Discrete Fractional Fourier Transform Based on New
Nearly Tridiagonal Commuting Matrices
Soo-Chang Pei, Fellow, IEEE, Wen-Liang Hsue, and Jian-Jiun Ding
Abstract—Based on discrete Hermite–Gaussian-like functions,
a discrete fractional Fourier transform (DFRFT), which provides sample approximations of the continuous fractional Fourier transform, was defined and investigated recently. In this paper, we propose a new nearly tridiagonal matrix, which commutes with the discrete Fourier transform (DFT) matrix. The eigen-vectors of the new nearly tridiagonal matrix are shown to be DFT eigenvectors, which are more similar to the continuous Her-mite–Gaussian functions than those developed before. Rigorous discussions on the relations between the eigendecomposition of the newly proposed nearly tridiagonal matrix and the DFT matrix are described. Furthermore, by appropriately combining two linearly independent matrices that both commute with the DFT matrix, we develop a method to obtain DFT eigenvectors even more similar to the continuous Hermite–Gaussian functions (HGFs). Then, new versions of DFRFT produce their transform outputs closer to the samples of the continuous fractional Fourier transform, and their applications are described. Related computer experiments are performed to illustrate the validity of the works in this paper.
Index Terms—Discrete fractional Fourier transform (DFRFT),
discrete Fourier transform, Hermite–Gaussian functions.
I. INTRODUCTION
T
RANSFORM operators have been frequently exploited for signal analysis, compression, and other applications in signal processing area. One of the most important transform operators is the Fourier transform. The continuous fractional Fourier transform (FRT) is a generalization of the continuous Fourier transform [8]–[10]. Some of the possible applications of the FRT are optical signal processing, quantum mechanics, time-frequency representation, optimal filtering [8]–[12], etc. Because of the importance of the FRT, the discrete fractional Fourier transform (DFRFT), which can be used for digitally computing the FRT, was defined and investigated recently [2]–[7], [21]. The basic requirements of a definition of the DFRFT are [2]–[4] 1) additive and 2) approximating the con-tinuous FRT.Most of the recently proposed DFRFTs in the open literature [2]–[4] were based on a nearly tridiagonal matrix, which com-mutes with the DFT matrix and was first introduced by Dick-inson and Steiglitz [13]. For example, in [2] and [3], Pei et al. Manuscript received July 21, 2004; revised October 11, 2005. The associate editor coordinating the review of this manuscript and approving it for publi-cation was Dr. Steven J. Grant. This work was supported by the National Sci-ence Council of Taiwan, R.O.C., under contracts 93-2219-E-002-004 and NSC 93-2752-E-002-006-PAE.
The authors are with the Department of Electrical Engineering and the Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, 10617, R.O.C. (e-mail: [email protected]; [email protected]; [email protected]).
Digital Object Identifier 10.1109/TSP.2006.879313
found that the eigenvectors of the nearly tridiagonal commuting matrix of the DFT matrix proposed by Dickinson et al. [13] are discrete Hermite–Gaussian-like functions. Based on these dis-crete Hermite–Gaussian-like functions, Pei et al. proposed an eigendecompostion-based definition of the DFRFT, of which its transform results approximate samples of the continuous frac-tional Fourier transform. In [4], Candan et al. consolidated and provided rigorous discussions on the DFRFT proposed by Pei
et al.
In [1], Grünbaum introduced an exactly tridiagonal matrix that commutes with the centered discrete Fourier transform ma-trix of even size. Grünbaum also showed that his exactly tridi-agonal matrix can be viewed as the discrete analogue of the second-order Hermite differential operator [1]. Inspired by the work of Grünbaum, we propose in this paper a new nearly tridi-agonal matrix commuting with the ordinary DFT matrix of any size, even or odd. Most of the eigenvectors of this new nearly tridiagonal commuting matrix will be better sample approxima-tions of the continuous Hermite–Gaussian funcapproxima-tions (HGFs), in the sense of smaller norms of approximation error vectors, than those of the Dickinson and Steiglitz matrix. Based on the eigen-decomposition of the new nearly tridiagonal matrix, we will pro-vide a new definition of DFRFT, which is closer to the contin-uous FRT. Furthermore, it will be shown that the new nearly tridiagonal matrix can be linearly combined with the Dickinson and Steiglitz matrix to generate several matrices which also commute with the DFT matrix and have Hermite–Gaussian-like eigenvectors even closer to the continuous HGFs.
II. PRELIMINARIES
A. Continuous Fractional Fourier Transform
The -order continuous FRT of is defined as [8], [9]
(1)
where the transform kernel is given by
if is not a multiple of if is a multiple of if is a multiple of
(2) 1053-587X/$20.00 © 2006 IEEE
in which . It is known that the transform kernel can also be written as [9], [10]
(3) where
(4)
is the -order HGF with being the -order Hermite
polynomial.
