Parseval Relationship of Samples in the Fractional Fourier Transform Domain

Abstract

This paper investigates the Parseval relationship of samples associated with the fractional Fourier transform. Firstly, the Parseval relationship for uniform samples of band-limited signal is obtained. Then, the relationship is extended to a general set of nonuniform samples of band-limited signal associated with the fractional Fourier transform. Finally, the two dimensional case is investigated in detail, it is also shown that the derived results can be regarded as the generalization of the classical ones in the Fourier domain to the fractional Fourier transform domain.

1. Introduction

As a generalization of the classical Fourier transform, the fractional Fourier transform (FrFT) has received much attention in recent years [1–5]. It has been shown that the FrFT can be applied to various applications, including optics, radar and sonar, communication signals and underwater signal processing, and so forth, [1–5]. The relationship between the Fourier transform and the FrFT is derived in [6–8]. The discretization and fast computation of FrFT have been proposed by researchers from different perspectives [9–14]. The generalization of the sampling formulae in the traditional Fourier domain to the FrFT domain has been deduced in [7, 8] and [15, 16]. The properties and advantages of the FrFT in signal processing community have been discussed in [17, 18]. For further properties and applications of FrFT in optics and signal processing community, one can refer to [1, 2].

The well-known operations and relations (such as Hilbert transform [19], convolution and product operations [20, 21], uncertainty principle [22], and Poisson summation formula [23]) in traditional Fourier domain have been extended to the fractional Fourier domain by different authors. The spectral analysis and reconstruction for periodic nonuniform samples is investigated in [24], and the short-time FrFT and its applications are studied in [25]. Recently, Lima and Campello De Souza give the definition and properties of FrFT over finite fields [26], Irarrazaval et al. investigates the application of the FrFT in quadratic field magnetic resonance imaging [27]. The relationship between the FrFT and the fractional calculus operators is studied and given in [28]. But, so far none of the research papers throw light on the extension of the traditional Parseval's relationship for band-limited signals associated with the fractional Fourier domain. It is, therefore, worthwhile and interesting to investigate the extension of the Parseval's relationship of band-limited signals in the FrFT domain.

Parseval relationship plays an important role in the Fourier transform domain [29–31], it relates the energy (or power) in the uniformly spaced sample values of a band-limited signal and the energy in the corresponding analog signal. Based on the relationship between the Fourier transform and the FrFT, this paper investigates the generalization of the traditional Parseval relationship of the Fourier domain to the FrFT domain.

The paper is organized as follows: the preliminaries are presented in Section 2, the main results of the paper are obtained in Section 3, and the conclusion and future working directions are given in Section 4.

2. Preliminaries

2.1. The Fractional Fourier Transform

The ordinary Fourier transform plays an important role in modern signal processing community, little need be said of the importance and ubiquity of the ordinary Fourier transform, and frequency domain concepts and techniques in many diverse areas of science and engineering. The Fourier transform of a signal 𝑓(𝑡) is defined as 𝐹(𝑢)=1√2𝜋^+∞−∞𝑓(𝑡)𝑒^{−𝑗𝑢𝑡}𝑑𝑡.(2.1)

The FrFT can be viewed as the generalization of the Fourier transform with an order parameter 𝛼, and the FrFT of a signal 𝑓(𝑡) is given by [1, 2] as 𝐹_𝛼(𝑢)=⎧⎪⎨⎪⎩1−𝑗cot𝛼2𝜋^+∞−∞𝑓𝑓((𝑢−)𝑢,),𝑓(𝑡)𝐾_𝛼(𝑢,𝑡)𝑑𝑡,𝛼𝛼𝛼≠==𝑛2(𝜋2𝑛𝑛,𝜋+,1)𝜋,(2.2) where 𝐾_𝛼(𝑢,𝑡)=exp{𝑗(1/2)[cot𝛼𝑡²−2csc𝛼𝑡𝑢+cot𝛼𝑢²]}. The original signal 𝑓(𝑡) can be derived by the inverse FrFT transform of 𝐹_𝛼 as 𝑓(𝑡)=⎧⎪⎨⎪⎩1+𝑗cot𝛼2𝜋^+∞−∞𝐹𝐹_𝛼(𝑢)𝐾_−𝛼(𝑡,𝑢)𝑑𝑢,𝛼≠𝑛𝜋,_𝛼𝐹(𝑡),𝛼=2𝑛𝜋,_𝛼(−𝑡),𝛼=(2𝑛+1)𝜋.(2.3)

It is easy to show that the FrFT reduces to the ordinary Fourier transform when 𝛼=𝜋/2. In order to obtain new results, this paper deals with the case of 𝛼≠𝑛𝜋.

