Relations between Gabor transforms and fractional Fourier transforms and their applications for Signal Processing

Relations Between Gabor Transforms and Fractional

Fourier Transforms and Their Applications for

Signal Processing

Soo-Chang Pei, Fellow, IEEE, and Jian-Jiun Ding

Abstract—Many useful relations between the Gabor transform

(GT) and the fractional Fourier transform (FRFT) have been derived. First, we find that, like the Wigner distribution function (WDF), the FRFT is also equivalent to the rotation operation of the GT. Then, we show that performing the scaled inverse Fourier transform (IFT) along an oblique line of the GT of ( ) can yield its FRFT. Since the GT is closely related to the FRFT, we can use it for analyzing the characteristics of the FRFT. Compared with the WDF, the GT does not have the cross-term problem. This advantage is important for the applications of filter design, sampling, and multiplexing in the FRFT domain. Moreover, we find that if the GT is combined with the WDF, the resultant operation [called the Gabor–Wigner transform (GWT)] also has rotation relation with the FRFT. We also derive the general form of the linear distribution that has rotation relation with the FRFT.

Index Terms—Fractional filter design, fractional Fourier

trans-form (FRFT), fractional multiplexing, fractional sampling, Gabor transform (GT), Gabor–Wigner transform (GWT), Wigner distri-bution function (WDF).

I. INTRODUCTION

T

(1) It has the additive property

(2) When , the FRFT becomes the identity operation. When

, the FRFT becomes the FT

(3)

Manuscript received July 9, 2006; revised January 4, 2007. The associate ed-itor coordinating the review of this manuscript and approving it for publication was Dr. Antonio Napolitano. This work was supported by the National Science Council, China, under contracts 93-2219-E-002-004 and NSC 93-2752-E-002-006-PAE.

The authors are with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C. (e-mail: [email protected]. edu.tw, [email protected]).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2007.896271

The FRFT can extend the utilities of the FT and is very useful for filter design, signal sampling, pattern recognition, optics anal-ysis, radar system analanal-ysis, and communication.

The FRFT is closely related to the Wigner distribution func-tion (WDF) [3], which is defined as

(4)

If is the WDF of a signal and is the

WDF of its FRFT , then they have the

fol-lowing clockwise-rotation relation [4]–[7]:

(5) Since the WDF and the FRFT have such a close relation, we often use the WDF to aid the FRFT for signal processing ap-plications. For example, when using the FRFT for filter design in the FRFT domain, the WDF is helpful for estimating the op-timal order [6]. For multiplexing and modulation, the WDF is helpful for assigning the time–frequency slot [7]. The fractional WDF, which generalizes the WDF by the FRFT, was developed in [24].

However, there is a problem for the WDF, i.e., “the cross-term

problem.” In other words, if and ,

, and are the WDFs of , , and ,

respectively, then

(6) This is because the WDF contains an autocorrelation term (see (4)) and is not a linear operation. The cross-term problem of the WDF can be seen from Fig. 2(d) or Fig. 5(c). This problem makes it difficult to distinguish the signal part, the noise part, and the cross-term part from the WDF. Using the WDF to analyze the FRFT characters is improper for the cases where the signal consists of several time–frequency components.

In [8], Ozaktas et al. found that Cohen’s class distributions also have clockwise-rotation relation with the FRFT. However, as the WDF, Cohen’s class distributions also have the cross-term problem.

In this paper, we derive many useful relations between the Gabor transform (GT) [9]–[11] and the FRFT (see Section II). The definition of the GT used in this paper is

(7) 1053-587X/$25.00 © 2007 IEEE

Fig. 1. GT ofF (u), where F (u) is the FRFT of f(t), f(t) = 1 for jtj  3, andf(t) = 0 otherwise. (a) G (t; !);  = 0. (b) G (t; !);  = =6. (c)G (t; !);  = 2=6. (d) G (t; !);  = =2.

We find that, like the WDF, the FRFT is also equivalent to the clockwise rotation of the GT (Section II-A). Thus, it is possible to use the GT instead of the WDF to perform signal processing in the FRFT domain. Compared with the WDF, the GT has the advantages of 1) avoiding the cross-term problem and 2) con-suming less computation time.

