A Generalized Convolution with a Weight Function for the Fourier Cosine and Sine Transforms

A generalized convolution with a weight function for the Fourier cosine and sine transforms is introduced. Its properties and applications to solving a system of integral equations are considered.

Computation of discrete cosine transform using Clenshaw's recurence formula

Clenshaw’s recurrenee formula is used to derive recursive algorithms for the discrete cosine transform @CT) and the inverse discrete cosine transform (IDCT). The recursive DCT algorithm presented here requires one fewer delay element per coefficient and one fewer multiply operation per coeflident compared with two recently proposed methods. Clenshaw’s recurrence formula provides a unified development for the recursive DCT and IDCT algorithms. The M v e al gorithms apply to arbitrary lengtb algorithms and are appropriate for VLSI implementation.

Digital signal processing and digital system design using discrete cosine transform [student course]

The discrete cosine transform (DCT) is an important functional block for image processing applications. The implementation of a DCT has been viewed as a specialized research task. We apply a micro-architecture based methodology to the hardware implementation of an efficient DCT algorithm in a digital design course. Several circuit optimization and design space exploration techniques at the register-transfer and logic levels are introduced in class for generating the final design. The students not only learn how the algorithm can be implemented, but also receive insights about how other signal processing algorithms can be translated into a hardware implementation. Since signal processing has very broad applications, the study and implementation of an extensively used signal processing algorithm in a digital design course significantly enhances the learning experience in both digital signal processing and digital design areas for the students.

Estimation of Symmetric Channels for Discrete Cosine Transform Type-I Multicarrier Systems: A Compressed Sensing Approach

The problem of channel estimation for multicarrier communications is addressed. We focus on systems employing the Discrete Cosine Transform Type-I (DCT1) even at both the transmitter and the receiver, presenting an algorithm which achieves an accurate estimation of symmetric channel filters using only a small number of training symbols. The solution is obtained by using either matrix inversion or compressed sensing algorithms. We provide the theoretical results which guarantee the validity of the proposed technique for the DCT1. Numerical simulations illustrate the good behaviour of the proposed algorithm.

Shift, scaling and derivative properties for the discrete cosine transform

A set of DCT domain properties for shifting and scaling by real amounts, and taking linear operations such as differentiation is described. The DCT coefficients of a sampled signal are subjected to a linear transform, which returns the DCT coefficients of the shifted, scaled and/or differentiated signal. The properties are derived by considering the inverse discrete transform as a cosine series expansion of the original continuous signal, assuming sampling in accordance with the Nyquist criterion. This approach can be applied in the signal domain, to give, for example, DCT based interpolation or derivatives. The same approach can be taken in decoding from the DCT to give, for example, derivatives in the signal domain. The techniques may prove useful in compressed domain processing applications, and are interesting because they allow operations from the continuous domain such as differentiation to be implemented in the discrete domain. An image matching algorithm illustrates the use of the properties, with improvements in computation time and matching quality.

On the Error-Free Computation of Fast Cosine Transform

We extend our previous work into error-free representations of transform basis functions by presenting a novel error-free encoding scheme for the fast implementation of a Linzer-Feig Fast Cosine Transform (FCT) and its inverse. We discuss an 8x8 L-F scaled Discrete Cosine Transform where the architecture uses a new algebraic integer quantization of the 1-D radix-8 DCT that allows the separable computation of a 2-D DCT without any intermediate number representation conversions. The resulting architecture is very regular and reduces latency by 50% compared to a previous error-free design, with virtually the same hardware cost.

Generalized Convolution Transforms and Toeplitz Plus Hankel Integral Equations

Mathematics Subject Classification: 44A05, 44A35

Developments and extensions of Roth's method of double Fourier series with applications to the solution of electromagnetic field problems

The first part of the thesis compares Roth's method with other methods, in particular the method of separation of variables and the finite cosine transform method, for solving certain elliptic partial differential equations arising in practice. In particular we consider the solution of steady state problems associated with insulated conductors in rectangular slots. Roth's method has two main disadvantages namely the slow rate of con­vergence of the double Fourier series and the restrictive form of the allowable boundary conditions. A combined Roth-separation of variables method is derived to remove the restrictions on the form of the boundary conditions and various Chebyshev approximations are used to try to improve the rate of convergence of the series. All the techniques are then applied to the Neumann problem arising from balanced rectangular windings in a transformer window. Roth's method is then extended to deal with problems other than those resulting from static fields. First we consider a rectangular insulated conductor in a rectangular slot when the current is varying sinusoidally with time. An approximate method is also developed and compared with the exact method.The approximation is then used to consider the problem of an insulated conductor in a slot facing an air gap. We also consider the exact method applied to the determination of the eddy-current loss produced in an isolated rectangular conductor by a transverse magnetic field varying sinusoidally with time. The results obtained using Roth's method are critically compared with those obtained by other authors using different methods. The final part of the thesis investigates further the application of Chebyshdev methods to the solution of elliptic partial differential equations; an area where Chebyshev approximations have rarely been used. A poisson equation with a polynomial term is treated first followed by a slot problem in cylindrical geometry.

Hand detection in cluttered scene images using Fourier-Mellin invariant features

This paper proposes an automatic hand detection system that combines the Fourier-Mellin Transform along with other computer vision techniques to achieve hand detection in cluttered scene color images. The proposed system uses the Fourier-Mellin Transform as an invariant feature extractor to perform RST invariant hand detection. In a first stage of the system a simple non-adaptive skin color-based image segmentation and an interest point detector based on corners are used in order to identify regions of interest that contains possible matches. A sliding window algorithm is then used to scan the image at different scales performing the FMT calculations only in the previously detected regions of interest and comparing the extracted FM descriptor of the windows with a hand descriptors database obtained from a train image set. The results of the performed experiments suggest the use of Fourier-Mellin invariant features as a promising approach for automatic hand detection.

