Closed-Form Discrete Fractional and Affine Fourier
Transforms
Soo-Chang Pei, Fellow, IEEE, and Jian-Jiun Ding
Abstract—The discrete fractional Fourier transform (DFRFT)
is the generalization of discrete Fourier transform. Many types of DFRFT have been derived and are useful for signal processing applications. In this paper, we will introduce a new type of DFRFT, which are unitary, reversible, and flexible; in addition, the closed-form analytic expression can be obtained. It works in performance similar to the continuous fractional Fourier transform (FRFT) and can be efficiently calculated by FFT. Since the continuous FRFT can be generalized into the continuous affine Fourier transform (AFT) (the so-called canonical transform), we also extend the DFRFT into the discrete affine Fourier transform (DAFT). We will derive two types of the DFRFT and DAFT. Type 1 will be similar to the continuous FRFT and AFT and can be used for computing the continuous FRFT and AFT. Type 2 is the improved form of type 1 and can be used for other applications of digital signal processing. Meanwhile, many important properties continuous FRFT and AFT are kept in closed-form DFRFT and DAFT, and some applications, such as the filter design and pattern recognition, will also be discussed. The closed-form DFRFT we introduce will have the lowest complexity among all current DFRFT's that are still similar to the continuous FRFT.
Index Terms—Affine Fourier transform, discrete affine Fourier
transform, discrete Fourier transform, discrete fractional Fourier transform, Fourier transform.
I. INTRODUCTION
T
HE continuous fractional Fourier transform (FRFT) [1], [2], which is the generalization of Fourier transform, is defined as(1)
where the phase of is constrained in the range of
. It has been discussed in recent years and used in many applications such as optical system analysis, filter design, soluiton of differential equations, phase retrieval, pattern recog-nition, etc. The continuous FRFT satisfies the additivity prop-erty as
(2)
Manuscript received January 15, 1999; revised November 20, 1999. This work was supported by the National Science Council, R.O.C., under Contract NSC89-2213-E-002-092. The associate editor coordinating the review of this paper and approving it for publication was Prof. Chin-Liang Wang.
The authors are with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. (e-mail: [email protected]).
Publisher Item Identifier S 1053-587X(00)03291-8.
The FRFT has been further generalized into the special affine Fourier transform (SAFT) [3] (the so-called canonical transform [4]). It is defined as
when (3)
when (4)
where must be satisfied. Special affine Fourier
transform has the additive property
(5) where
and it has the reversible property
(5a) We will call this special affine Fourier transform the affine Fourier transform (AFT). The affine Fourier transform can extend the utilities of FRFT and is a useful tool for the optical system analysis. The effect of the FRFT and AFT can be interpreted by the Wigner distribution function (WDF). After doing the FRFT, the WDF of will be the rotation of the WDF of with angle [23], and after doing the AFT,
the WDF of will be the twisting of the WDF
of .
After the continuous fractional Fourier transform has been derived, many researchers have tried to derive their discrete counterpart, that is, the discrete fractional Fourier transform (DFRFT). We briefly review DFRFT's below. The name for each type of DFRFT is not recalled by the original authors. We give their names for easy classification.
1) Direct form of DFRFT. The simplest way to derive the DFRFT is sampling the continuous FRFT and computing it directly, but when we sample the continuous FRFT directly, then the resultant discrete transform we obtain will lose many important properties. The most serious problem is the DFRFT of this type will not be unitary and reversible. Besides, lacks closed-form properties, and not additive, so its applications are very limited.
2) Improved sampling-type DFRFT. In [5], a way to sample the continuous FRFT properly is introduced, and then, the resultant DFRFT will have the similar transform results as 1053–587X/00$10.00 © 2000 IEEE
the continuous FRFT. Although, in this case, the DFRFT can work very similarly to the continuous case and has a fast algorithm, but the transform kernel will not be orthog-onal and additive. Besides, many constraints, including the input signal constraint, should be satisfied.
3) Linear combination-type DFRFT. In [6]–[8], and [24], the discrete fractional Fourier transform is derived by using the linear combination of identity operation, DFT, time inverse operation, and IDFT. In this case, the transform matrix is orthogonal, and the additivity property and the reversibility property will satisfy for this type of DFRFT. However, the main problem is that the transform results will not match to the continuous FRFT. Besides, it will work very similarly to the original Fourier transform or the identity operation and lose the important character-istic of “fractionalization.”
4) Eigenvectors decomposition-type DFRFT. In [9]–[11], and [16], the authors derive another type of discrete frac-tional Fourier transform by searching the eigenvectors and eigenvalues of the DFT matrix and then compute the fractional power of the DFT matrix. This type of DFRFT will work very similarly to the continuous FRFT and will also have the properties of orthogonal, additivity, and reversibility. In [11], they have further improved this type of DFRFT by modifying their eigenvectors more similarly to the continuous Hermite functions, which are the eigenfunctions of the FRFT. These types of DFRFT’s lack the fast computation algorithm, and the eigenvectors cannot be written in a closed form.