A signal 𝑓(𝑡) is said to be band-limited with respect to Ω_𝛼 in FrFT domain with order 𝛼, if 𝐹_𝛼(𝑢)=0,|𝑢|>Ω_𝛼.(2.4)

For a signal 𝑓(𝑡) bandlimited in the LCT domain, the following lemma reflects the relationship between the band-limited signals in Fourier domain and the FrFT domain.

Lemma 2.1. Suppose that a signal 𝑓(𝑡) is band-limited with respect to Ω_𝛼 in FrFT domain with order 𝛼, and let𝑔(𝑡)=^Ω𝛼_−Ω𝛼𝐹_𝛼(𝑢)exp−𝑗12cot𝛼𝑢²+𝑗csc𝛼𝑢𝑡𝑑𝑢,(2.5) then the Fourier transform of signal 𝑔(𝑡) can be represented by the FrFT of signal 𝑓(𝑡) as 𝐺(𝑢)=12𝜋𝐹^𝛼(sin𝛼𝑢)𝑒^{−𝑗(1/4)sin2𝛼𝑢2},(2.6) and 𝑔(𝑡) is a |csc𝛼|Ω_𝛼 band-limited signal in the ordinary Fourier transform domain.

Proof. Performing the Fourier transform to (2.5), we obtain that 𝐺(𝑤)=1√2𝜋^+∞−∞=1𝑔(𝑡)𝑒^{−𝑗𝑤𝑡}𝑑𝑡√2𝜋^+∞−∞^Ω𝛼_−Ω𝛼𝐹_𝛼(𝑢)exp−𝑗12cot𝛼𝑢²=1+𝑗csc𝛼𝑢𝑡𝑑𝑢𝑒^{−𝑗𝑤𝑡}𝑑𝑢𝑑𝑡√2𝜋^Ω𝛼−Ω_𝛼𝐹_𝛼(𝑢)exp−𝑗12cot𝛼𝑢²^+∞−∞=1𝑒^{𝑗(csc𝛼𝑢−𝑤)𝑡}𝑑𝑢𝑑𝑡√2𝜋^Ω𝛼−Ω_𝛼𝐹_𝛼(𝑢)exp−𝑗1=12cot𝛼𝑢²√12𝜋𝛿(csc𝛼𝑢−𝑤)𝑑𝑢2𝜋𝐹^𝛼(sin𝛼𝑤)𝑒^{−𝑗(1/4)sin2𝛼𝑤2}.(2.7) This proves the relationship between the Fourier transform of 𝑔(𝑡) and the FrFT of 𝑓(𝑡). Because 𝑓(𝑡) is band-limited with respect to Ω_𝛼 in FrFT domain with order 𝛼, so it is easy to show that signal 𝑔(𝑡) is a |csc𝛼|Ω_𝛼 band-limited in the ordinary Fourier transform domain.

From the definition of signal 𝑓(𝑡) and 𝑔(𝑡), the relationship between the signal 𝑓(𝑡) and 𝑔(𝑡) can be derived as 𝑔(𝑡)=𝑓(𝑡)2𝜋1+𝑗cot𝛼exp𝑗21cot𝛼𝑡².(2.8)

