Browsing by Subject "Fourier transformations."

Now showing 1 - 8 of 8

Open Access
Design of application specific instruction set processors for the EFT and FHT algorithms
(2006) Atak, Oğuzhan
Orthogonal Frequency Division Multiplexing (OFDM) is a multicarrier transmission technique which is used in many digital communication systems. In this technique, Fast Fourier Transformation (FFT) and inverse FFT (IFFT) are kernel processing blocks which are used for data modulation and demodulation respectively. Another algorithm which can be used for multi-carrier transmission is the Fast Hartley Transform algorithm. The FHT is a real valued transformation and can give significantly better results than FFT algorithm in terms of energy efficiency, speed and die area. This thesis presents Application Specific Instruction Set Processors (ASIP) for the FFT and FHT algorithms. ASIPs combine the flexibility of general purpose processors and efficiency of application specific integrated circuits (ASIC). Programmability makes the processor flexible and special instructions, memory architecture and pipeline makes the processor efficient. In order to design a low power processor we have selected the recently proposed cached FFT algorithm which outperforms standard FFT. For the cached FFT algorithm we have designed two ASIPs one having a single execution unit
Open Access
The discrete fractional Fourier transform
(1998) Candan, Çağatay
In this work, the discrete counterpart of the continuous Fractional Fourier Transform (FrFT) is proposed, discussed and consolidated. The discrete transform generalizes the Discrete Fourier Transform (DFT) to arbitrary orders, in the same sense that the continuous FrFT generalizes the continuous time Fourier Transform. The definition proposed satisfies the requirements of unitarity, additivity of the orders and reduction to DFT. The definition proposed tends to the continuous transform as the dimension of the discrete transform matrix increases and provides a good approximation to the continuous FrFT for the finite dimensional matrices. Simulation results and some properties of the discrete FrFT are also discussed.
Open Access
Feature extraction with the fractional Fourier transform
(1998) Güleryüz, Özgür
In this work, alternative design and implementation techniques for feature extraction applications are proposed. The proposed techniques amount to decomposing the overall feature extraction problem into a global linear system followed by a local nonlinear system. Different output representations for representation of input features are also allowed and used in these techniques. These different output representations bring cui additional degree of freedom to the feature extraction problems. The systems provide multi-outputs consisting of different features of the input signal or image. Efficient implementation of the linear part of the .system is obtained by using fractional Fourier filtering circuits. Expressions for the proposed techniques are derived and several illustrative examples cxre given in which efficient implementations for feature extraction applications are obtained.
Open Access
Image processing with the fractional Fourier transform: synthesis, compression and perspective projections
(2000) Yetik, I Şamil
In this work, first we give a summary of the fractional Fourier transform including its definition, important properties, generalization to two-dimensions and its discrete counterpart. After that, we repeat the concept of filtering in the fractional Fourier domains and give multi-stage and multi-channel filtering configurations. Due to the nonlinear nature of the problem, the transform orders in fractional Fourier domain filtering configurations have usually not been optimized but chosen uniformly up to date. We discuss the optimization of orders in the multi-channel filtering configuration. In the next part of this thesis, we discuss the application of fractional Fourier transform based filtering configurations to image representation and compression. Next, we introduce the fractional Fourier domain decomposition for continuous signals and systems. In the last part, we analyse perspective projections in the space-frequency plane and show that under certain conditions they can be approximately modeled in terms of the fractional Fourier transform.
Open Access
Repeated filtering in consecutive fractional Fourier domains
(1997) Erden, M. Fatih
In the first part of this thesis, relationships between the fractional Fourier transformation and Fourier optical systems are analyzed to further elucidate the importance of this transformation in optics. Then in the second part, the concept of repeated filtering is considered. In this part, the repeated filtering method is interpreted in two different ways. In the first interpretation the linear transformation between input and output is constrained to be of the form of repeated filtering in consecutive domains. The applications of this constrained linear transformation to signal synthesis (beam shaping) and signal restoration are discussed. In the second interpretation, general linear systems are synthesized with repeated filtering in consecutive domains, and the synthesis of some important linear systems in signal processing and the .synthesis of optical interconnection architectures are considered for illustrative purposes. In all of the examples, when our repeated filtering method is compared with single domain filtering methods, significant improvements in performance are obtained with only modest increases in optical or digital implementation costs. Similarly, when the proposed method is compared with general linear systems, it is seen that acceptable performance may be possible with significant computational savings in implementation costs.