5) Group theory-type DFRFT. In [13], the concept of group theory [15] is used, and the DFRFT as the multiplication of DFT and the periodic chirps are derived. The DFRFT derived will satisfy the rotation property on the Wigner distribution, and the additivity and reversible property will also be satisfied. However, this type of DFRFT can be derived only when the fractional order of the DFRFT equals some specified angles, and when the number of points is not prime, it will be very complicated to de-rive.
6) Impulse train-type DFRFT. Recently, in [14], another type of DFRFT is derived. This type of DFRFT can be viewed as a special case of the continuous FRFT. In this case, the input function is a periodic, equally spaced impulse train, and if the number of impulses in a period
is , and the period is , then . Besides, the
value of is limited and must be a rational number ( is the order of FRFT). Because this type of DFRFT can be viewed as a special case of continuous FRFT, many properties of the FRFT will also exist and have the fast algorithm. However, this type of DFRFT has many constraints and cannot be defined for all values of . Although many types of the discrete fractional Fourier trans-form (DFRFT) have been derived recently, no discrete affine Fourier transform (DAFT) has yet been derived.
In this paper, we will derive a new type of DFRFT, and then extend it into the discrete affine Fourier transform (DAFT). The DFRFT and DAFT we derived come from the proper sampling of the continuous FRFT and AFT. The DFRFT introduced in
[5] is also derived from the sampling of the continuous FRFT. Here, however, we will sample the continuous FRFT and affine Fourier transform by some proper intervals, and therefore, the transform matrix will be orthogonal and reversible. It can be written in the closed form so that many properties can be de-rived, and the fast algorithms can be achieved. Our idea comes from the [12] and [22]. In these papers, when we sample the fractional Fourier transform properly, we will obtain an unitary transform. We will improve upon these ideas.
In this paper, our focus is on the practical applications. Thus, although our DFRFT/DAFT seem neat in concepts and sacrifice the additivity property, they are very suitable for the practical applications due to the simpler and closed form of discrete fractional convolution and correlation introduced in Section II-D and the advantages listed in Section II-E. Our DFRFT/DAFT will have the lowest complexity among all the current the DFRFT/DAFT's that still have the similar properties as the continuous FRFT/AFT.
Due to the orientation of practical usage, we will derive two types of DFRFT/DAFT. These two types of DFRFT/DAFT are essentially the same but different in parameterizations. The first type we derive has the parameters that are more directly linked to the continuous FRFT/AFT and suits the applications of com-puting the continuous FRFT/AFT. On the other hand, type 2 has the simpler parameters set and allows more elegant expres-sion for the operator kernels. It is suitable for other applications of DFRFT/DAFT, such as the filter design, pattern recognition (described in Section IV), and the use for the phase retrieval discussed in [12] and [22] can also be improved by the type 2 DFRFT/DAFT proposed in this paper.
In Section II, we will give the derivation and definitions of our new types of DFRFT and DAFT. For different applications, we will use different parameterizations to define 2 types of DFRFT/DAFT. In Section III, we will discuss their properties and their transform results for some special signals. In Section IV, we discuss their applications. Finally, in Section V, we make a conclusion.
II. DERIVATION OFCLOSED-FORMDISCRETEFRACTIONAL ANDAFFINEFOURIERTRANSFORMS
A. The Closed-Form Discrete Fractional Fourier Transform of Type 1
1) The Derivation: To derive the DFRFT, we first sample
the input function and the output function of the
FRFT [see (1)] by the interval , as
(6)
where , and
. Here, we do not start our sampling at and . Instead, we try to make the DC component in the center. From (6), we can convert (1) as
The above equation can be written as the form of transformation matrix
(8)
where
(9) in order for (8) to be reversible. We will try to make the inverse transform to be the Hermitian (conjugation and transpose) of
when , i.e.,
for (10)
Then, from (8) and (9)
(11)
If we want the summation for in (11) to become ,
then
(12) where is some integer prime to . In this case, (9) becomes
(13) and
(14) We then normalize to satisfy (11) and obtain the
trans-form matrix as
sgn
(15)
For simplicity, we can choose sgn , and
rewrite (15) as
sgn
(16) Then, we obtain the following two formulas of discrete
frac-tional Fourier transforms (DFRFT) for 1) and 2)
DFRFT of type 1: 1) when is integer i.e., (17) 2) when is integer i.e., (18)
Additionally, the constraints that
are the number of points in the time, frequency domain
(19) must also be satisfied. We note that when and
, (17) will become the DFT, and when , (18) will become the inverse DFT. We also note
that when and is some integer, there is no
proper choice for and that satisfies this constraint of (15), and we cannot use (17) or (18) as the definition of DFRFT when . In fact, in these cases, we can just use the following definitions:
3)
when (20)
4)
when (21)
Equations (17)–(21) are the definition of the DFRFT.
