Discrete Fractional Fourier Transform

This paper presents a new method for the detection and parameter estimation of multicomponent LFM signals based on the fractional Fourier transform. For the optimization in the fractional Fourier domain, an algorithm based on Quasi-Newton method is proposed which consists of two steps of searching, leading to a reduction in computation without loss of accuracy. And for multicomponent signals, we further propose a signal separation technique in the fractional Fourier domain which can effectively suppress the interferences on the detection of the weak components brought by the stronger components. The statistical analysis of the estimate errors is also performed which perfects the method theoretically, and finally, simulation results are provided to show the validity of our method.

We present the detailed process of converting the classical Fourier Transform algorithm into the quantum one by using QR decomposition. This provides an example of a technique for building quantum algorithms using classical ones. The Quantum Fourier Transform is one of the most important quantum subroutines known at present, used in most algorithms that have exponential speed up compared to the classical ones. We briefly review Fast Fourier Transform and then make explicit all the steps that led to the quantum formulation of the algorithm, generalizing Coppersmith's work.

ABSTRAK Fast Fourier Transform (FFT) adalah suatu algoritma untuk menghitung transformasi Fourier diskrit (Discrete Fourier Transform, DFT) dengan cepat dan efisien. Transformasi Fourier cepat diterapkan dalam beragam bidang, mulai dari pengolahan sinyal digital, memecahkan persamaan diferensial parsial, dan untuk algoritma untuk mengalikan bilangan bulat besar. Ada pun kelas dasar dari algoritma FFT yaitu decimation in time (DIT) dan decimation in frequency (DIF). Garis besar dari kata fast diartikan karena formulasi FFT jauh lebih cepat dibandingkan dengan metode perhitungan algoritma Fourier Transform sebelumnya. Pada penelitian tugas akhir ini telah dibuat suatu aplikasi sistem pengolahan suara untuk menampilkan output frequency FFT, gelombang sinyal suara dan sederetan nilai waktu pada file data suara berbentuk .wav. Tool yang digunakan untuk membuat aplikasi sistem pengolahan suara yaitu Matlab 2013a. Metode yang digunakan pada aplikasi ini metode waterfall.

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Course Name:  MATRICES, VECTORS and FOURIER ANALYSIS
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Credits: AIUB
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It is a sample Assignment with faculty given marks. It is directed towards student who need solved sample maths with accurate formatting. All topic names that match to each math problem is given at 2nd page.
Anyone from CSE, EEE etc and anyone who is interested in the topic can view and enjoy the topic.
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[ ! ]These sample maths are not too descriptive and requires a basic understanding before scrolling the sections in it.
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In this paper, by investigating the definitions of the fractional power spectrum and the fractional correlation for the deterministic process, we consider the case associated with the random process in an explicit manner. The fractional power spectral relations for the fractional Fourier domain filter are derived, and the expression for the fractional power spectrum in terms of the fractional correlation is obtained. In addition, the definitions and the properties of the fractional white noise and the chirp-stationary process are presented. Simulation results verify the theoretical derivations and demonstrate the potential applications, such as detection and parameter estimation of chirp signals, fractional power spectral estimation and system identification in the fractional Fourier domain.

Encryption technology has been developed quickly and many image encryption methods have been used to protect confidential image data from unauthorized access.
In this paper, we introduce general quick brief introduction about cryptography, and then  propose a technique for image encryption/decryption by using the nature of FrFT in signals analysis, without use of special methods of encryption algorithms, based on multi-order Fractional Fourier Transform. Taking into account the security of the method used in the encryption work. In this research was to identify indicators to measure the security of the encryption Techniques: (i) sensitivity proposed Techniques for the encryption key can not get the original data only in the presence of this key, (ii) The complexity of the processes that are form internal proposed methods for encryption and decryption processes, and (iii) statistical analysis has been performed by calculating the histograms analysis, comparing with traditional Fourier transform the security system based on the fractional Fourier transform is protected by order of FrFT, it is can provides additional keys for encryption to make it more difficult to break. The keg is formed by combination of order of Fractional Fourier Transform and the matrix. Then, the encrypted image is obtained by the summation of different orders. Numerical simulation results are given to demonstrate this proposed method.

