Sampling and Sampling Rate Conversion of Band Limited Signals in ...

158

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 1, JANUARY 2008

Sampling and Sampling Rate Conversion of Band Limited Signals in the Fractional Fourier Transform Domain Ran Tao, Senior Member, IEEE, Bing Deng, Wei-Qiang Zhang, Student Member, IEEE, and Yue Wang

Abstract—The fractional Fourier transform (FRFT) has become a very active area in signal processing community in recent years, with many applications in radar, communication, information security, etc., This study carefully investigates the sampling of a continuous-time band limited signal to obtain its discrete-time version, as well as sampling rate conversion, for the FRFT. Firstly, based on product theorem for the FRFT, the sampling theorems and reconstruction formulas are derived, which explain how to sample a continuous-time signal to obtain its discrete-time version for band limited signals in the fractional Fourier domain. Secondly, the formulas and significance of decimation and interpolation are studied in the fractional Fourier domain. Using the results, the sampling rate conversion theory for the FRFT with a rational fraction as conversion factor is deduced, which illustrates how to sample the discrete-time version without aliasing. The theorems proposed in this study are the generalizations of the conventional versions for the Fourier transform. Finally, the theory introduced in this paper is validated by simulations. Index Terms—Fractional Fourier transform (FRFT), sampling rate conversion, sampling theorem.

FT FRFT DTFRFT

NOMENCLATURE Fourier transform Fractional Fourier transform. Discrete-time FRFT. Notation for the FRFT. Transform result of for the FRFT. Notation for the DTFRFT. Transform result of for the DTFRFT.

I. INTRODUCTION HE fractional Fourier transform (FRFT) has a history dating back to the 1930s [1]. It was then employed by Namias to solve some differential and partial differential

T

Manuscript received May 20, 2006; revised April 27, 2007. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Antonia Papandreou-Suppappola. This work was supported in part by the National Science Foundation of China under Grants 60232010 and 60572094, and in part by the National Science Foundation of China for Distinguished Young Scholars under Grant 60625104. R. Tao and Y. Wang are with the Department of Electronic Engineering, Beijing Institute of Technology, Beijing 100081, China (e-mail: [email protected]. cn). B. Deng is with the Department of Electronic Engineering, Beijing Institute of Technology, Beijing 100081, China, and also with the Department of Electronic Engineering, Naval Aeronautical Engineering Institute, Yantai City 264001, China (e-mail: [email protected]). W.-Q. Zhang is with the Department of Electronic Engineering, Tsinghua University, Beijing 100084, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2007.901666

equations in quantum mechanics from classical quadratic Hamiltonians [2]. The results were later improved by McBride and Kerr [3]. They developed operational calculus to define the FRFT. The FRFT is a generalization of the conventional Fourier transform (FT) with potential applications. But, without proper physical illumination and fast digital computation algorithm, the methodology had remained unknown to the signal processing community until the introduction of the efficient digital computational algorithms of the FRFT and the interpretation as rotation in the time-frequency plane [4]–[10]. The FRFT processes signals in a unified time-frequency domain. Comparing with the FT, the FRFT is more flexible and suitable for processing nonstationary signals due to an additional degree of freedom. Furthermore, the fast algorithm of the discrete FRFT has also been proposed. Therefore, the FRFT has been widely applied in radar, communication, information security, etc. [5]–[18]. The FRFT of the order can be interpreted as a rotation in the time-frequency plane with an angle , and the time domain and frequency domain are the special cases of the FRFT domain and , respectively, where is an with being integer [4]. Hence, the conventional Shannon sampling theorem for the FT can also be considered as a special case of the sampling theorem for the FRFT [19]–[23]. Based on the relationship between the FT and FRFT, i.e., the three decomposition steps of the FRFT [4], Xia [19], Zayed [20] and Erseghe et al. [21], independently generalized the classical Shannon theorem from the frequency domain to the FRFT domain. Based on chirp-periodicity Erseghe generalized the FT for continuous-time, periodic continuous-time, discrete-time, periodic discrete-time signals to four corresponding versions of the FRFT [21]. In [22], Shannon’s interpolation theorem was generalized for the FRFT. It was concluded that a signal limited in a certain FRFT domain can be represented by its samples in any other FRFT domain. In [23], Zayed derived two sampling formulas to reconstruct a band limited or time limited signal, which use samples from both the signal and its Hilbert transformation sampled at half the Nyquist rate. Since signals are always processed within a finite time interval and a finite bandwidth in practical engineering applications, signal sampling based on the conventional Shannon sampling theorem can meet the criteria of ideal reconstruction. However, the sampling theorem for the FRFT shows that the sampling method is not always efficient with the possibility of unnecessary computational cost. In other words, a signal can potentially be sampled with a rate less than the Nyquist rate without aliasing of signal’s FRFT.

