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Before giving a more detailed question below, the basic one is: can anyone recommend a good signal-processing reference which would be maximally readable by a pure mathematician (who nevertheless wants to use the techniques for actual data analysis)?

Actually, the specific thing I want to understand is how to analyze a small segment of a mystery signal, which should just be the sum of a few sinusoids, but for which the available segment is potentially much shorter than the period of the sinusoids. The goal is to recover the underlying frequencies. I'd appreciate a general reference anyway, but if someone can address my specific problem, that would help too, of course!

More detail: I am a pure mathematician, but I need to learn some signal processing techniques for a side-project in biology. At first I was excited by this, since I am already comfortable with all the "underlying" background material (essentially just Fourier transforms, etc.) and thought it would be easy enough and fun to grasp what engineers and scientists were actually doing.

However, all references I've found are written for someone with the opposite background, or at least, they are written not just to deemphasize math, but sort of to avoid it as much as possible. This makes it very hard and a bit frustrating for me to read, since firstly a bunch of terminology is thrown at me without an underlying theory (so many specific window functions with different names!) and the fact that I am very comfortable with analysis hasn't helped much. My first instinct was just to go at it alone from first principles (the definition of the DFT...) but it seems silly to ignore the vast history in this subject.

I did not mean that to sound as ranty as it did. I am not complaining, I am just wondering if anyone can recommend a reference more amenable to my background, which could take advantage of it or at least, ease me into the subject.

Thanks!

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  • $\begingroup$ Why do you put yourself in a "tour d'ivoire" (ivory tower) ! Be a mathematcician ! a physician ! both !! $\endgroup$
    – user36539
    Commented Sep 28, 2013 at 12:53

6 Answers 6

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I do not know if this one will be "maximally readable" for you:

MR2883645 Damelin, Steven B.(1-GSO-NDM); Miller, Willard, Jr.(1-MN) The mathematics of signal processing. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2012. xii+449 pp. ISBN: 978-1-107-60104-8

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I am sorry I probably can say nothing on the "detailed question", but let me comment on the "basic" one.

I spent some time on the book

Fundamentals of Wireless Communication

David Tse and Pramod Viswanath

Here is link to pdf with lectures by D. Tse based on it.

The book is intended for wireless signal processing, with some examples related to GSM, CDMA. It covers basic ideas: the transmission chain, fading channel; Shannon's channel's capacity theory. Advanced topics like MIMO.

I think it is quite good for mathematician learning something about wireless communication.

One of the authors - D. Tse is leading expert in this field and his approach is quite mathematical,

see e.g. paper

L. Zheng and D. Tse, ``Communicating on the Grassmann Manifold: A Geometric Approach to the Non-coherent Multiple Antenna Channel'', IEEE Transactions on Information Theory, vol. 48(2), February 2002, pp. 359-383.

The geometry of Grassmann manifold appears to be related with capacity of MIMO channels.

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    $\begingroup$ (The link to lecture notes by D. Tse is broken.) $\endgroup$
    – Calvin Khor
    Commented Jun 27, 2019 at 10:42
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    $\begingroup$ @CalvinKhor A copy of the PDF is saved on the Wayback Machine. $\endgroup$
    – The Amplitwist
    Commented Nov 27, 2023 at 13:00
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It's clearly too late, but my own bible for Bayesian spectrum analysis is here:

http://bayes.wustl.edu/glb/book.pdf

It's written by a physicist, not a pure mathematician, but it's fairly self-contained.

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Goyal, Kovačević and Vetterli have made their books freely available online:

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Martin Vetterli wrote two books: http://fourierandwavelets.org/

  • Foundations of Signal Processing

  • Fourier and Wavelet Signal Processing

My concern, these may not be theoretical enough.

A lot signal processing has uncanny resemblance to just plain Fourier analysis. Stein-Shakarchi doesn't tell you what a high-pass filter or low-pass filter is, specifically. He just figures you'll find out on your own. One cannot forget Daubechies' "Ten Lectures on Wavelets"

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There is a Prony method that can find the frequencies of uniformly sampled data which are sums of sinusoids.

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