The fractional Fourier transform (FrFT) [1, 2, 3, 4] is an efective mathematical tool that has been applied in quantum mechanics, neural networks, diferential equations, optics, radar, sonar, and other communication systems [5, 6, 7, 8, 9]. It can be viewed as an expansion of the Fourier transform (FT). In the recent work, the authors of [10, 11] have proposed an extended form of the FrFT to the coupled fractional Fourier transform (CFrFT). The utility of the CFrFT kernel in constructing the other general transformations, such as the short-time coupled fractional Fourier transform and the coupled fractional Wigner-Ville distribution were discussed in [12, 13, 14, 15]. Especially, the authors of [10] studied in detail various uncertainty inequalities associated with the CFrFT. However, some properties of the CFrFT like convolution and correlation theorems have not yet studied.

Therefore, in the present research we first establish some basic results pertaining to the CFrFT, which are missing in the existing literature. We then study convolution and correlation operators related to the generalized transformation. Finally, we also explore an uncertainty inequality associated with the proposed CFrFT.

The paper is outlined as follows. In Section 2 we recall the definition of the FrFT and some existing results related to the transformation. The definition of the CFrFT is also introduced in this section. Section 3 is devoted to definition and properties of the CFrFT. We also present convolution and correlation theorems related to the CFrFT. Section 4 deals with the derivation of an uncertainty principles associated with the CFrFT. Lastly, Section 5 contains a brief conclusion of the work.

First of all, we introduce the basic results related to the FrFT and the CFrFT. We begin with the following definition.

Definition 2.1.The space is the set of measurable functions on satisfying

Here . Also for every with if and only if .

In particular, when we define -norm

and if f is continuous, then

The inner product of is defined by

Definition 2.2.Let function . The 2-D FrFT with parameter ϑ is defined by

where is given by

Here, and

It is not dificult to check that the FrFT kernel fulfills the following basic properties:

and

where is the conjugation of .

Definition 2.3.Let and . The inverse formula of the 2-D FrFT for the function is given by

Formula (8) tells us how to generate the original function from its FrFT. Next, we introduce a definition of the CFrFT.

Definition 2.4.The CFrFT of any function belongs to is defined as [10, 11]

where

In this case,

with and .

Relation (9) may be expressed in the form

Denoting

we easily get

Now formula (11) may be represented as

which explains the direct interaction between the CFrFT and the 2-D Fourier transformation. Here denotes the 2-D Fourier transform of given by (see [16, 17, 18])

where

The original function can be obtained in terms of the CFrFT using the definition as below.

Definition 2.5.For any and f in , the inverse of the CFrFT is calculated by

As a particular case, when , then one obtains

Substituting this into (9) yields

In this case, represents the 2-D FrFT described by identity (5). It is not dificult to verify that for , the CFrFT becomes the 2-D FT defined in relation (15).

The next result will be needed to derive the uncertainty principle involving the CFrFT in the next section.

Theorem 2.6 (Parseval’s formula). For all , the following relation holds true:

and

Proof. With the help of Parseval’s relation concerning the 2-D Fourier transform, we find that

Applying equation (14) to relation (20) results in

and the proof is complete.

To verify the use of the CFrFT properties, we give an illustration with a simple example.

Example 1.Consider the Gaussian function of the form

Thus, with the help of equation (9), we get

Further, we obtain

Hence,

as shown in Figure 1.

In this part, we investigate in detail several properties of the CFrFT. The results are modifications of the corresponding properties of the 2-D FrFT.

Theorem 3.1 (Linearity). Let two functions with , then we have

Proof. Due to the definition of the CFrFT (9), we get

The above equation can be rewritten in the form

Thus, the theorem was to be proved.

Theorem 3.2 (Shifting). Let f belongs to and , then we have

where

Proof. With the help of equation (9), we immediately obtain

Substituting in the above identity results in

and the proof is complete.

Theorem 3.3 (Modulation). Assume that . Let , then we have

Proof. It follows from equation (9) that

This finishes the proof of the theorem.

Definition 3.4.For any functions , the convolution operator for the is defined by

As an easy consequence we get the following theorem.

Theorem 3.5 (CFrFT Convolution). Let , one has

Proof. In fact, we have

The change of variables , yields

This equation implies that

This ends the proof of the theorem.

Definition 3.6.For any functions . The correlation operator for the CFrFT is defined by

Theorem 3.7. Let , we have

Proof. Applying (35) results in

Setting , we find that

which completes the proof.

Recently, the authors of [13, 14, 19, 20] have proposed several uncertainty principles related to various transformations. In this part, we establish an uncertainty principle related to the CFrFT. The following theorem will be useful for deriving the main result in this section.

Theorem 4.1. Let , one has[10]

The extension of equation (37) mentioned above is given by the following result.

Theorem 4.2.Let with , the following inequality is satisfied:

Proof. It follows from Holder’s inequality for the CFrFT that

Applying Plancherel’s formula (19) to equation (39), we get

Hence, assuming that ,

In a similar way, we find that

By combining equations (41) and (42), we easily obtain

Applying equation (37) in Theorem 4.1 results in

This equation yields

We finally arrive at

which completes the proof.

In this research paper, we have demonstrated several properties of the CFrFT including convolution and correlation theorems. We also established a generalized uncertainty inequality related to the proposed CFrFT. All these results are nontrivial extensions of the 2-D FT and 2-D FrFT properties.