2) Some Important Discussion About the DFRFT of Type 1: We also note, from (1) and (2), the inverse of the forward continuous FRFT with order is just the forward continuous FRFT with order . In fact, this property will also exist for the DFRFT defined as (17)–(21). Since, from (11), the inverse
of is just its Hermitian, i.e., , and if we
de-fine as (16), then we find
(22) In the above equation, we notice that the sampling interval of the input (the second subscript) and the sampling interval of the output (the third subscript) are exchanged. Then, we can conclude that
(23) that is, the DFRFT of order with the sampling interval in the input and at the output will be the inverse of the DFRFT of order with the sampling interval in the input and at the output. This is the reversible property of the DFRFT of type 1.
As the continuous FRFT, the DFRFT of type 1 also has the periodic property, that is
(24) The DFRFT of type 1 will have the period of as the contin-uous FRFT. The DFRFT’s of type 1 also have the conjugation property that
if is real (25)
The rest important properties will be derived in the Section III. Although the DFRFT introduced here has no additivity prop-erty, it can be convertible, that is, we can convert the DFRFT with some set of parameters into the DFRFT with another set of
parameters. Suppose we use for the DFRFT with the
parameter and use for the DFRFT with the
param-eter (The sampling interval in both time domains is the same and fixed). Then, from (16), we find
where is the DFT or IDFT of
(26) In addition
where
is some integer such that (27)
Therefore, we obtain
(28) Although the DFRFT defined in (17)–(21) has no additive prop-erty, if we fix , then we can convert the transform result of the DFRFT with order into order by two chirp multiplications and one convolution operation.
We note, in (19), that if is very small, then and must also be very small, and the number of points must increase.
This will increase the computation time of the DFRFT because for the continuous FRFT
so when is very small, we can first do the forward DFT for the sampling of and do the DFRFT defined as (17) or (18) with the order . Thus, we can change the DFRFT of type 1 to
(29) where
(29a) Then, because
for the case that is small, we can define the DFRFT as follows.
Modification form of the DFRFT of type 1 when
(30) where is the same as (29a), and the constraint for becomes (31) When is small, we can use (30) as the DFRFT.
The DFRFT of type 1 has a very important advantage, that is, it is efficient to calculate and implement. Because there are two chirp multiplications and one FFT, the total number of the
mul-tiplication operations required is , where
is the length of the output. Among all types of DFRFT, the linear combination type DFRFT [6]–[8], [24] will
have the least complexity and only require
multi-plication operations. However, it does not match the continuous FRFT and lacks many of the characteristics of the continuous FRFT. For example, it is hard to filter out the chirp noise with this type of DFRFT. For most of the other types of DFRFT, such as the improved directly sampling-type DFRFT [5], we need multiplication operations because there is one convolution operation and two chirp multiplication operations required. The DFRFT we introduce will have the lowest com-plexity among all types of DFRFT that still work similarly to the continuous FRFT.
3) Applications for Calculating the Continuous FRFT: We can use the DFRFT of type 1 to calculate the continuous FRFT. When using the DFRFT for this application, we first sample the input continuous function into a discrete sequence, do the for-ward DFRFT, and get the output of DFRFT as the sampling of the transform results of continuous FRFT. We note that because when we derive the DFRFT of type 1, we have normalized the
unitary [from (14) to (15)]. Thus, when using the DFRFT of type 1 to implement the continuous FRFT, we must consider this nor-malization factor, that is, if
(32) then
(33) We will use some examples to discuss this.
In Figs. 1 and 2, we will give some examples to illustrate the application of DFRFT for calculating the continuous FRFT. The original continuous input functions are:
Fig. 1: (rectangular)
Fig. 2: (triangular).
Then, we sample the input function with the sampling interval
Fig. 1: (rectangular)
Fig. 2:
and use the DFRFT of order and
. The value of is chosen as for
for for
We can compare the transform results for the rectangular func-tion with the results of the continuous FRFT in [2]. We find that the transform results of the DFRFT are similar to the ones of the continuous FRFT in these two examples. The closed form of the continuous FRFT of rectangular function is derived in [24], and the continuous FRFT of the triangular function can be calculated from the numerical method. We use these results to calculate our errors of the transform results in Figs. 1 and 2. The error is calculated from
err
(34)
where and are defined as (32), and is the
largest integer such that (we just consider the
interval that for simplification). Then, we obtain the error as in Table I. In fact, when choosing the same value of sampling interval in the time and frequency domains and the same number of points, the error of our DFRFT when used to calculate the continuous FRFT will be the same as the DFRFT introduced in [5]. However, our DFRFT will only require about half of the computation of the DFRFT introduced in [5], and many of the constraints in [5], such as the original signal, must
be bandlimited in all the domains ( when
for all the value of ) will be not required here.
Not all the input functions of the FRFT will be time limited, as is the above experiment. If the input function has very long time duration, we will modify the above process a little. We will
cut the input function into several subsections with short time duration and sample them
(35a)
where (35b)
and input them into DFRFT. We will use the shifting property for the continuous FRFT [2]
(36)
Thus, we require that must be the multiple
integers of
where is some integer.
