The Generalization of the Poisson Sum Formula Associated with the Linear Canonical Transform

Abstract

The generalization of the classical Poisson sum formula, by replacing the ordinary Fourier transform by the canonical transformation, has been derived in the linear canonical transform sense. Firstly, a new sum formula of Chirp-periodic property has been introduced, and then the relationship between this new sum and the original signal is derived. Secondly, the generalization of the classical Poisson sum formula to the linear canonical transform sense has been obtained.

1. Introductions

As a generalization of the classical Fourier transform and the fractional Fourier transform (FrFT), the linear canonical transform (LCT) receives much interest in recent years [1–3]. Many important transforms, for example, the Fourier transform, the Fresnel transform, and the scaling operations are all special cases of the LCT. It has been shown to be one of the most useful tools in several areas [4–6], including optics, quantum physics, and in the signal processing community. Its relationship with the Fourier transform and the fractional Fourier transform can be found in [7, 8]. The well-known operations, for example, the Hilbert transform, the Parseval relationship, the convolution and product operations, and the spectral analysis, in traditional Fourier domain have been extended to the linear canonical transform domain by different authors [9–13]. For further properties and applications of LCT in optics and signal processing community, one can refer to [1, 2]. The classical sampling theorems associated with the LCT have also been investigated and studied in the LCT domain in various literatures. The extensions of the classical Shannon sampling theorem for band-limited or time-limited signals in the LCT domain have been deduced in [13, 14].

However, for the best of our knowledge, none of the research papers throw light on the study of the traditional Poisson sum formula [15–21] associated with the LCT have been reported as yet. The Poisson summation formula is a very useful tool not only in many branches of the mathematics, but also it finds many applications in various fields, for example, mechanics, signal processing community, and many scientific fields. It is therefore, worthwhile as well as interesting to investigate the Poisson sum formula associated with the LCT.

The objective of this paper is to study and investigate the Poisson formula associated with the LCT. In other words, we want to generalize the classical Poisson sum formula by replacing the ordinary Fourier transform by the canonical transform. In order to obtain the desired results for a signal 𝑥(𝑡), we first deduce a new sum formula for the signal 𝑥(𝑡) and then achieve the innovative results in the LCT domain. The paper is organized as follows, the preliminaries are proposed in Section 2, the main results of the paper are investigated in Section 3, and the conclusion is given in Section 4.

2. Preliminaries

2.1. The Linear Canonical Transform

The linear canonical transform (LCT) of a signal 𝑥(𝑡) with parameter matrix 𝐴 is defined as [1–3] 𝑋_𝐴(𝑢)=^∞_−∞𝐾_𝐴(𝑢,𝑡)𝑥(𝑡)𝑑𝑡, (2.1) where 𝐾_𝐴(𝑢,𝑡)=1𝑒𝑗2𝜋𝑏^{𝑗((𝑎/2𝑏)𝑡2−(1/𝑏)𝑢𝑡+(𝑑/2𝑏)𝑢2)},𝑏≠0, (2.2) and 𝑋_𝐴(𝑢)=√𝑑𝑒^{𝑗(𝑐𝑑/2)𝑢2}𝑥(𝑑𝑢), 𝑏=0. The parameter 𝐴=^𝑎𝑏_𝑐𝑑, and satisfy det(𝐴)=1, 𝑎,𝑏,𝑐,𝑑∈𝑅. In this case, the linear canonical transform is a unitary transform [1], therefore we can derive the inverse transform of LCT as another LCT transform.

The inverse transform of LCT can be derived by an LCT with the parameter of 𝐴⁻¹ as following: 𝑥(𝑡)=^∞_−∞𝐾_𝐴⁻¹(𝑡,𝑢)𝑋_𝐴(𝑢)𝑑𝑢. (2.3)

The LCT can be looked at as the generalization of the well-known operations in the science and engineering community [4–6]. The relationship between the LCT and the Fourier transform, the fractional Fourier transform have been derived in [1–3].