2.2. The Two Dimensional FrFT

In [32], the-two dimensional FrFT of a signal 𝑓(𝑥,𝑦) is defined as 𝐹_𝛼,𝛽(𝑢,𝑣)=^+∞−∞𝑓(𝑥,𝑦)𝐾_𝛼,𝛽(𝑥,𝑦,𝑢,𝑣)𝑑𝑥𝑑𝑦,(2.9)

where the FrFT kernel 𝐾_𝛼,𝛽(𝑥,𝑦,𝑢,𝑣) can be written as 𝐾_𝛼,𝛽(𝑥,𝑦,𝑢,𝑣)=√1−𝑗cot𝛼√1−𝑗cot𝛽2𝜋𝑒^{𝑗((𝑥2+𝑢2)/2)cot𝛼−𝑗𝑢𝑥csc𝛼}𝑒^{𝑗((𝑦2+𝑣2)/2)cot𝛽−𝑗𝑦𝑣csc𝛽}.(2.10)

The original signal 𝑓(𝑥,𝑦) can be recovered by a two-dimensional FrFT with backward angles (−𝛼,−𝛽) as follows: 𝑓(𝑥,𝑦)=^+∞−∞𝐹_𝛼,𝛽(𝑢,𝑣)𝐾_{−𝛼,−𝛽}(𝑢,𝑣,𝑥,𝑦)𝑑𝑢𝑑𝑣.(2.11)

The definition of bandlimited two-dimensional signals can be similarly defined as the one-dimensional signal; following the prove of Lemma 2.1, the two-dimensional cases can be summarized as Lemma 2.2.

Lemma 2.2. Suppose that a signal 𝑓(𝑥,𝑦) is band-limited with respect to (Ω_𝛼,Ω_𝛽) in FrFT domain with order 𝛼 and 𝛽, and let 𝑔(𝑥,𝑦)=^Ω𝛼_−Ω𝛼^Ω𝛽_−Ω𝛽𝐹_𝛼,𝛽(𝑢,𝑣)𝑒^{−𝑗(1/2)cot𝛼𝑢2+𝑗csc𝛼𝑢𝑥}𝑒^{−𝑗(1/2)cot𝛽𝑣2+𝑗csc𝛽𝑣𝑦}𝑑𝑢𝑑𝑣,(2.12) then the Fourier transform of signal 𝑔(𝑥,𝑦) can be represented by the FrFT of signal 𝑓(𝑥,𝑦) as 𝐺(𝑢,𝑣)=1(2𝜋)²𝐹_𝛼(sin𝛼𝑢,sin𝛽𝑣)𝑒^{−𝑗(1/4)sin2𝛼𝑢2}𝑒^{−𝑗(1/4)sin2𝛽𝑣2},(2.13) and 𝑔(𝑥,𝑦) is a (|csc𝛼|Ω_𝛼,|csc𝛽|Ω_𝛽) band-limited signal in the ordinary Fourier transform domain.

Proof. Similar with the proof of Lemma 2.1, the results can derived easily.

From (2.12) and the definition of the two-dimensional FrFT, the relationship between the signal 𝑓(𝑥,𝑦) and 𝑔(𝑥,𝑦) is 𝑔(𝑥,𝑦)=2𝜋1+𝑗cot𝛼2𝜋1+𝑗cot𝛽𝑓(𝑥,𝑦)𝑒^{𝑗(1/2)cot𝛼𝑥2}𝑒^{𝑗(1/2)cot𝛽𝑦2}.(2.14)

2.3. The Parseval Relationship

The Parseval's relation states that the energy in time domain is the same as the energy in frequency domain, which can be expressed as follows [29]: ^+∞−∞𝑓(𝑡)𝑔^∗(𝑡)𝑑𝑡=^+∞−∞𝐹(𝑢)𝐺^∗(𝑢)𝑑𝑢,(2.15) where 𝐹(𝑢) and 𝐺(𝑢) are Fourier transforms of 𝑓(𝑡) and 𝑔(𝑡), respectively. This formula is called Parseval's relation and holds for all members of the Fourier transform family.