Then, from and the relation of (19), we see
that must satisfy
where sgn (37)
Thus, if we choose as(37), then together with the shift prop-erty, we obtain
(38a)
(38b) and we can obtain the approximated value of (the
contin-uous FRFT of ) from
(38c) Therefore, for very long input, we can also use the DFRFT of type 1 to compute the transform results of the continuous FRFT [but the sampling interval must be chosen as (37)]. In Fig. 3, we will show an example. Here, we use
(39) as the input of continuous FRFT. The transform result of for continuous FRFT is
(39a)
We will choose , , and . In (37),
we choose
and then, . We then use the
DFRFT to compute the transform result of for continuous FRFT by the method from (35a)–(38c), and in (35a), we choose and . We plot the result in the Fig. 3. Then, we also use (34) to compute the error and obtain
err for
Fig. 1. DFRFT for the rectangular functionx(n) = 5(n=225), i.e., f(t) =
5(t=4:5). Upper left: = 0:05. Upper right: = 0:2. Lower left: = 0:4,
Lower right: = =4.
Fig. 2. DFRFT for the triangular functionx(n) = 3(n=125) , i.e., f(t) =
3(t=2:5)). Upper left: = 0:05. Upper right: = 0:2. Lower left: = 0:4.
Lower right: = =4.
When we use the DFRFT to calculate the continuous FRFT, we should consider its precision. There are two main constraints that must be satisfied for precision. First, the value of can be ignored outside
(constraint 1):
where
Second, we must consider the aliasing effect of the sampling. Consider first the bandwidth of the term
(40) Because
Max Max
if the bandwidth of is , then the bandwidth of (40) is . Then, from the sampling theory
(constraint 2):
There are some remarks about the above two constraints.
1) From (40), we find that if the value of (the effective width of the input signal) increases, then must de-crease, that is, the sampling interval in the time domain will also depend on the effective width of the input signal. 2) When (the bandwidth of ) increases and is fixed, then must be decreased, and should be increased.
3) When increases, i.e., , and is fixed,
then must be decreased, and will be increased. B. Closed-Form Discrete Affine Fourier Transform of Type 1
We can use a similar way to derive the DAFT that is analogous to the continuous case in (3). Similar to the process to derive the DFRFT, we find that if and satisfy
(41)
and , then the transform matrix will be
reversible, where
(42)
and . Thus, the DAFT of type 1 is
as follows. DAFT of type 1 1) when (43) 2) when (44)
The above DFRFT defined as (17), and (18) is a special case of
the DAFT wherein .
The reversible property for the DAFT of type 1 is
(45)
where is the DAFT of for
parame-ters and where the sampling interval is the input, and is the output. This reversible property is the same as the continuous AFT.
As for the case of DFRFT, when , we also find that the DAFT fails to be defined from (43) and (44) because the right side of (41) will be 0. However, for , the continuous affine Fourier transform results will be
(46) or
(47) where is the Fourier transform of . Therefore, we can define the DAFT, when , as follows.
1)
TABLE I
ERRORS FORUSINGDFRFTOFTYPE1TOCOMPUTE THECONTINUOUSDFRFT
Fig. 3. Experiment for using DFRFT to compute the transform result of continuous FRFT for the signal with nonfinite time duration. Upper-left: Input [see (39)]. Upper-right: Exact transform result [see (39a)]. Lower left: Transform results calculated from DFRFT withh = 022. Lower right: Transform results calculated from DFRFT withh = 055.
2)
when is not an integer. (49)
In (49), , and
(50) We note that it is no problem for (48) to be reversible. Besides, (49) is also reversible
(51) From the constraint of (41), we find that (43) and (44) can also be written as
(52) Thus, if we fix and , then only the values of sgn
will affect the transform result. Some important equality relations for the DAFT of type 1 are
(53) (54) Here we have assumed that the values of and are fixed.
The number of computations for the DAFT will also be
pro-portional to , where . This is
the same as the case of the DFRFT.
The DAFT has no additivity property, but it is convertible. That is, we can convert the DAFT with some set of parameters into another set of parameters. Suppose we use for
the DAFT with the parameter and use
for the DAFT with the parameter ( and
are fixed). Then, as for the case of DFRFT, we find that the following relation can be satisfied:
(55)
where is defined as (27), and can be obtained
from the DFT of a chirp
(56) In addition, we can use the DAFT to compute the continuous affine Fourier transform, and then, as in the case of the DFRFT, the following two constraints must be satisfied:
constraint 1:
where (57)
constraint 2:
(58) The above three remarks for using DFRFT to calculate the con-tinuous FRFT listed in Section II-A will also be applied here. C. Closed-Form Fractional and Affine Fourier Transform of Type 2
In general, we can use the discrete transform to do the fol-lowing: 1) Compute the continuous transform for spectral anal-ysis, and 2) suit for processing the discrete data signals. For the former case, the input of the discrete transform is the sampling of some continuous function. For the later case, the input is just a pure discrete sequence. For example, we can use DFT for com-puting the continuous Fourier transform, and the input is the sampling of some continuous function. Meanwhile, we can also use DFT for other digital signal processing applications, and in this case, the input is inherently a discrete sequence itself, such as the daily stock market or checking account, etc. Similarly, ex-cept for computing the continuous FRFT/AFT, we can also use DFRFT/DAFT for some other applications and just use them as the discrete data transforms.