B. DFRFT Based on Commuting Matrix
In this subsection, the definition of the DFRFT discussed in [2]–[4] is briefly summarized. The -point DFT matrix is defined by
(5) It is known that the DFT matrix has only four distinct eigen-values 1, -1, , and [14]. In [13], Dickinson and Steiglitz introduced a nearly tridiagonal matrix as shown in (6), which commutes with the DFT matrix . See (6) shown at the bottom
of the page, where . That is, with defined above,
. Therefore, the DFT matrix and the above matrix share a common eigenvector set, and we can find the eigen-vectors of from those of the matrix [4]. Furthermore, the matrix has the following properties [2]:
1) the eigenvectors of approximate samples of the contin-uous HGFs, and thus eigenvectors of can be seen as the discrete Hermite–Gaussian-like functions;
2) the eigenvector of with zero-crossings can approximate the -order continuous HGF.
In [2] and [3], Pei et al. defined the -order DFRFT matrix by
for odd
for even
(7)
where for odd and
for even , is a diagonal matrix with its diagonal entries corresponding to the eigenvalues for each column eigenvectors in , and is the -order discrete Her-mite–Gaussian-like function with zero-crossings and is ob-tained from the corresponding normalized eigenvector of . The
-based DFRFT of input data is obtained by .
III. NEWNEARLYTRIDIAGONALCOMMUTINGMATRIX In this section, we propose a novel nearly tridiagonal matrix which commutes with the ordinary DFT matrix of any size. Moreover, we will demonstrate that using the eigenvectors of the new matrix to approximate the samples of continuous HGFs is always better than using the eigenvectors of matrix, in the sense of smaller norms of approximation error vectors.
A. Definition and Some Basic Properties
Let us define an nearly tridiagonal matrix as (8), shown at the bottom of the page. That is, the nonzero entries of
are (9) .. . ... ... . .. ... ... (6) .. . ... ... . .. ... ... (8)
Note that except for the two 0.5 entries at the upper-right and lower-left corners, is tridiagonal, which is similar to the matrix of (6). Thus, we call them nearly tridiagonal. Since is real and symmetric, has real and orthogonal eigenvectors.
The rationale that we define in the form of (8) is described as follows. In [1], Grünbaum introduced an exactly tridiagonal matrix which commutes with the centered discrete Fourier transform matrix of even size. For example, we first consider the case that the sizes of these two matrices are both . The centered discrete Fourier transform matrix and its exactly tridiagonal commuting matrix defined in [1] are then, respectively
(10)
(11) Let us define a permutation matrix as
(12)
It can be shown that, after performing row and column
permu-tations on , the resultant matrix is the
in-verse DFT matrix . From and
, we have that commutes with .
Therefore, commutes with and thus also
com-mutes with . We conclude that the following matrix is a new commuting matrix of the four-point DFT matrix
(13)
The above discussion is about the four-point case. We then gen-eralize (13) to the -point case and define the nearly tridiagonal matrix in (8). We find that, whenever is even or odd, has the following important property.
Property 1: The matrix in (8) commutes with the
DFT matrix defined in (5), i.e., .
Proof: See Appendix I.
To explore the relation of eigenspaces between and , we need the following property.
Fig. 1. Continuous HGFs (solid line), the discrete Hermite–Gaussian-like func-tions based onS (‘*’), and the discrete Hermite–Gaussian-like functions based onT (“o”), with N = 25. (a) Eighth order: The error norms of S and T are 0.2637 and 0.0959, respectively. (b) Tenth order: The error norms ofS and T are 0.4965 and 0.1472. (c) Eighteenth order: The error norms ofS and T are 0.9312 and 0.5795.
Property 2: Let be the eigenspace of corresponding to
eigenvalue , i.e., . If , then
.
Proof: Since , . From Property 1, .
From Property 2, we can find the eigenvector set of from that of . Note the following.
1) From Property 2, it can be seen that if is the eigenvector of corresponding to an eigenvalue with multiplicity 1, then is also an eigenvector of .
2) If is an eigenvector of corresponding to an eigenvalue with multiple multiplicities, may not be an eigenvector of .
In Appendix II, we show that the entries of the eigenvectors of are solutions of a discrete version of the second-order dif-ferential equation of the continuous HGFs [4]. Therefore, the eigenvectors of approximate the continuous HGFs. To moti-vate our further discussions, we perform some computer exper-iments to show that the differences between the DFT eigenvec-tors derived from and the samples of continuous HGFs are usually smaller than those between the DFT eigenvectors ob-tained from and the samples of continuous HGFs. Its reason is also illustrated in Appendix II.
