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Comment by dbetteridge | Hacker News Reader

Fantastic project!

For someone looking to learn more about signals and audio processing DFTs FFTs etc any good material people can recommend? I've completely forgotten my Uni CompEng Signals and want to revisit it.

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I don't have links, but I have some advice that may be contrary to what a CS major might say. I recommend against a CS-first approach. Focus on theoretical fundamentals. Take "DFT" and "FFT" out of your lexicon -- those are not important right now. Forget about the the D and the P in DSP. Focus on the S: signals. Signals are just waves. Understand the mathematical basics of waves -- bonus points if you study differential equations whose solutions are waves.

Once you're pretty confident with your understanding of wave math, next focus on linear algebra. Linear algebra sounds like a strange requirement for understanding signals, but it actually is fundamental. Fourier series should then just "click" for you -- a Fourier transform is basically a change of basis in a vector space.

Bonus points if you learn about infinite-dimensional vector spaces, what Dirac delta functions are and what they mean. For example, why is it that a delta function has infinite spectral energy? Either you have no idea, or you find the answer obvious. There is no in between. Good DSP folks can answer theoretical questions like this without thinking. Once you understand how to think about waves in the time basis and the frequency basis, you will be well equipped to understand pretty much everything in DSP.

What’s wave math?

Fourier transforms and more generally convolutions.

So remove DFT but use Fourier still?

No, real wave maths is continuous wavelets and those are not as useful in processing. (DFT is a kind of wavelet transform, just limited.)

And then you have the advanced tensor wave math which is used in physics but rarely in sound processing. I bet you didn't have evaluating Schrodinger's and symmetric solutions was what you had in mind.

You'll get much more mileage out of statistics and discrete mathematics, plus control theory and optimization theory.

There are many different types of Fourier transforms (i.e. generalisations and specifications(?)) useful for signal analysis, not just the (classical full spectrum) FT. E.g. the fractional FT (useful for signal separation), laplace transform (used all over the place in control for linear(ised) systems and analysis).

> Schrödinger

Well momentum is the FT of position so...

Also note the OP's question is "Whats wave math?" not "why don't use the (D|F)FT?" which I'm still surprised was suggested: Its a tool, use it when appropriate, use something else when its not.

What would be a good practical project to learn about DSP? Preferably in Python, and preferably relevant to music somehow.

Well I did most of my signals analysis with an oscilloscope circuitry and a function (read: waveform) generator. You could very easily get computer versions of those and substitute the function generator for some music. But I'd consult the elders of the internet for better ideas.

Coincidentally, that's how I came around to it, from a physics undergrad, to music software, to cs.

Julius Smith @ Stanford has a couple of great free books online:

Introduction to Digital Filters with Audio Applications

https://ccrma.stanford.edu/~jos/filters/

Mathematics of the Discrete Fourier Transform

https://ccrma.stanford.edu/~jos/mdft/

http://dspguide.com/ is a practical, not too mathematical guide. Really, I recommend it to anyone that knows math and computer science. Understanding linearity, what an impulse response is, and convolution is really useful, and intuitive.

https://ocw.mit.edu/resources/res-6-007-signals-and-systems-...

"Understanding Digital Signal Processing" by Richard Lyons is a wonderful book. Very readable, heavier on intuitive explanations/diagrams and lighter on formulas.

If you like audio, Will Pirkle's books on the subject are great although he gets some flak in professional circles for the code quality.

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