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

I think that "Group representations in probability and statistics" by Persi Diaconis is a very good choice. It is really intended for nonalgebraists 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). 


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 TimeFrequency 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 "TimeFrequency Representations"  to anyone who wants to learn signal processing the "right" way. (Incidentally, these authors have a new book called Ideal Sequence Design in TimeFrequency Space, but I have not read it.) 


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


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. 


I am responding to the question, "What is a good reference for introducing noncommutative 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 


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 3dimensional cube, and even obtains some quantitative estimates using the representation theory of the group of symmetries of the cube. 

