Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Successive applications of the FRFT of various orders is <s> The functional Fourier transform (FRFT), which is a generalization of the classical Fourier transform, was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community, to which it may, however, be potentially useful. The FRFT depends on a parameter /spl alpha/ and can be interpreted as a rotation by an angle /spl alpha/ in the time-frequency plane. An FRFT with /spl alpha/=/spl pi//2 corresponds to the classical Fourier transform, and an FRFT with /spl alpha/=0 corresponds to the identity operator. On the other hand, the angles of successively performed FRFTs simply add up, as do the angles of successive rotations. The FRFT of a signal can also be interpreted as a decomposition of the signal in terms of chirps. The authors briefly introduce the FRFT and a number of its properties and then present some new results: the interpretation as a rotation in the time-frequency plane, and the FRFT's relationships with time-frequency representations such as the Wigner distribution, the ambiguity function, the short-time Fourier transform and the spectrogram. These relationships have a very simple and natural form and support the FRFT's interpretation as a rotation operator. Examples of FRFTs of some simple signals are given. An example of the application of the FRFT is also given. > <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Successive applications of the FRFT of various orders is <s> Preface. Acknowledgments. Introduction. Signals, Systems, and Transformations. Wigner Distributions and Linear Canonical Transforms. The Fractional Fourier Transform. Time-Order and Space-Order Representations. The Discrete Fractional Fourier Transform. Optical Signals and Systems. Phase-Space Optics. The Fractional Fourier Transform in Optics. Applications of the Fractional Fourier Transform to Filtering, Estimation, and Signal Recovery. Applications of the Fractional Fourier Transform to Matched Filtering, Detection, and Pattern Recognition. Bibliography on the Fractional Fourier Transform. Other Cited Works. Credits. Index. <s> BIB002
equal to a single application of the FRFT whose order is equal to the sum of individual orders (e.g., F a1 F a2 ¼ F a 1 þ a 2 Þ. 6. Inverse FRFT is obtained by applying F Àa to the transformed signal (i.e., F a F Àa ¼ F 0 Þ. 7. FRFT satisfies the Parseval theorem: /xðtÞ,yðtÞS ¼ /X a ðuÞ,Y a ðuÞS. Additional properties of the FRFT and some transform pairs are listed in Tables 1 and 2 . These properties clearly indicate that the FRFT is an extension of the ordinary Fourier transform. More extensive descriptions and proofs of these properties, along with additional transform pairs, can be found in other references (e.g., BIB002   BIB001 ). The FRFTs of a few sample signals are shown in Fig. 1 . A particularly interesting case of the above listed properties is the fact that it can be shown that the FRFT of a product of two signals, y(t) = x(t) w(t), can be shown to be equal to  : where WðuÞ is the Fourier transform of w(t). In other words, the resultant FRFT is equal to convolution of the FRFT of x(t) with the Fourier transform of w(t) multiplied by a chirp function. A similar expression can be derived for convolution of two signals ðyðtÞ ¼ xðtÞ Ã wðtÞÞ. In particular, convolution of two signals in time domain equals to  : which shows that the FRFT of a convolution can therefore be obtained by taking the FRFT of one of the signals, multiplying by a chirp, convolving with a scaled version of the other signal, and multiplying again by a chirp and by a scale factor. These two rules are of particular interests when developing a time-frequency representation based on the FRFT. Our next task is to understand the relationship between the FRFT and other mathematical representations. First, we begin by establishing the relationships with the ordinary Fourier transform. Then, we establish relationships with various time-frequency representations and linear canonical analysis.