# Discrete time Fourier and related transforms

Also, chirplets, z-transforms, chromatic derivatives…

October 17, 2019 — October 17, 2019

Care and feeding of Discrete Fourier transforms (DTFT), especially Fast Fourier Transforms, and other operators on discrete time series. Complexity results, timings, algorithms, properties. These are useful in a vast number of applications, such as filter design, time series analysis, various nifty optimisations of other algorithms etc.

## 1 Chirp *z*-transform

Chirplets, one-sided discrete Laplace transform related to damped sinusoid representation. (Bluestein 1970; Rabiner, Schafer, and Rader 1969)

A recent publication (Sukhoy and Stoytchev 2019) shows that these are as tractable as FFTs to invert, which is to say, very. I will read the paper and see if that is as useful to me as it seems like it might be. (The paper has a lot of elementary proofreading errors, which is a bad start.)

🏗

## 2 Windowing the DTFT

(Harris 1978; Cooley, Lewis, and Welch 1970; Rafii 2018)

🏗

## 3 Chromatic derivatives

(Narasimha, Ignjatovic, and Vaidyanathan 2002; Aleksandar Ignjatovic 2007; A. Ignjatovic 2009; Aleksandar Ignjatovic, Wijenayake, and Keller 2018b, 2019, 2018b)

🏗

## 4 References

*Digital signal processing: signals, systems and filters*.

*IEEE Transactions on Audio and Electroacoustics*.

*Time Series Analysis: Forecasting and Control*. Wiley Series in Probability and Statistics.

*Proceedings of the IEEE*.

*Journal of Sound and Vibration*.

*An Introduction to Statistical Signal Processing*.

*IEEE Transactions on Acoustics, Speech, and Signal Processing*.

*Proceedings of the IEEE*.

*Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing*. STOC ’12.

*Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms*. Proceedings.

*Journal of Fourier Analysis and Applications*.

*IEEE Transactions on Signal Processing*.

*IEEE Transactions on Signal Processing*.

*IEEE Transactions on Signal Processing*.

*Fundamentals of Statistical Signal Processing*. Prentice Hall Signal Processing Series.

*Digital Spectral Analysis with Applications*.

*Physical Review Letters*.

*Mathematical Methods and Algorithms for Signal Processing*.

*IEEE Signal Processing Letters*.

*Discrete-Time Signal Processing*.

*Introduction to Signal Processing*. Prentice Hall Signal Processing Series.

*arXiv:1501.00320 [Cs, Math]*.

*Signal processing for communications*. Communication and information sciences.

*IEEE Transactions on Audio and Electroacoustics*.

*IEEE Signal Processing Magazine*.

*Introduction to Digital Filters with Audio Applications*.

*Spectral Analysis of Signals*.

*Scientific Reports*.

*Discrete Random Signals and Statistical Signal Processing*.

*arXiv:1701.01595 [Cs, Math]*.