On the Eigenstructure of DFT Matrices

The discrete Fourier transform (DFT) not only enables fast implementation of the discrete convolution operation, which is critical for the efficient processing of analog signals through digital means, but it also represents a rich and beautiful analytical structure that is interesting on its own. A typical senior-level digital signal processing (DSP) course involves a fairly detailed treatment of DFT and a list of related topics, such as circular shift, correlation, convolution operations, and the connection of circular operations with the linear operations. Despite having detailed expositions on DFT, most DSP textbooks (including advanced ones) lack discussions on the eigenstructure of the DFT matrix. Here, we present a self-contained exposition on such.


On higher order approximations for hermite-gaussian functions and discrete fractional Fourier transforms
Candan, Çağatay (Institute of Electrical and Electronics Engineers (IEEE), 2007-10-01)
Discrete equivalents of Hermite-Gaussian functions play a critical role in the definition of a discrete fractional Fourier transform. The discrete equivalents are typically calculated through the eigendecomposition of a commutator matrix. In this letter, we first characterize the space of DFT-commuting matrices and then construct matrices approximating the Hermite-Gaussian generating differential equation and use the matrices to accurately generate the discrete equivalents of Hermite-Gaussians.
Exact Relation Between Continuous and Discrete Linear Canonical Transforms
Öktem, Sevinç Figen (Institute of Electrical and Electronics Engineers (IEEE), 2009-08-01)
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 p...
Properly Handling Complex Differentiation in Optimization and Approximation Problems
Candan, Çağatay (Institute of Electrical and Electronics Engineers (IEEE), 2019-03-01)
Functions of complex variables arise frequently in the formulation of signal processing problems. The basic calculus rules on differentiation and integration for functions of complex variables resemble, but are not identical to, the rules of their real variable counterparts. On the contrary, the standard calculus rules on differentiation, integration, series expansion, and so on are the special cases of the complex analysis with the restriction of the complex variable to the real line. The goal of this lect...
An efficient filtering structure for Lagrange interpolation
Candan, Çağatay (Institute of Electrical and Electronics Engineers (IEEE), 2007-01-01)
A novel filtering structure with linear complexity is proposed for Lagrange interpolation. The structure is similar to the Farrow structure in principle, but it is more efficient and has the additional feature of being order updatable on-the-fly. The main application for the proposed structure is the implementation of fractional delay filters to mitigate the symbol synchronization errors in digital communications. Some other applications are time-delay estimation, echo cancellation, acoustic modeling, and a...
Design and Analysis of Frequency-Tunable Amplifiers using Varactor Diode Topologies
Nesimoglu, Tayfun (Springer Science and Business Media LLC, 2011-08-01)
The design of frequency-tunable amplifiers is investigated and the trade-off between linearity, efficiency and tunability is revealed. Several tunable amplifiers using various varactor diode topologies as tunable devices are designed by using load-pull techniques and their performances are compared. The amplifier using anti-series distortion-free varactor stack topology achieves 38% power added efficiency and it may be tuned from 1.74 to 2.36 GHz (about 35% tunable range). The amplifier using anti-series/an...
Citation Formats
Ç. Candan, “On the Eigenstructure of DFT Matrices,” IEEE SIGNAL PROCESSING MAGAZINE, pp. 105–108, 2011, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/38444.