When we use DFRFT and DAFT to compute the continuous FRFT/AFT, the mathematical form of the DFRFT and DAFT
should be almost the same as the continuous FRFT and AFT. Thus, in Sections II-A and II-B, we derive the DFRFT and DAFT from the sampling of kernels of the continuous FRFT and AFT. However, when we use DFRFT and DAFT for some other applications, the above requirement, such as phase alignment, is not necessary. We make DFRFT and DAFT remain the simple basic structures of FRFT and AFT, but they have the same abilities and are easier to compute and design.
We will derive the DAFT of type 2 from the transform matrix of the DAFT of type 1 and then simplify it into the DFRFT of type 2. The DAFT of type 1 defined as (43) and (44) has too many parameters. We can try to simplify it and set
. Then
(59)
Then, from , we find
(60) Because can be any real value, there will be no constraint for , and can be any real value. Thus, the DFRFT matrix defined as (59) will have three parameters without any constraint and has the free dimension of 3.
We note that the continuous affine Fourier transform has four parameters plus one constraint and has the free di-mension of 3 in total. Although, in (59), the free didi-mension is also 3, but the value of sgn can only be , in fact, the free dimension is near 2. Thus, as in (12), which has a parameter in the Fourier term, we can also put a parameter into the Fourier transform term of (59)
(61) where is prime to . Then, we find that the reversible property will be satisfied:
(62) Therefore, we can define the discrete affine Fourier transform (DAFT) as follows.
DAFT of type 2
(63) where
are the number of points in the time, frequency domain
is prime to (64)
and its inverse transform is the following.
Inverse DAFT of type 2
(65) We note that when , the inverse transform with the parameters is just the same as forward transform with
the parameters . Thus, when , the DAFT
with the parameters is the inverse of the DAFT
with the parameters
when (66)
In this paper, we use to denote the DAFT of type 2
(67) We note, from (1) and (3), that the continuous FRFT is a spe-cial case of the continuous AFT in that the inner and outer chirp
terms have the same parameters . In the
similar way, we can define the DFRFT from the DAFT by
set-ting and and obtain the following.
DFRFT of type 2
where (68)
We call the above DFRFT/DAFT defined in Sections II-A and B the DFRFT/DAFT of type 1. They are suitable for calculating the continuous FRFT/AFT. In addition, we call the DFRFT and DAFT defined in (68) and (63) the DFRFT/DAFT of type 2. The DFRFT/DAFT of type 2 are simple and suitable for other digital signal processing applications.
For the DAFT of type 2, we can use and to control the variation of the chirp in the frequency and time domains. When , the DAFT defined as (63) will be similar to a chirp multi-plication operation followed by a DFT. In addition, when , the DAFT will be similar to the DFT followed by a chirp mul-tiplication operation. When , then the transform matrix
will be a symmetry matrix, i.e.,
, and the transform matrix for the inverse DAFT
is just the conjugate of .
The DAFT/DFRFT of type 2 also need
multiplication operations. They also have no additive property, but they are convertible. We can calculate the DAFT with the
parameters from the DAFT with the parameters
from
(69)
where is the modulo symbol defined as (27), and
is defined as
can also be calculated from the DFT of a chirp
(71) After two chirp multiplications and one convolution, we can convert the DAFT with some parameters into the DAFT with other parameters. In the case where
(72) In this case, we can even save the convolution operation. Other important properties of the DFRFT/DAFT of type 2 are intro-duced in Section III.
We show the relations between DAFT of type 2 and its special cases in Table II.
D. Discrete Fractional/Affine Convolutions and Correlations Since the discrete fractional and affine Fourier transforms have been defined, we can use them to define the discrete frac-tional and affine convolutions and correlations. We only discuss the affine case, and the rest of the discrete fractional convolu-tion and convoluconvolu-tion can be obtained by substituting as
.
The discrete affine convolution can be defined as follows: Discrete affine convolution
(73) We must remember that the DAFT with the parameters is the inverse of the DAFT with the parameters
. If , then (73) can be
rewritten as
(74)
We note that the term has been cancelled. The
above equation can also be written as
(75) where
(76)
TABLE II
RELATIONSBETWEEN THEDAFTOFTYPE2ANDITSSPECIALCASES
In the case where
Thus, when , the discrete affine convolution can be written as follows.