Computer Experiment 1: Fig. 1(a)–(c) shows the eighth-,
tenth-, and eighteenth-order continuous HGFs, the discrete mite–Gaussian-like functions based on , and the discrete Her-mite–Gaussian-like functions based on , with . From Fig. 1, we observe that the discrete Hermite–Gaussian-like func-tions based on are closer to the continuous HGFs than those
Fig. 2. Error norms of the discrete Hermite–Gaussian-like functions based on S ( ), T ( ), and S + 15T ( ) of various orders, where (a)N = 25, (b) N = 50, and (c) N = 100.
based on . The error norms, which are the Euclidean norms of the error vectors between the discrete Hermite–Gaussian-like functions based on (or ) and samples of the continuous HGFs, are plotted in Fig. 2. The error norms of the discrete Hermite–Gaussian-like functions based on both and tend to increase for higher order ones because of the aliasing effects. We also show in Fig. 2 the error norms for the discrete Her-mite–Gaussian-like functions based on , which will be explained in detail in Section IV.
Fig. 2 shows that the discrete Hermite–Gaussian-like func-tions based on approximate the continuous HGFs with smaller error norms than those based on except for the case where the order is very high. We did Matlab experiments for
each of (from to ) and found that when the
order is smaller than , where when
when (14)
the error norm of the discrete Hermite–Gaussian-like function based on is smaller than that based on . When the order is larger than , sometimes the discrete Hermite–Gaussian-like function based on has larger error norm but sometimes its error norm is still less. In fact, for orders larger than , the dis-crete Hermite–Gaussian-like functions based on both and fail to approximate the continuous HGFs well. Since the relation between the discrete Hermite–Gaussian-like function (denoted
Fig. 3. Error norms of discrete Hermite–Gaussian-like functions based onS andT for different N.
by ) based on (or ) and the continuous HGF (denoted by ) is (proven in Appendix II)
(15)
and the effect that using to approximate may
get worse when is large (this fact can also be seen from Appendix II), thus if the time support of is large, it is dif-ficult to use to approximate well. Here, we define the time support of the continuous HGF as the threshold such that for . The time support of the continuous HGF grows with its order. This is a possible interpretation as to why we cannot use the discrete Hermite–Gaussian-like function based on or to approximate it well when the order is high. From our experiments, we find that when the time supports of the continuous HGFs exceed , where
for
for (16)
the error norms of the discrete Hermite–Gaussian-like functions will be larger than 0.8.
From these experiments, we expect that, if we define a new DFRFT based on , the transform outputs will be closer to the samples of the continuous FRT than those of the DFRFT based on , especially for input signals with their spectra concentrated mostly at low frequencies. Therefore, the performances of the DFRFT based on will be similar to the continuous FRT. Many useful properties of the continuous FRT (such as the rotation in the WDF and the property that the shifting operation in the time domain corresponds to the mixture of the shifting and the modulation operations in the DFRFT domain) also apply for the
DFRFT based on .
In Fig. 3, we vary and compare the error norms of the eigenvectors obtained from and . We find that the error norms of with points correspond to the error norms of with points for lower order discrete HGFs. We have known that, when increases, the DFT eigenvectors obtained from and will converge to continuous HGFs. From the experiment in Fig. 3, we can conclude that the convergence rate of the eigen-vectors derived from will be twice of that of the eigenvectors derived from . This interesting phenomenon will be helpful for the further exploration of the DFT eigenvectors.
B. Computing DFT Eigenvectors Using Commuting Matrix
In [4], with defined in (6) being commutative with , Candan et al. introduced a method to find the common eigen-vector set of and . In this subsection, we develop a method to compute eigenvectors of completely from those of .
Property 3: For the matrix defined in (8), the trans-formed matrix
(17) is a block diagonal matrix, where and are two square
matrices of sizes and , respectively.
denotes the largest integer less than or equal to , and is the unitary symmetric matrix defined by [4]
if is odd
if is even
(18) where
. .. i.e., the exchange matrix
(19)
Proof: (a) If is odd
(20)
where is of size , is an
submatrix of , and is an
submatrix of . Entries of are zeros except for the entry at the lower-left corner.
(b) If is even
(21)
where is the vector of size and
is an submatrix of . The sizes of
and are and ,
respectively. Combining (a) and (b), we complete the proof. From the proof of Property 3, we have that the submatrices
and of the transformed matrix are
if is odd
if is even (22)
if is odd
if is even (23)
where is the zero vector, and is the
exchange matrix of size . We have the following two com-ments on eigenvalue multiplicities of and .
1) If is odd: From the definition of in (8), we see that in (22) is symmetric and exactly tridiagonal with nonzero subdiagonal entries, and is a matrix whose entries are all zero except a nonzero entry at the lower-right corner. From (22), we conclude that is symmetric and exactly tridiagonal with nonzero subdiagonal entries. Similarly, we have that in (23) is also symmetric and exactly tridi-agonal with nonzero subditridi-agonal entries. From [15], we know that any symmetric and exactly tridiagonal matrix with nonzero subdiagonal entries has distinct eigenvalues. Therefore, if is odd, the diagonal block matrices and
of in (17) have distinct eigenvalues.