In particular, because it can avoid the cross-term problem, the GT is more effective than the WDF for filter design, sampling, multiplexing, and signal compression in the FRFT domain (see Section IV). Without being misled by the cross term, the optimal parameter of the FRFT can be more easily determined. (This problem has perplexed the researchers in the field of the FRFT for many years. Now, we can use the GT to solve it successfully.) Moreover, with the GT, the cutoff criterion and the location of time-frequency slot can also be obtained more easily. As the WDF, the GT can also be generalized into the fractional GT [25], [26]. However, the fractional GT does not satisfy the constraint in (37). A linear operation that does not satisfy the constraint will have no rotation relation with the FRFT.

Although sometimes the GT may not have very good clarity, we can combine the GT with the WDF, i.e., the Gabor–Wigner transform (GWT) (see Section III). The GWT also has rotation relation with the FRFT. Moreover, the GWT combines the ad-vantages of the GT (no cross term) and the WDF (high clarity).

II. RELATIONSBETWEENGABORTRANSFORMS ANDFRFTS A. Clockwise-Rotation Relation

Theorem 1: If is the FRFT of , is the GT

of , and is the GT of , then and

have the following relation:

(8) That is, the FRFT with parameter is equivalent to rotating the GT in the clockwise direction with angle . Its proof is shown in the Appendix. We perform an experiment in Fig. 1 to show that the FRFT is equivalent to the rotation operation for the GT.

for for (9)

Note that the standard definition of the GT in [9]–[11] is

(10)

Fig. 2. GTs and the WDFs ofs(t), r(t), and f(t) = s(t) + r(t). Note that the WDF has the “cross-term problem” but not the GT. (a) GT ofs(t). (b) GT ofr(t). (c) GT of f(t) = s(t) + r(t). (d) WDF of f(t).

In this paper, we modify the definition slightly as in (7) for the consideration of rotation relation. When using the standard def-inition, the amplitude has rotation relation with the FRFT, but the phase does not.

(11) Thus, we use (7) instead of (10) as the definition of the GT in this paper. With the modification, the GT has rotation relation with FRFTs both in amplitude and in phase, as (8).

B. Comparison With Wigner Distribution Function

We have proven that the clockwise rotation of the GT is equiv-alent to performing the FRFT. Although the WDF also has such similar property, it has the problem of “cross term.” If we use the GT instead of the WDF, because the GT is a linear operator and

need not calculate the autocorrelation ,

the cross-term problem can be avoided. That is, if

and , and are their GTs,

then

(12) In Fig. 2, we show the GTs and the WDFs of , , and

, where

for otherwise

(13) Obviously, the cross-term problem can be avoided if we use the GT instead of the WDF, see Fig. 2(c) and (d). This advan-tage is very important for filter design and other applications (Section IV).

Then, we discuss the implementation of the GT. Note that, although from (4) and (7), the ranges of integration for the WDF and the GT are both , from the fact that

the GT in (7) can be well approximated by

(15)

where (16)

to avoid the aliasing effect (17)

and is the bandwidth of , i.e.,

when

(18)

If in (16), the range of and are and

, then, to implement the GT, we need to perform the -point DFTs

times where means the least integer is no smaller than (19) In comparison, the digital implementation of the WDF is

(20)

where and (21)

is the bandwidth of . Note that, from (4),

. Thus,

from (18), the bandwidth of is about

twice the bandwidth of . Therefore, to avoid the

aliasing effect, of the WDF should be even smaller than that of the GT (about half). Moreover, although the WDF also needs times of the -point DFTs, as in (19), however, if is not time limited, then in (20), should be chosen as

(22) which is much larger than (17). Even when is time limited but the time duration is many times larger than 4.2919, the value of of the WDF is much larger than that of the GT.

Thus, because the integration range of the WDF is

and that of the GT can be reduced into ,

as in (15), the computation time of the GT will be much less than that of the WDF. It is another advantage of the GT.

Then, what is the disadvantage of the GT? The disadvantage is that the GT may have worse clarity than the WDF for the

case where has the form of . However,

the GT does not always have worse clarity. For example, when

Fig. 3. GT and the WDF forf(t) = exp(j0:15t ). (a) GT of exp(j0:15t ). (b) WDF ofexp(j0:15t ).