A signal 𝑥(𝑡) is said to be band-limited with respect to Ω_𝐴 in linear canonical transform domain, when 𝑋_𝐴(𝑢)=0for|𝑢|>Ω_𝐴, (2.4) where Ω_𝐴 is called the bandwidth of signal 𝑥(𝑡) in the linear canonical transform domain. At the same time, a signal 𝑥(𝑡) is called Chirp-periodic with period 𝑇 and of parameter 𝐴 if it satisfies the following equation: 𝑒^{𝑗((𝑎𝑡2)/2𝑏)}𝑥(𝑡)=𝑒^{𝑗(𝑎(𝑡+𝑇)2/2𝑏)}𝑥(𝑡+𝑇). (2.5)

The following identities will be used in the following sections.

Lemma 2.1. The inverse linear canonical transform of signal 𝑍_𝐴(𝑢)=∑^+∞_𝑛=−∞𝛿(𝑢−𝑛𝐵)𝑒^{𝑗(𝑎/2𝑏)𝑢2} for parameter 𝐴=^𝑎𝑏_𝑐𝑑 is 𝑧(𝑡)=1𝐵−1𝑒𝑗2𝜋^{−𝑗(𝑎/2𝑏)𝑡2∞}_𝑛=−∞𝛿𝑡𝑏−𝑛𝐵. (2.6) And the inverse linear canonical transform of signal 𝛿(𝑢−𝑢₀) is 𝑠(𝑡)=−1𝑒𝑗2𝜋𝑏^{−𝑗(𝑎/2𝑏)(𝑡2+𝑢20)+𝑗𝑢0𝑡𝑏}. (2.7)

Proof. These results can be derived easily by the definition of the LCT and the inverse transform of LCT.

Assuming a signal 𝑥(𝑡) is band-limited to Ω_𝐴 in the linear canonical transform domain, then from the results derived in [13], 𝑥(𝑡) is not band-limited in the traditional Fourier domain. Therefore, the classical results of bandlimited signal processing method in Fourier domain can be used in the LCT domain to obtain the novel results associated with the LCT.

2.2. The Poisson Sum Formula

The Poisson sum formula demonstrates that the sum of infinite samples in time domain of a signal 𝑥(𝑡) is equivalent to the sum of infinite samples of 𝑋(𝑢) in the Fourier domain. Mathematically, the Poisson sum formula can be represented as follows: ^∞_𝑛=−∞𝑥()=1𝑡+𝑘𝜏𝜏^∞_𝑛=−∞𝑋𝑛𝜏𝑒^{𝑗(𝑛/𝜏)𝑡}, (2.8) or ^∞_𝑘=−∞𝑥()=1𝑘𝜏𝜏^∞_𝑛=−∞𝑋𝑛𝜏, (2.9) where 𝑋(𝑢) is the traditional Fourier transform of signal 𝑥(𝑡). It is well known that it will be valid only if 𝑥(𝑡) and its Fourier transform 𝑋(𝑢) are regular enough and only if both series of (2.9) converge [20].

In order to obtain the new results associated with the linear canonical transform, a new summation associated with the signal 𝑥(𝑡) is introduced as following: 𝑦(𝑡)=^∞_𝑘=−∞𝑥(𝑡+𝑘𝜏)𝑒^{𝑗(𝑎/2𝑏)(2𝑘𝜏𝑡+𝑘2𝜏2)}, (2.10) where 𝜏 is a constant. From (2.10), the function 𝑦(𝑡) can be seen as a periodic phase-shift replica of the original function 𝑥(𝑡). The signal 𝑦(𝑡) will be used in the following sections to investigate the Poisson sum formula associate with the linear canonical transform.

3. The Main Results

Suppose a signal 𝑥(𝑡) is band-limited to Ω_𝐴 in linear canonical transform domain of parameter 𝐴, then, from (2.10) a new function 𝑦(𝑡) can be deduced from signal 𝑥(𝑡). Firstly, the properties of the signal 𝑦(𝑡) associated with the linear canonical transform can be derived from Theorem 3.1.

Theorem 3.1. Suppose a signal 𝑥(𝑡) is band-limited to Ω_𝐴 in the linear canonical transform domain of parameter 𝐴, and 𝑦(𝑡)=∑^∞_𝑘=−∞𝑥(𝑡+𝑘𝜏)𝑒^{𝑗(𝑎/2𝑏)(2𝑘𝜏𝑡+𝑘2𝜏2)}, then the following results about 𝑦(𝑡) is true.(a)𝑦(𝑡) is a Chirp-periodic signal with period 𝜏.(b)𝑥(𝑡) is a band-limited signal in linear canonical transform domain with parameter 𝐴, if and only if 𝑦(𝑡) has a finite number of nonzero linear canonical series coefficients for any 𝜏.

