Nov 1, 1997 - Yan Zhang, Ben-Yuan Gu, Bi-Zhen Dong, and Guo-Zhen Yang. Institute of Physics, Academia Sinica, P.O. Box 603, Beijing 100080, China.
November 1, 1997 / Vol. 22, No. 21 / OPTICS LETTERS
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Fractional Gabor transform Yan Zhang, Ben-Yuan Gu, Bi-Zhen Dong, and Guo-Zhen Yang Institute of Physics, Academia Sinica, P.O. Box 603, Beijing 100080, China
Hongwu Ren, Xueru Zhang, and Shutian Liu Harbin Institute of Technology, Harbin 150001, China Received May 14, 1997 A fractional Gabor transform (FRGT) is proposed. This new transform is a generalization of the conventional Gabor transform (GT) based on the Fourier transform to the windowed fractional Fourier transform (FRFT). The FRGT provides analyses of signals in both the real space and the FRFT frequency domain simultaneously. The space – FRFT frequency pattern can be rotated as the fractional order changes. The FRGT has an additional freedom, compared with the conventional GT, i.e., the transform order. The FRGT may offer a useful tool for guiding optimal filter design in the FRFT domain in signal processing. 1997 Optical Society of America
The well-known Gabor transform (GT) is performed on space – frequency signal representations that can be used to analyze transient signals. It has proved useful in a wide range of applications, including radar systems, sonar systems, communication, and space science.1 – 3 The mathematical definition of the GT is given by1,2 Z Gs s p, qd ssxdgp sx 2 qdexps2i2ppxddx , (1) where ssxd is a signal; p stands for the complex conjugate; p and q are the coordinates in the space and the frequency domains, respectively; and gsxd is a selected window function. The signal ssxd is f irst multiplied by a window function, and this synthesizing signal is then analyzed by the Fourier transform. The position of the window is translated along the spatialcoordinate axis. If the window function satisf ies Z (2) jgsxdj2 1 , the signal function ssxd can be reconstructed from its GT spectrum by the inverse GT: Z Z ssxd Gs sp, qdgsx 2 qdexpsi2ppxddpdq . (3) Let p mDx2 and q nDx1 , where m, n 0, 61, 62, . . .; the discrete GT can then be expressed as Z ssxdgp sx 2 nDx dexps2i2pmDx xddx , G sm, nd s
1
2
(4) where Dx1 and Dx2 are the spatial resolution and the frequency resolution, respectively. Since 1993, when the fractional Fourier transform (FRFT) was introduced into optical information processing, it has attracted extensive interest because of its special relationship with the conventional Fourier transform. The optical definition of the FRFT,4,5 its optical implementation architecture,6,7 fractional convolution and correlation operations,8,9 and its fast algorithm10 have been proposed. Applications of FRFT include shift-variant filtering, pattern recognition,8,9,11 the study of beam propagation, optical diffraction,8,12 lens design,13 and phase retrieval.14 The mathematical definition of the FRFT with order P is given by 0146-9592/97/211583-03$10.00/0
Fs a sx2 d Bsx1 , x2, ad
Z p
Bsx1 , x2 , adssx1 ddx1 ,
1 2 i cot a exphipfsx1 2 1 x2 2 dcot a
2 2x1 x2 csc agj ,
(5)
where Bsx1 , x2, ad is a transform kernel, a P py2 is the transform angle, P is the transform order, and x1 and x2 are the space and the FRFT domain coordinates, respectively. When P 1, the FRFT reduces to the conventional Fourier transform. We now generalize the original GT to a new version, replacing the conventional Fourier transform with the FRFT in the def inition of the GT. The new version is called the fractional Gabor transform (FRGT). Under this FRGT, the original signal is first multiplied by a window function and then analyzed with the FRFT. Similarly to the GT, this new transform can be def ined as Z Gs a s p, qd s1 2 i cot ad1/2 ssxdgp sx 2 qd 3 