Hand detection in cluttered scene images using Fourier-Mellin invariant features

This paper proposes an automatic hand detection system that combines the Fourier-Mellin Transform along with other computer vision techniques to achieve hand detection in cluttered scene color images. The proposed system uses the Fourier-Mellin Transform as an invariant feature extractor to perform RST invariant hand detection. In a first stage of the system a simple non-adaptive skin color-based image segmentation and an interest point detector based on corners are used in order to identify regions of interest that contains possible matches. A sliding window algorithm is then used to scan the image at different scales performing the FMT calculations only in the previously detected regions of interest and comparing the extracted FM descriptor of the windows with a hand descriptors database obtained from a train image set. The results of the performed experiments suggest the use of Fourier-Mellin invariant features as a promising approach for automatic hand detection.

Parallel vector processing of multidimensional orthogonal transforms for digital signal processing applications

The performance of the parallel vector implementation of the one- and two-dimensional orthogonal transforms is evaluated. The orthogonal transforms are computed using actual or modified fast Fourier transform (FFT) kernels. The factors considered in comparing the speed-up of these vectorized digital signal processing algorithms are discussed and it is shown that the traditional way of comparing th execution speed of digital signal processing algorithms by the ratios of the number of multiplications and additions is no longer effective for vector implementation; the structure of the algorithm must also be considered as a factor when comparing the execution speed of vectorized digital signal processing algorithms. Simulation results on the Cray X/MP with the following orthogonal transforms are presented: discrete Fourier transform (DFT), discrete cosine transform (DCT), discrete sine transform (DST), discrete Hartley transform (DHT), discrete Walsh transform (DWHT), and discrete Hadamard transform (DHDT). A comparison between the DHT and the fast Hartley transform is also included.(34 refs)

On the use of discrete cosine transforms for multicarrier communications

In this correspondence, the conditions to use any kind of discrete cosine transform (DCT) for multicarrier data transmission are derived. The symmetric convolution-multiplication property of each DCT implies that when symmetric convolution is performed in the time domain, an element-by-element multiplication is performed in the corresponding discrete trigonometric domain. Therefore, appending symmetric redun-dancy (as prefix and suffix) into each data symbol to be transmitted, and also enforcing symmetry for the equivalent channel impulse response, the linear convolution performed in the transmission channel becomes a symmetric convolution in those samples of interest. Furthermore, the channel equalization can be carried out by means of a bank of scalars in the corresponding discrete cosine transform domain. The expressions for obtaining the value of each scalar corresponding to these one-tap per subcarrier equalizers are presented. This study is completed with several computer simulations in mobile broadband wireless communication scenarios, considering the presence of carrier frequency offset (CFO). The obtained results indicate that the proposed systems outperform the standardized ones based on the DFT.

Codificador JPEG baseado em FPGA

O presente trabalho consiste na implementação em hardware de unidades funcionais dedicadas e optimizadas, para a realização das operações de codificação e descodificação, definidas na norma de codificação com perda Joint Photographic Experts Group (JPEG), ITU-T T.81 ISO/IEC 10918-1. Realiza-se um estudo sobre esta norma de forma a caracterizar os seus principais blocos funcionais. A finalidade deste estudo foca-se na pesquisa e na proposta de optimizações, de forma a minimizar o hardware necessário para a realização de cada bloco, de modo a que o sistema realizado obtenha taxas de compressão elevadas, minimizando a distorção obtida. A redução de hardware de cada sistema, codificador e descodificador, é conseguida à custa da manipulação das equações dos blocos Forward Discrete Cosine Transform (FDCT) e Quantificação (Q) e dos blocos Forward Discrete Cosine Transform (IDCT) e Quantificação Inversa (IQ). Com as conclusões retiradas do estudo e através da análise de estruturas conhecidas, descreveu-se cada bloco em Very-High-Speed Integrated Circuits (VHSIC) Hardware Description Language (VHDL) e fez-se a sua síntese em Field Programmable Gate Array (FPGA). Cada sistema implementado recorre à execução de cada bloco em paralelo de forma a optimizar a codificação/descodificação. Assim, para o sistema codificador, será realizada a operação da FDCT e Quantificação sobre duas matrizes diferentes e em simultâneo. O mesmo sucede para o sistema descodificador, composto pelos blocos Quantificação Inversa e IDCT. A validação de cada bloco sintetizado é executada com recurso a vectores de teste obtidos através do estudo efectuado. Após a integração de cada bloco, verificou-se que, para imagens greyscale de referência com resolução de 256 linhas por 256 colunas, é necessário 820,5 μs para a codificação de uma imagem e 830,5 μs para a descodificação da mesma. Considerando uma frequência de trabalho de 100 MHz, processam-se aproximadamente 1200 imagens por segundo.

Many-core approach to 2D-DCT calculation using an FPGA

Trabalho Final de Mestrado para obtenção do grau de Mestre em Engenharia de Electrónica e Telecomunicações

On the bicanonical map of irregular varieties

From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the natural higher-dimensional generalization to this context of curves of genus $2$: varieties birationally equivalent to the theta-divisor of an indecomposable principally polarized abelian variety. The proof is based on the (generalized) Fourier-Mukai transform.