Discrete Inequalities on LCT

Linear canonical transform (LCT) is widely used in physical optics, mathematics and information processing. This paper investigates the generalized uncertainty principles, which plays an important role in physics, of LCT for concentrated data in limited supports. The discrete generalized uncertainty relation, whose bounds are related to LCT parameters and data lengths, is derived in theory. The uncertainty principle discloses that the data in LCT domains may have much higher concentration than that in traditional domains.


Introduction
In physics, the uncertainty principle plays an important role in elementary fields, and data concentration is often considered carefully via the uncertainty principle [1]- [8].In continuous signals, the supports are assumed to be ( ) , −∞ +∞ , based on which various uncertainty relations [1] [2] [9]- [21] have been presented.However, in practice, both the supports of time and frequency are often limited.In such case, the support ( ) , −∞ +∞ fails to hold true.In limited supports, some papers such as [22]- [25] have discussed the uncertainty principle in conventional time-frequency domains for continuous and discrete cases and some conclusions are achieved.However, none of them has covered the linear canonical transform (LCT) in terms of Heisenberg uncertainty principles that have been widely used in various fields [4]- [6].Therefore, there has a great need to discuss the uncertainty relations in LCT domains.As the generalization of the traditional FT, FRFT [5] [6] [26]- [28] and so on, LCT has some special properties with more transform parameters (or freedoms) and sometimes yields the better result [29].Readers can see more details on LCT in [6] and so on.

Definition of LCT
Before discussing the uncertainty principle, we will introduce some relevant preliminaries.Here, we first briefly review the definition of LCT.For given continuous signal ( ) ( ) ( ) x t = , its LCT [6] is defined as where Z n ∈ and i is the complex unit, ( ) , , , a b c d are the transform parameters defined as that in [6].In addition, ( • are the LCT transform pairs, i.e., , we have the following equations: However, unlike the discrete FT, there are a few definitions for the DLCT (discrete LCT), but not only one.In this paper, we will employ the definition defined as follows [6]: e e e , , 1 , .
Clearly, if ( ) ( ) reduces to the traditional discrete FT [6].Also, we can rewrite definition (2) as ˆA , , ,  A a b c d = and For DLCT, we have the following property [5] [6]: More details on DLCT can be found in [6].

Frequency-Limiting Operators
Definition 1: Let ( ) x t be a complex-valued signal with , then definition 1 reduces to the case in time domain [22] [23].If ( ) , then definition 1 reduces to the case in traditional frequency domain [22] [23].

Definition 2: Generalized frequency-limiting operator
, then definition 2 is the time-limiting operator [22] , then definition 2 is the traditional frequency-limiting operator [22] [23].Definitions 1 and 2 disclose the relation between A ε and A W .For the discrete case, we have the following definitions.
x n = and its DLCT ( ) Here, 0 ⋅ is the 0-norm operator that counts the non-zero elements.
Definition 4: Generalized discrete frequency-limiting operator x k is the DLCT of ( ) x n and Clearly, definitions 3 and 4 are the discrete extensions of definitions 1 and 2. They have the similar physical meaning.These definitions are introduced for the first time, the traditional cases [22] [23] are only their special cases.Definition 3 and 4 disclose the relation between A ε and A N .

The Uncertainty Principle
First let us introduce a lemma.Lemma 3: , , Exchange the locations of the sum operators, we obtain Hence, according to the definition of the Frobenius matrix norm [1] and the definition of DLCT, we have ( ) In the similar manner with the continuous case, we can obtain , thus, we get ( ) . Therefore, we can obtain the following theorem 2.
Theorem 2: Let ( ) be the DLCT of the time sequence ( ) ( ) be the numbers of nonzero entries in ( )

Extensions
Set 0 in theorem 2, we can obtain the following theorem 3 directly.
be the DLCT of the time sequence ( ) ( ) counts the numbers of nonzero entries in ( )
Clearly, theorem 3 is a special case of theorem 2. Also, this theorem can be derived via theorem 1 in [25].
Differently, we obtain this result in a different way.Here we note that since = , there is at least one non-zero element in every LCT domain for 1 1 a b a b = .Through setting special value for ( ) in theorem 3, we have Corollary 1: We can obtain the following more general uncertainty relation associated with DLCT.x n = .
Using the triangle inequality, we have x n = and Parseval's principle [6], we obtain: ( ) ( ) Adding all the above inequalities, we have x n = and Parseval's principle [6], we obtain T 1 X X = , hence ( ) ( ) From the definition and property of DLCT [6] we have

Conclusion
In practice, for the discrete data, not only the supports are limited, but also they are sequences of data points whose number of non-zero elements is countable accurately.This paper discussed the generalized uncertainty relations on LCT in terms of data concentration.We show that the uncertainty bounds are related to the LCT parameters and the support lengths.These uncertainty relations will enrich the ensemble of uncertainty principles and yield the potential illumination for physics.

whereF⋅
is the Frobenius matrix norm.Proof: From the definition of the operator

Theorem 4
is the extension of theorem 3 and discloses the uncertainty relation between multiple signals.