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The Fourier Transform and its Applications

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Re: The Fourier Transform and its Applications

#31

Earlier quoted context omitted.

the other somewhat confusing thing is that algorithms like discrete fourier transform/fft are actually fourier series.

The discrete fourier transform is actually finite-dimensional linear algebra. No infinite series nor convergence problems here.

> The discrete fourier transform is actually finite-dimensional linear algebra.

Not only that, it's the most interesting kind of linear algebra: it shows that, though every (finite-dimensional if you don't like choice) vector space has a basis—so that we can pretend that all n-dimensional vector spaces are the same for a fixed n—the simple act of changing that basis (which is all the DFT is doing) can have a profound impact.

Re: The Fourier Transform and its Applications

#32

Earlier quoted context omitted.

> electrical engineering, physics and computer science books mostly relegated the topics to a few pages in an appendix. Out of curiosity, is this really true? Perhaps I misunderstand you, but I have always been under the impression that Fourier Transform was a fundamental building block covered extensively in all EE curriculum? I took Computer Engineering with some classes shared between Comp. Eng. and EE students. I…

I don't know about physics, but it's definitely not true about electrical engineering. The Fourier transform is one of those things that is absolutely fundamental to any branch of electrical engineering, you run into it when doing anything, from microelectronics to power distribution. And there's typically no way you get an EE diploma without a soul-wrenching course on signals & systems. As a personal anecdote: I end…

And there's typically no way you get an EE diploma without a soul-wrenching course on signals & systems.

Similar to the person you're replying to, my education is in computer engineering rather than electrical engineering. I took the "signals and systems" course which was a treatment of Fourier series, Fourier transforms and convolutions. Since the final exam was open book, I did well by primarily being able to quickly look up the information relevant to each question on the exam. Convolutions did seem a little bit convoluted but the course didn't seem particularly soul-wrenching (compared to, say, the intro circuits course or the intro electronics course). Perhaps I was spared the soul-wrenching bits (e.g. I took a stats course instead of the course on random signals) or perhaps my university's electrical engineering department is subpar.

Re: The Fourier Transform and its Applications

#33
post #17

There is this hard question that is often glossed over in a first course on Fourier analysis: We learn that the Fourier transform of f(t) is an integral of exp(iwt)f(t) over all t. But when we look at a table of Fourier transform pairs, we find things like the Fourier transform of a constant function (is the Dirac delta). Integral does not converge? WTF? I really think Prof. Osgood hits a sweet spot of not sweeping t…

the other somewhat confusing thing is that algorithms like discrete fourier transform/fft are actually fourier series.

There are three of them.

1. Fourier transform: unbounded real domain unbounded real domain

2. Fourier series: bounded real domain unbounded integer domain

3. Discrete Fourier transform (DFT): bounded integer domain bounded integer domain

FFT is an algorithm for quick calculation of DFT, which is not Fourier series.

Re: The Fourier Transform and its Applications

#34
post #32

Earlier quoted context omitted.

I don't know about physics, but it's definitely not true about electrical engineering. The Fourier transform is one of those things that is absolutely fundamental to any branch of electrical engineering, you run into it when doing anything, from microelectronics to power distribution. And there's typically no way you get an EE diploma without a soul-wrenching course on signals & systems. As a personal anecdote: I end…

And there's typically no way you get an EE diploma without a soul-wrenching course on signals & systems. Similar to the person you're replying to, my education is in computer engineering rather than electrical engineering. I took the "signals and systems" course which was a treatment of Fourier series, Fourier transforms and convolutions. Since the final exam was open book, I did well by primarily being able to quick…

Or, and just guessing by the open-book part, that program didn't have you derive convolutions from scratch, but just use formulaic derivations for common cases. Which is useful and all, but not the same level of soul torture.

Re: The Fourier Transform and its Applications

#35
post #32

Earlier quoted context omitted.

I don't know about physics, but it's definitely not true about electrical engineering. The Fourier transform is one of those things that is absolutely fundamental to any branch of electrical engineering, you run into it when doing anything, from microelectronics to power distribution. And there's typically no way you get an EE diploma without a soul-wrenching course on signals & systems. As a personal anecdote: I end…

And there's typically no way you get an EE diploma without a soul-wrenching course on signals & systems. Similar to the person you're replying to, my education is in computer engineering rather than electrical engineering. I took the "signals and systems" course which was a treatment of Fourier series, Fourier transforms and convolutions. Since the final exam was open book, I did well by primarily being able to quick…

The soul-wrenching part depends a lot on the professor, I suppose. My Signals teacher was absolutely stellar and put a lot of effort into giving us a solid understanding of the theoretical underpinnings of her course, so it was pretty math-intensive. The exam, as I remember it, was pretty modest in terms of difficulty, but the course was pretty hard. I aced every homework, lab and exam, but that's largely because I loved that course. Everyone I knew struggled with it. It ate a good 4-6 hours off every week, in addition to the time spent on assignments & co. (sometimes more, but by my own making, i.e. because I'd hack on something alluded to, but not taught in that course. Like I said, I really liked that class).

My Circuits and Systems professor, on the other hand, wasn't too interested in this. He obviously cared about us knowing the math, but insisted that he was teaching a Circuits class, not a Mathematics course. His course seemed a lot easier as a consequence. I was never that good at math. I learned it because you can't do engineering without it, but it has always been my weak spot.

