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Full-Text Articles in Physical Sciences and Mathematics
An Introduction To Fourier Analysis With Applications To Music, Nathan Lenssen, Deanna Needell
An Introduction To Fourier Analysis With Applications To Music, Nathan Lenssen, Deanna Needell
Journal of Humanistic Mathematics
In our modern world, we are often faced with problems in which a traditionally analog signal is discretized to enable computer analysis. A fundamental tool used by mathematicians, engineers, and scientists in this context is the discrete Fourier transform (DFT), which allows us to analyze individual frequency components of digital signals. In this paper we develop the discrete Fourier transform from basic calculus, providing the reader with the setup to understand how the DFT can be used to analyze a musical signal for chord structure. By investigating the DFT alongside an application in music processing, we gain an appreciation for …
Greedy Signal Recovery Review, Deanna Needell, Joel A. Tropp, Roman Vershynin
Greedy Signal Recovery Review, Deanna Needell, Joel A. Tropp, Roman Vershynin
CMC Faculty Publications and Research
The two major approaches to sparse recovery are L1-minimization and greedy methods. Recently, Needell and Vershynin developed Regularized Orthogonal Matching Pursuit (ROMP) that has bridged the gap between these two approaches. ROMP is the first stable greedy algorithm providing uniform guarantees.
Even more recently, Needell and Tropp developed the stable greedy algorithm Compressive Sampling Matching Pursuit (CoSaMP). CoSaMP provides uniform guarantees and improves upon the stability bounds and RIC requirements of ROMP. CoSaMP offers rigorous bounds on computational cost and storage. In many cases, the running time is just O(NlogN), where N is the ambient dimension of the signal. This …