Open Access
Signal recovery from partial fractional fourier domain information and pulse shape design using iterative projections
(2005) Güven, H. Emre
Signal design and recovery problems come up in a wide variety of applications in signal processing. In this thesis, we first investigate the problem of pulse shape design for use in communication settings with matched filtering where the rate of communication, intersymbol interference, and bandwidth of the signal constitute conflicting themes. In order to design pulse shapes that satisfy certain criteria such as bit rate, spectral characteristics, and worst case degradation due to intersymbol interference, we benefit from the wellknown Projections Onto Convex Sets. Secondly, we investigate the problem of signal recovery from partial information in fractional Fourier domains. Fractional Fourier transform is a mathematical generalization of the ordinary Fourier transform, the latter being a special case of the first. Here, we assume that low resolution or partial information in different fractional Fourier transform domains is available in different intervals. These information intervals define convex sets and can be combined within the Projections Onto Convex Sets framework. We present generic scenarios and simulation examples in order to illustrate the use of the method.
Open Access
Signal representation and recovery under partial information, redundancy, and generalized finite extent constraints
(2009) Öktem, Sevinç Figen
We study a number of fundamental issues and problems associated with linear canonical transforms (LCTs) and fractional Fourier transforms (FRTs). First, we study signal representation under generalized finite extent constraints. Then we turn our attention to signal recovery problems under partial and redundant information in multiple transform domains. In the signal representation part, we focus on sampling issues, the number of degrees of freedom, and the timefrequency support of the set of signals which are confined to finite intervals in two arbitrary linear canonical domains. We develop the notion of bicanonical width product, which is the generalization of the ordinary time-bandwidth product, to refer to the number of degrees of freedom of this set of signals. The bicanonical width product is shown to be the area of the time-frequency support of this set of signals, which is simply given by a parallelogram. Furthermore, these signals can be represented in these two LCT domains with the minimum number of samples given by the bicanonical width product. We prove that with these samples the discrete LCT provides a good approximation to the continuous LCT due to the underlying exact relation between them. In addition, the problem of finding the minimum number of samples to represent arbitrary signals is addressed based on the LCT sampling theorem. We show that this problem reduces to a simple geometrical problem, which aims to find the smallest parallelogram enclosing a given time-frequency support. By using this equivalence, we see that the bicanonical width product provides a better fit to the actual number of degrees of freedom of a signal as compared to the time-bandwidth product. We give theoretical bounds on the representational efficiency of this approach. In the process, we accomplish to relate LCT domains to the time-frequency plane. We show that each LCT domain is essentially a scaled FRT domain, and thus any LCT domain can be labeled by the associated fractional order, instead of its three parameters. We apply these concepts knowledge to the analysis of optical systems with arbitrary numbers of apertures. We propose a method to find the largest number of degrees of freedom that can pass through the system. Besides, we investigate the minimum number of samples to represent the wave at any plane in the system. In the signal recovery part of this thesis, we study a class of signal recovery problems where partial information in two or more fractional Fourier domains are available. We propose a novel linear algebraic approach to these problems and use the condition number as a measure of redundant information in given samples. By analyzing the effect of the number of known samples and their distributions on the condition number, we explore the redundancy and information relations between the given data under different partial information conditions.
Open Access
Two-dimensional fractional Fourier transform and its optical implementation
(1996) Şahin, Ayşegül
The IVactional Fourier transform of order a is defined in a manner sucli that tlu' common Fourier transform is a special case with order a = 1. Tlie definition is easil}^ extended to two dimensions just repeating the transibrm in x and y directions independently. The properties of the separable two dimensional fractional Fourier transform defined in this manner are derived and several oj)- tical implementations are given. However, this definition, ibr certain purposes, motivatcxi us to look for a new, non-separabhi definition. We ])resent sucli a d('iinition of the two dimensional fractional Fourier transform with its optical implementation. The usefulness of the new definition is justified with a noise filtering example.