# A path to get the foundation of SP and DSP for a pure math

I have a Ph.D. in pure math (interested in Harmonic analysis and operator theory). I am looking forward some proper references to lead me get the foundation of discrete/signal processing more and more. Actually, I had a review of the Heppenheim's books (both signal and digital ones) and (rather) got what he is saying in these (very nice) book. Now, I am going to develop my knowledge concerning this context and do need to cover some more advanced ones. Thanks in advances for you suggestions.

Probably based on my own field, I would like to read on some texts whose approaches are focused on theoretical bases. However having some (proper) references which make me feel (realize) some real applications are in priority.

You have already read those Oppenheim's Signals & Systems, and Discrete-Time Signal Processing books.

I'm not sure what you mean by foundations but in some sense these two are also the foundations on signal processing. In other words, there are no (popular & successful) graduate level DSP books that discuss at an advanced level the same topics that are covered on them.

However the following books (or subjects) will enhance your understanding, or broaden your appreciation of the subject.

First of all, the very first graduate level course on DSP, Communications, and Control is called Linear System Theory which brings together all the undergraduate mathematical stuff from a new, advanced, deeper, and foundational point of view of the Hilbert (linear vector) Spaces, Linear Mappings, and Matrix theory. It does not have a definite book but a bunch of books on Linear Algebra & Matrices, Measure Theory, and Diferential Equations were used. Note that there's a control theory oriented bunch of Linear System Theory books (Desoer's crew) that I do not recommend for DSP, unless you will be designing control systems on the field. Signal processing does not make much use of state-space approach, unless it's absolutely necessary.

Then the second refresher/deepener is on Probability, Statistics and Random Processes. Fortunately it has two very strongly recomended books though:

• Statistical Digital Signal Procesing _ Monson HAYES
• Discrete RaNdom Signals and Statistical Signal Processing _ THERRIEN

The first book is a must read, the second one is following the style of Oppenheim series but is harder to follow, and less practical than the first.

Then the following books / subjects will be awating you :

• Adaptive Filter Theory _ HAYKIN
• Multiresolution Signal Decomposition _ AKANSU
• Multirate Digital Signal Processing _ RABINER
• Estimation & Detection Theory _ KAY
• Pattern Classification _ DUDA
• Theory and Applications of Digital Signal Processing _ RABINER & GOLD

Then the following applications will make your day :

• Speech and Hearing for Communication - FLETCHER

• Digital Processing of Speech _ Rabiner

• Speech and Audio Signal Processing _ GOLD

• Discrete-Time Processing of Speech _ PROAKIS

• Advances in Speech Coding _ GERSHO

• Two-Dimensional Signal and Image Processing _ LIM

• Digital Image Processing _ GONZALES

• Fundamentals of Image Processing _ JAIN

• Signal Compression_JAYANT

• Introduction to Data compression _ SAYOOD

Of course the list is by no means complete...

• Just a note: What we used to call "Linear System Theory" in the olden days is now more often called "Signals and Systems". I guess Alan Oppenheim gets to set the semantics here. Must be nice to be Alan Oppenheim. Oct 23 '20 at 17:50
• @robertbristow-johnson So what appears to be Signals & Systems is actually a refined and stripped down version of Linear System Theory for undergraduates, leaving the latter for graduate work... Must be nice to be him of course! Yet, his books are criticised for being too verbose (agreed!), and lacking some important mathematical tools that'r required for solving end-of-chapter problems, and most importantly lacking any practical interest whatsoever. Of course a single book cannot meet all, and he's chosen his aim as establishing the theoretical (mathematical) body of signals & systems. Oct 23 '20 at 18:14

I found these books to be very good in their respective field:

J.R. Ohm - Multimedia Communication Technology This has focus on representation and transmission of signals. It follows a practical approach hands down and features very good, informative illustrations. In general, Ohm's books are recommendable. His newest one is about feature extraction, but I have not read it yet.

Cover and Thomas - Elements of Information Theory As the title suggests, this has more of a theoretical approach. It is not about signal processing per se, but I found it very insightful. It covers the mathematical and stochastic aspects of any kind of information transmission. As I said, this is not about DSP, but more about how to get signals over a channel of some kind. As theoretical as it is, it provides extremely useful background knowledge in my everyday work as an engineer.

Vary and Martin - Digital Speech Transmission As the title says, this is all about speech communication. If you are interested in this particular field, this is a good overview.

All these are quite special and not directly about DSP, but with Oppenheim you already have the basics and to dive deeper into everything, the field is just to broad for one book to cover it all.

• Thanks a lot, Valuable information Oct 23 '20 at 9:54