Simplification form of the discrete affine convolution
(77) This is just the conventional discrete fractional convolution of
and and with the extra
multiplication of . We note that in this case,
will have no effect. Thus, for the simplification, when using the discrete affine convolution, we often set . In fact, for practical applications, control of the parameter is quite suf-ficient to control performance. The discrete affine convolution can be used in digital filter design, discrete fractional Hilbert transforms, etc.
From the definition of the continuous fractional correlation [18], we can use the similar way to define the discrete affine cor-relation. If is the fractional correlation of and , then we have the following. Discrete affine correlation
conj (78)
We use conj to denote the conjugation operation. We will de-note it as
(79) The original discrete correlation is the special case of (78) that
. Equation (78) can also be written as
For simplification, we set , and . Then, the above equation becomes
(81) We obtain the following.
Simplification form of the discrete affine correlation
(82)
where are defined in (27) and (70).
Equation (81) will be much simpler than (80). Thus, for the
simplification, we can set and
for discrete affine correlation. The discrete affine correlation can be used for pattern recognition. In Section IV-B, we will illustrate this.
The simplification form of discrete affine convolution/corre-lation defined as (77) and (81) only require one conventional dis-crete convolution, and the relations between its input and output are very clear. Therefore, the discrete affine convolution/corre-lation is easier to implement and analyze. It further enhances the proposed DAFT/DFRFT as a useful tool for digital signal processing.
E. Comparison of Closed-Form DFRFT and DAFT with Other Types of DFRFT
At the end of this section, we will compare the DFRFT and the DAFT introduced in this paper with other types of DFRFT.
The name for each type of DFRFT is as follows. • Direct: direct form of DFRFT;
• Improved: improved sampling type DFRFT [5]; • Linear: linear combination type DFRFT [6]–[8], [24]; • Eigenfxs.: eigenvectors decomposition-type DFRFT
[9]–[11], [16];
• Group: group theory-type DFRFT [13]; • Impulse: impulse train-type DFRFT [14];
• Proposed: the DFRFT/DAFT we derive in this paper, i.e., the closed-form DFRFT/DAFT.
Each term of the comparison means the following. • Reversible: whether the DFRFT is reversible;
• Similarity: whether the DFRFT is similar to its continuous counterparts;
• Closed form: whether the DFRFT can be written in the closed form;
• Complexity: the number of multiplication operations re-quired ( is the number of points);
• FFT: whether the DFRFT can be implemented by FFT and the number of FFT's required;
• Constraints: the number of constraints used for calculating the continuous FRFT;
• All orders: whether the DFRFT can be defined for all the order without constraints;
• Properties: the number of properties that can be derived; • Addv./Convt.: whether the DFRFT is additive or
convert-ible (the convertibility means that we can convert the DFRFT with some parameters into the DFRFT with other parameters);
• DSP: whether the DFRFT is suitable for the digital signal processing applications.
From Table III, we have seen that there are many advantages for the DFRFT/DAFT defined in this paper. The only disadvan-tage is that the additivity property is not satisfied. However, this disadvantage will have only a small affect on the practical usage of the DFRFT/DAFT. Besides, we can use the chirp multiplica-tion and convolumultiplica-tion to convert the DFRFT/DAFT with some parameters into another set of parameters, i.e., convertible. We give the proposed DFRFT/DAFT “OO” for last the term. This is because of their advantages of reversibility, less complexity, and the characteristic of “fractionalization;” in addition, many properties can be derived. Meanwhile, the DFRFT/DAFT de-fined here is very simple, and each of the parameters and will have the clear roles; thus, the design and the analysis will be very easy for practical applications.
III. PROPERTIES OF THEDISCRETEFRACTIONAL ANDAFFINE
FOURIERTRANSFORMS
A. Properties of the DFRFT and DAFT
Because the DFRFT/DAFT we derived are reversible, simple, and can be written in the closed form, their properties can be easily derived. We will discuss the properties of the DFRFT and DAFT in this subsection. Only the properties of the DAFT of type 2 are listed in Table IV and discussed here. The properties of the DFRFT/DAFT of type 1 and the properties of the DFRFT of type 2 can be obtained by the parameters substituting listed in Table I. We will use to represent the transform matrices of the DAFT, use to represent the transform operation, use to represent the input, and use
to represent the transform results of . Some of the properties are proved as follows.
TABLE III
COMPARISONS FORDIFFERENTTYPES OFDFRFT/DAFT
b) Proof of Property 10:
From the conjugation property for the transform matrix, we find that the inverse transform
(83)
can be rewritten as
Then applying property 1, we find that
(84) Then using the conjugation property for the transform matrix again, we obtain
(85)
From (85), we can calculate the inverse DAFT from the forward DAFT with the same parameters. (Remember that we can cal-culate the DFT from the IDFT).
