Here, we first briefly review the definition of FRFT. For given continuous signal and, its FRFT [6] is defined as

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] :

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] .

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

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.

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

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.

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

Here, we find that when, (5) reduce to the traditional case in Theorem 3 [(3.9), 25].

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

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:

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.