The FrFT can be regarded as the generalization of the Fourier transform, and the similar relation of (2.15) in the FrFT sense can be obtained as [1, 2] ^+∞−∞𝑓(𝑡)𝑔^∗(𝑡)𝑑𝑡=^+∞−∞𝐹_𝛼(𝑢)𝐺^∗𝛼(𝑢)𝑑𝑢,(2.16) where 𝐹_𝛼(𝑢) and 𝐺_𝛼(𝑢) are FrFT of 𝑓(𝑡) and 𝑔(𝑡) with order 𝛼, respectively. When 𝑓(𝑡)=𝑔(𝑡), the relation of (2.16) can be written as ^+∞−∞||𝑓(𝑡)||²𝑑𝑡=^+∞−∞||𝐹_𝛼(𝑢)||²𝑑𝑢.(2.17)

Equations (2.15)–(2.17) are the Parseval's relationship between the continuous signal and its fractional Fourier transform (or Fourier transform) and can be derived by the Parseval theorem for 𝐿² signals.

In practical situations, we often encounter the calculation of Parseval relations between the discrete signal and the analog signal. Marvasti and Chuande in [30], and Luthra in [31] investigate the Parseval relations of band-limited signal in the traditional Fourier transform domain. The Parseval relation for band-limited discrete uniformly sampled signal 𝑓(𝑡) in the Fourier domain is [30, 31] ⁺^∞_𝑛=−∞||𝑓(𝑛𝑇)||²=1𝑇^+∞−∞||𝑓(𝑡)||²𝑑𝑡=1𝑇^+𝑊−𝑊||𝐹(𝑢)||²𝑑𝑢,(2.18)

where 𝐹(𝑢) is the ordinary Fourier transform of 𝑓(𝑡), and 𝑇 is the sampling interval that satisfies 1/𝑇≥𝑊/𝜋, and 𝑓(𝑡) is band-limited to (−𝑊,𝑊) in the ordinary Fourier transform domain. Similarly, the Parseval relationship for bandlimited two-dimensional signal 𝑓(𝑥,𝑦) associated with the Fourier transform can be written as follow: ^+∞_{𝑛=−∞+∞𝑚=−∞}||𝑓𝑛𝑇₁,𝑚𝑇₂||²=1𝑇₁𝑇₂^+∞−∞=1||𝑓(𝑥,𝑦)||²𝑑𝑡𝑇₁𝑇₂^+𝑊1_−𝑊1^+𝑊2_−𝑊2||𝐹(𝑢,𝑣)||²𝑑𝑢𝑑𝑣.(2.19)

It is proved in [30] that if a set of samples 𝑡_𝑛(𝑛=…,−1,0,1,…) is a sampling set, then the associated Parseval relation for the nonuniformly sampled signals can be written as ⁺^∞_𝑛=−∞||𝑓𝑡_𝑛||²=1𝑇^+∞−∞𝑓(𝑡)𝑓^∗𝑙𝑝(𝑡)𝑑𝑡=1𝑇^+𝑊−𝑊𝐹(𝑤)𝐹^∗𝑙𝑝(𝑤)𝑑𝑤,(2.20)

where 𝑓_𝑙𝑝(𝑡) is the low-pass filtered version of the nonuniformly samples, and 𝑇=𝑊/𝜋. 𝐹(𝑤) and 𝐹_𝑙𝑝(𝑤) are the corresponding Fourier transforms of 𝑓(𝑡) and 𝑓_𝑙𝑝(𝑡).

The objective of this paper is to obtain the corresponding Parseval relationship for a set of uniform and nonuniform samples of a band-limited signal in the FrFT domain. It is shown that the derived results can be seen as the generalization of the classical results in the Fourier domain.

3. The Main Results

Suppose that a signal 𝑓(𝑡) is band-limited to (−Ω_𝛼,Ω_𝛼) in the FrFT domain for order 𝛼, and 𝑇_𝛼 is the sampling interval that satisfies the uniform sampling theorem of signal in the FrFT domain [1, 2]; for example, 1/𝑇_𝛼≥Ω_𝛼|csc𝛼|/𝜋. The objective of this section is to investigate the Parseval relationship for uniform and nonuniform samples of signal 𝑓(𝑡) in the FrFT domain.