, the GT has better clarity than the WDF, as the example in Fig. 3. In fact, if

(23)

when , the WDF has higher clarity. When , the GT

will have better clarity.

In summary, compared with the WDF, the GT has the advan-tages of 1) avoiding the cross-term problem (due to linearity, ), 2) less computa-tion time for digital implementacomputa-tion ( in (16) is smaller than that in (20)), and 3) suitable for analyzing the signal

remained terms , where .

The disadvantage of the GT is blurring. However, this problem can be solved by combining the GT with the WDF. The new distribution can also avoid the cross-term problem, and its clarity is as good as that of the WDF. See Section III-A. C. Other Important Relations Between GTs and FRFTs

We then derive other relations between the GT and the FRFT in Table I. We can compare them with those of the WDF [6], [19]. The new materials here are not limited in deriving the relations between the GT and the FRFT. Since, from what we know, many relations, such as the 1) recovery, 2) projection, and 3) power inte-gration relations in Table I, between GTs and the original function

shown here have not been reported in the literature. The proofs of recovery, power integration, and power-decayed relations are in the Appendix. From the “recovery relation,” we

can obtain from the GT of if we perform the

half-scaled inverse Fourier transform (IFT) along the direction of (24) From the “power integration relation,” the integration of

along the oblique line of

where varies from (25)

is equal to the local energy of around . The

“en-ergy sum property” is a generalization of that in [10]. From

the “power-decayed relation,” if for , the

average power of

de-cays with when and the decayed rate is faster than .

TABLE I

RELATIONSBETWEEN THEGABORTRANSFORM AND THEFRFT

III. OTHER OPERATIONSTHATHAVEROTATION RELATIONSWITHFRFT

A. Combination of Gabor Transforms and WDFs

We have compared the performances of the WDF and the GT in Section II. The most important advantage of the WDF is its high clarity, but it has the cross-term problem. In contrast, the most significant merit of the GT is its ability to avoid the cross-term problem, but its clarity is not as good as that of the WDF. Then, one may ask how to achieve the goals of i) higher clarity and ii) avoiding the cross-term problem at the same time. This problem can be solved by the GWT.

Theorem 2: Gabor-Wigner Transform: If we define a new

time frequency (called the GWT) that has the

fol-lowing relation with the GT and the WDF :

where is

function with two variables (26)

then also has rotation relation with the FRFT: (27)

where and are GWTs of and its FRFT,

, respectively. Equation (26) can be further generalized as

(28)

where is any function with

vari-ables, and are any rotation-symmetry functions that

de-pend only on . The GWT defined in (28) also

satisfies (27) and has rotation relation with the FRFT. Note that, in (26) and (28), there are infinite ways to choose

and . For example, if

Fig. 4. GWT off(t) (defined the same as that in Fig. 2) and the GWT we use are defined in (29)–(31).

. We can even choose as

the logic operation of AND and set

as AND ]. From

Theorem 2, all the GWTs shown above have rotation

relation [see (27)] with the FRFT.

Moreover, if in (26) or in (28)

is chosen properly, the resultant GWT can avoid the cross-term problem while maintaining the clarity as good as that of the WDF. That is, the GWT can combine the advantages of the GT and the WDF and will be a powerful tool for analyzing the character of a signal in the FRFT domain.

In Fig. 4, we perform several experiments. For each of the subfigures, the GWT we choose are

in Fig. 4(a):

in Fig. 4(b): (29)

in Fig. 4(c): (30)

in Fig. 4(d): (31)

Comparing Fig. 4 with Fig. 2, we find that, like the GT, the GWT can also avoid the cross-term problem. Moreover, the clarity of the GWT is obviously better than that of the GT. Thus, the GWT combines the advantages of the GT and the WDF. In particular, using the GWT defined in (31) will achieve better performance than other types of GWT.

Corollary 1: Real Gabor Transform and Real-GWT: To process a real signal, in (26), we can choose real . That is,

real

(32) We can call it the real GT. From Theorem 2, it has rotation relation with the FRFT. It has the property of input– real-output and are suitable for processing a real signal. Generally,

we can choose real , and the resultant GWT

will be suitable for processing real signals.

