Support-Limited Generalized Uncertainty Relations on Fractional Fourier Transform ()
1. 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.
2. Preliminaries
2.1. Definition of DFRFT
Here, we first briefly review the definition of FRFT. For given continuous signal
and
, its FRFT [6] is defined as
(1)
where
and
is the complex unit,
is the transform parameter defined as that in [6] . In addition,
. If
,
, i.e., the inverse FRFT
.
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] :
(2)
Clearly, if
, (2) reduces to the traditional discrete FT [6] [32] . Also, we can rewrite definition (2) as
,
where
,
.
For DFRFT, we have the following property [5] [6] [32] :
.
More details on DFRFT can be found in [6] and [32] .
2.2. Frequency-Limiting Operators
Definition 1: Let
be a complex-valued signal with
and its FRFT
, if there is a function
vanishing outside
(
is a measurable set) such that
(![]()
is a small value with
), then
is
-concentrated.
Specially, if
, then definition 1 reduces to the case in time domain [23] [24] . If
, then definition 1 reduces to the case in traditional frequency domain [23] [24] . The
can be calculated after the
is
fixed because
and
. Therefore, ![]()
Definition 2: Generalized frequency-limiting operator
is defined as
(3)
If
, then definition 2 is the time-limiting operator [23] [24] . If
, then definition 2 is the traditional frequency-limiting operator [23] [24] . Definitions 1 and 2 disclose the relation between
and
. For the discrete case, we have the following definitions.
Definition 3: Let
(
with) be a discrete sequence with
and its DFRFT
, if there is a sequence
satisfying
such that
(
is a small value with
), then
is
-concentrated.
Here,
is the 0-norm operator that counts the non-zero elements.
Definition 4: Generalized discrete frequency-limiting operator
is defined as
with
is the DFRFT of
and
is the character function
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 [23] [24] are only their special cases. Definition 3 and 4 disclose the relation between
and
.
2.3. 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.
Lemma 1:
, where
denotes the Frobenius norm operator.
Proof: From the definition of the operator
in definition 2, we have
![]()
Exchange the locations of the integral operators, we obtain
,
so that
.
Set
, we have
.
Now, we know that [see the proof of (3.1) in 25]
.
Let
, then
![]()
where
is the character function of the set
. Therefore, via Parseval’s theorem [6] and the definition of
FRFT in (1) we have
![]()
Hence, we obtain the final result
![]()
Now we give the first theorem.
Theorem 1: Let
be a measurable set and suppose
is the FRFT of
for transform parameter
, such that
is
?concentrated on
. Then
. (4)
Proof: Since
, therefore we can find such
that makes
.
Meanwhile, via triangle inequality and the definitions of concentration we have
![]()
At the same time, we know
,
so that
,
i.e.,
.
Therefore,
.
From [24] [27] , we know that
.
Use the above two results, we obtain
,
i.e.,
.
Hence,
. The special case
is trivial. Here, we find that when
, (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 (
,
) are involved, which is instructive
for the discrete case in the next section. If
, what will happen? Clearly, it is impossible. From the conclusion [33] , if
, then
, otherwise
, which is in conflict with that
is measurable and limited. Therefore, in the continuous case,
cannot hold true. However, what about
the discrete case? The next section will answer.
3. The Discrete Heisenberg Uncertainty Principles
3.1. The Uncertainty Relation
First let us introduce a lemma.
Lemma 3:
, where
is the Frobenius matrix norm.
Proof: From the definition of the operator
in definition 4, we have
.
Exchange the locations of the sum operators, we obtain
![]()
Hence, according to the definition of the Frobenius matrix norm [27] [34] and the definition of DFRFT, we have
.
In the similar manner with the continuous case, we can obtain
. Since
, we have
, thus, we get
. Therefore, we can obtain the following theorem 2.
Theorem 2: Let
be the DFRFT of the time sequence
for transform parameter
, with
-concentrated on index set
. Let
be the numbers of nonzero entries in
(
respectively). Then
. (5)
Here, we find that when
, (5) reduce to the traditional case in Theorem 3 [(3.9), 25].
3.2. The Extensions
Set
in theorem 2, we can obtain the following theorem 3 directly.
Theorem 3: Let
be the DFRFT of the time sequence
with length N. ![]()
counts the numbers of nonzero entries in
(
respectively). Then
. (6)
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
. Therefore,
for
.
Through setting special value for
in theorem 3, we have
Corollary 1:
. (7)
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
(the continuous version of
) is band-limited, then
is obtained through sampling
. From the sequence length N in the definition of DFRFT in (2), we know the sampling period defined as
:
(
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
are limited within the scope
[32] [35] , i.e., all the energy of
must be within
. Without
loss of generality, we assume
based on the shifting property of FRFT [6] [32] , i.e., all the energy of
must be within
. Let
be the sites where
is nonzero, and ![]()
be the corresponding nonzero elements of
. Accordingly, from the definition of DFRFT
[6] [32] , we have
and. (8)
We rewrite (8) in terms of matrices and vectors. Define the matrix
, where
, then we obtain
,
where
,
and
.
Clearly,
is a
matrix, which includes
matrixes with dimensions of ![]()
so that we can rewrite matrix
as
and
, where
.
From the definition of DFRFT, we know that the bases
(for different
ks and
) are mutually orthogonal [6] [32] . Therefore, the different rows are not correlated so that ![]()
is nonsingular and
can be rewritten as
. Since every ele-
ment in
is not zero and
is nonsingular, then there must be a non-zero element in
at least. Other
wise,
, which is in conflict with
. Therefore, in every
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.
Clearly, if
and
, then the generalized uncertainty bounds are lower than the tradi-
tional cases. Therefore, the generalized uncertainty principles show that the resolution will be higher.
Theorem 4: Let
be the DFRFT of the time sequence
(
and
) with length N and
.
counts the number of nonzero elements in
. Then
, where
. (9)
Proof: From the assumption and the definition of DFRFT [6] [32] , we know
for.
where
,
.
Therefore, let
, we have [26]
![]()
where
and
with
and
with
.
Hence, we obtain
.
Set
, then
![]()
Using the triangle inequality, we have
, hence
![]()
From
and Parseval’s principle [6] , we obtain
.
Hence
.
Therefore, we obtain
![]()
Adding all the above inequalities, we have
with.
Similarly, from
and Parseval’s principle [6] , we obtain
, hence
.
From the definition and property of DFRFT [6] [32] we have
![]()
with
.
Hence, we finally obtain the proof
with.
4. 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
with
and sampling period
,
(see Figure 1(a)).
Clearly, we can obtain from Figure 1 that
,
,
. Therefore, we have
. 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 Figure 1(b) and Figure 1(c)).
5. 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
![]()
![]()
(a) (b) (c)
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).
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 [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.
Acknowledgements
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.
NOTES
*Corresponding author.