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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.

In information processing, the uncertainty principle plays an important role in elementary fields, and data concentration is often considered carefully via the uncertainty principle [

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.

Here, we first briefly review the definition of FRFT. For given continuous signal

where

However, unlike the discrete FT, there are a few definitions for the DFRFT [

Clearly, if

where

For DFRFT, we have the following property [

More details on DFRFT can be found in [

Definition 1: Let

is a small value with

Specially, if

fixed because

Definition 2: Generalized frequency-limiting operator

If

Definition 3: Let

Here,

Definition 4: Generalized discrete frequency-limiting operator

on

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 [

As shown in introduction, the existed continuous generalized uncertainty relations [

Lemma 1:

Proof: From the definition of the operator

Exchange the locations of the integral operators, we obtain

so that

Set

Now, we know that [see the proof of (3.1) in 25]

Let

where

FRFT in (1) we have

Hence, we obtain the final result

Now we give the first theorem.

Theorem 1: Let

Proof: Since

Meanwhile, via triangle inequality and the definitions of concentration we have

At the same time, we know

so that

i.e.,

Therefore,

From [

Use the above two results, we obtain

i.e.,

Hence,

Obviously, this bound is different from that [

for the discrete case in the next section. If

the discrete case? The next section will answer.

First let us introduce a lemma.

Lemma 3:

Proof: From the definition of the operator

Exchange the locations of the sum operators, we obtain

Hence, according to the definition of the Frobenius matrix norm [

In the similar manner with the continuous case, we can obtain

Theorem 2: Let

Here, we find that when

Set

Theorem 3: Let

Clearly, theorem 3 is a special case of theorem 2. Also, this theorem can be derived via theorem 1 in [

Differently, we obtain this result in a different way. Here we note that since

Through setting special value for

Corollary 1:

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

Theorem, we know that all the energy of

loss of generality, we assume

[

We rewrite (8) in terms of matrices and vectors. Define the matrix

where

Clearly,

so that we can rewrite matrix

From the definition of DFRFT, we know that the bases

ks and

ment in

wise,

domain in total. Thus, theorem 3 is verified.

Furthermore, we can obtain the following more general uncertainty relation associated with DFRFT.

Clearly, if

tional cases. Therefore, the generalized uncertainty principles show that the resolution will be higher.

Theorem 4: Let

Proof: From the assumption and the definition of DFRFT [

where

Therefore, let

where

Hence, we obtain

Set

Using the triangle inequality, we have

From

Hence

Therefore, we obtain

Adding all the above inequalities, we have

Similarly, from

From the definition and property of DFRFT [

with

Hence, we finally obtain the proof

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

Clearly, we can obtain from

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 finding the optimal filtering parameters [

We will thank Professor R. Tao very much for his valuable suggestions in improving our work. This work was fully supported by the NSFCs (61002052 and 61471412) and partly supported by the NSFC (61250006) and Third Term of 2110 in Dalian Navy Academy.

XiaotongWang,GuanleiXu, (2015) Support-Limited Generalized Uncertainty Relations on Fractional Fourier Transform. Journal of Signal and Information Processing,06,227-237. doi: 10.4236/jsip.2015.63021