The problem of constructing an orthogonal set of eigenvectors for a DFT matrix is well studied. An elegant solution is mentioned by Matveev in [1]. In this paper, we present a distilled form of his solution including some steps unexplained in his paper, along with correction of typos and errors using more consistent notation. Then we compare the computational complexity of his method with the m...

a Nearly Tridiagonal Matrix Magdy Tawfik Hanna1 ABSTRACT A fully-fledged definition for the fractional discrete Fourier transform of type IV (FDFT-IV) is presented and shown to outperform the simple definition of the FDFT-IV which is proved to be just a linear combination of the signal, its DFT-IV and their flipped versions. This definition heavily depends on the availability of orthonormal eig...

We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of...

The implementation of OFDM transmitters typically consists of a discrete DFT matrix and a digital-to-analog (DAC) converter. Many existing results on the analysis of OFDM systems, e.g., spectral roll-off, are based on a convenient analog representation. In this paper, we show that the analog representation and the DFT-based OFDM transmitters are equivalent only in special cases. Using the analo...

We consider the minimization of hit error rate (BER) for OFDM systems with orthogonal precoders. For low S N Q we show that the conventional OFDM system is the optimal solution and the optimal precoder is the identity matrix. For high SNR, corresponding to a relatively practical range of BER, there are channel independent solutions for optimal precoders. One such solution is the DFT matrix and ...

We study the problem of finding unitary submatrices of the discrete Fourier transform matrix. This problem is related to a diverse set of questions on idempotents on ZN , tiling ZN , difference graphs and maximal cliques. Each of these is related to the problem of interpolating a discrete bandlimited signal using an orthogonal basis.

Many existing results on the analysis of OFDM systems are based on an analog representation. The actual implementation of OFDM transmitters typically consists of a discrete DFT matrix and a digital-to-analog (DAC) converter. In this paper, we show that the analog representation admits the implementation of a DFT matrix followed by a DAC converter only in special cases. Necessary and sufficient ...

Determining orthonormal eigenvectors of the DFT matrix, which is closer to the samples of Hermite-Gaussian functions, is crucial in the definition of the discrete fractional Fourier transform. In this work, we disclose eigenvectors of the DFT matrix inspired by the ideas behind bilinear transform. The bilinear transform maps the analog space to the discrete sample space. As jω in the analog s-d...

The recent emergence of the discrete fractional Fourier transform (DFRFT) has caused a revived interest in the eigenanalysis of the discrete Fourier transform (DFT) matrix F with the objective of generating orthonormal Hermite-Gaussian-like eigenvectors. The Grünbaum tridiagonal matrix T – which commutes with matrix F – has only one repeated eigenvalue with multiplicity two and simple remaining...