1053-587X/$25.00 © 2007 IEEE

TAO et al.: SAMPLING AND SAMPLING RATE CONVERSION OF BAND LIMITED SIGNALS

The sampling theorems mentioned earlier explain how to sample a band limited signal without aliasing. The advances in digital signal processing necessitates performing more complex signal processing operations such as coding, transmitting and storing. In order to reduce the computational load as well as saving the storage space, different sampling rates and the conversion between them are typically required in a signal processing system. Under these circumstances, the theory of multirate signal processing was introduced and improved [24]. This theory explains how to implement the sampling rate conversion from a discrete-time signal to another. The conventional sampling rate conversion theorem is operated in the FT domain, which helps to eliminate spectral aliasing due to decimation and mirror because of interpolation. If a signal is sampled by using the sampling theorem for the FRFT, the conventional sampling rate conversion theorem can not guarantee undistorted sampling rate conversion. Therefore, several questions are raised, such as 1) how to generalize the conventional sampling rate conversion theorem? and 2) how to achieve undistorted sampling rate conversion by eliminating the influences on signal’s FRFT caused by interpolation and decimation? In this paper, the sampling theorem for band limited signals in the FRFT domain is deduced from the viewpoint of signals and systems. Then, some discussions are presented to demonstrate its implementation. Next, we propose the generalization of conventional sampling rate conversion theorem, i.e., the sampling rate conversion theorem with a factor of rational fraction for the FRFT. After that, simulations are presented. Finally, conclusions are given. II. PRELIMINARIES The continuous-time FRFT with angle defined as [4]

of a signal

is

(1) where indicates the rotation angle in the time-frequency plane, is the kernel function, shown in (2) at the bottom of the . page, where The discrete-time FRFT (DTFRFT) is defined as [21]

159

Its inverse is

(4) is the sampling period, denotes the integral inwhere terval with the width . The FRFT has the following four special cases: (5) (6) (7) (8) where denotes the FT of . The time domain is the , while the frequency domain is FRFT domain with . Since the FRFT is the FRFT domain with periodic with the period of , can be limited in the interval . In this paper, the FRFT with is not taken into consideration, and if , the following result is obtained: (9) Therefore, it can be assumed that , i.e., , to analyze the fractional power spectrum, i.e., the squared modulus of the FRFT. In Sections III–V, this assumption is employed. According to [4] and [25], the th order FRFT can be interpreted as a rotation in the time frequency plane having an angle . Considering the Wigner distribution, we can write (10) where and indicate the Wigner distributions of and , respectively, and indicates the operator which rotates a two dimensional function counterclockwise by an angle of . Lohmann generalized (10) and obtained the relationship between the fractional power spectrum and the Radon–Wigner transform [26] as (11)

(3)

where denotes the Radon transform, i.e., the operator having the integral projection of a two dimensional function onto an axis making angle of with the axis. Another interpretation

(2)

160


Fig. 1. Relationship between the FRFT and the Radon–Wigner transform of a signal ( ).

xt

of (11) is the marginal integral after rotating the coordinates by an angle , i.e.,

X u

where signal (12) The above mentioned relationship expressed by (11) and (12) is shown in Fig. 1. The product theorem for the FRFT given below is utilized in Section III. Product theorem [27] tells that if , then

xt

xt

Fig. 2. Effect of sampling. (a) Continuous-time signal ( ). (b) FRFT of ( ), ( ). (c) Sampled signal ( ). (d) FRFT of ( ), denoted by denoted by ( ).