2) If is even: From the definition of in (8), we can see that in (22) is symmetric and tridiagonal with nonzero subdiagonal entries, thus in (22) is symmetric and ex-actly tridiagonal. Besides, except that the entries at the last row and last column are all zero, subdiagonal entries of are nonzero. We can see directly from (22) that has two
independent eigenvectors corresponding to
the zero eigenvalue: and
(24) Therefore, with the exception that the zero eigenvalue of is of multiplicity two, has distinct eigenvalues. As to the matrix in (23), we can easily see that it is sym-metric and tridiagonal with nonzero subdiagonal entries. Thus, has distinct eigenvalues.
From (17), the relations of eigenvalues and eigenvectors
be-tween and the diagonal block matrices of , and ,
can be stated as the following two facts:
1) if is an eigenvector of corresponding to eigenvalue
, then , which is the even extension
of , is an eigenvector of corresponding to eigenvalue ;
2) if is an eigenvector of corresponding to eigenvalue
, then , which is the odd extension
of , is an eigenvector of corresponding to eigenvalue .
Therefore, we can compute eigenvalues of from the union
of eigenvalues of and , and compute even and odd
eigenvectors of from even extensions and odd extensions of eigenvectors of and , respectively. From previous dis-cussions, and have distinct eigenvalues with the ex-ception that, when is even, the zero eigenvalue of is of multiplicity two.
Property 4: a) If is an eigenvector, corresponding to any eigenvalue if is odd or to a nonzero eigenvalue if is even,
of in (17), then its even extension, , is an
eigenvector of . (b) If is an eigenvector of in (17), then
its odd extension, , is an eigenvector of .
Proof: a) Assume that is an eigenvector of corre-sponding to eigenvalue , which is any value if is odd or is nonzero if is even. From (17), is also an eigenvalue of . Since we do not know whether can be also an eigenvalue of
, we divide the discussion into two cases:
a.1) If the multiplicity of of is 1, is not an eigenvalue of and thus the eigenspace of corresponding to , , is of dimension 1. From Property 2, the even extension of is an eigenvector of .
a.2) When is an eigenvalue of multiplicity one of both and , if the corresponding eigenvectors of of and are and , respectively, then the eigenspace of corresponding to is of dimension two and is
with and
. From Property 2, with
, we have . That is
(25) with being even and being odd. Since is even, in (25) is even [5]. Using the fact that any even vector is orthogonal to the odd vector, we have . Therefore, is an eigenvector of
(26) b) The proof is similar to a). The proof of Property 4 is completed.
From Property 4, we can compute most of the eigenvectors of from even extensions and odd extensions of eigenvectors
of submatrices and of , respectively. But
if is even, zero is an eigenvalue of of multiplicity two and Property 4 does not apply for this case. Thus, if is even, the even extensions of eigenvectors corresponding to the zero eigenvalue of are not necessarily eigenvectors of . For
even, we then need to develop a method to compute the eigen-vectors of in the even subspace spanned by eigenvectors of of the zero eigenvalue.
Property 5: If is even, the two orthogonal eigenvectors of in the subspace spanned by even eigenvectors of of
eigen-value zero are , where
is the -length column vector with zero entries except
a 1 at the entry.
Proof: It is not difficult to verify from definition of in (8)
that, if is even, and are
two independent even eigenvectors of corresponding to the zero eigenvalue. The eigenspace of corresponding to the zero eigenvalue is then
, where and are any constants. From Property 2, it is reasonable to assume that
is an eigenvector of in , with being a constant. Since even eigenvectors corresponds to eigenvalue 1 for [4],
we assume , which results in if
. Therefore, the eigenvector of in corresponding
to eigenvalue is
(27) and the eigenvector of in corresponding to eigenvalue
is
(28) It can be easily seen that and are orthogonal, and thus the proof is completed.
In this subsection, we develop a method to find the orthog-onal common eigenvector set of and from the eigen-de-composition of . To define DFRFT based on , we also need to determine the orders of the discrete Hermite–Gaussian-like functions based on .
C. Determining the Hermite–Gaussian Orders of Matrix Eigenvectors
In [4], Candan et al. proposed a rule to assign orders for the Hermite–Gaussian eigenvectors of in (6). Similar to the pro-cedures in [4], we determine the orders of the eigenvectors of as follows.
We have shown that the diagonal block matrices and in (17) are both tridiagonal. Therefore, the eigenvector of or of the largest eigenvalue has no zero-crossing, the eigenvector of or of the second largest eigenvalue has one zero-crossing, etc. [4]. Thus, the eigenvectors of obtained from even extensions of the eigenvectors of has
zero-crossings. The eigenvectors of obtained from odd extensions of the eigenvectors of
has zero-crossings. We can
then assign the eigenvector of with zero-crossings as the -order discrete Hermite–Gaussian-like function. It should be noticed that, if is even, has two even eigenvectors corresponding to the zero eigenvalue. They have the largest and the second largest zero-crossings and should be determined separately from the following.
