## Live Q&A with Chris Bore - Geometric Representation of Signals

Chris Bore - **Watch Now** - Duration: 38:31

**JulesP**

**ChrisBore**Speaker

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.

**ChrisBore**Speaker

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"

**Brewster**

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

**Darkphibre**

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.

**Navneet**

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

**ChrisBore**Speaker

I'm not sure I grasped the question but no - bandwidth and time duration alone do not define a unique signal: and indeed both constraints are formally not possible. But even if they weer that would not be unqiue: however, a sgnal defined in both frequency and time would be, in a sense, unique.

**Brewster**

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)

13:03:01 From Darkphibre / Amity : That was an AMAZING talk, tyvm. 13:04:21 From rakhel : it's like an insightful TED talk 13:04:53 From JulianPenketh : I have ONE slightly negative comment.... ;-) 13:07:50 From Emanuele Ziglioli : They have 4 courses on Coursera 13:08:01 From Ibrahim Khalife : At EPFL, yes I have seen it. In fact it is all public 13:08:02 From Emanuele Ziglioli : it's all about Hilbert spaces, yeah 13:08:16 From Darkphibre / Amity : Could you type in the name of the authors again? I missed it in the conversation. 13:08:21 From Emanuele Ziglioli : Vetterli 13:08:23 From Michael Kirkhart : Julian: if you have some links you can place in the chat, it would be helpful. 13:08:26 From Emanuele Ziglioli : Prandoni, EPFL 13:08:53 From Emanuele Ziglioli : https://www.coursera.org/specializations/digital-signal-processing 13:14:15 From Radu Pralea : I've read a comment on a forum some 20 years ago, as an EE student: there are two types of communication engineers in the world: those who read Wozencraft (& Jacobs) and those who haven't. I still haven't read it yet :), but at least I browsed it, and that's when I first encountered "Geometric Interpretation of Signals" (probably directly influenced by Shannon) 13:14:50 From rakhel : https://www.youtube.com/c/3blue1brown 13:15:04 From Michael Kirkhart : 3Blue1Brown is very good - have watched several of his videos. 13:15:08 From Ibrahim Khalife : Wozencraft, yes! 13:33:44 From JulianPenketh : 3Blue1Brown is helpful for this: he showed me that when we multiply a vector by a matrix, we are actually just representing that vector in a new space - the matrix is not just some random columns of random numbers (as I used to think once!); each column of the matrix is an axis - or basis vector - of this new space (rather than the one the vector started in.....). So the new vector can be thought of as the same one as before, but how it would look using these new basis vectors.. so all matrices are really transformation matrices...... or something like that ;-). This insight for me (from 3b1b), really made this matrix-vector stuff concrete. 13:34:26 From JulianPenketh : (which yes, can be viewed as a rotation) 13:37:15 From Darkphibre / Amity : Quick follow-up to my question: Was the choice of the three time-domain samples in the talk arbitrary? Or is there a useful/common technique (such as 2nd order analysis) that was implied? 13:38:12 From Michael Kirkhart : If I had to guess, it was limited to 3 time domain samples because humans are limited to 3 spatial dimensional analysis? 13:38:58 From Darkphibre / Amity : Yup, thank you! 13:39:05 From Radu Pralea : 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. 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)