3.1. The Parseval Relationship for Uniform Samples

Theorem 3.1. Suppose that a signal 𝑓(𝑡) is Ω_𝛼 band-limited in the FrFT domain with order 𝛼, then the Parseval relationship associated with the signal 𝑓(𝑡) in the FrFT domain can be expressed as ⁺^∞_𝑛=−∞||𝑓𝑛𝑇_𝛼||²=1𝑇_𝛼^+∞−∞||𝑓(𝑡)||²𝑑𝑡=|csc𝛼|𝑇_𝛼2𝜋^{|csc𝛼|Ω𝛼}_{−|csc𝛼|Ω𝛼}||𝐹_𝛼(sin𝛼𝑢)||²𝑑𝑢,(3.1) where 𝑇_𝛼 is sampling interval, and 𝐹_𝛼(𝑢) is the FrFT of signal 𝑓(𝑡) with order 𝛼.

Proof. Let 𝑔(𝑡)=^Ω𝛼_−Ω𝛼𝐹_𝛼(𝑢)exp−𝑗12cot𝛼𝑢²+𝑗csc𝛼𝑢𝑡𝑑𝑢,(3.2) then 𝑔(𝑡) is a band-limited signal in the traditional Fourier domain. Applying (2.18) to signal 𝑔(𝑡), we obtain that ⁺^∞_𝑛=−∞||𝑔𝑛𝑇_𝛼||²=1𝑇_𝛼^+∞−∞||𝑔(𝑡)||²𝑑𝑡=1𝑇_𝛼^{+|csc𝛼|Ω𝛼}_{−|csc𝛼|Ω𝛼}||𝐺(𝑢)||²𝑑𝑢.(3.3) Substituting (2.8) into (3.3), we obtain that ⁺^∞_𝑛=−∞||||𝑓𝑛𝑇_𝛼2𝜋1+𝑗cot𝛼exp𝑗21cot𝛼𝑛𝑇^𝛼²||||²=1𝑇_𝛼^+∞−∞||||𝑓(𝑡)2𝜋1+𝑗cot𝛼exp𝑗21cot𝛼𝑡²||||²=1𝑑𝑡𝑇_𝛼^{+|csc𝛼|Ω𝛼}_{−|csc𝛼|Ω𝛼}|||𝐹_𝛼(sin𝛼𝑢)𝑒^{−𝑗(1/4)sin2𝛼𝑢2}|||²𝑑𝑢.(3.4) It is easy to verify that, |√2𝜋/(1+𝑗cot𝛼)|²=2𝜋|√(1−𝑗cot𝛼)/(1+cot²𝛼)|²=2𝜋|1−𝑗cot𝛼|/(1+cot²𝛼)²=2𝜋/|csc𝛼|, and the magnitude of exponential function is |||exp𝑗12cot𝛼𝑛𝑇^𝛼²|||=1,|||exp𝑗12cot𝛼(𝑡)²|||=1.(3.5) Substituting these results in (3.4), we obtain the final result as follows: ⁺^∞_𝑛=−∞||𝑓𝑛𝑇_𝛼||²=1𝑇_𝛼^+∞−∞||𝑓(𝑡)||²𝑑𝑡=|csc𝛼|𝑇_𝛼2𝜋^{|csc𝛼|Ω𝛼}_{−|csc𝛼|Ω𝛼}||𝐹_𝛼(sin𝛼𝑢)||²𝑑𝑢.(3.6)

Equation (3.1) can be seen as the generalization of the Parseval relations for the uniformly sampled signals associated with the FrFT. The next subsection focus on the generalization of the Parseval relations for the nonuniform sampling sets in the FrFT domain.

3.2. The Parseval Relationship of Nonuniform Samples

Suppose that a general nonuniform sampling set {𝑡_𝑛,𝑛=…,−1,0,1,…} is obtained from the Ω_𝛼 bandlimited signal 𝑓(𝑡) in the FrFT domain. If this sampling set satisfies the condition proposed in [30], then the Parseval relationship for this nonuniform sampling set can be derived as the following Theorem 3.2.