exphipfsp2 1 x2 dcot a 2 2px csc agjdx , (6) where gsxd is a window function and p and q are the coordinates in the space and the FRFT domains, respectively. It is evident that the FRGT reduces to the conventional GT when P 1. If the window function gsxd satisf ies Eq. (2), it is easy to prove that the target signal can be reconstructed from its FRGT spectrum by use of the inverse FRGT: Z Z Gs a s p, qdgsx 2 qd ssxd s1 1 i cot ad1/2 3 exph2ipfs p2 1 x2 dcot a 2 2px csc agjdpdq . (7) Similarly to the GT, the FRGT also has the corresponding discrete form. Let q nDx1 , p mDx2 , and m, n 0, 61, 62, . . .; the discrete FRGT reads as Z ssxdgp sx 2 nDx1 dU ap sx, mDx2 ddx , Gs a sm, nd (8a) 1997 Optical Society of America
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OPTICS LETTERS / Vol. 22, No. 21 / November 1, 1997
U a sx, mDx2 d s1 1 i cot ad1/2 exps2iphfsmDx2 d2 1 x2 g 3 cot a 2 2mDx2 x csc ajd ,
s1 sxd sins4pxd,
25 # x # 5 ,
(10)
(8b)
where Dx1 and Dx2 are the resolutions in the space and the FRFT domains, respectively. Equations (8a) and (8b) imply that the signal is projected by a set of basis functions of hmn sxd gsx 2 nDx1 dU a sx, mDx2d. When P 1, i.e., the conventional GT, the corresponding basis is a packet of plane waves: U sx, mDx2d expsi2pmDx2 xd multiplying a window function gsx 2 nDx1 d. In the space-frequency representation a monochromatic plane wave is represented by a straight line parallel to the space axis. The vertical distance of this line to the space axis determines the frequency of the plane wave. For a given m, when horizontally shifting the window we can get the mth frequency component for the individual space position in the target signal. This result implies that the signal is first decomposed into a series of plane waves with different frequencies for a given position in space. When the window is scanned along the space axis, we obtain the entire space-frequency spectrum of the target signal. So, in the space-frequency representation, the sampling grids of the GT are rectangular lattices. In contrast, the basis of the FRGT is a packet of modif ied chirp waves in the form of U a sx, mDx2 d multiplied by a window function gsx 2 nDx1 d. The modif ied chirp wave function can be written as U a sx, mDx2 d s1 1 i cot ad1/2 expfipsmDx2 d2 tan ag 3 expf2ipsx 2 mDx2 sec ad2 cot ag . (9) It can clearly be seen that U a sx, mDx2 d is composed of a chirp function, s1 1 i cot ad1/2 expf2ipsx 2 mDx2 sec ad2 cot ag , multiplied by a pure phase function expfipsmDx2 d2 3 tan ag, independently of the argument x, which can be extracted from the integral in Eq. (8a). It is worth emphasizing that the spatial frequency of this chirp function varies linearly with the space coordinate; the so-called chirp-rate parameter is given by bs 2cot ad, which describes the rate of change of frequency. In the space-frequency representation each chirp function is represented by one oblique line with a def inite slope b and an intersection with the space axis that depends on the index m.8 It is noted that the slope of the oblique line depends on the transform angle a of the FRFT. For a given m, when the window is moved horizontally the signal is analyzed along the oblique line with index m. This analysis implies that the signal is first decomposed into the chirpwave components with different indices m. When the window scans along the space axis, we can obtain the entire space –FRFT frequency spectrum of the signal. So, the sampling grids of the FRGT are of parallelograms in the space-frequency representation. The presence of the pure-phase function in the modified chirp wave just shears the sampling grids along the spatial-frequency axis direction. To demonstrate the character of the FRGT, we now consider two signals:
s2 sxd exps2ipx2 d,
25 # x # 5 .