Proof. (a) By the definition of the Chirp-periodicity, we obtain 𝑦(𝑡+𝜏)𝑒^{𝑗(𝑎/2𝑏)(𝑡+𝜏)2}=^∞_𝑘=−∞𝑥(𝑡+𝜏+𝑘𝜏)𝑒^{𝑗(𝑎/2𝑏)[2𝑘𝜏(𝑡+𝜏)+𝑘2𝜏2]}𝑒^{𝑗(𝑎/2𝑏)(𝑡+𝜏)2}=𝑒^{𝑗(𝑎/2𝑏)𝑡2∞}_𝑘=−∞𝑥[𝑡+(𝑘+1)𝜏]𝑒^{𝑗(𝑎/2𝑏)[2(𝑘+1)𝜏𝑡+(𝑘+1)2𝜏2]}=𝑦(𝑡)𝑒^{𝑗(𝑎/2𝑏)𝑡2}. (3.1) This proves the Chirp-periodicity of the signal 𝑦(𝑡).
(b) To prove the necessary condition, the 𝑛th coefficient 𝑐_𝑛,𝐴 of signal 𝑦(𝑡) can be deduced from the linear canonical series definition proposed in [4] as 𝑐_𝑛,𝐴=𝑏−𝑗𝑎𝜏^𝜏₀𝑦(𝑡)𝑒^{𝑗(𝑎/2𝑏)[(𝑛𝑏2𝜋/𝜏)2+𝑡2]−𝑗(𝑛2𝜋/𝜏)𝑡}𝑑𝑡. (3.2) Equation (3.2) can be rewritten as 𝑐_𝑛,𝐴=𝑏−𝑗𝑎𝜏𝑒^{𝑗(𝑎/2𝑏)(𝑛2𝜋𝑏/𝜏)2𝜏}₀^∞_𝑘=−∞𝑥(𝑡+𝑘𝜏)𝑒^{𝑗(𝑎/2𝑏)(2𝑘𝜏𝑡+𝑘2𝜏2)}𝑒^{𝑗(𝑎/2𝑏)𝑡2−𝑗(𝑛(2𝜋/𝜏))𝑡}=𝑑𝑡𝑏−𝑗𝑎𝜏𝑒^{𝑗(𝑎/2𝑏)(𝑛2𝜋𝑏/𝜏)2+∞}_𝑘=−∞^(𝑘+1)𝜏_𝑘𝜏𝑥(𝜆)𝑒^{𝑗(𝑎/2𝑏)𝜆2−𝑗(𝑛(2𝜋/𝜏))(𝜆−𝑘𝜏)}=𝑑𝜆𝑏−𝑗𝑎/𝜏𝐾_𝐴(𝑢,𝑡)^+∞_−∞𝐾_𝐴(𝑢,𝑡)𝑥(𝜆)𝑒^{𝑗(𝑎/2𝑏)(𝑛𝐹′)2}𝑒^{𝑗(𝑎/2𝑏)𝜆2−𝑗(1/𝑏)(𝑛𝐹′)𝜆}=𝑑𝜆𝑏−𝑗𝑎/𝜏𝐾_𝐴()𝑋𝑢,𝑡^𝐴𝑛2𝜋𝜏𝑏. (3.3) Since 𝑥(𝑡) is a Ω_𝐴 band-limited signal in linear canonical transform domain of parameter 𝐴, that is to say 𝑋_𝐴(𝑢)=^+∞_−∞𝑥(𝑡)𝐾_𝐴(𝑢,𝑡)𝑒^{𝑗(𝑎/2𝑏)(𝑡2+𝑢2)−𝑗(1/𝑏)𝑢𝑡}𝑑𝑡=0,|𝑢|>Ω_𝛼. (3.4) Comparing (3.3) and (3.4), we obtain 𝑐_𝑛,𝐴=0,when𝑛>𝜏Ω_𝐴.2𝜋𝑏 (3.5) Therefore, the necessary condition is proved.
To prove the sufficient condition, let us assume that 𝑐_𝑛,𝐴=0 for 𝑛>𝑁, where 𝑁 is any finite integer. From (3.5), the 𝑋_𝐴(𝑢)=^+∞_−∞𝑥(𝑡)𝐾_𝐴(𝑢,𝑡)𝑑𝑡=0,|𝑢|>2𝜋𝑏𝑁𝜏. (3.6) Hence, 𝑥(𝑡) is band-limited signal having bandwidth as following: Ω_𝐴≤2𝜋𝑏𝑁𝜏. (3.7) This proves the sufficient condition of the theorem.