X u

x t

x t

denotes unit impulse. Then we obtain the sampled

(17) Based on the product theorem expressed by (13), we get the following results:

(18)

(13)

is If substituted into (18), after the corresponding manipulations, we obtain

III. SAMPLING THEOREM OF THE FRFT FOR A BAND LIMITED SIGNAL Definition of the Band Limited Signal in the FRFT Domain Definition 1: If the FRFT of signal lowing condition: when

(19)

satisfies the fol-

or

(14) then is the band limited signal in the FRFT domain, whose bandwidth is defined as (15) Note that the band limited signal refers to the one in the FRFT domain with , i.e., the FT domain, which is one of the special cases of our definition in this paper. A. FRFT of Sampled Signals The uniform impulse train is defined as

(16)

Since we have

,

(20) Equation (20) shows that replicates with a period of , along with linear phase modulation depending on the harmonic order (as shown in Fig. 2). When


The expression above shows that the part of modulates the amplitude of by modulate the phase. When

161

with but it does not

The equation above shows that the part of with modulates linearly the phase of in addition to amplitude modulation by . Since the main interval is in the digital frequency domain, the linear phase modulation is depicted as the polygonal line in Fig. 2(d). For general cases

X u

Fig. 3. Reconstruction of a signal. (a) FRFT of the original signal, ( ). (b) FRFT of the sampled signal, ( ). (c) Ideal bandpass filter in the FRFT ( ). domain. (d) FRFT of the reconstructed signal,

X u

X u

The sampling theorem for the band limited signal for the FRFT , we have , then the is expressed by (22b). Let lowpass version of sampling theorem for the FRFT is expressed as follows: (23) Furthermore, assuming , hence , in (22b) and (23), respectively, we obtain the conventional bandpass and lowpass sampling theorems [28].

The formula shows that the absolute value of the rate of linear increases, which means that phase modulation increases as the slope of the polygonal line also increases correspondingly. B. Sampling Theorem Since

replicates with a period of , and from the definition of band limited signal in the FRFT domain, we have , when or . Then, as illustrated in Fig. 3(b), we can determine an appropriate value for , , that satisfies

C. Reconstruction Formulas According to the sampling formulas mentioned earlier in this paper, the ideal reconstruction of the signal from its sampled version can be realized through a bandpass filter in the FRFT domain, as shown in Fig. 3. The transfer function of the bandpass filter is

otherwise.

(24)

Thus, the reconstructed signal is derived as follows: (21a) (21b) where rounds elements to the nearest integers towards zero. Therefore, there will be no overlapping between the shifted versions of , when the sampling rate is determined by (22a) , which corresponds to the lowpass case, (21b) When always holds true. Therefore, from (21a) we have (22b)

(25)

162


Also

(26) Substitute (26) into (25)

(27) According to (17), (28) follows, shown at the bottom of the page. For a lowpass signal, let , and . Then, (28) becomes

Fig. 4. Superposition of the FRFT domain on the time-frequency domain of the chirp signal x(t). and indicate the transform orders of the FRFT; is the counterclockwise angle between the Wigner distribution of the signal and u are the projections from the top of the Wigner and the time axis. u distribution W (t; ! ) on u and u axes, respectively.

can be achieved by time shift and frequency shift. In this case, (where the sampling rate does not have to be greater than , is the highest frequency of the signal in the frequency domain) according to the classical sampling theorem. On the other hand, the sampling rate should not to be less than (where , is the highest “frequency” of the signal in the th FRFT domain) according to the sampling theorem for the FRFT. The reconstruction filters (the lowpass filters) must be designed in the frequency domain or the th FRFT domain depending on what sampling theorem is used. Obviously, the sampling rate can be less than only if the inequality is satisfied as