From the modulation property, we find, as the continuous FRFT, after DAFT, that the modulation will partially remain as the modulation and partially become the shifting operation. Similarly, from the time-shifting property, we find that after the DAFT, the shifting operation will partially remain as the shifting operation and partially become the modulation operation. We note that for the original DFT and the IDFT, the shifting opera-tion will totally become the modulaopera-tion operaopera-tion, and the mod-ulation operation will totally become the shifting operation.
Thus, from the above discussion, we can say that many of the properties of continuous FRFR/AFT in [2] are also kept in our closed-form DFRFT/DAFT.
We also discuss the scaling property that only exists for the DFRFT and DAFT of type 1. In these cases, the input is the sampling of some continuous function.
Scaling property for the DFRFT and DAFT of type 1 Suppose is the sampling of the continuous function with the interval of
(86) and is the scaling of
(87) Now, if we sample with the interval of
(88) then we will find . Because the constraint of (19) or (41) must be satisfied, the sampling interval in the fractional
TABLE IV PROPERTIES OFDAFT
domain now is , where is the original sampling interval in the fractional domain. Thus
(89)
We can conclude that if
(90)
(91)
where we use to denote sampling with the
in-terval of , then from (43) and (44)
TABLE IV
PROPERTIES OFDAFT (Continued)
For the special case of DFRFT
where (93)
This scaling property is very similar to the continuous FRFT case in [2]. The scaling property of the continuous FRFT is
(93a)
where .
B. Transform Results for Some Special Signals
In Table V, we just list the transform results of some spe-cial signals for the DAFT of type 2. The transform result for comes from the first transform result and (85).
IV. APPLICATIONS OF THEDFRFTANDDAFT Because of the advantages of the DFRFT/DAFT listed in Sec-tion II-E, there are many signal processing applicaSec-tions for the DFRFT/DAFT. The main application for the DFRAT/DAFT of
type 1 is useful for computing the continuous FRF/AFT. In addi-tion, for the DFRAT/DAFT of type 2, there are also some prac-tical applications. We will introduce two examples, that is, the filter design (a special case of discrete fractional convolution) and the pattern recognition (use discrete fractional correlation). In fact, except for these applications, there are also some po-tential applications for the DFRFT/DAFT. For example, because the DFRFT/DAFT are the unitary transform constructed from the orthogonal chirp basis [see (94)] if a function is similar to the combination of several chirps, then it is convenient to use the DFRFT/DAFT to expand this function. It is also possible to use the DFRFT/DAFT for the phase retrieval [12], discrete frac-tional Hilbert transform, and beam shaping, etc.
According to our experiment, among the three parameters of DAFT, is the most important, next is , and will have the least importance. Besides, from the discussion in Section II-D, we find that when using the discrete affine convolution and cor-relation, it is convenient to set for the simplification. Thus, for digital signal processing applications, we have the following. 1) We usually use the parameter to control the
perfor-mance, and set and .
2) Sometimes (the applications about scaling), we also ad-just and still set .
3) In lesser conditions (such as the phase retrieval), we also adjust the value of .
The design method of the digital signal processing applications when using DAFT can be simplified by the above principles.
TABLE V
TRANSFORMRESULTS OFDAFTFORSOMESPECIALFUNCTIONS
A. Filter Design
For the inverse formula of DAFT of (63), we find
(94) where
(95) Therefore, the DAFT is an unitary transform with the or-thonormal chirp basis of as above. Thus, if a discrete function is a chirp or the combination of several chirps, then it is convenient to use the DAFT to analyze so that DAFT can be used for filter design to remove the chirp noise. If the chirp noise has the form
(96) then we can use the DAFT to filter out this chirp
(97) where represents the inverse of the DAFT with param-eters represents the width of bandstop filter, and
(98) We will give an example in the following. In this example, the number of points for the function is . We use the Gaussian function as the input
(99) Suppose it is interfered by the chirp noise so that the received signal becomes
(100) Then, from (96) and (98), we find that
(101)
Then, we set and and obtain the center of the
bandstop filter as
(102)
We also choose the width of the bandstop filter as 9. There-fore, the filter in the fractional domain is
(103) Then we substitute them into (97) to obtain the recovered signal. In Fig. 4, we will show the results, and the recovered signal will be very similar to the original signal . We also show the conventional DFT of defined in (100) and plot the result in the middle-left of Fig. 4. It is clear that the conventional DFT cannot separate the signal from the chirp interference.
Except for filtering out the chirp noise, since the DAFT system is time (space) variant, we can also use the DAFT to de-sign the time-varying filter to remove some noise or distortions with spectrum varying with time (space). The general formula for the filter design by DAFT is
(104) For simplification of computation, when we use the DAFT for the application of filter design, it is convenient to set and , as in the above case. It is even simpler than using
the DFRFT ( and ) because we save two chirp
multiplications (one for the forward DAFT and another one for the inverse).
B. Pattern Recognition
Because of the space-variant properties of the DFRFT and DAFT, we can use them for the pattern recognition to determine the same object located in a different place.
