Geometric Representation of Signals

hi. Here is the link to the Comms book (and other stuff ; you can also download the PDF of the book for free how nice) from the italian guys at Lausanne who teach on Coursera also
https://www.sp4comm.org/getit.html
And here is the website for their incredible 'foundations' book - that starts with the geometric perspective (and in my view, is better and easier than the comms book above)
https://fourierandwavelets.org/
amazingly, that book (and a sister book on wavelets) are both on there as free PDFs also. (you just need a free 2 years to study them)
PS Yes, 3b1b is brilliant on youtube - he also presents a section on Khan Academy on multivariate calculus/vector stuff (another amazing free/donatable resource for general maths and sci)

Yes, many things happen in time, and can be adequately o rwell modelled as sequential time operations: but staying in the domain of original measurement can cramp our style. We would not have mp3 adio or mp4 vieo compression without out-of-time-order processing to enciode those signals efficiently: nor mobile streaming without reoredering of data for energy efficiency; or frequency domain analysis or filtering. And many systems are measured in other than time domains: cameras for instance encode spatially even though the sensor array may be read out partly or wholly seqentially: Vector Network Analyzers measure phase and amplitude directly in frequency domain; so to a large extent do our ears; MRI codes spatial infomation by frequency.. So I would say time, or the related sequence order, is an easy way to visualize and often a useful way to process but to go beyond it is useful.. With any signal processing we can ask what is the input, and what is the desired output or outcome: what comes between these can be freely chosen and should not be constrained by sequentiality (or any other feature of the input and output spaces) but by coniderations of efficiency, however expressed.

A comment from th chat, to which I did not have time to respond:
"The time domain view is not just a "computationally-orientational" artefact... things really happen this way (a signal that passes a linear system gets convoluted with its impulse response), and if we discretize this, we get the classical DSP representation (the time-domain). Audio or radio signals are really functions of a time variable... there are other types of signals which aren't time functions but the time-domain view is fundamental"

Hi Chris,
Great way to simplify things. For others interested in resources to help visual signals (and the math), I like this website:
https://www.jezzamon.com/fourier/index.html
and the videos in the series on YouTube (3 blue 1 brown Season 4):
https://www.youtube.com/playlist?list=PLZHQObOWTQDNPOjrT6KVlfJuKtYTftqH6
Brewster

Thank you for this talk. I loved your humor (4 dimensional words). Using the letters as axis is brilliant to decouple our desire to leap ahead.
Using Cuboid-sphere-of-confusion. OMG, the definition of RMS in dimensionality reduction.
"A filter projects onto a lower-dimensional space," So many deep concepts explained so simply.
A fourier transform is a rotation, as is a PCA?! (I come from a data science background)
One thing that I didn't understand throughout it all: I'm used to working in audio... I'm used to thinking of samples in terms of time domain. It hurts my brain a bit to think of each sample being a dimension, and I feel like I'm missing something there.

Does it mean that there exist only one signal in universe with bandwidth B and time constraint T?

I'm expecting Mr. Bill to come marching across your paper and straws. (for those outside of US: reference to old Saturday Night Live shows)