1) From Property 5, when , since is
odd, the eigenvector with the second most
and the eigenvector with the most zero-crossings is .
2) When , since is even,
the eigenvector with the second most
zero-cross-ings is ,
and the eigenvector with the most zero-crossings is .
IV. LINEARCOMBINATIONS OFMATRICES AND From previous discussions, the DFT matrix has two lin-early independent commuting matrices, and , which is de-fined in (6) and (8), respectively. Since the eigenvectors of both
and are HGF-like, we can expect that eigenvectors of are also discrete Hermite–Gaussian-like functions. More-over, from our experiments, we observed that the following phe-nomena always occur:
if then
if then (29)
where and are the DFT eigenvectors derived from and and is the continuous HGF. Thus, we can conjecture that if we combine and properly, we can obtain the DFT eigenvectors that are very close to . From (29), we can conclude that both and should be positive. Moreover, since, compared with , the eigenvectors of are more similar to the samples of HGFs, should be much larger than . In this section, we show that, with proper choices of and , the DFT eigenvector set derived from can very well ap-proximate the samples of continuous HGFs. The approximation errors are even less than those of the case where we use the DFT eigenvectors derived from or to approximate the samples of continuous HGFs.
A. New Versions of Discrete Hermite–Gaussian-Like Functions
Property 6: If and are any two constants, then
commutes with the DFT matrix , where and are
defined in (6) and (8), respectively.
Proof: Using the fact that both and commute with , we have
(30)
From Property 6, we can compute the eigenvectors of DFT
matrix using . Since and
have the same eigenvectors if is nonzero, we discuss in the following linear combinations of and of the form
and assume . We can easily observe that if
ap-proaches zero, eigenvectors of approaches those of
. On the other hand, if , eigenvectors of
ap-proaches those of . We next show through computer experi-ments that eigenvectors of are new versions of discrete Hermite–Gaussian-like functions and, with appropriate choice of , these new discrete Hermite–Gaussian-like functions ap-proximate samples of the continuous HGFs better than those
Fig. 4. Total error norms of discrete Hermite–Gaussian-like functions based on S + kT. (a) N = 25. (b) N = 145.
obtained from both and , in the sense that most of these new discrete Hermite–Gaussian-like functions have smaller norms of approximation error vectors.
Computer Experiment 2: To determine the optimal choice of
, in the sense of the minimal total error norm, we first com-pute the eigenvectors of , which are new versions of discrete Hermite–Gaussian-like functions. All of the resulting eigenvectors are compared with samples of the continuous HGFs of the corresponding orders and the total error norms are
calculated. For and , the total error norms
are plotted versus various values of (from to
with spacing 1) in Fig. 4(a) and (b). From these results and other experiments for different values of (up to 145), we find that the optimal in the sense of the minimal total error norm is ap-proximately 15. For 25, 50, and 100, in Fig. 2, we plot the error norms of discrete Hermite–Gaussian-like functions of
various orders based on , , and with . It is
obvious that discrete Hermite–Gaussian-like functions based on outperform those based on both and .
B. Computing DFT Eigenvectors from Those of Property 7: For the matrix defined in (6)
(31) is a block diagonal matrix [4], where is the unitary matrix defined in (18).
Proof: a) If is odd, similar to the proof a) of Property 3, we have
(32)
where is of size , is an
submatrix of , and is an
submatrix of . The entries of are zeros except a 1 at the lower-left corner.
b) If is even, similar to the proof b) of Property 3, we have
(33)
with being the vector of size and
being the submatrix of . Combining
(a) and (b), we complete the proof.
From the proof of Property 7, we conclude that
if is odd
if is even (34)
if is odd
if is even (35)
It is easy to see that is nearly tridiagonal, in the sense that is tridiagonal except the two nonzero entries at the upper-right and lower-left corners. From (17) and (31), we have
(36) Therefore, we have the following comments.
1) can be transformed to a matrix of block diagonal form from (36).
2) From (22), (23), (34), and (35), , , , and are all symmetric tridiagonal and have positive subdiag-onal entries, except that the last subdiagsubdiag-onal entry of is zero if is even. Therefore, if is nonnegative, the
di-agonal submatrices and are both
symmetric and tridiagonal with nonzero subdiagonal
en-tries. Thus, both and have distinct
eigenvalues.
3) Similar to the proof of Property 4 and from the fact that commutes with , we have that the even extensions
of eigenvectors of and the odd extensions of
the eigenvectors of are eigenvectors of . We
can therefore compute all of the eigenvectors of from the
eigenvectors of and . The orders
of the discrete Hermite–Gaussian-like functions based on can be determined similarly using the procedure similar to that proposed in [4].