(29) The result is the same as that obtained in [19], [20]. Correspondingly, (28) and (29) are changed to the conventional bandpass . and lowpass reconstruction formulas when

i.e., (30) From Fig. 4,

D. Discussion The sampling theorem derived in Section III-C is verified in this subsection by using a chirp signal as an example. Chirplike signals are often encountered in signal processing, existing as both real-world and artificial signals. Also, chirp-like signals can be interpreted as the first order approximation of frequency varying signals. Therefore, analysis and processing of chirp-like signals gained considerable importance in signal processing [29]. According to the relationship between the FRFT and time-frequency distribution, we examine the FRFT domain superimposed on the time-frequency plane with the 0th FRFT th FRFT domain as the domain as the time domain and the frequency domain. It is assumed that the time-frequency distribution of a chirp signal that is symmetrical around the origin as shown by the highlighted line in Fig. 4. The symmetry around the origin

and

(31)

or

and where above

(32)

equals the chirp rate. Combine the two equations

(33)

(28)


163

Substitute (33) into (30)

Fig. 5. Increment of the sampling rate by an integer factor L.

Fig. 5 shows a block diagram of a system which achieves the augment of the sampling rate by an integer factor of . Proof: Firstly we analyze the DTFRFT of the discrete-time signal after zero-valued interpolation

Finally, we obtain

(34) The inequality of (34) holds true if and are in the same quadrant. Thus, it is feasible to sample a chirp signal with the without aliasing in the th sampling frequency less than FRFT domain only if and are in the same quadrant. The relative cut-off frequency of the lowpass reconstruction filter in the th FRFT domain is limited in the interval as follows:

(35) Although Fig. 4 shows the case that the chirp rate is less than zero, the same conclusion can be drawn for the case that the chirp rate is greater than zero.

(38) Let

IV. SAMPLING RATE CONVERSION BASED ON THE FRACTIONAL FOURIER TRANSFORM Section III explains how to sample a continuous-time band limited signal to obtain its discrete-time version without overlapping in its fractional power spectrum. Section III gives a demonstration by using a chirp signal as an example. But, the realization of the ideal sampling rate conversion of this discrete-time signal has remained a challenge. This study proposes the generalization of the conventional sampling rate conversion theorem, i.e., the sampling rate conversion theorem with rational fraction times for the FRFT, describing how to convert the sampling rate of a lowpass signal to obtain another discrete-time version without any distortion. The derivation is detailed as below. Theorem 1: To increase the sampling rate by an integer factor , first pass the original discrete-time signal through an -fold expander, and filter the output of this expander with a lowpass filter called interpolation filter for suppressing the images produced by the expander. The -fold expander takes an input and produces an output sequence as

(39) From the assumption in Section II, it is known that . Substitute for and put (39) into (38)

(40) (36) The transfer function of the interpolation filter is

otherwise.

Because

, leads to

(37)

(41)

164


Substitute (41) into (40) (see equation at the bottom of the page). Finally, we obtain

TABLE I SCALING FACTOR OF ZERO-VALUED INTERPOLATION

(42) Since the FT is a special case of the FRFT with an order of , by inserting in (42) we can obtain the following version for the FT Fig. 6. Reduction of the sampling rate by an integer factor

(43) Equation (42) shows that the th FRFT of discrete-time signal after zero-valued interpolation by an integer factor can be obtained through the th FRFT of the original discrete-time signal, along with phase and amplitude modulation. From the viewpoint of the fractional power spectrum, and are both periodic. Since the period of is , the period of is . Then, it is deduced that the period of is . Therefore, we can achieve this sampling rate conversion by filtering with the lowpass filter designed as in (37). This FRFT domain lowpass filter is defined as in [30]. The proof of Theorem 1 is completed. The scaling factor does not always compress the spectrum in the version for the FRFT of zero-valued interpolation, which is different from the conventional FT. If , then , since the signs of and are the same. Based on and , we obtain , which leads to (44) If

,

; If . Therefore,

, varies from

M.

to as varies from 0 to . Let , we obtain , where denotes the transform order to hold . When , then . Hence, (44) is obtained again. Since is symmetrical around , approaches from as changes from to . The variation of scaling factor versus is listed in Table I. Theorem 2: To reduce the sampling rate by an integer factor , first utilize a lowpass digital filter called the decimation filter to ensure that the signal being decimated is band limited in the th FRFT domain, whose transfer function is

otherwise.