Based on the derived results of Theorem 3.1, the following Theorem 3.2 can be deduced.

Theorem 3.2. Suppose a signal 𝑥(𝑡) is band-limited to Ω_𝐴 in the linear canonical transform domain of parameter 𝐴, and 𝑦(𝑡)=∑^∞_𝑘=−∞𝑥(𝑡+𝑘𝜏)𝑒^{𝑗(𝑎/2𝑏)(2𝑘𝜏𝑡+𝑘2𝜏2)} is derived by shifting signal 𝑥(𝑡) to left and right, then the following conclusions can be deduced.(a)When 1/𝜏>Ω_𝐴, 𝑦(𝑡) can be deduced from the following formula: 𝑦(𝑡)=𝑏𝜏𝑋_𝐴(0)𝑒^−𝑗(𝑎2𝑏^)𝑡2. (3.8)(b)When Ω_𝐴/2<1/𝜏<Ω_𝐴, 𝑦(𝑡) can be deduced from the following formula: 𝑦(𝑡)=𝑏𝜏𝑒^−𝑗(𝑎2𝑏^)𝑡2𝑋_𝐴(0)+𝑒^−𝑗(𝑎2𝑏^{)(𝑏/𝜏)2}𝑋_𝐴𝑏𝜏𝑒^{𝑗𝑡/𝜏}+𝑋_𝐴−𝑏𝜏𝑒^{−𝑗𝑡/𝜏}. (3.9)(c)When Ω_𝐴/𝑛<1/𝜏<Ω_𝐴/(𝑛−1), 𝑦(𝑡) can be deduced from the following formula: 𝑦(𝑡)=𝑏𝜏𝑒^{−𝑗(𝑎/2𝑏)𝑡2}𝑋_𝐴(0)+^𝑛_𝑘=1𝑒^{−𝑗(𝑎/2𝑏)(𝑘(𝑎/𝜏))2}𝑋_𝐴𝑘𝑏𝜏𝑒^{𝑗𝑘𝑡/𝜏}+𝑋_𝐴𝑏−𝑘𝜏𝑒^{−𝑗𝑘𝑡/𝜏}. (3.10)