V. DISCRETEFRACTIONALFOURIERTRANSFORMBASED ON
OR ANDITSAPPLICATIONS
From the results in previous sections, we know that we can
use the matrix or to derive the eigenvectors of
and the resultant eigenvectors can approximate the continuous HGFs better than those obtained from , in the sense of smaller
Fig. 5. Comparing the transform results of the continuous FRT and the DFRFTs based on S, T, and S + 15T for a rectangular function (real parts: solid lines, imaginary parts: dashes, ordera = 0:25). (a) Continuous FRT(a = 0:25); (b) DFRFT based on S (RMSE = 0:0913); (c) DFRFT based onT (RMSE = 0:0647); (d) DFRFT based on S + 15T (RMSE = 0:0526).
norms of approximation error vectors. Since the obtained DFT eigenvectors is very close to the eigenfunctions of the con-tinuous Fourier transform, it is possible to define the DFRFT whose performance is very similar to the continuous FRT. The
DFRFT based on (or ) is
for odd
for even
(37)
where for odd ,
for even , and is the -order
normalized DFT eigenvectors computed from (or ).
Computer Experiment 3: To compare the performance, we
first compute the continuous FRT and the DFRFTs based on , and of the following rectangular function
when elsewhere (38)
The continuous FRT is computed by numerical integration of the definition of FRT in (1). The DFRFTs based on , , and for the samples of in (38) are computed with and sampling interval is 1/8. The transform results are plotted in Fig. 5 with order . We find that the transform results of the DFRFTs based on and are more similar to those of the continuous FRT. Their root-mean-square errors (RMSEs) are less than that of the DFRFT based on .
In Fig. 6, we change the transform order from 0.1 to 1 and compare the RMSE. We find that, for any order , the DFRFTs
Fig. 6. RMSE of DFRFTs based onS, T and S + 15T of the samples of the rectangular function in (38).
Fig. 7. RMSE of DFRFTs based onS, T and S + 15T of the samples of the triangular function in (39).
FRT and the RMSEs are less than those of the cases where we apply the DFRFT based on . The experiment in Fig. 7 is similar to that in Fig. 6, except that the input is a triangular function
when elsewhere (39)
Since the performances of the DFRFTs based on and are very close to those of the continuous FRT, the tions of the continuous FRT can also be treated as the applica-tions of the DFRFTs based on and . For example, it is known that, after doing the continuous FRT of order , the WDF is rotated clockwise by the angle of [9], [16]–[18]:
where (40)
means the WDF of
(41)
and means the WDF of FRT . In
addition
if
(42)
Fig. 8. DWDF ofF [n] where F [n] is the DFRFT based on S + 15T for a rectangular function. (a)a = 0; (b) a = 0:2; (c) a = 0:4; (d) a = 0:6; (e)a = 0:8; and (f) a = 1.
Due to (40) and (42), we can use the continuous FRT together with space-translation and modulation for generalized signal multiplexing and space–frequency slot adaptation [17], [19].
Similarly, we can also use the DFRFT based on or for fractional multiplexing and space–frequency slot adaptation, since the properties in (40) and (42) also apply for the DFRFT based on or . The discrete form of the Wigner distri-bution function (DWDF) is
(43)
If is the DWDF of where is the DFRFT
of , then it can be shown that
where (44)
In Fig. 8, we show an example. The input is a rectangular function:
when otherwise (45)
Then we do the 256-point DFRFTs based on of orders 0, 0.2, 0.4, 0.6, 0.8, and 1 for and compute the DWDF. The results are plotted in Fig. 8(a)–(f). We use the gray-level to show the magnitude of the DWDF. From the results, it can be seen that, after doing the DFRFT of order , the DWDF is rotated clockwise by the angle of , as the continuous case.
In Fig. 9, we give an example that uses the DFRFT based on to do space-frequency slot adaptation (i.e., making the space–frequency distribution of a signal suit for some spe-cific region). Here, the input function is a chirp-modulated Gaussian function. Its DWDF is plotted in Fig. 9(a):
(46) We want to multiplex such that its energy is concentrated on the region of
Fig. 9. Doing time-frequency slot adaptation forf[n] such that the energy is concentrated on the rectangular region ofn 2 [10; 30] and m 2 [20; 60].
First, we do the DFRFT for to rotate and make
it align with -axis. We first compute the principal direction , which can be determined from the eigenvector cor-responding to the larger eigenvalue of the moment covariance
matrix :
(48)
(49) After some calculation, we find that the principal axis is
(50) Since
(51) thus, from (44), to make the DWDF align with axis, we should do the DFRFT of order where
(52) After doing the DFRFT of order 0.6428, the DWDF, is aligned with axis, as in Fig. 9(b). Then, to shift the DWDF into the desired time-frequency slot, we do space-translation and fre-quency modulation for the result of the DFRFT (denoted by
)
(53) After doing it, the energy of the DWDF is concentrated on the
slot of and , as in Fig. 9(c). Using the
DFRFT based on or together with space translation
and modulation, we can do time-frequency slot adaptation and hence do fractional multiplexing.