We will use the discrete fractional correlation described in Section II-D. We will apply the result of (81), that is, is the fractional correlation of and , and
. We will further simplify it by setting
and , i.e.,
DFT (105)
Then
(106) Suppose the input signal is time limited as
Fig. 4. Example of filter design by DAFT in Section IV-A. Upper left: Original signal. Upper right: Interfered signal (with chirp noise). Middle left: DFT of the interfered signal. Middle right: DAFT(p = 0; q = 0:025; s = 1) of the interfered signal. Lower left: Noise part filtered out. Lower right: Recovered signal.
and we only consider in the range that so
that (106) can be written as
(108) If the magnitude response is only considered, then
(109)
Now, we use as the reference template and suppose
that is a space-shifted version of the reference pattern, i.e., (110) and suppose is real. Then
(111) then the peak of is located at , and its value is
peak of
(112) This is an ideal peak at . Because the width of is , the peak will only distort a little when the phase of the
exponential term is in the range of when , i.e.,
(113) If is outside the above range, then the peak will distort seri-ously, and we can identify that the object is out of some region.
Fig. 5. Reference and the shifted object of the example in Section IV-B. Upper left: Reference. Upper right, Lower left, and Lower right: Shifted object with
n = 14; 45; 65; respectively.
Fig. 6. Discrete fractional correlation between the shifted object and reference in the example in Secion IV-B. Upper left to lower right:n = 0; 14; 45; and
65, respectively.
Thus, the discrete fractional correlation and, hence, the DAFT, can be used for the space variant pattern recognition to detect a pattern in a certain region.
We will give an example in the following. The total number of points is . The reference here is a rectangular function
(114) and the object is the space-shifted version of
(115)
We try four cases for , , and . The fractional
order is , and from the criterion of (113), . Thus, the criterion of (113) is satisfied
when , but for and , (113) is violated. We
find that for , the peak of will almost have
no attenuation, and for , the peak of will
attenuate a lot. We show the results in Figs. 5 and 6. We also calculate the conventional discrete correlation of and for comparison
DFT where
DFT DFT (116)
We change the last operation to be DFT instead of IDFT to avoid the time reverse. The results are shown in Fig. 7. We find, for the original discrete correlation, that the peak will never distort, no matter how much displacement there is. The conventional dis-crete correlation can be used for space-invariant pattern recog-nition, and the discrete fractional correlation can be used for space-variant pattern recognition.
Fig. 7. Conventional discrete correlation between the shifted object and reference in the example in Section IV-B. Upper left to lower right:n = 0;
14; 45; and 65, respectively.
V. CONCLUSION
In this paper, we have introduced new types of the discrete fractional Fourier transform (DFRFT) and the discrete affine Fourier transform (DAFT). The first type comes from sampling the continuous transforms directly, and the second type is the simplification of the former. We also discuss their applications for computing the continuous FRFT and affine Fourier trans-form, discrete filter design, and pattern recognition.
The DFRFT and DAFT we derive in this paper keep many of the important properties that the continuous FRFT and affine Fourier transform have. For example, they have similar formulas, are all unitary, reversible, partial time variant, use the chirp functions as their transform basis (see Sections III and IV-A), and they all transform a signal into the intermediate domain between time and frequency. Thus, as the continuous fractional and affine Fourier transforms are useful tools for continuous signal processing, we think the DFRFT and DAFT will also be very useful for digital signal processing. Since the DFRFT and DAFT can be efficiently implemented by FFT, and many important properties of FRFT and AFT can be kept, we believe they will have many signal processing applications in the future.
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Soo-Chang Pei (M’86–SM’89–F’00) was born in
Soo-Auo, Taiwan, R.O.C., in 1949. He received the B.S.E.E. degree from National Taiwan University (NTU), Taipei, in 1970 and the M.S.E.E. and Ph.D. degrees from the University of California, Santa Barbara UCSB, in 1972 and 1975, respectively.
He was an Engineering Officer in the Chinese Navy Shipyard from 1970 to 1971. From 1971 to 1975, he was a Research Assistant with UCSB. He was Professor and Chairman with the Department of Electrical Engineeering of Tatung Institute of Tech-nology and NTU from 1981 to 1983 and 1995 to 1998, respectively. Presently, he is a Professor with the Department of Electrical Engineeering, NTU. His research interests include digital signal processing, image processing, optical information processing, and laser holography.
Dr. Pei is a member of Eta Kappa Nu and the Optical Society of America.
Jian-Jiun Ding was born in 1973 in Pingdong,
Taiwan, R.O.C. He received the B.S. and M.S. degrees in electrical engineering from National Taiwan University (NTU), Taipei, in 1995 and 1997, respectively.
He is currently pursuing the Ph.D. degree under the supervision of Prof. S.-C. Pei in the Department of Electrical Engineering at NTU. He is also a Teaching Assistant. His current research areas include fractional and affine Fourier transforms, other fractional transforms, orthogonal polynomials, integer transforms, quaternion Fourier transforms, pattern recognition, fractals, filter design, etc.