The fractional Fourier transform (FRFT) is a potent tool to analyze the chirp signal. However, it fails in locating the fractional Fourier domain (FRFD)-frequency contents which is required in some applications. The short-time fractional Fourier transform (STFRFT) is proposed to solve this problem. It displays the time and FRFD-frequency information jointly in the short-time fractional Fourier domain (STFRFD). Two aspects of its performance are considered: the 2-D resolution and the STFRFD support. The time-FRFD-bandwidth product (TFBP) is defined to measure the resolvable area and the STFRFD support. The optimal STFRFT is obtained with the criteria that maximize the 2-D resolution and minimize the STFRFD support. Its inverse transform, properties and computational complexity are presented. Two applications are discussed: the estimations of the time-of-arrival (TOA) and pulsewidth (PW) of chirp signals, and the STFRFD filtering. Simulations verify the validity of the proposed algorithms.

A technique for image encryption using fractional Fourier transform (FRT) and radial Hilbert transform (RHT) is proposed. The spatial frequency spectrum of the image to be encrypted is first segregated into two parts/channels using RHT, and image subtraction technique. Each of these channels is encrypted independently using double random phase encoding in the FRT domain. The different fractional orders and random phase masks used during the process of encryption and decryption are the keys to enhance the security of the proposed system. The algorithms to implement the proposed encryption and decryption scheme are discussed, and results of digital simulation are presented.

Abstract The extension of the Fourier transform operator to a fractional power has received much attention in signal theory and is finding attractive applications. The paper introduces and develops the fractional discrete cosine transform (DCT) on the same lines, discussing multiplicity and computational aspects. Similarities and differences with respect to the fractional Fourier transform are pointed out

J. Indian Inst. Sci., Jan.–Feb. 2005, 85, 11–26 © Indian Institute of Science. ... †Author for correspondence. *On lein from MITS, Gwalior 474 005, India. ... Fractional Fourier transform: A novel tool for signal ... RAJIV SAXENA* AND KULBIR SINGH † *Principal, Rustamji ...

In questa tesi si studia l’utilizzo della tecnica OFDM in trasmissione per generare diverse sottoportanti in parallelo; sfruttando la modulazione di frequenza introdotta è possibile passare dalla multiplazione OFDM alla multiplazione OTDM in ricezione, in modo tale da compensare gli effetti della dispersione e ridurre la complessità del ricevitore.
La multiplazione OFDM si basa sulla trasformata di Fourier frazionaria (FRFT), che è una generalizzazione della trasformata convenzionale. Recentemente sono stati implementati sistemi tutto-ottico basati su questo approccio [6]; lo scopo di questa tesi è quello di valutare le prestazioni di un sistema che implementi questa trasformata in forma digitale, valutando il Bit Error Rate (BER) simulando la trasmissione in fibra ottica.

Data hiding is the key factor in secure communication today, Because of duplication and data manipulation. With the help of data hiding, that we can make our personal and important data will ensure against activities without authentication. Due to the availability of low-cost editing tools, any user can easily copy, modify, and retransmit digital data over the network. It could also cause a social disorder if digital media content, such as confidential government documents, court evidence or other important information, is maliciously manipulated. To effectively support the growth of multimedia communications, it is essential to develop tools that protect and authenticate digital information. Steganography, watermarking and cryptography of such types of methods are used to hide data in secret communications. In this proposed work use a lossless data hiding technique using watermarking and steganography together, so that achieve high security that provides a higher level of security and authentication of data without loss. Combination of watermarking and steganography techniques used to get multiple layers for hiding personal information. With the help of watermarking algorithm we are embedding the user personal information within the image. The embedding and detection method of the watermark technique used the DWT transformation. Watermarked image and secret image hide within the original cover image through steganography. The most Common use of steganography is to hide a file in another file. Discrete Wavelet Transform (DWT) method is used to obtain frequency domain analysis of the image.