Theorem 3.2. The Parseval relationship of nonuniform samples can be written as ⁺^∞_𝑛=−∞||𝑓𝑡_𝑛||²=1𝑇_𝛼^+∞−∞=|csc𝛼|𝑓(𝑡)𝑓^∗𝑙𝑝(𝑡)𝑑𝑡𝑇_𝛼(2𝜋)²^{+|csc𝛼|Ω𝛼}_{−|csc𝛼|Ω𝛼}𝑒^{−𝑗(1/4)sin2𝛼𝑢2}𝐹_𝛼(𝑢sin𝛼)𝐺^∗𝑙𝑝(𝑢)𝑑𝑢,(3.7) where 𝑓_𝑙𝑝(𝑡)=∑^{+∞𝑛=−∞}𝑓(𝑡_𝑛)exp(𝑗(1/2)cot𝛼(𝑡²−𝑡^2𝑛))(sin𝑐[csc𝛼Ω_𝛼(𝑡−𝑡_𝑛)]), 𝐹_𝛼(𝑢) is the FrFT of 𝑓(𝑡), and 𝐺_𝑙𝑝(𝑢) is the Fourier transform of 𝑔_𝑙p(𝑡).

Proof. Let 𝑔(𝑡)=^Ω𝛼_−Ω𝛼𝐹_𝛼(𝑢)exp−𝑗12cot𝛼𝑢²+𝑗csc𝛼𝑢𝑡𝑑𝑢(3.8) then 𝑔(𝑡) is a |csc𝛼|Ω_𝛼 bandlimited signal in the Fourier domain. Applying the classical Parseval relationship of (2.20) for the bandlimited signals in the Fourier domain to signal 𝑔(𝑡), we obtain ⁺^∞_𝑛=−∞||𝑔𝑡_𝑛||²=1𝑇_𝛼^+∞−∞𝑔(𝑡)𝑔^∗𝑙𝑝(𝑡)𝑑𝑡=1𝑇_𝛼^{+|csc𝛼|Ω𝛼}_{−|csc𝛼|Ω𝛼}𝐺(𝑢)𝐺^∗𝑙𝑝(𝑢)𝑑𝑢,(3.9) where 𝑔_𝑙𝑝 is the signal obtained after low-pass filtering of the sampled signal 𝑔_𝑙𝑝(𝑡)=^+∞_𝑛_=−∞=𝑔𝑡_𝑛sin𝑐csc𝛼Ω_𝛼𝑡−𝑡_𝑛_+∞𝑛_=−∞𝑓𝑡_𝑛2𝜋1+𝑗cot𝛼exp𝑗21cot𝛼𝑡^2𝑛sin𝑐csc𝛼Ω^𝛼𝑡−𝑡_𝑛.(3.10) From the relationship between 𝑔(𝑡) and 𝑓(𝑡), the following relations can be obtained: ⁺^∞_𝑛=−∞||𝑔𝑡_𝑛||²=^+∞_𝑛_=−∞||||𝑓𝑡_𝑛2𝜋1+𝑗cot𝛼exp𝑗12cot𝛼𝑡^2𝑛||||²=2𝜋|csc𝛼|^+∞_𝑛=−∞1||𝑓𝑡_𝑛||²,𝑇_𝛼^+∞−∞𝑔(𝑡)𝑔^∗𝑙𝑝(𝑡)𝑑𝑡=2𝜋|csc𝛼|𝑇_𝛼^+∞−∞𝑓(𝑡)^+∞_𝑛_=−∞𝑓^∗𝑡_𝑛exp𝑗12cot𝛼𝑡²×sin𝑐csc𝛼Ω_𝛼−𝑡^2𝑛=2𝜋𝑡−𝑡_𝑛𝑑𝑡|csc𝛼|𝑇_𝛼^+∞−∞𝑓(𝑡)𝑓^∗𝑙𝑝(𝑡)𝑑𝑡.(3.11)
From (3.11), the first part of (3.9) can be rewritten as⁺^∞_𝑛=−∞||𝑓𝑡_𝑛||²=1𝑇_𝛼^+∞−∞𝑓(𝑡)𝑓^∗𝑙𝑝(𝑡)𝑑𝑡.(3.12) From Lemma 2.1, the following relationship holds for 𝐺(𝑢) and 𝐹_𝛼(𝑢): 𝐺(𝑢)=12𝜋𝐹^𝛼(sin𝛼𝑢)𝑒^{−𝑗(1/4)sin2𝛼𝑢2}.(3.13) Substitute (3.13) in to the final part of (3.9), we obtain that 1𝑇_𝛼^{+|csc𝛼|Ω𝛼}_{−|csc𝛼|Ω𝛼}𝐺(𝑢)𝐺^∗𝑙𝑝(𝑢)𝑑𝑢=12𝜋𝑇_𝛼^{+|csc𝛼|Ω𝛼}_{−|csc𝛼|Ω𝛼}𝑒^{−𝑗(1/4)sin2𝛼𝑢2}𝐹_𝛼(𝑢sin𝛼)𝐺^∗𝑙𝑝(𝑢)𝑑𝑢.(3.14) The final result follows from (3.11) and (3.14).