Moreover, since the partially space-invariant property is also preserved for the DFRFTs based and , therefore, as the case of the continuous FRT [9], [20], we can also use them for space-variant pattern recognition. This application is described in detail in our recent paper [24].
In addition to the above applications, other applications of the continuous FRT, such as filter design [11], [12], asymmetric edge detection [22], and matching pursuit [23] are also potential
applications of the DFRFTs based on and .
VI. CONCLUSION
In this paper, we proposed a new nearly tridiagonal com-muting matrix of the DFT. Its eigenvectors were found to be new discrete Hermite–Gaussian-like functions. We showed that most of the eigenvectors of the proposed nearly tridiagonal matrix well approximates the continuous HGFs. The approximation error is smaller than that of the case where we use the eigenvec-tors derived from S matrix to approximate continuous HGFs. We also provided rigorous discussions on the properties of eigen-vectors and eigenvalues of the new nearly tridiagonal matrix, and gave a method to compute the DFT eigenvectors completely from those of the new nearly tridiagonal matrix. Furthermore, by properly combining two nearly tridiagonal matrices, a new set of commuting matrices, whose DFT eigenvectors are even more similar to the continuous HGFs, were also obtained. Fi-nally, based on these new nearly tridiagonal matrices, new ver-sions of the DFRFT were defined, and their applications were illustrated.
APPENDIXI
MATRIX COMMUTESWITH THEDFT MATRIX (i.e.,
) 1) The entry of [1] is (A1) (A2) where (A3)
Substituting (A3) into (A2), we obtain
(A4)
2) Similarly, we can see that the entry of
is
(A5) 3) Comparing (A5) with (A1), we find that (A5) can be obtained by interchanging and in (A1). Since inter-changing and in (A4) does not change its value, we
prove that .
APPENDIXII
EIGENVECTORS OF AREDISCRETEHERMITE–GAUSSIAN-LIKE FUNCTIONS AND THEAPPROXIMATIONERRORISSMALLER
THANTHAT OF THEEIGENVECTORS OF MATRIX The second-order differential equation for the continuous Hermite–Gaussian function (HGF) is
is the order of (A6)
The central concept of our proof is that, instead of using
, we use ) ( is any positive integer) to
approximate the second-order differentiation in (A6) more ac-curately, as follows:
where
(A7) It is known that the relation between the HGF and the Hermite-Gaussian-like discrete Fourier transform (DFT) eigenvector is (the case of matrix has been proved in [13] and the case of matrix will be shown in (A22)–(A26))
i.e. and (A8)
Although for the relation between and , the interval is , however, to approximate the second-order differenti-ation in (A7) more accurately, it is proper to choose an even
smaller instead of (i.e., ). The value of should
be as large as possible to make very small. Then we apply (A6), (A7) and obtain a recursive relation for , , and
, where
(A9) We then do linear combination to convert it into a recursive re-lation for , , and (the detail of conversion can be seen from (A10)–(A20) that is an example of ). Then, we compare the result with the difference equations corresponding to and . We can use whose difference equations is closer to the relation among , , and to conclude whether the eigenvectors of or of can approximate the continuous HGF with smaller approximation error.
We show an example that uses , i.e., .
From (A6) and (A7)
where (A10)
We can replace by , , , and
and obtain
(A11)
(A12)
In (A10) and (A11), there are seven terms, , ,
, , , , and . To compare with
and , we should do linear combinations for (A10) and (A11)
to eliminate the terms of , , , and
(i.e., preserve only the terms of and .) It can be done by
(A13) After some calculation, we obtain
(A14) Then we try to simplify the right side of (A14). From
, we can eliminate the terms of and . Moreover, if the order of the HGF is not large,
will be small. Thus, the right side of (A14) becomes
(A15) where
. From (A11), . Thus, (from (A14) and (A15))
from A14 and A15 (A16)
Then, we apply (A12) to express the coefficients of , , and in terms of and . It requires very complicated
com-putations. Here we show only the case of in
detail.
(A17)
(A18)
Similarly
(A19) Therefore, (A16) and hence (A6) can be approximated by
(A20) Now, we have approximated (A6) by a recursive relation among
, , and . We compare it with matrix and
ma-trix. From [13], it is known that, if is an eigenvector of with
eigenvalue , then and it can be
approxi-mated as (using the fact that
(A21) For matrix, let be an eigenvector of with eigenvalue .