Corollary 2: From (28), if is the GT of , then

a) ( is a constant) also has rotation relation

with the FRFT; it is a special case of (28) where

and (33)

b) and also have rotation

re-lation with the FRFT, where depends only on .

B. General Form of Linear Distribution With Rotation Relation We have known that the GT, the WDF, and their combination (the GWT) all have rotation relations with the FRFT. The GWT

can avoid the cross-term problem if is

chosen properly. However, the GWT is not always a linear op-eration. The following question is raised: Is there any other time–frequency distribution that a) has rotation relation with the FRFT and b) is a linear operation in addition to the GT? Then, we try to answer this question.

Any linear time–frequency distribution can be expressed as the following form:

(34)

If we replace by (the FRFT of ), then, from (1)

(35) Theorem 3: General Form of Linear Distribution That has Rotation Relation With the FRFT: For a linear time–frequency distribution defined in (34), if it has rotation relation with the FRFT, i.e.,

(36) then, from (35), the kernel should satisfy

(37) Moreover, because is a linear operation, the cross-term problem is avoided, as in the GT. The general form of quadratic distributions with rotation relation is discussed in [8]. Here, we derive the general form of linear distributions that have rotation relation with the FRFT.

Theorem 3 is helpful for finding new linear distributions that can analyze the character of a signal in the FRFT domain. For

example, suppose that has the form of

After some calculation (see the Appendix), we obtain the following.

Theorem 4: If the linear time–frequency distribution has the form (39), shown at the bottom of the page,

where and , then

has rotation relation with the FRFT, as in (36). Moreover,

from and , (39) can be rewritten as

(40)

where and . Equation (40) is

the general form of the second-order exponential distribution that has rotation-symmetry relation with the FRFT. The proof is shown in the Appendix.

Note that the GT is the special case of (40) where

and (41)

IV. APPLICATIONS FORSIGNALPROCESSING IN THE FRFT DOMAIN

Since the GT and the GWT are closely related to the FRFT, we can use it as an assistant tool for signal processing in the FRFT domain. Compared with the WDF, the GT and the GWT can avoid the cross-term problem. This advantage is very impor-tant for the applications that are related to multiple component analysis, such as filter design, sampling, multiplexing, multi-path problem analysis, communication for multiple users, and system modeling.

Suppose that the signal is composed of many time–fre-quency components

(42)

In many applications, we should decompose into

, i.e., perform multiple component analysis. However, this is hardly achieved by the WDF. Due to

cross terms, the WDFs of , and usually

mix together. In contrast, when using the GT (or the GWT), without the interference of the cross term, multiple component analysis will be easier to do.

A. Fractional Filter Design

It is known that we can use FRFTs instead of FTs for filter design [1], [13], [14], [20]–[23], as follows:

(43)

where and are the input and the output of the filter, and is the transfer function. Equation (43) can be gener-alized into the multiple-output form, i.e., trying more than one

[20]–[22]:

(44) When using the FRFT for filter design, if ’s are chosen prop-erly, many noises that are difficult to be removed by the filter designed by the FT will be eliminated successfully, especially for the noises whose instantaneous frequencies vary with time.

However, there are two important issues regarding the filter design in the FRFT domain:

A) how to choose the parameter properly;

B) how to choose the cutoff criteria when designing the low-pass, high-low-pass, or bandpass filter.

Although the method for searching the optimal transfer func-tion has been developed when the original signal is a random process and we have known some statistical proper-ties of [13], [14], there is no efficient way for determining the optimal order . Although we can use an iterative method to determine , it is cumbersome. Moreover, it is not suitable for the case where we should try more than one to filter out the noises [i.e., the case in (44)]. In addition, we still need a way to determine cutoff criteria when is deterministic and is the form of low-pass, high-pass, or bandpass filter.

It seems that the WDF is helpful for determining and the cutoff line. However, due to cross terms, using the WDF may not be suitable when consists of several time–frequency components.

In this paper, we find that the GT and the GWT also have rotation relations with the FRFT. This hints that we can use them instead of the WDF for fractional filter design. Moreover, since the cross term can be avoided, the signals and the noises are easy to separate by the GT and the GWT. We can apply the following theorem.