Proof. (1)Proof of (a). Since 𝑥(𝑡) is a Ω_𝐴 band-limited signal in the linear canonical transform domain, 𝑋_𝐴(𝑢) if sampled in the linear canonical transform domain of order 𝐴 at a rate of 𝐵>Ω_𝐴, then the samples can be represented as follows: 𝑌_𝑠,𝐴(𝑢)=𝑋_𝐴(𝑢)^+∞_𝑛=−∞𝛿(𝑢−𝑛𝐵)=𝑋_𝐴(0)𝛿(𝑢),𝐵>Ω_𝐴. (3.11) The first part of (3.11) can be reorganized as 𝑌_𝑠,𝐴(𝑢)=𝑋_𝐴(𝑢)^+∞_𝑛=−∞𝛿(𝑢−𝑛𝐵)𝑒^{𝑗(𝑎/2𝑏)𝑢2}𝑒^{−𝑗(𝑎/2𝑏)𝑢2} (3.12) If we let 𝑍_𝐴(𝑢)=∑^+∞_𝑛=−∞𝛿(𝑢−𝑛𝐵)𝑒^{𝑗(𝑎/2𝑏)𝑢2}, then (3.12) can be rewritten as 𝑌_𝑠,𝐴(𝑢)=𝑋_𝐴(𝑢)𝑍_𝐴(𝑢)𝑒^{−𝑗(𝑎/2𝑏)𝑢2}. (3.13) Applying the convolution and product theorem proposed in [7] and Lemma 2.1 to (3.13), the inverse linear canonical transform of formula (3.13) can be represented as 𝑦_𝑠(𝑡)=1𝑒𝑗2𝜋𝑏^{−𝑗(𝑎/2𝑏)𝑡2}𝑥(𝑡)𝑒^{(𝑎/2𝑏)𝑡2}∗𝑧(𝑡)𝑒^{𝑗(𝑎/2𝑏)𝑡2}=1𝑒𝑗2𝜋𝑏^{−𝑗(𝑎/2𝑏)𝑡2}𝑥(𝑡)𝑒^{𝑗(𝑎/2𝑏)𝑡2}∗1𝐵^+∞_𝑛=−∞𝛿1𝑏𝑛𝑡−𝐵=11𝑗2𝜋𝑏𝐵^+∞_𝑛=−∞𝑥𝑛𝑡−𝐵𝑏𝑒^{𝑗(𝑎/𝑏)[2(𝑛/𝐵)𝑏𝑡−((𝑛/𝐵)𝑏)2]}. (3.14) From the second part of (3.11), the inverse linear canonical transform of 𝑌_𝑠,𝐴(𝑢) can be derived as 𝑦_𝑠(𝑡)=1𝑋𝑗2𝜋𝑏^𝐴(0)𝑒^{−𝑗(𝑎/2𝑏)𝑡2}. (3.15) If we select 𝜏=𝑏/𝐵, then from (3.14)-(3.15) 𝑦(𝑡)=𝑏𝜏2𝜋()𝑦1−𝑗𝑎/𝑏^𝑠(𝑡)=𝑏𝜏𝑋_𝐴(0)𝑒^{−𝑗(𝑎/2𝑏)𝑡2}. (3.16) This proves (a).
(2)Proof of (b). Similar to the method of proving (a), if 𝑋_𝐴(𝑢) is sampled in the linear canonical transform domain of parameter 𝐴 at a rate of Ω_𝐴/2<𝐵<Ω_𝐴, there are essentially three nonzero samples of 𝑋_𝐴(𝑢): 𝑌_𝑠,𝐴(𝑢)=𝑋_𝐴(𝑢)^+∞_𝑛=−∞𝛿(𝑢−𝑛𝐵)=𝑋_𝛼(0)𝛿(𝑢)+𝑋_𝛼(𝐵)𝛿(𝑢−𝐵)+𝑋_𝛼(−𝐵)𝛿(𝑢+𝐵). (3.17) In this condition, (3.14) is also correct, and from (3.13), the relationship between 𝑦(𝑡) and 𝑦_𝑠(𝑡) can be derived as follows: 𝑦(𝑡)=𝐵𝑗2𝜋𝑏𝑦_𝑠(𝑡). (3.18) While 𝑦_𝑠(𝑡) can be deduced from (3.18) and Lemma 2.1 as 𝑦_𝑠(𝑡)=1/𝑗2𝜋𝑏𝑋_𝐴(0)𝑒^{−𝑗(𝑎/2𝑏)𝑡2}+𝑋_𝐴(𝐵)𝑒^{−𝑗(𝑎/2𝑏)(𝑡2+𝐵2)+𝑗𝐵𝑡(1/𝑏)}+𝑋_𝐴(−𝐵)𝑒^{−𝑗(𝑎/2𝑏)(𝑡2+𝐵2)−𝑗𝐵𝑡(1/𝑏)}. (3.19) If 𝜏=𝑏/𝐵 is chosen, then from (3.18)-(3.19). 𝑦(𝑡)=𝐵𝑒^{−𝑗(𝑎/2𝑏)𝑡2}𝑋_𝐴(0)+𝑋_𝐴(𝐵)𝑒^{−𝑗(𝑎/2𝑏)𝐵2+𝑗𝐵𝑡(1/𝑏)}+𝑋_𝐴(−𝐵)𝑒^{−𝑗(𝑎/2𝑏)𝐵2−𝑗𝐵𝑡(1/𝑏)}=𝐵𝑒^{−𝑗(𝑎/2𝑏)𝑡2}𝑋_𝐴(0)+𝑒^{−𝑗(𝑎/2𝑏)𝐵2}𝑋_𝐴(𝐵)𝑒^{𝑗𝐵𝑡/𝑏}+𝑋_𝐴(−𝐵)𝑒^{−𝑗𝐵𝑡/𝑏}=𝑏𝜏𝑒^{−𝑗(𝑎/2𝑏)𝑡2}𝑋_𝐴(0)+𝑒^{−𝑗(𝑎/2𝑏)(𝑏/𝜏)2}𝑋_𝐴𝑏𝜏𝑒^{𝑗𝑡/𝜏}+𝑋_𝐴−𝑏𝜏𝑒^{−𝑗𝑡/𝜏}. (3.20) Thus, (b) is also proved.