(Edit: I don't mean it in the "how much is 12% out of 45 USD? zomg I suck at math" way, that's 5.4 USD, I'm not an idiot. I mean it in the "vector calculus made sense, I think, but I could never have made a career out of it as I did with writing software and designing electronic devices" kind of way.)

My Systems professor was smart but had long stopped caring about that course and his students, so I learned most of the material by myself. At the time, I thought he was an idiot, but I've come to understand him in time. His course wasn't too torturous, largely because I skipped most of the classes.

Most of my colleagues (particularly those in CompEng) didn't take separate courses on Signals and Systems though. They had a single, bigger Signals & Systems course, that was taught by a couple of hardcore professors, both well into their sixties and both of them very draconian. One of them also taught an introductory Electronic Devices and Circuits course that I took, so I knew him firsthand: he was a very good professor, but he was about as flexible as a chopstick. An year when less than 1/3rd of his students failed the final exam was a really good year.

Re: The Fourier Transform and its Applications

#36

30 lectures on Fourier transforms and not one mention of the Fourier transform over compact groups? (Which is arguably the most generalized/abstract formulation of the concept of a Fourier transform.) A lot of recent research has been done on FFTs over groups like the rotation group and the symmetric group, and this work has lead to significant progress on problems that were previously considered intractable.

> the Fourier transform over compact groups … is arguably the most generalized/abstract formulation of the concept of a Fourier transform.

A good-spirited correction: anyone who knows enough to know this much generality knows that calling anything mathematical the "most generalised" version of a concept is begging to be corrected. Source: I do harmonic analysis on non-compact groups, and suspect that the folks over at the n-category café probably regard my kind of work as dangerously pedestrian (see, for example, https://golem.ph.utexas.edu/category/2010/11/integral_transf...).

Re: The Fourier Transform and its Applications

#37
post #9

I 'took' the stanford NLP course online it was great. Doing this one now. Funny - I did Fourier in Comp. Eng. - but it was jammed together with so many other things, I felt overwhelmed by all of it, I did enough to do decently on the exam, but I feel a lot of it was lost on me, and I forgot much of it. Using fourier in most computer applications is very rare! I'm excited to have a project I need to use it for - and w…

The prof noted that it's almost all EE folks, with a few others, but didn't mention Comp Sci. Can anyone comment on why there were seemingly no Comp Sci people in this class? Wouldn't this be 'core' to Comp Sci?

>The prof noted that it's almost all EE folks, with a few others, but didn't mention Comp Sci. Can anyone comment on why there were seemingly no Comp Sci people in this class? Wouldn't this be 'core' to Comp Sci?

Why would it? Core CS deals primarily with discrete mathematics. Outside of a DSP or image processing course, there's little reason for a CS student to encounter the Fourier or other transforms.

Re: The Fourier Transform and its Applications

#38
post #9

I 'took' the stanford NLP course online it was great. Doing this one now. Funny - I did Fourier in Comp. Eng. - but it was jammed together with so many other things, I felt overwhelmed by all of it, I did enough to do decently on the exam, but I feel a lot of it was lost on me, and I forgot much of it. Using fourier in most computer applications is very rare! I'm excited to have a project I need to use it for - and w…

Which NLP course are you referring to? Did it have any videos available?

Re: The Fourier Transform and its Applications

#39

This is an exceptional course. In my CS undergrad days, some 7 years ago, when I wanted to make sense of Fourier Series and Transform, this course was my savior. The mathematics text book had a dry and unintuitive treatment of the topics and electrical engineering, physics and computer science books mostly relegated the topics to a few pages in an appendix. I found this course and printed out the PDF of the lecture n…

> electrical engineering, physics and computer science books mostly relegated the topics to a few pages in an appendix. Out of curiosity, is this really true? Perhaps I misunderstand you, but I have always been under the impression that Fourier Transform was a fundamental building block covered extensively in all EE curriculum? I took Computer Engineering with some classes shared between Comp. Eng. and EE students. I…

> is this really true?

Not in my experience.

As an undergrad MechE, I spent a lot of quality time with Fourier transforms in control theory, vibrations, solving pde's, and math classes. As a graduate CS student, the FFT got a lot of attention in algorithms (for the DFT), computer vision, and robotics (signals filtering) classes.

Re: The Fourier Transform and its Applications

#40
post #36

30 lectures on Fourier transforms and not one mention of the Fourier transform over compact groups? (Which is arguably the most generalized/abstract formulation of the concept of a Fourier transform.) A lot of recent research has been done on FFTs over groups like the rotation group and the symmetric group, and this work has lead to significant progress on problems that were previously considered intractable.

> the Fourier transform over compact groups … is arguably the most generalized/abstract formulation of the concept of a Fourier transform. A good-spirited correction: anyone who knows enough to know this much generality knows that calling anything mathematical the "most generalised" version of a concept is begging to be corrected. Source: I do harmonic analysis on non-compact groups, and suspect that the folks over a…

Haha, fair point (although in my defense, I did qualify my statement with "arguably").

That's interesting that you do harmonic analysis on non-compact groups; I've been meaning to learn more about that actually. I've been working on group synchronization problems related to the ideas in this paper (http://arxiv.org/pdf/1505.03840.pdf), which briefly mentions how one might extend the concept to non-compact groups, but doesn't elaborate further.

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