Elsevier

Signal Processing

Volume 93, Issue 5, May 2013, Pages 1027-1043
Signal Processing

Heisenberg's uncertainty principles for the 2-D nonseparable linear canonical transforms

https://doi.org/10.1016/j.sigpro.2012.11.023Get rights and content

Abstract

The uncertainty principles of the 1-D Fourier transform (FT), the 1-D fractional Fourier transform (FRFT), and the 1-D linear canonical transform (LCT) have already been derived. In this paper, we extend the previous works and derive the uncertainty principles for the two-dimensional nonseparable linear canonical transform (2-D NSLCT), including the complex input case, the real input case, and the case where det(B)=0 where B is a parameter subset of the 2-D NSLCT. Since the 2-D NSLCT is a generalization of many operations, with the derived uncertainty principles, the uncertain principles of many 2-D operations, such as the 2-D Fresnel transform, the 2-D FT, the 2-D FRFT, the 2-D LCT, and the 2-D gyrator transform, can also be found. Moreover, we find that the rotation, scaling, and chirp multiplication of the 2-D Gaussian function can minimize the product of the variances in the space and the transform domains.

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 isOF(A,B,C,D)[g(x,y)]=G(A,B,C,D)(u,v)=(2πdet(B))1exp[j(k1u2+k2uv+k3v2)/2det(B)]exp[j((b22u+b12v)x+(b21ub11v)y)/det(B)]exp[j(p1x2+p2xy+p3y2)/2det(B)]g(x,y)dxdywhendet(B)0where A, B, C, D are all 2×2 matrices, which represent the 16 parameters of the 2-D NSLCT:A=[a11a12a21a22],B=[b11b12b21b22],C=[c11c12c21c22],D=[d11d12d21d22]

The values of k1, k2, k3, p1, p2, and p3 in (1) arek1=d11b22d12b21,k2=2(d11b12+d12b11),k3=d21b12+d22b11,p1=a11b22a21b12,p2=2(a12b22a22b12),p3=a12b21+a22b11.

Moreover, the following constraints should be satisfied:ATC=CTA,BTD=DTB,ATDCTB=Ii.e.,a11c12+a21c22=a12c11+a22c21,b11d12+b21d22=b12d11+b22d21,a11d11+a21d21(b11c11+b21c21)=1,a12d12+a22d22(b12c12+b22c22)=1,a11d12+a21d22=c11b12+c21b22,a12d11+a22d21=c12b11+c22b21.

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:OF(A1,B1,C1,D1){OF(A,B,C,D)[g(x,y)]}=OF(A2,B2,C2,D2)[g(x,y)]where[A2B2C2D2]=[A1B1C1D1][ABCD]

The 2-D NSLCT is a generalization of several well-known transform operations. For example, whenb11=b22=1,c11=c22=1,others=0,the 2-D NSLCT becomes the 2-D Fourier transform (2-D FT).G(u,v)=FT[g(x,y)]=(j2π)1g(x,y)exp[j(ux+vy)]dxdy.

Whena12=a21=b12=b21=c12=c21=d12=d21=0,a11=d11=cosα,b11=c11=sinα,a22=d22=cosβ,b22=c22=sinβ,the 2-D NSLCT becomes the 2-D fractional Fourier transform (2-D FRFT) [5], [6]:Ga,b(u,v)=(2p)1(1cotα)(1cotβ)exp[j(u2cotα+v2cotβ)/2]×exp[j(uxcscα+vycscβ)]exp[j(x2cotα+y2cotβ)/2]g(x,y)dxdy.

Whena12=a21=b12=b21=c12=c21=d12=d21=0,a11=a,a22=a1,b11=b,b22=b1,c11=c,c22=c1,d11=d,d22=d1,the 2-D NSLCT becomes the 2-D separable linear canonical transform (2-D LCT) [7]:G(a,b,c,d,a1,b1,c1,d1)(u,v)=(2πbb1)1exp(jdu2/2b+jd1v2/2b1)×exp(jux/b+vy/b1)exp(jax2/2b+ja1y2/2b1)g(x,y)dxdy.

Whena12=a21=b11=b22=c11=c22=d12=d21=0,a11=d11=a22=d22=cosα,b12=b21=c21=c12=sinα,the 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 C0, 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):X(ω)=(2π)1/2x(t)exp(jωt)dt,and the variances of x(t) in the time and the frequency domains areΔt2=(tt0)2|x(t)|2dt/|x(t)|2dt,Δω2=(ωω0)2|X(ω)|2dω/|X(ω)|2dωwheret0=t|x(t)|2dt/|x(t)|2dt,ω0=ω|X(ω)|2dω/|X(ω)|2dω,then the following inequality is satisfied:Δt2Δω21/4.

Moreover, in the case where x(t) is a Gaussian function:x(t)=Kexp[(tt0)2/2]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 Δx,y2Δu,v2 whereΔx,y2=((xx0)2+(yy0)2)|g(x,y)|2dxdy/|g(x,y)|2dxdy,Δu,v2=((uu0)2+(vv0)2)|G(A,B,C,D)(u,v)|2dudv/|G(A,B,C,D)(u,v)|2dudv,x0=x|g(x,y)|2dxdy/|g(x,y)|2dxdy,y0=y|g(x,y)|2dxdy/|g

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 G(A1,B1,C1,D1)(u,v) 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

Δw,s2=((ww0)2+(ss0)2)|G(A1,B1,C1,D1)(w,s)|2dwd

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] becomesOF(A,B,C,D)[g(x,y)]=G(A,B,C,D)(u,v)=det(D)g(d11u+d21v,d12u+d22v)×exp{j[(c11d

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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