Theorem 5: The cutoff line on the – plane of the GT (or the GWT) is equivalent to the low-pass, high-pass, or bandpass filter in the FRFT domain. In fact, the following FRFT filter:

where for for (45)

(or for , for , dependent

on which part is the noise), is equivalent to the following cutoff line in the plane of the GT (or the GWT):

(46)

After applying the FRFT filter, the time–frequency components on one side of is removed, and those on the other side are preserved.

Theorem 5 provides an efficient way for designing the filter in the FRFT domain. With it, we can use the following process to design the FRFT filter by the GT (or the GWT).

Step 1) First, we perform the GT (or the GWT) for the re-ceived signal.

Step 2) Then, we determine the cutoff lines on the – plane of the GT (or the GWT) that can well separate the noise region from the signal region. The cutoff lines can be determined by the method of “clustering” and “segmentation,” which are usually used in image processing [27]. We first use clustering to segment the – plane into the signal regions, the noise re-gions, and the regions that are neither signal nor noise. Then, we try to find the straight lines that can well separate the noise regions from the signal re-gions. These straight lines are the cutoff lines we want. If the noise region is not adjacent to the signal region, the cutoff line can be placed between the two regions. If the noise region is adjacent to the signal region, the cutoff line should approximate to the boundaries of the two regions.

Step 3) Then, we determine the orders of the FRFT and the transfer functions by the cutoff lines.

i) The slope of the cutoff line determines the order of the FRFT.

ii) The distance between the cutoff line and the origin determines the criterion of the transfer function. See (45) and (46) in Theorem 5. Step 4) Then, using (44) and applying the parameters

deter-mined by Step 3), we can filter the noises success-fully by no more than times of the FRFTs ( is the number of cutoff lines).

We give an example in Fig. 5. The input signal is

show in Fig. 5(a) (47) It is interfered by the following three noises:

(48)

and is plotted in Fig. 5(b). We

want to recover from by the FRFT filters.

The WDFs of are plotted in Fig. 5(c).

In Fig. 5(c), the signals, the noises, and cross terms are mixed together. It is hard to know how to separate the signals from the noises after observing Fig. 5(c). In contrast, when we perform the GT, the cross-term problem can be avoided. In Fig. 5(d), the signals and the noises are separable. Moreover, we can use the GWT to further improve the clarity. The GWT [applying the definition in (30)] of is shown in Fig. 5(e), which shows the signal parts and the noise parts are very clearly separable.

Fig. 5. Using the FRFT together with the GT for filter design. (a) Signals(t). (b)f(t) = s(t) + noise. (c) WDF of f(t). (d) GT of f(t). (e) GWT of f(t). (f) Cutoff lines on the GWT. (g) FRFT of f(t) = s(t) + n(t)( = 01:1397). (h) High-pass filter. (i) Applying high-pass filter for (g). (j) GWT of (g). (k) Cutoff lines corresponding to the high-pass filter. (l) GWT of (i). (m) Re-covered signal. (n) ReRe-covered signal (real part) and the original signal.

From Fig. 5(f), we can use the following four cutoff lines to separate the noises and the signals, as follows:

(49) We can use them together with Theorem 5 to design the FRFT filters and remove the noises. For example, for cutoff lines

and , . From (45) and (46), it means that we should perform the FRFT with order -1.1397 for . The result is shown in Fig. 5(g) and its GWT is shown in Fig. 5(j). Then, in (49), since the values of for and are –0.65, and 0.65, from (46), to remove the component between and , we must multiply Fig. 5(g) by the high-pass filter in Fig. 5(h)

( for , otherwise).

The result is shown in Fig. 5(i). The GWT of Fig. 5(i) is shown in Fig. 5(l), which shows that the noise between and is filtered successfully.

Therefore, from (49) and Theorem 5, we can use the fol-lowing process to filter noises and recover the original signal

from : a b for for c d for for e f for for g (50)

(a) and (b) come from and , (c) and (d) come from , and (e) and (f) come from in Fig. 5(f). The recovered signal is plotted in Fig. 5(m). It is very close to the original signal [The recovered signal and are plotted together in Fig. 5(n).] The mean-square error (MSE) is only

MSE (51)

where MSE

and recovered signal

(52) In comparison, when using the GT, the MSE is 0.2105%. When using the WDF, only two of the three chirp noises are possible to be removed. In Fig. 5(c), due to the cross term, the central chirp noise part cannot be detected and removed. We can only remove the outer two chirp noises, and the MSE is as high as 63.18%. Therefore, when we use the GT and the GWT for filter design, the performance gains (compared with the case of the WDF) are

for Gabor transform performance gain

for the GWT performance gain (53)

Thus, using the GT and the GWT for the filter design in the FRFT domain has the following two significant advantages.

