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I'm an electronics engineering student with high inclination to analysis and pure mathematics. I was just wondering if there was any book ( or any resource ) that treats signal and systems and signal processing with a lot of mathematics rigour ( actually doing proper complex analysis, using functional analysis and linear algebra rigorously to explain convolution, fourier, laplace and z transforms for example ).

I'm very disapointed with the books i've read ( Oppenhein, Lathi and related ) because it actually throws a lot of the beauty of analysis and algebra away, focusing on the computational side.

Thanks a lot

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  • $\begingroup$ One place to look is in the control systems area, rather than specifically looking at signal processing. The control theory dudes tend to be much more mathematically rigorous than most in the signal processing literature. I don't have any recommendations yet, but I will look up and see if anything looks promising. $\endgroup$ – Peter K. May 14 '13 at 15:16
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    $\begingroup$ there was a book by Papoulis called Signal Analysis that was pretty heavy on the math. also Oppenhiem and Schafer is pretty rigorous. $\endgroup$ – robert bristow-johnson Jul 14 '14 at 3:00
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    $\begingroup$ Read the lifes of Norbert Wiener and Claude Shannon I would rather choose to be creative than to be rigorous. $\endgroup$ – Fat32 Feb 5 '15 at 23:48
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Oppenheim and Willsky's Signals and Systems or Lathi's Linear Systems and Signals are intended for Sophomores who have only a single semester of differential equations under their belts, so it is a bit unfair to criticize them for leaving out the functional analysis and the conformal mapping. At the sophomore level my favorite book is Siebert's Circuits, Signals, and Systems. It won't give you the mathematical rigor you desire either, but you can see his great love for the mathematics and he has these wonderful, witty, footnotes that provide a great historical perspective.

There is a great book (which I love, but do not recommend to you) by the (applied) mathematician Richard Hamming (of the "Hamming window", "Hamming code", "Hamming distance", "Hamming bound" and "Hamming problem") called Digital Filters. In it he makes a number of snarky comments like:

Since we are interested mainly in using mathematics, we are obliged in our turn to be ambiguous with respect to mathematical rigor. Those who believe that mathematical rigor justifies the use of mathematics in applications are referred to Lighthill and Papoulis for rigor; those who believe that it is the usefulness in practice that justifies the mathematics are referred to the rest of this book. (1998 Dover edition, page 72.)

So in addition to the book by Papoulis that @Matt_L recommended I will add Hamming's (and Siebert's!) recommendation of M.J. Lighthill, Fourier Analysis and Generalized Functions, Cambridge University Press, 1958.

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    $\begingroup$ Hamming was known as someone who liked to fly by the seat of his pants in regards to signal processing. He was a practitioner of Einstein's "imagination is more important than knowledge". I like him. $\endgroup$ – Spacey May 14 '13 at 15:05
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    $\begingroup$ Added quote to math.stackexchange.com/a/67461/2206 :) $\endgroup$ – endolith May 15 '13 at 15:34
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You should take a look at "Mathematical Methods and Algorithms for Signal Processing" by Moon and Stirling. The only downside is its long list of errata, so hopefully there will be a new edition soon.

A beautiful book about the Fourier transform as it's used in signal and system theory is "The Fourier Integral and its Applications" by Papoulis. It's quite old but I think it's an excellent book. It only treats continuous-time systems. You can find a pdf on the web.

If you're interested in filter banks and transforms you should check out Strang's "Wavelets and Filter Banks" and Vaidyanathan's "Multirate Systems and Filter Banks".

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There aren't many general signal processing books that follow a strictly mathematical (i.e. theorem/proof) style. The Mathematics of Signal Processing by Damelin and Miller might be exactly what you have in mind though. From the preface:

Basically, this is a book about mathematics...where ideas from signal processing are used to motivate much of the material, and applications of the theory to signal processing are featured. It is meant for math students who are interested in potential applications of mathematical structures and for students from the fields of application who want to understand the mathematical foundations of their subject.

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