibit comparisons between the approximations of f using f D N (·) and H h,N f (·). In Fig. 3 we restrict ourselves to the case when N = 2, t ∈ (−4, 4) is real, while in Fig. 4 we illustrate the case when |t| < 2, t ∈ C, N = 10. It is noted from Table 1 that while the obtained error estimates are topping the absolute (exact) error, they are pretty close to the exact error. Moreover, the regularized Gaussian sampling of derivative representation is giving a more accurate reconstruction of signals than the derivative sampling theorem. Furthermore, as N increases, the gap between the exact error | f (·) − H h,N f (·)| and its bound narrows noticeably.

Example 2 Consider the B ∞
π -function   Here, a = 1/2, b = √ 3/2, = π . Tables 2, 3, and 4 demonstrate the absolute error f (·) − H h,N f (·) and its associated bound B(H h,N ; ·) + A(ε, H h,N ; ·), where ε = 10 −7 , N = 5, 10, and h = 1, 1/2, 1/4, respectively. We notice from Tables 2, 3, and 4 that, as predicted by the theory, the number of correct digits increases when N doubles. Moreover, the precision increases when N is fixed but h decreases, as expected in the over sampling case. Furthermore, the error bounds are quite realistic; that is, they do not overestimate the absolute error very much. Graphs of the real and imaginary parts of f (t) and H 1 2 ,10f (t) on the interval

Conclusion
This paper regularizes the derivative sampling theorem for signals in the LCT domain through incorporating the reconstruction sampling representation with a carefully scaled    Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/.