(3)Proof of (c). If 𝑋_𝐴(𝑢) is sampled in linear canonical transform domain of order 𝐴 at a rate of Ω_𝐴/𝑛<𝐵<Ω_𝐴/(𝑛−1), there are essentially 2𝑛−1 nonzero samples remaining 𝑌_𝑠,𝐴(𝑢)=𝑋_𝐴(𝑢)^+∞_𝑛=−∞𝛿(𝑢−𝑛𝐵)=𝑋_𝐴(0)𝛿(𝑢)+𝑋_𝐴(𝐵)𝛿(𝑢±𝐵)+⋯+𝑋_𝐴[(𝑛−1)𝐵]𝛿[𝑢±(𝑛−1)𝐵]. (3.21) Again, (3.14) is also correct in this case, and using similar method in proving (a) and (b), 𝑦(𝑡) can be deduced as 𝑦(𝑡)=𝐵𝑗2𝜋𝑏𝑦_𝑠(𝑡)=𝐵𝑒^{−𝑗(𝑎/2𝑏)𝑡2}𝑋_𝐴(0)+𝑋_𝐴(𝐵)𝑒^{−𝑗(𝑎/2𝑏)𝐵2+𝑗𝐵𝑡/𝑏}+𝑋_𝐴(−𝐵)𝑒^{−𝑗(𝑎/2𝑏)𝐵2−𝑗𝐵𝑡/𝑏}+⋯+𝑋_𝐴(𝑛𝐵)𝑒^{−𝑗(𝑎/2𝑏)(𝑛𝐵)2+𝑗𝑛𝐵𝑡/𝑏}+𝑋_𝐴(−𝑛𝐵)𝑒^{−𝑗(𝑎/2𝑏)(𝑛𝐵)2−𝑗𝑛𝐵𝑡/𝑏}=𝐵𝑒^{−𝑗(𝑎/2𝑏)𝑡2}𝑋_𝐴(0)+^𝑛_𝑘=1𝑒^{−𝑗(𝑎/2𝑏)(𝑘𝐵)2}𝑋_𝐴(𝑘𝐵)𝑒^{𝑗𝑘𝐵𝑡/𝑏}+𝑋_𝐴(−𝑘𝐵)𝑒^{−𝑗𝑘𝐵𝑡/𝑏}. (3.22) If we select 𝜏=𝑏/𝐵, then 𝑦(𝑡)=𝑏𝜏𝑒^{−𝑗(𝑎/2𝑏)𝑡2}𝑋_𝐴(0)+^𝑛_𝑘=1𝑒^{−𝑗(𝑎/2𝑏)(𝑘(𝑏/𝜏))2}𝑋_𝐴𝑘𝑏𝜏𝑒^{𝑗𝑘𝑡/𝜏}+𝑋_𝐴𝑏−𝑘𝜏𝑒^{−𝑗𝑘𝑡/𝜏}. (3.23) Part (c) of Theorem 3.1 is proved.

4. Conclusion

In this paper, the generalization of the classical Poisson sum formula to the linear canonical transform domain is investigated, by replacing the ordinary Fourier transform by the canonical transform, we firstly derived a new Chirp-periodic sum, and then the classical Poisson summations are generalized to the linear canonical transform domain based on the relationship derived. The classical results can be looked at as the special cases of the derived results. The applications of the derived results in sampling theories, signal analysis will be investigated in the linear canonical transform domain in the future.

Acknowledgments

This work is supported by National Natural Science Foundations of China (no. 60901058 and no. 61171195) and the Beijing Natural Science Foundation (no.1102029).