Browsing by Subject "Fractional fourier transform"

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Open Access
Anamorphic fractional Fourier transform: optical implementation and applications
(Optical Society of America, 1995-11-10) Mendlovic, D.; Bitran, Y.; Dorsch, R. G.; Ferreira, C.; Garcia, J.; Özaktaş, Haldun M.
An additional degree of freedom is introduced to fractional-Fourier-transform systems by use of anamorphic optics. A different fractional Fourier order along the orthogonal principal directions is performed. A laboratory experimental system shows preliminary results that demonstrate the proposed theory. Applications such as anamorphic fractional correlation and multiplexing in fractional domains are briefly suggested.
Open Access
Chirp filtering in the fractional Fourier domain
(Optical Society of America, 1994-11-10) Dorsch, R. G.; Lohmann, A. W.; Bitran, Y.; Mendlovic, D.; Özaktaş, Haldun M.
In the Wigner domain of a one-dimensional function, a certain chirp term represents a rotated line delta function. On the other hand, a fractional Fourier transform (FRT) can be associated with a rotation of the Wigner-distribution function by an angle connected with the FRT order. Thus with the FRT tool a chirp and a delta function can be transformed one into the other. Taking the chirp as additive noise, the FRT is used for filtering the line delta function in the appropriate fractional Fourier domain. Experimental filtering results for a Gaussian input function, which is modulated by an additive chirp noise, are shown. Excellent agreement between experiments and computer simulations is achieved.
Open Access
Exact relation between continuous and discrete linear canonical transforms
(Institute of Electrical and Electronics Engineers, 2009-08) Oktem, F. S.; Özaktaş, Haldun M.
Linear canonical transforms (LCTs) are a family of integral transforms with wide application in optical, acoustical, electromagnetic, and other wave propagation problems. The Fourier and fractional Fourier transforms are special cases of LCTs. We present the exact relation between continuous and discrete LCTs (which generalizes the corresponding relation for Fourier transforms), and also express it in terms of a new definition of the discrete LCT (DLCT), which is independent of the sampling interval. This provides the foundation for approximately computing the samples of the LCT of a continuous signal with the DLCT. The DLCT in this letter is analogous to the DFT and approximates the continuous LCT in the same sense that the DFT approximates the continuous Fourier transform. We also define the bicanonical width product which is a generalization of the time-bandwidth product.
Open Access
Fractional fourier transform meets transformer encoder
(Institute of Electrical and Electronics Engineers, 2022-10-28) Şahinuç, Furkan; Koç, Aykut
Utilizing signal processing tools in deep learning models has been drawing increasing attention. Fourier transform (FT), one of the most popular signal processing tools, is employed in many deep learning models. Transformer-based sequential input processing models have also started to make use of FT. In the existing FNet model, it is shown that replacing the attention layer, which is computationally expensive, with FT accelerates model training without sacrificing task performances significantly. We further improve this idea by introducing the fractional Fourier transform (FrFT) into the transformer architecture. As a parameterized transform with a fraction order, FrFT provides an opportunity to access any intermediate domain between time and frequency and find better-performing transformation domains. According to the needs of downstream tasks, a suitable fractional order can be used in our proposed model FrFNet. Our experiments on downstream tasks show that FrFNet leads to performance improvements over the ordinary FNet.
Open Access
Fractional fourier transform pre-processing for neural networks and its application to object recognition
(Elsevier, 2002-01) Barshan, Billur; Ayrulu, Birsel
This study investigates fractional Fourier transform pre-processing of input signals to neural networks. The fractional Fourier transform is a generalization of the ordinary Fourier transform with an order parameter a. Judicious choice of this parameter can lead to overall improvement of the neural network performance. As an illustrative example, we consider recognition and position estimation of different types of objects based on their sonar returns. Raw amplitude and time-of-flight patterns acquired from a real sonar system are processed, demonstrating reduced error in both recognition and position estimation of objects. (C) 2002 Elsevier Science Ltd. All rights reserved.
Open Access
Sparse representation of two-and three-dimensional images with fractional fourier, hartley, linear canonical, and haar wavelet transforms
(Elsevier Ltd, 2017) Koç A.; Bartan, B.; Gundogdu, E.; Çukur, T.; Özaktaş, Haldun M.
Sparse recovery aims to reconstruct signals that are sparse in a linear transform domain from a heavily underdetermined set of measurements. The success of sparse recovery relies critically on the knowledge of transform domains that give compressible representations of the signal of interest. Here we consider two- and three-dimensional images, and investigate various multi-dimensional transforms in terms of the compressibility of the resultant coefficients. Specifically, we compare the fractional Fourier (FRT) and linear canonical transforms (LCT), which are generalized versions of the Fourier transform (FT), as well as Hartley and simplified fractional Hartley transforms, which differ from corresponding Fourier transforms in that they produce real outputs for real inputs. We also examine a cascade approach to improve transform-domain sparsity, where the Haar wavelet transform is applied following an initial Hartley transform. To compare the various methods, images are recovered from a subset of coefficients in the respective transform domains. The number of coefficients that are retained in the subset are varied systematically to examine the level of signal sparsity in each transform domain. Recovery performance is assessed via the structural similarity index (SSIM) and mean squared error (MSE) in reference to original images. Our analyses show that FRT and LCT transform yield the most sparse representations among the tested transforms as dictated by the improved quality of the recovered images. Furthermore, the cascade approach improves transform-domain sparsity among techniques applied on small image patches.