Heisenberg's uncertainty principles for the 2-D nonseparable linear canonical transforms
graphical abstract
Highlights
► The Heisenberg Uncertainty Principles for the 2-D NSLCT are derived. ► The derived principles can be applied to 2-D FRFTs, 2-D LCTs, and gyrator transforms. ► The principles for the real input case and the case where det(B)=0 are also derived. ► The functions that can achieve the lower bound of the inequality are also found.
Introduction
The two-dimensional nonseparable linear canonical transform (2-D NSLCT) [1], [2], [3], [4] is the 2-D counterpart of the 1-D linear canonical transform. If we use OF(A,B,C,D) to denote the operation of the 2-D NSLCT and use G(A,B,C,D)(u, v) to denote its output, then the definition of the 2-D NSLCT iswhere A, B, C, D are all 2×2 matrices, which represent the 16 parameters of the 2-D NSLCT:
The values of k1, k2, k3, p1, p2, and p3 in (1) are
Moreover, the following constraints should be satisfied:
Therefore, the 2-D NSLCT has 16 parameters (from (2)) and 6 constraints (from (3)). Its degree of freedom is 10. The 2-D NSLCT has the additivity property as follows:where
The 2-D NSLCT is a generalization of several well-known transform operations. For example, whenthe 2-D NSLCT becomes the 2-D Fourier transform (2-D FT).
Whenthe 2-D NSLCT becomes the 2-D fractional Fourier transform (2-D FRFT) [5], [6]:
Whenthe 2-D NSLCT becomes the 2-D separable linear canonical transform (2-D LCT) [7]:
Whenthe 2-D NSLCT becomes the gyrator transform [8]. When A=D=I, C=0, and B is a diagonal matrix, the 2-D NSLCT becomes the Fresnel transform [9], which can describe the propagation of electromagnetic waves in the free space. When B=C=0, the 2-D NSLCT becomes the affine transformation operation in the space domain. When B=0 but C≠0, the 2-D NSLCT becomes the combination of chirp multiplication and the affine transformation operation.
The 2-D NSLCT is very flexible and is a generalization of many well-known operations. As the FT and the FRFT, the 2-D NSLCT is also useful for signal processing applications, such as filter design, pattern recognition, optics, and analyzing the propagation of electromagnetic waves [1], [2], [3], [4].
In this paper, we derive Heisenberg's uncertainty principles for the 2-D NSLCT. The Heisenberg uncertainty principles of the 1-D FT, the 1-D FRFT, and the 1-D LCT have already been derived in the literature (reviewed in Section 2). In this paper, we must generalize these results and derive the uncertainty principles for the 2-D NSLCT (see Section 3, Theorem 2, and Corollary 1). We find that the scaling, rotation, and chirp multiplication of the Gaussian function will satisfy the lower bound of the uncertainty inequality of the 2-D NSLCT (see Section 4). Moreover, since the 2-D FRFT, the 2-D LCT, the 2-D Fresnel transform, and the gyrator transform are all the special cases of the 2-D NSLCT, using the uncertainty principle of the 2-D NSLCT, the uncertainty principles of these 2-D operations can also be obtained (see Section 5). Furthermore, we also derive the uncertainty principle of the 2-D NSLCT for the case where det(B)=0 (see Section 6) and the case where the input is a real function (see Section 7). In Section 8, we make a conclusion.
Section snippets
Preliminary
The well-known Heisenberg uncertainty principle [10], [11] is that, if X(ω) is the 1-D FT of x(t):and the variances of x(t) in the time and the frequency domains arewherethen the following inequality is satisfied:
Moreover, in the case where x(t) is a Gaussian function:where K and t0 are
Heisenberg's uncertainty principle of the 2-D NSLCT
The formula of the 2-D NSLCT has 16 parameters. It seems very complicated. However, with some simplification techniques, the Heisenberg uncertainty principle of the 2-D NSLCT can still be derived successfully.
We will try to find the lower bound of where
Some theorems related to Heisenberg's uncertainty principle and examples
In Section 3, we successfully derived the Heisenberg uncertainty principle for the 2-D NSLCT. In this section, we further extend the principle and derive some theorems that are closely related to the inequality in (72). Theorem 3 Suppose that and G(A,B,C,D) (u, v) are the 2-D NSLCTs of g(x, y) with parameters {A1, B1, C1, D1} and {A, B, C, D}, respectively. If Δ2u,v is the variance of G(A,B,C,D) (u, v) (defined the same as in (26)) and
Some important special cases
Since the 2-D NSLCT is the generalization of many 2-D operations, we can use (72) to find their Heisenberg uncertainty principles. We use Table 1 to summarize the Heisenberg uncertainty principles of these 2-D operations and the functions that can minimize the product of Δ2x,y and Δ2u,v. These uncertainty principles can be proved by substituting the relations among the 2-D NSLCT and these 2-D operations (see Section 1) into (72) and (76). Note that the functions in the 3rdcolumn of Table 1 can
Uncertainty principles for the case where det(B)=0
Note that the Heisenberg uncertainty principle in Section 3 was derived based on the formula of the 2-D NSLCT in (1). However, this formula is valid only for the case where det(B)≠0. In this section, we discuss how the Heisenberg uncertainty principle changes for the case where det(B)=0. The case can be further divided into two sub-cases: (1) B=0, (2) B≠0 but det(B)=0.
When B=0, the definition of the 2-D NSLCT [1] becomes
Heisenberg's uncertainty principles for real signals
In [12], [13], [14], [17], [18], they showed that when the input function is real, the Heisenberg uncertainty principle of the 1-D LCT is different from that of the case where the input function is complex. Similarly, it is reasonable to think that when the input function is real, the Heisenberg uncertainty principle of the 2-D NSLCT should be somewhat different from that in Theorem 2.
Suppose that the input function g(x, y) is real. For simplifying the derivation, we first assume that x0, y0, u0
Conclusions
The Heisenberg uncertainty principle of the 2-D NSLCT is derived in this paper (see Theorem 2). Although the 2-D NSLCT has 16 parameters and is very complicated, with several converting techniques, its Heisenberg uncertainty principle can be derived successfully. We also showed that the lower bound can be achieved by the 2-D Gaussian function with affine transformation and chirp multiplication (see Theorem 4). Moreover, since the 2-D FT, the 2-D FRFT, the 2-D LCT, the 2-D Fresnel transform, and
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