(45)

Subsequently pass the filtered signal through an -fold decimator, which takes an input and produces an output sequence as

(46) Fig. 6 shows the block diagram of a system that reduces the sampling rate by an integer factor .


165

Proof: If

(47)

otherwise then

(51)

(48)

otherwise.

Since the discrete-time FT of

is

Also

(49) If

we

replace

we obtain

by ,

we

ob-

tain

(52) From the sampling theorem for the FRFT in Section III, can be assumed in the range of . Thus, substitute (52) into (51)

(50) As is well known, if with the period of , then

is a periodic discrete-time signal

where expresses that the sum is obtained by terms. Since is a periodic discrete-time signal with the period of , we can obtain

Substitute the equation above into (50)

(53) Now we derive the DTFRFT of based on . Note that we use instead of in order to conveniently intro-

166


duce how to achieve the fractional sampling rate conversion in a cascade mode, which will be detailed at the end of this section

Finally, we have

(56) Substitute (53) into (56)

(54) Compare (54) with (40), and assume ,

, (57) Let in (57), the conventional version for the FT of -fold decimation is derived as

Rewrite (57) as (55) Because and , we obtain (58)

Thus

, Equation (58) demonstrates that the th DTFRFT, of the discrete-time signal after decimation is obtained by 1) scaling the th FRFT of the original discrete-time signal with the factor of , and 2) shifting the th FRFT domain by . This process is accompanied by phase shift, amplitude, and phase modulation. In order to avoid overlapping, the lowpass filter has to be used before decimation in the th FRFT domain. The transfer function of the lowpass filter is (59)

otherwise. The proof of Theorem 2 is also accomplished. Similarly, we can obtain

(60) The variation rule of where

,

is shown in Table II, .


167

where denotes the signal duration. For example, in the simulation mentioned above. Now the original signal is reconstructed with a chirp rate of 1.5 through the lowpass filter in the th fractional Fourier domain with the unchanged parameters as above. Obviously, the sampling frequency of 20 Hz satisfies the sampling theorem derived earlier in this paper. According to (33) and (35), we have

TABLE II SCALING FACTOR OF THE DECIMATION

i.e., Fig. 7. Sampling rate conversion by a factor L=M ((cot =L ) = cot = (cot =M )).

(62) The sampling rate conversion with an integer factor is described above. But, the sampling rate conversion with only an integer factor does not meet the requirement of an actual system, which requires the sampling rate to be converted with a rational fraction factor. We can achieve the sampling rate with a rational fraction factor as shown in Fig. 7 according to the sampling rate conversions as shown in Figs. 5 and 6. Corollary 1: To achieve the sampling rate conversion by a ra: first, pass the original discrete-time signal tional fraction through an -fold expander; second, filter the output of this expander with a lowpass filter whose transfer function is otherwise. Finally, pass the filtered signal through an The block diagram is shown in Fig. 7.

Since

and

,

, we have

(61) -fold decimator.

V. SIMULATION RESULTS A. Sample and Reconstruction of a Chirp Signal According to the Sampling Theorems for the FRFT Firstly the sampling theorem for the FRFT is demonstrated with a chirp signal. Let , Hz and the , then the sampled signal sampling interval being is

Then (63) Thus, by substituting (63), , and into (62), the original signal can be recovered without distortion using an as appropriate filter having the cut-off frequency (64)

Assume ( , ), according to (20) and (33), under this condition the critical chirp rate is to avoid overlapping in the th fractional power spectrum of . The value of the critical chirp rate is obtained by solving the following equation as

Determining and substituting it to (29), we obtain the reconstruction expression as follows:

(65) Fig. 8 gives the partial plots of the original signal and the reconstructed signal expressed by (65). It can be seen that the reconstructed signal approaches the original signal.