A novel quaternion color representation tool is proposed to the images and videos efficiently. In this work, we consider a full model for representation and processing color images in the quaternion algebra. Color images are presented in the threefold complex plane where each color component is described by a complex image. Our preliminary experimental results show significant performance improvements of the proposed approach over other well-known color image processing techniques. Moreover, we have shown how a particular image enhancement of the framework leads to excellent color enhancement (better than other algorithms tested). In the framework of the proposed model, many other color processing algorithms, including filtration and restoration, can be expressed.

An important issue with oversampled FIR analysis filter banks (FBs) is to determine inverse synthesis FBs, when they exist. Given any complex oversampled FIR analysis FB, we first provide an algorithm to determine whether there exists an inverse FIR synthesis system. We also provide a method to ensure the Hermitian symmetry property on the synthesis side, which is serviceable to processing real-valued signals. As an invertible analysis scheme corresponds to a redundant decomposition, there is no unique inverse FB. Given a particular solution, we parameterize the whole family of inverses through a null space projection. The resulting reduced parameter set simplifies design procedures, since the perfect reconstruction constrained optimization problem is recast as an unconstrained optimization problem. The design of optimized synthesis FBs based on time or frequency localization criteria is then investigated, using a simple yet efficient gradient algorithm.

Color in an image is resolved to 3 or 4 color components and 2-Dimages of these components are stored in separate channels. Most of the color image enhancement algorithms are applied channel-by-channel on each image. But such a system of color image processing is not processing the original color. When a color image is represented as a quaternion image, processing is done in original colors. This paper proposes an implementation of the quaternion approach of enhancement algorithm for enhancing color images and is referred as the modified alpha-rooting by the two-dimensional quaternion discrete Fourier transform (2-D QDFT). Enhancement results of this proposed method are compared with the channel-by-channel image enhancement by the 2-D DFT. Enhancements in color images are quantitatively measured by the color enhancement measure estimation (CEME), which allows for selecting optimum parameters for processing by thegenetic algorithm. Enhancement of color images by the quaternion based method allows for obtaining images which are closer to the genuine representation of the real original color.

The main objective of the paper is to study the three-dimensional fractional Fourier Mellin transforms (3DFRFMT), their basic properties and applicability due to mainly use in the radar system, reconstruction of grayscale images, in the detection of the human face, etc. Only the fractional Fourier transform is based on time-frequency distribution, whereas only the fractional Mellin transform is on scale covariant transformation. Both transforms can discover action in the definite assortment. The fractional Fourier transform is applicable for controlling the range of shift, whereas the fractional Mellin transform is accustomed to managing the range of rotation and scaling of the function. So, combining both transformations, we get an elegant expression for 3DFRFMT, which can be used in several fields. The paper introduces the concept of three-dimensional fractional Fourier Mellin transforms and their applications. Modulation property is the most useful concept in the signal system, radar technology, pattern reorganization, and many more in the integral transform. Parseval's identity applies to the conservation of energy in the universe. Thus we establish the modulation theorem, Parseval's theorem, scaling theorem, analytic theorem for three-dimensional fractional Fourier Mellin transform. We also give some examples of three-dimensional fractional Fourier-Mellin transform on some functions. Finally, we provide three-dimensional fractional Fourier-Mellin transform applications for solving homogeneous and non-homogeneous Mboctara partial differential equations that we can apply with advantages to solve the different types of problems in signal processing systems. The transform is beneficial in a maritime strategy as a co-realtor to control moments in any specific three-dimensional space. The concept is the most powerful tool to deal with any information system problems. After obtaining the generalization, we can explore many more ideas in applying three-dimensional fractional Fourier-Mellin transformations in many real word problems.