The Spectral Proper Orthogonal Decomposition #3b1b

I made this video in an attempt to popularize the Spectral POD technique. It is an incredibly powerful analysis tool for understanding the data coming from a multitude of sensors. It elevates the Fourier Transform to a whole new level; hence I call it "The Mother of All Fourier Transforms"!

I'm linking below the works of the greatest minds in this topic. I attempted to keep the mathematical terminology to a minimum, as I believe the qualitative understanding precedes the mathematical rigor. For rigorous definitions, read the papers below!

Towne, Schmidt and Colonius. "Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis" Journal of Fluid Mechanics (2018). https://arxiv.org/pdf/1708.04393.pdf

Taira, K., S. L. Brunton, S. Dawson, C. W. Rowley, T. Colonius, B. J. McKeon, O. T. Schmidt, S. Gordeyev, V. Theofilis, and L. S. Ukeiley. “Modal analysis of fluid flows: an overview.” Aiaa journal (2017). https://flowphysics.ucsd.edu/wp-content/papercite-data/pdf/tairaetal_2017_aiaaj.pdf

Schmidt, O. and Colonius, T. "Guide to Spectral Proper Orthogonal Decomposition". AIAA Journal (2020). https://arc.aiaa.org/doi/10.2514/1.J058809

Also, the implementation of the Streaming SPOD by Schmidt and Towne (2019) is incredibly powerful to enable the analysis of huge data sets, as it uses a minimum of memory. Check out:

An efficient streaming algorithm for spectral proper orthogonal decomposition. Schmidt and Towne. Computer Physics Communications (2019)

https://www.sciencedirect.com/science/article/abs/pii/S0010465518304016#:~:text=A%20streaming%20algorithm%20to%20compute,stationary%20random%20processes%20is%20presented.&text=The%20algorithm's%20low%20memory%20requirement,arbitrarily%20long%20streams%20of%20data.

Here's their Matlab Exchange code:

https://www.mathworks.com/matlabcentral/fileexchange/69963-streaming-spectral-proper-orthogonal-decomposition

Timestamps: 0:00 Intro + Prereqs 0:29 Example of sensors in a medium propagating waves 1:32 Shortcomings of POD 2:09 Traditional Fourier Transform to multiple sensors 3:28 The journey of a grad student 4:46 The Welch method for power spectrum estimation 7:55 Will the student win? 8:53 Multi-sensor FFT recap 9:30 Welch averaging loses phase information 10:22 The SPOD algorithm for discrete data 12:31 Interpreting POD modes for complex matrices 14:28 SPOD modes are simply spatial amplitude-phase relationships 15:33 Application examples and outro ... https://www.youtube.com/watch?v=OhyksL-1vew