168


Fig. 8. Partial plots of the original signal and the reconstructed signal with the chirp rate = 1:5 Hz/s.

Fig. 10. Fractional power spectrum of the discrete signal decimated with M = < .

3, = 0:959, = 0:1571

B. Sampling Rate Conversion for the FRFT

Fig. 9. RMSE of signal reconstruction versus the chirp rate.

Subsequently, we verify the sampling and reconstruction theory for the FRFT using the root mean square error (RMSE), whose definition is as follows:

(66) denotes the sample number. Thus, the signal reconwhere struction error is

(67)

where is chosen as 0.001 s, . Fig. 9 shows the RMSE of signal reconstruction versus the chirp rate with the same cut-off frequency, . In this case, the RMSE of signal reconstruction is insignificant if the chirp rate is less than the critical value 2. When the chirp rate increases greater than , the RMSE increases correspondingly. This phenomenon is consistent with our derivation.

First, simulation analysis for decimation in the FRFT domain is presented. Figs. 10–13 demonstrate the fractional power spectrum of the discrete time signal decimated by the factors of 2 , and are and 3, in which the modulus of located in the sequence from the top to the bottom. Figs. 10–13 show that the variation rule revealed by this simulation is consistent with Table II. Second, the simulation of analyzing the zero-valued interpolation in the FRFT domain is performed. Assuming , the sampling rate is 40 Hz, and the sampling interval is . The th fractional power spectrum of the disis illustrated in Fig. 14. Fig. 15 shows crete-time signal fractional power spectrum after the th is manipulated by the zero-valued interpolation with factor of 2 as well as lowpass filtering in the th FRFT domain. The upper subfigure shows the fractional power spectrum after the zero-valued interpolation, while the lower subfigure shows the fractional power spectrum processed by a following lowpass filter in the FRFT domain. According to (66), the error of sampling rate conversion is defined as

(68) where denotes the discrete copy of that is the DTFRFT of the true signal sampled with the desired sampling rate, while is the discrete copy of . The DTFRFT , thus obtained by sampling rate conversion, only . Fig. 16 demonstrates has a small RMSE, the corresponding fractional power spectrum with the factor of interpolation increased to a value of 3. The RMSE turns out to be equal to 2.2731 10 in this case. Based on the sampling rate conversion method with a factor of shown in Fig. 7, simulation of sampling rate conversion with factor of 3/2 and 3/4 in turn have been demonstrated using the same original signal as shown in Fig. 14. In the former case the RMSE of sampling rate conversion is 7.6743 10 and in


Fig. 11. Fractional power spectrum of the discrete signal decimated with 3, = 1:249, = = 0:3218.

M=

Fig. 12. Fractional power spectrum of the discrete signal decimated with M = 2, = 1:1071, = = 0:4636.

Fig. 14.

169

th fractional power spectrum of the sampled signal. = 2:5536.

Fig. 15. th fractional power spectrum of the zero-valued interpolation with factor of 2. = 1:9296.

Fig. 13. Fractional power spectrum of the discrete signal decimated with M = 2, = 1:3258, = 0:7854 > .

Fig. 16. th fractional power spectrum of the zero-valued interpolation with factor of 3. = 1:7359.

the latter case its value is 3.3748 10 . Figs. 17 and 18 show the fractional power spectrum of sampling rate conversion with a factor of 3/2 and 3/4, respectively, (as shown in Fig. 7), where is the th fractional power spectrum after zero-valued

interpolation with factor of , is the fractional power spectrum processed by following lowpass filter in the FRFT domain, and is the final th fractional power spectrum after sampling rate conversion with factor of .

170


Fig. 17. Sampling rate conversion with 3/2.

Fig. 18. Sampling rate conversion with 3/4.

