# Non-commutative Fourier transform

What is a good reference for introducing non-commutative fourier transform for Electrical Engineers and Theoretical Computer Scientists in an explicit way?

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What do you mean by non-com. FT ? Expansion by group characters or matrix elements ? What is the use of this in EE ? –  Alexander Chervov Jun 5 '12 at 5:57
I join Alexander's petition for clarification. Unless representation theory is studied in EE, the question is not very clear to me. –  Juan Bermejo Vega Jun 5 '12 at 9:39
@Juan Representation Theory is useful in coding theory (space time codes) –  Turbo Jun 5 '12 at 21:52
I think representation theory of finite groups is exactly what's needed. For some strange reason the actual Fourier transform formulas are never given (e.g. in Serre's book) –  Igor Rivin Jun 5 '12 at 23:09
@Igor what is use of finite groups in EE ? I worked in RT before and now work in EE (wireless com.), I would be happy to know the connections, but do not know. –  Alexander Chervov Jun 6 '12 at 5:47

I think that "Group representations in probability and statistics" by Persi Diaconis is a very good choice. It is really intended for non-algebraists and is well motivated by real world examples. And it is available for free at project Euclid.

You could also try my book which is intended for a 4th year math undergrad or first year grad course and addresses some applications (sorry to plug my own book). It is very explicit (you won't see the word module).

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I think the book Engineering Applications of Noncommutative Harmonic Analysis by Chirikjian and Kyatkin might be exactly what you are looking for. Although I haven't read very much of it, the sections that I have read are very nice and seem mathematically rigorous, yet geared toward applications.

Also, there are a couple of excellent books by Myoung An and Richard Tolimieri about harmonic analysis over finite groups. The first is called Time-Frequency Representations, and it's about harmonic analysis over finite abelian groups with applications to audio signal processing. The second is called Group Filters and Image Processing, and it's about harmonic analysis over finite nonabelian groups with applications to image processing. I can't say enough about these books. The mathematical presentation is rigorous and elegant, and the applied examples are very explicit, including Matlab code and demonstrating how the authors have applied the techniques in work they have done for defense contractors. An overview of some of the material presented in these two book appears in this paper.

I highly recommend the books by Tolimieri and An -- especially "Time-Frequency Representations" -- to anyone who wants to learn signal processing the "right" way. (Incidentally, these authors have a new book called Ideal Sequence Design in Time-Frequency Space, but I have not read it.)

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This looks interesting. –  Turbo Jun 5 '12 at 21:53
Looks like Myong An is fromPrometheus:) –  Turbo Jun 13 '12 at 22:04
Actually, Prometheus is a company that Myoung and Richard started in order to publish their books and make them available at much more reasonable prices than the big publishers were willing to accept. –  William DeMeo Jun 22 '12 at 5:02

Barry Simon's representation theory of finite and compact groups is good.

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The title is actually "Representations of Finite and Compact Groups" –  Stopple Jun 4 '12 at 21:35
Hi Igor: Is the book explicit enough for a computer scientist or an EE to understand and work through to get maturity? –  Turbo Jun 4 '12 at 23:49
I think it is pretty concrete, though I have not had any guinea pigs, er, electrical engineers, to test it on. –  Igor Rivin Jun 5 '12 at 0:19
@Igor guinea pigs nice –  Turbo Jun 5 '12 at 21:56
@Igor Is there a notion of Fourier transform in incidence algebras? –  Turbo Jun 6 '12 at 0:23

Perhaps you want to have a look at Lang "$SL_2(\mathbb{R})$" - Chapter V Spherical transform and Chapter VIII Plancherel formula: His approach is a classical, global one, which is probably more digestive than an infinitesimal approach at first encounter. Global approach means that you work only on the group (and not on its Lie algebra), and prove everything via integral transforms, Mellin transforms and special functions, so only a little bit of advanced calculus is needed.

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Would this help in non-commutative setting? –  Turbo Jun 13 '12 at 22:02
What is a non-commutative setting and by what should it help? SL(2, R) is non-commutative, and the Harish-Chandra transform (for reductive groups only) and the Plancherel theorems are the only generalization of abelian Fourier analysis, I know. –  Marc Palm Jun 15 '12 at 10:19

I am responding to the question, "What is a good reference for introducing non-commutative fourier transform for Electrical Engineers"

As an introduction to the subject, electrical engineers might find it helpful to look at the book, "Symmetries and Groups in Signal Processing", authored by me, and published by Springer.

Virendra P. Sinha

emailmvps@gmail.com

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I think $\S$ 16 of the book "Elements of the Theory of Representations" by A.A. Kirillov might be relevant. In $\S$ 16 he discusses concrete examples of representations of finite groups. Thus in $\S$ 16.1 he studies basic harmonic analysis on 3-dimensional cube, and even obtains some quantitative estimates using the representation theory of the group of symmetries of the cube.

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