1) The design process becomes easier. The orders of the FRFTs and the criteria can be easily determined by the cutoff lines and the iterative process that often used in the literature can be avoided.

2) The ability of noise removing is much improved. In addi-tion to the examples in Fig. 5 and (47)–(53), we also per-form many other experiments and show that the GT and the GWT always have higher ability than the WDF for fil-tering the noise.

In fact, we can conclude as in Theorem 6.

Theorem 6: Using the FT can only filter the noises that do not overlap with the signals in the frequency domain. In contrast, using the FRFT can filter the noises that do not overlap with the signals on the – plane of the GT or the GWT.

Note that the frequency domain is 1-D but the – plane is 2-D. Two objects are easier to separate on a 2-D plane than on a 1-D axis. Thus, with the aid of the GT or the GWT, many noises that cannot be removed by the conventional filter will be filtered by the FRFT successfully.

B. Fractional Sampling

In [15], Xia introduced a way that used the FRFT instead of the FT for signal sampling. When using FRFTs for signal sam-pling, we first try to find such that the supporting of

is minimal, as follows:

supporting if

optimal width is minimal (54)

Then, we can sample as

where width (55)

To recover from , we can apply

(56) Some signals have narrower bandwidth in the FRFT domain than that in the frequency domain, as follows:

width

width (57)

In this case, if we use the FRFT with parameter instead of the FT to do signal sampling, the sampling interval in (55) can be larger, which means that the number of sampling points can be reduced. Alternatively, if the number of sampling points remains the same, the approximating error is reduced.

When has only one time–frequency (T–F) component, we can use the WDF to estimate the optimal value of . How-ever, when has two or more T–F components, as Fig. 2(d), since the “orient” for each of the T-F component is different, it is proper to separate into several T–F components and de-termine the optimal for each of the components. Owning to cross terms, it is difficult for the WDF to achieve this. For ex-ample, in Fig. 2(d), it is difficult to conclude whether the central region around (0, 0) is “the third T–F component” of or just the cross term of the left and right parts.

In contrast, when using the GT, since there is no cross-term problem, we can easily determine the number of the

time–fre-Fig. 6. Using the FRFT together with time–frequency component decompo-sition by the GT to sample the signalf(t) [see (13) and Fig. 2] in the FRFT domain.

quency components of and decompose into the

sum-mation of these components. For the example in Fig. 2, we can do the following.

a) First, we decompose in Fig. 2(c) into the left part (near to ) and right part (near to ).

b) Then, we use the power integration relation (see Table I) to estimate the optimal order [defined by (54)] for the left and right parts, which are

(58) c) Then, we can use the process in Fig. 6 to perform frac-tional sampling for . When the number of sampling points is constrained to 10, the reconstruction errors are as in (59) and (60), shown at the bottom of the page. C. Modulation and Multiplexing in FRFT Domain

The conventional modulation and multiplexing are

(61) In [7], it was generalized into the fractional modulation and mul-tiplexing, which modulates the FRFT of instead of :

where

(62) (63) The conventional multiplexing consider only whether there is unfilled space in the frequency domain. However, when using the FRFT for multiplexing, if there is some unfilled slot in the – plane, we can place a new signal in this slot and multiplex it with . Using the fractional multiplexing can improve the efficiency of time-frequency slot usage and has higher capacity for data transmission.

Fig. 7. Experiments using the FRFT together with the GWT for modulation and multiplexing. (a)G(u), consisted of seven components. (b) WDF of G(u). (c) GWT ofG(u). (d) GWT after f(t) is multiplexed.

However, the cross-term problem makes it difficult to know which T–F slot is spared by the WDF. Thus, to perform signal modulation and multiplexing in the FRFT domain, it is better to use the GT or the GWT to find the spare T–F slots.