VI. CONCLUSION In this paper, the sampling of a continuous-time band limited signal is discussed to obtain its distortion-free discrete-time version and achieve the perfect sampling rate conversion for the discrete-time version. The two problems in the FRFT domain are analyzed. This study derives the sampling theorem for the FRFT of band limited signals from the viewpoint of signals and systems. Since chirp-like signals are common in signal processing, by sampling a chirp signal we demonstrate that a signal can be sampled without aliasing at a rate less than the Nyquist rate by using the sampling theorem for the FRFT. Furthermore, this study derives the sampling rate conversion theory with a factor of rational fraction, and validates its feasibility by performing required simulations. Since the FT is a special case of the FRFT, the results obtained in this paper generalizes the conventional theorems of sampling a continuous-time signal and converting the sampling rate for the FT. The results obtained in this paper extend the theories of the FRFT, which can advance corresponding applications such as filter banks theorem for the FRFT. REFERENCES [1] H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform. West Sussex, U.K.: Wiley, 2001.

[2] V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl., vol. 25, pp. 241–265, 1980. [3] A. C. McBride and F. H. Kerr, “On Namias’ fractional Fourier transform,” IMA J. Appl. Math., vol. 39, pp. 159–175, 1987. [4] L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process., vol. 42, no. 11, pp. 3084–3091, Nov. 1994. [5] D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. B, vol. 38, pp. 209–219, 1996. [6] H. M. Ozaktas et al., “Convolution, filtering, and multiplexing in fractional Fourier domains,” J. Opt. Soc. Amer. A, vol. 11, pp. 547–559, 1994. [7] H. M. Ozaktas et al., “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process., vol. 44, no. 9, pp. 2141–2150, Sep. 1996. [8] B. Santhanam and J. H. McClellan, “The discrete rotational Fourier transform,” IEEE Trans. Signal Process., vol. 44, no. 4, pp. 994–998, Apr. 1996. [9] S.-C. Pei, M.-H. Yeh, and C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process., vol. 47, no. 5, pp. 1335–1348, May 1999. [10] S.-C. Pei and J.-J. Ding, “Closed-form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process., vol. 48, no. 5, pp. 1338–1353, May 2000. [11] I. S. Yetik and A. Nehorai, “Beamforming using the fractional Fourier transform,” IEEE Trans. Signal Process., vol. 51, no. 6, pp. 1663–1668, Jun. 2003. [12] M. A. Kutay et al., “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process., vol. 45, no. 5, pp. 1129–1143, May 1997. [13] L. Qi et al., “Detection and parameter estimation of multicomponent LFM signal based on the fractional Fourier transform,” Science in China, vol. 47, pp. 184–198, Apr. 2004. [14] H.-B. Sun et al., “Application of the fractional Fourier transform to moving target detection in airborne SAR,” IEEE Trans. Aerosp. Electron. Syst., vol. 38, no. 4, pp. 1416–1424, Apr. 2002. [15] M. Martone, “A multicarrier system based on the fractional Fourier transform for time-frequency-selective channels,” IEEE Trans. Commun., vol. 49, no. 6, pp. 1011–1020, Jun. 2001. [16] I. Djurovic, S. Stankovic, and I. Pitas, “Digital watermarking in the fractional Fourier transformation domain,” J. Netw. Comput. Appl., vol. 24, pp. 167–173, 2001. [17] B. Hennelly and J. T. Sheridan, “Fractional Fourier transform-based image encryption: Phase retrieval algorithm,” Opt. Commun., vol. 226, pp. 61–80, 2003. [18] R. Tao, B. Deng, and Y. Wang, “Research progress of the fractional Fourier transform in signal processing,” Science in China, vol. 49, pp. 1–25, Feb. 2006. [19] X. G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Process. Lett., vol. 3, no. 1, pp. 72–74, Mar. 1996. [20] A. I. Zayed, “On the relationship between the Fourier and fractional Fourier transforms,” IEEE Signal Proces. Lett., vol. 3, no. 4, pp. 310–311, Dec. 1996. [21] T. Erseghe, P. Kraniauskas, and G. Cariolaro, “Unified fractional Fourier transform and sampling theorem,” IEEE Trans. Signal Process., vol. 47, no. 12, pp. 3419–3423, Dec. 1999. [22] C. Candan and H. M. Ozaktas, “Sampling and series expansion theorems for fractional Fourier and other transforms,” Signal Process., vol. 83, pp. 2455–2457, 2003. [23] A. I. Zayed and A. G. Garcia, “New sampling formulae for the fractional Fourier transform,” Signal Process., vol. 77, pp. 111–114, 1999. [24] P. P. Vaidyanathan, “Multirate digital filters, filter banks, polyphase networks, and applications: A tutorial,” Proc. IEEE, vol. 78, no. 1, pp. 56–93, Jan. 1990. [25] H. M. Ozaktas, N. Erkaya, and M. A. Kutay, “Effect of fractional Fourier transformation on time-frequency distributions belonging to the Cohen class,” IEEE Signal Process. Lett., vol. 3, no. 2, pp. 40–41, Feb. 1996. [26] A. W. Lohmann, “Relationships between the Radon-Wigner and fractional Fourier transforms,” J. Opt. Soc. Amer. A, vol. 11, pp. 1398–1401, Jun. 1994. [27] L. B. Almeida, “Product and convolution theorems for the fractional Fourier transform,” IEEE Signal Process. Lett., vol. 4, no. 1, pp. 15–17, Jan. 1997. [28] R. G. Vaughan, N. L. Scott, and D. R. White, “The theory of bandpass sampling,” IEEE Trans. Signal Process., vol. SP-39, no. 9, pp. 1973–1984, Sep. 1991. [29] A. Bultan, “A four-parameter atomic decomposition of chirplets,” IEEE Trans. Signal Process., vol. 47, no. 3, pp. 731–745, Mar. 1999. [30] A. I. Zayed, “A convolution and product theorem for the fractional Fourier transform,” IEEE Signal Process. Lett., vol. 5, no. 4, pp. 101–103, Apr. 1998.