From (8), is
(A22) Assume that is large and . We can apply the Taylor
series for cosine and secant functions, and
. Then, in (A22), the coefficient function of becomes
(A23) Therefore, (A22) becomes
(A25) In (A23) we apply the fact that
when . It is not difficult to see that, when , (A25) approximates (A20), where
(A26) Thus, the eigenvector of is close to the samples of the contin-uous HGF and their eigenvalues also have a close relation. Then, we compare (A25) and (A21) with (A20) to conclude whether the eigenvectors of or those of will be more similar to the continuous HGF. We find the following:
(I) coefficient differences between (see (A21)) and the recursive formula for HGFs (see (A20)):
2nd term 3rd term
right-sided term (A27)
(II) coefficient differences between (see (A25)) and the recursive formula for HGFs (see (A20)):
2nd term 3rd term
right-sided term (A28)
From (A27) and (A28), it can be seen that the coefficient differences ratios for all the three terms are
for for i.e.
when is large (A29)
That is, the rows of are 16/11 times closer to the discrete form of the differential equation of the contin-uous HGF in (A6) than the rows of when choosing
. If in (A12) we choose
(A30) using the similar process from (A10)–(A29), we find that the coefficient difference ratio becomes
for for i.e. (A31)
When , it becomes 16 (for ):9 (for ) (i.e., 1.778:1). When , it becomes 35 (for ):19 (for ) (i.e., 1.842:1). In
general, when choosing , the coefficient
differ-ence ratio is
for for (A32)
When , it becomes
for for (A33)
That is, the matrix is two times closer to the differential equa-tion in (A6) than matrix if in (A7) we use a very small to approximate the second-order differentiation. Therefore, the eigenvectors of can approximate the continuous HGF with less error norm.
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Soo-Chang Pei (SM’89–F’00) was born in Soo-Auo, Taiwan, R.O.C., in 1949. He received B.S.E.E. de-gree from National Taiwan University, Taipei, Taiwan, R.O.C., in 1970 and M.S.E.E. and Ph.D. degrees from the University of California, Santa Barbara, in 1972 and 1975, respectively.
From 1970 to 1971, he was an engineering officer in the Chinese Navy Shipyard. From 1971 to 1975, he was a Research Assistant at the University of California, Santa Barbara. He was the Professor and Chairman in the Electrical Engineering department of Tatung Institute of Technology and National Taiwan University from 1981 to 1983 and 1995 to 1998, respectively. Presently, he is the Dean of Electrical Engineering and Computer Science College and the Professor of Electrical Engineering department at National Taiwan University. His research interests include digital signal processing, image processing, optical information pro-cessing, and laser holography.
Dr. Pei received the National Sun Yet-Sen Academic Achievement Award in Engineering in 1984, the Distinguished Research Award from the National Sci-ence Council from 1990 to 1998, the Outstanding Electrical Engineering Pro-fessor Award from the Chinese Institute of Electrical Engineering in 1998, the
Academic Achievement Award in Engineering from the Ministry of Education in 1998, the Pan Wen-Yuan Distinguished Research Award in 2002, and the Na-tional Chair Professor Award from Ministry of Education in 2002. He has been President of the Chinese Image Processing and Pattern Recognition Society in Taiwan from 1996 to 1998, and is a member of Eta Kappa Nu and the Optical Society of America (OSA). He became an IEEE Fellow in 2000 for contribu-tions to the development of digital eigenfilter design, color image coding and signal compression, and to electrical engineering education in Taiwan.
Wen-Liang Hsue was born in Taipei, Taiwan, R.O.C., in 1966. He received the B.S. and M.S. degrees, both in electrical engineering, from the National Taiwan University, Taipei, Taiwan, R.O.C., in 1988 and 1993, respectively. He is currently working towards the Ph.D. degree in communication engineering at the National Taiwan University.
From 1995 to 2000, he was with the Directorate General of Telecommunications, Taiwan, R.O.C. Since 2000, he has been a Lecturer in the Department of Electronic Engineering, Lan-Yang Institute of Technology, I-Lan, Taiwan. His current research interests include fractional Fourier transform, digital signal processing, and array signal processing.
Jian-Jiun Ding was born in Taiwan, R.O.C., in 1973. He received the B.S., M.S., and Ph.D. degrees, all in electrical engineering, from the National Taiwan University (NTU), Taipei, Taiwan, R.O.C., in 1995, 1997, and 2001, respectively.
Currently, he is a Postdoctoral Researcher with the Department of Electrical Engineering, NTU. His cur-rent research areas include fractional Fourier trans-forms, linear canonical transtrans-forms, orthogonal poly-nomials, fast algorithms, quaternion algebra, pattern recognition, and filter design.
![Fig. 9. Doing time-frequency slot adaptation for f[n] such that the energy is concentrated on the rectangular region of n 2 [10; 30] and m 2 [20; 60].](https://thumb-ap.123doks.com/thumbv2/9libinfo/8851680.242490/10.891.66.436.97.357/fig-doing-frequency-adaptation-energy-concentrated-rectangular-region.webp)
