Support-Limited Generalized Uncertainty Relations on Fractional Fourier Transform

This paper investigates the generalized uncertainty principles of fractional Fourier transform (FRFT) for concentrated data in limited supports. The continuous and discrete generalized uncertainty relations, whose bounds are related to FRFT parameters and signal lengths, were derived in theory. These uncertainty principles disclose that the data in FRFT domains may have much higher concentration than that in traditional time-frequency domains, which will enrich the ensemble of generalized uncertainty principles.


Introduction
In information processing, 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 infinite, based on which various uncertainty relations [1] [2] [9]- [21] [22] have been presented.However, in practice, both the supports of time and frequency are often limited for N-point discrete signals.In such case, the infinite support fails to hold true.In limited supports, some papers such as [23]- [26] have discussed the uncertainty principle in conventional time-frequency domains for continuous and discrete cases and some conclusions are achieved that can be taken as our special cases in the following sections.However, none of them has covered the FRFT 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 FRFT domains.As the rotation of the traditional FT [27], FRFT [5] [6] [28]- [30] has some special properties with its transform parameter and sometimes yields the better results such as the detection of LFM signal [31].Readers can see more details on FRFT in [6] and [32] and so on.
In this paper, we extend the Heisenberg uncertainty principle in FRFT domain for both discrete and continuous cases for the ε-concentrated signals or the signals with finite supports.It is shown that these bounds are connected with lengths of the supports and FRFT parameters.In a word, there have been no reported papers covering these results and conclusions, and most of them are new or novel.

Definition of DFRFT
Here, we first briefly review the definition of FRFT.For given continuous signal ( ) ( ) ( ) x t L R L R ∈  and ( ) 2 1 x t = , its FRFT [6] is defined as cot cot sin 2 2 , d 1 cot e e e d π 2π where Z n ∈ and i is the complex unit, α is the transform parameter defined as that in [6].In addition, ( . However, unlike the discrete FT, there are a few definitions for the DFRFT [32], but not only one.In this paper, we will employ the definition defined as follows [6] [32]: ˆ1 cot e e e , , 1 , . Clearly, if π 2 α = , (2) reduces to the traditional discrete FT [6] [32].Also, we can rewrite definition (2) as For DFRFT, we have the following property [5] More details on DFRFT can be found in [6] and [32].
with) be a discrete sequence with ( ) ( ) and its DFRFT ( ) x k , if there is a sequence ( ) Here, 0 ⋅ is the 0-norm operator that counts the non-zero elements.
Definition 4: Generalized discrete frequency-limiting operator N P α is defined as x k is the DFRFT of ( ) . 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 [23] [24] are only their special cases.Definition 3 and 4 disclose the relation between α ε and N α .

The Continuous Heisenberg Uncertainty Principles
As shown in introduction, the existed continuous generalized uncertainty relations [9]- [21] are mainly for the infinite supports.Here, we discuss the case of finite support.First we introduce the following lemma.
Exchange the locations of the integral operators, we obtain . Now, we know that [see the proof of (3.1) in 25] ( ) is the character function of the set W α .Therefore, via Parseval's theorem [6] and the definition of FRFT in (1) we have Hence, we obtain the final result , d d d .sin sin

∫ ∫ ∫
Now we give the first theorem.
W β be a measurable set and suppose ( ) is the FRFT of ( ) x t for transform parameter α ( ) Proof: Since , therefore we can find such ( ) Meanwhile, via triangle inequality and the definitions of concentration we have .
x t P P x t x t P x t P x t P x t x t P x t P x t P x t At the same time, we know x t P P x t x t P P x t P P x t From [24] [27], we know that Use the above two results, we obtain Here, we find that when π 2 π k α β − = + , (4) reduce to the traditional case in Theorem 2 [(3.1), 25].Obviously, this bound is different from that [20] of infinite case.In [20], the main involved objects are the variances of the signal in infinite supports.Here the measurable sets ( W α , W β ) are involved, which is instructive for the discrete case in the next section.If 0 what will happen?Clearly, it is impossible.From the conclusion [33], if 0

The Uncertainty Relation
First let us introduce a lemma.
Proof: From the definition of the operator N N P P α β in definition 4, we have , , Exchange the locations of the sum operators, we obtain , , ˆ, .
Hence, according to the definition of the Frobenius matrix norm [27] [34] and the definition of DFRFT, we have ( ) ( ) , sin In the similar manner with the continuous case, we can obtain . Therefore, we can obtain the following theorem 2.

The Extensions
in theorem 2, we can obtain the following theorem 3 directly.
x k β be the DFRFT of the time sequence ( ) ( ) N β 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 [26].Differently, we obtain this result in a different way.Here we note that since , there is at least one non-zero element in every FRFT domain for Proof: Now we prove corollary 1 in the sense of sampling and mathematical solution for better understanding these relations.Without loss of generality, we often assume that the continuous signal ( ) x t (the continuous version of ( ) x n  ) is band-limited, then ( ) x n  is obtained through sampling ( ) x t .From the sequence length N in the definition of DFRFT in (2), we know the sampling period defined as s T :  1 implies this result).We assume there is no aliasing after sampling in the FRFT domain, then from the sampling Theorem, we know that all the energy of ( ) i.e., all the energy of ( ) loss of generality, we assume 0 m = based on the shifting property of FRFT [6] [32], i.e., all the energy of ( ) n n n  be the sites where ( ) We rewrite (8) in terms of matrices and vectors.Define the matrix  ( ) , , , , From the definition of DFRFT, we know that the bases ( ) ks and l n s ) are mutually orthogonal [6] [32].Therefore, the different rows are not correlated so that s Z can be rewritten as ( ) Since every ele- ment in b is not and is nonsingular, then there must be a non-zero element in , ˆs  there is at least one non-zero element.Therefore, there are at least non-zero elements in the DFRFT domain in total.Thus, theorem 3 is verified.Furthermore, we can obtain the following more general uncertainty relation associated with DFRFT.
Using the triangle inequality, we have x n =  and Parseval's principle [6], we obtain ( ) ( ) Therefore, we obtain Hence, we finally obtain the proof

The Simulation
In this section we give an example to show that the data in FRFT domains may have much higher concentration than that in traditional time-frequency domains.Now considering the chirp signal ( ) This verifies that the data in FRFT domains may have much higher concentration than that in traditional time-frequency domains.(Note here that if the transformed coefficient is less than 0.1, then we take it as zero value.See

Conclusion
In practice, we often process the data with limited lengths for both the continuous (ε-concentrated) and discrete signals.Especially 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 FRFT in term of data concentration.We show that the uncertainty bounds are related to the FRFT parameters and the support lengths.These uncertainty relations will enrich the ensemble of FRFT.Moreover, these uncertainty relations will help find the optimal filtering parameters [31] such as [6] [34] [36].Our simulation also shows that the data in FRFT domains may have much higher concentration than that in traditional time-frequency domains.
which is in conflict with that W β is measura- ble and limited.Therefore, in the continuous case, hold true.However, what about the discrete case?The next section will answer.

Figure 1 .
Figure 1.The simulation of a signal with its FRFT and FT.(a) The original signal in time domain; (b) The FT of the signal (i.e., the traditional frequency domain); (c) The FRFT of the signal (i.e., the FRFT domain).