(11)
Although the window function can be chosen arbitrarily, for simplicity we prefer to employ a Gaussian function of gsxd exps2px2 d. Figure 1 shows equalintensity contour plots of the spectra of the FRGT with two different orders for signal s1 sxd. The vertical axis represents the fractional Fourier-transform domain coordinates with the same order as that of the FRGT. Signal s1 sxd is a sinusoidal wave. It can be seen from Fig. 1 that the spectrum of s1 sxd is composed of two bright parallel bands. However, the declination degree with respect to the space axis depends on the order of the FRGT. As the order increases, the bands rotate counterclockwise to the space axis. Thus the FRGT provides an additional freedom, i.e., the transform order, compared with the conventional GT, which should provide some advantages. To achieve optimal filter design in the FRFT domain requires that the signal or the noise spectrum of the FRFT have a narrow bandpass and a condensed distribution.11 The corresponding FRGT spectrum distributions are bands that are parallel to the space axis. By changing the order of the FRGT and searching for the best spectral pattern, we can implement the optimal f ilter design. For this specif ic signal the FRFT with the order of P 1 is the best candidate. The corresponding spectrum consists of two bright bands that are parallel to the space axis. This spectrum possesses the desired spectral pattern appropriate to the optimal filter design. Figure 2 displays equal-intensity contour plots of the spectrum of the FRGT with two different orders for the second
Fig. 1. Equal-intensity contour plots of the spectra of the FRGT with different orders for the signal of s1 sxd sins4pxd. Solid curves, P 1; dashed curves, P 0.5.
November 1, 1997 / Vol. 22, No. 21 / OPTICS LETTERS
Fig. 2. Equal-intensity contour plots of the spectra of the FRGT with different orders for the signal of s2 sxd exps2ipx2 d. Solid curves, P 1; dashed curves, P 0.5.
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FRGT: A mask with the selected window function (a wide slit on the mask) is placed on the input plane in a typical optical configuration of the FRFT.5 A film-recorded one-dimensional signal is placed just before the mask, and another film that is used for recording the FRGT information is put on the output plane. When the transform is performed, the object film moves perpendicular to the slit, and the detection f ilm moves parallel to the slit with the same velocity v. The transform spectrum can then be recorded on the f ilm in the output plane in the twodimensional format. In summary, we have generalized the conventional Gabor transform to the fractional Gabor transform. When the fractional order P is 1, the new transform reduces to the conventional GT. The FRGT can provide distinguishable information in real space and its FRFT domain simultaneously. Therefore FRGT is expected to be quite useful for signal processing. Another application is Doppler radar signal processing. The radar signal of a f loating object is a chirplike function,15 and the basis of the FRGT is a packet of modified chirp waves with parameter P adjusted to match closely the underlying physics of f loating objects. Therefore the FRGT can be powerful for detection and characterization of the radar signals of f loating objects, for instance, small iceberg fragments that are hazardous to navigating vessels. The authors are grateful to the reviewers for their useful comments and suggestions on revising the manuscript. This project was supported by the National Natural Science Foundation of China and partially done at the Harbin Institute of Technology. S. Liu thanks the Educational Committee of China and a research foundation of the Harbin Institute of Technology for support.
Fig. 3. Schematic of the optical setup for performing the FRGT. f , focal length of the lens; L f s1 2 cos ad, distance between two consecutive elements.
signal, s2 sxd. It is evident that the spectrum is a bright band with different declination degrees with respect to the space axis, depending on the order of P . As P increases, the band is rotated counterclockwise toward the space axis as a whole. For the optimal filter design, the domain of the FRFT with P 0.5 is best because the corresponding FRGT spectrum consists of only a band that is parallel to the space axis with a narrower bandpass and a condensed distribution. This spectrum provides an ideal spectrum pattern for f ilter design. From these two specif ic examples we have clearly shown the character of the FRGT and its applications to f ilter design. For different signals, by analyzing the spectrum of the FRGT with various orders we can obtain the most favorable spectrum pattern for the signal and guide the design of the optimal f ilter in the FRFT domain. The FRGT can be implemented in an optical system conveniently. Figure 3 is a simple schematic of the optical system for performing the one-dimensional
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