We show an example in Fig. 7. In Fig. 7(a), consists of nine components. We want to multiplex the scaled Gaussian

signal into . When using the WDF,

as in Fig. 7(b), due to the interference of the cross term, it seems that there is no spare T–F slot for placing . When using the GWT, the spare slot can be detected, as the region surrounded by the dashed lines in Fig. 7(c). We then multiplex into the unfilled T–F slot in Fig. 7(c) by shifting, modulation, and the FRFT:

(64) Remember that in the – plane of the GWT, the FRFT has the effect of rotation, shifting has the effect of horizontal displace-ment, and modulation has the effect of vertical displacement.

Then, we add by . The GWT of the result of

mul-tiplexing is shown in Fig. 7(d). Note that, in Fig. 7(d), is successfully placed into the unfilled T–F slot in Fig. 7(c). D. Other Applications

The relations between the GT (or the GWT) and the FRFT are also useful for fractional matching pursuit [16],

super-res-Conventional method err 2.763

Sampling in the FRFT domain err 1.306 (59)

olution [17], fractional wavelet transform [18], data compres-sion, and spread-spectrum signal analyzing in the FRFT do-main, since these applications are closely related to the time–fre-quency character of a signal, and the character can be effectively analyzed by the GT or the GWT.

V. CONCLUSION

We have derived several interesting relations between the FRFT and the GT, including rotation relation, recovery relation, and power integration relation. Moreover, because the GT can avoid the cross term, which is a serious problem of the WDF, we can use the GT instead of the WDF to perform signal processing in the FRFT domain, such as fractional sampling and fractional filter design. In addition, we have also introduced the GWT, i.e., the combination of the GT and the WDF. The GWT combines the advantages of the GT (no cross term) and the WDF (high clarity) and also has rotation relation with the FRFT.

APPENDIX

Proof of Clockwise-Rotation Relation in Theorem 1:

Applying (7) and (1)

(65)

Then, applying the fact that (from [12])

(66) we can rewrite (65) as (67), shown at the bottom of the page.

Proof of Theorem 2: From (28) and (8)

Proof of Recovery Relation in Table I: From (7) and , we obtain

(68) Then, we generalize it into the case of the FRFT. If we replace

by , then

(69) After applying (8), we obtain the recovery relation.

Proof of Power Integration Relation in Table I: If we

inte-grate along the axis, then

(70)

(74)

(75)

(71)

After replacing by and applying (8) and (14), we

obtain the power integration relation.

Proof of Power-Decayed Relation: From the fact that (72) we obtain

when

(73) Then, together with the power integration relation in Table I, we obtain

It is equivalent to the power-decayed relation in Table I. Proof of Theorem 4: From (38), (37) can be rewritten as (74), shown at the top of the page. From (74), to eliminate the

two square root terms, should be . Then, (74) is

simplified as (75), shown at the top of the page. Comparing the coefficients of in (75), we obtain

i.e., where

(76)

Moreover, comparing the constant terms in (75), we obtain

(77)

(78) Thus, we can define as the following rotation symmetry function:

(79) We can show that, no matter how we choose and as, (78) will be satisfied for any , , and . Therefore, we obtain Theorem 4.

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Soo-Chang Pei (SM’89–F’00) was born in Soo-Auo,

Taiwan, R.O.C., in 1949. He received the B.S.E.E. degree from the 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 the Professor and Chairman in the Elec-trical Engineering Department, Tatung Institute of Technology and NTU, 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, 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 (OSA). He received the National Sun Yet-Sen Academic Achievement Award in Engineering in 1984, the Distinguished Research Award from the National Science 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 the Ministry of Education in 2002. He was President of the Chinese Image Processing and Pattern Recognition Society in Taiwan from 1996 to 1998. He became an IEEE Fellow in 2000 for contribu-tions to the development of digital eigenfilter design, color image coding and signal compression, and electrical engineering education in Taiwan.

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.

From 2001 to 2006, he was a Postdoctoral Re-searcher in the Department of Electrical Engineering, NTU. He is currently an Assistant Professor with the Department of Electrical Engineering, NTU. His current research areas include fractional Fourier transforms, linear canonical transforms, time–fre-quency analysis, orthogonal polynomials, fast algorithms, bioinformatics, quaternion algebra, number theoretic transform, data compression, pattern recognition, and filter design.