Ran Tao (M’00–SM’04) was born in Nanling County, Anhui Province, China, in 1964. He received the Ph.D. degree in electrical engineering from Harbin Institute of Technology, Harbin, China, in 1993. He has been a Professor at Beijing Institute of Technology since 1999. From March 2001 to April 2002, he was a Visiting Scholar in the University of Michigan, Ann Arbor. His research interests are in the areas of the fractional Fourier transform, time-frequency signal processing, and signal processing for radar and communications. Dr. Tao was the recipient of National Science Foundation of China for Distinguished Young Scholars in 2006, and of Teaching and Research Award for Outstanding Young Teachers in Higher Education Institutions of MOE, P.R.C., in 2000. He is the Vice-President of Chinese Radar Industry Association, and a Fellow of Chinese Institute of Electronics.

Bing Deng was born in Hengyang City, Hunan Province, China, in 1975. He received the M.S. degree in underwater acoustic engineering from Naval University of Engineering, Hubei, China, in 2000, and the Ph.D. degree in communication and information systems from Beijing Institute of Technology, Beijing, China, in 2006. He is currently a Lecturer in the Department of Electronic and Information Engineering, Naval Aeronautical Engineering Institute, Shandong, China. His main research interests are in fractional Fourier transform, time-frequency analysis, and nonstationary signal processing.

171

Wei-Qiang Zhang (S’04) was born in Hebei, China, in 1979. He received the M.S. degree in communication and information systems from Beijing Institute of Technology, Beijing, in 2005. He is currently pursuing the Ph.D. degree in information and communication engineering at Tsinghua University, Beijing. His research interests include time-frequency analysis, parameter estimation, higher-order statistics, radar signal processing, and speech signal processing.

Yue Wang received the B.S. degree in radar engineering from Xidian University, Shanxi, China, in 1956. He was the President of Beijing Institute of Technology (BIT), Beijing, China, from 1993 to 1997. He is currently a Professor with BIT. His research interests include information system theory and technology, and signal processing. Prof. Wang is a Fellow of the Chinese Academy of Sciences and the Chinese Academy of Engineering.