3.3. The Parseval Relationship for Two-Dimensional Case

Based on the definitions of two-dimensional FrFT and bandlimited signals, the Parseval relationship of the one-dimensional cases can be generalized to 2-D signals based on the Lemma 2.2. We would like to give the following Theorem 3.3.

Theorem 3.3. Suppose that a signal 𝑓(𝑥,𝑦) is (Ω_𝛼,Ω_𝛽) band-limited in the FrFT domain with order 𝛼 and 𝛽, and then the Parseval relationship associated with the signal 𝑓(𝑥,𝑦) in the FrFT domain can be expressed as ^+∞_{𝑛=−∞+∞𝑚=−∞}||𝑓𝑛𝑇₁,𝑚𝑇₂||²=1𝑇_𝛼𝑇_𝛽^+∞−∞=1||𝑓(𝑥,𝑦)||²𝑑𝑡𝑇_𝛼𝑇_𝛽^+Ω𝛼_−Ω𝛼^+Ω𝛽_−Ω𝛽||𝐹_𝛼,𝛽(sin𝛼𝑢,sin𝛽𝑣)||²𝑑𝑢𝑑𝑣,(3.15) where 𝑇_𝛼 and 𝑇_𝛽 are sampling interval, and 𝐹_𝛼,𝛽(𝑢,𝑣) is the two-dimensional FrFT of signal 𝑓(𝑥,𝑦) with order 𝛼 and 𝛽.

Proof. Similar with the proof of Theorem 3.1, let 𝑔(𝑥,𝑦)=2𝜋1+𝑗cot𝛼2𝜋1+𝑗cot𝛽𝑓(𝑥,𝑦)𝑒^{𝑗(1/2)cot𝛼𝑥2}𝑒^{𝑗(1/2)cot𝛽𝑦2}.(3.16) Then, from Lemma 2.2  𝑔(𝑥,𝑦) is a (|csc𝛼|Ω_𝛼,|csc𝛽|Ω_𝛽) band-limited signal in the ordinary Fourier transform domain. By applying the classical two-dimensional Parseval relationship of (2.19) to signal 𝑔(𝑥,𝑦), we can obtain the final result.

4. Conclusions

Based on the relationship between the Fourier transform and the FrFT, this paper investigates the Parseval's relationship of sampled signals in the FrFT domain. We firstly investigate the Parseval relationship for the uniformly samples of bandlimited signal associated with the FrFT. Then, we extend this relationship to a general set of nonuniform samples of band-limited signal in the FrFT domain. Finally, we studied the Parseval relations for uniformly sampled bandlimited two-dimensional signals, and it is also shown that the derived results can be seen as the generalization of the classical results in the Fourier domain to the FrFT domain. Future works includes the derivation of the Parseval's relations in the linear canonical transform domain for one- and two-dimensional uniformly (nonuniformly) sampled signals, and the applications of the derived results in the sampling theories and other related areas.

Acknowledgments

The authors would like to thank the anonymous reviewers and the handing editor for their valuable comments and suggestions for the improvements of this manuscript. The authors also thank Dr. Hai Jin of Beijing Institute of Technology for the proofreading of the paper. This work was partially supported by the National Natural Science Foundation of China (no. 60901058 and no. 61171195) and Beijing Natural Science Foundation (no. 1102029).