I am very interested in learning about audio from a signal processing standpoint. However, whenever I try to further my education by reading books, I get extremely frustrated because the books use all kinds of crazy math notation that makes absolutely no sense to me...

An example would be this book:

The first chapter (1.1) starts out completely clear... A nice diagram showing analog -> digital -> analog.. Great.. got it..

Then 1.2 goes to show $X(\Omega) = \int_{-\infty}^\infty x(t)e^{-i\Omega t} dt\ldots$

Uhhh.. yeah.. sorry.. I am totally lost and frustrated at this point.

Can anyone guide me to some sort of book or online site that will help me make sense of this notation?

EDIT: Some of your comments asked about my background. Unfortunately, I've only got a background in algebra, and that itself is slightly shaky as I haven't done a lot with it since high school.

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    $\begingroup$ What is your background so far? What math have you taken? Without some course(s) or sufficient background to understand some things about linear time-invariant systems, it may be a bit tough going at first. You'll need to be reasonably proficient in calculus, have a little background in complex arithmetic and the Fourier transform, and know at least enough differential equations to understand constant-coefficient linear ordinary differential equations. $\endgroup$ – cardinal Sep 24 '11 at 22:44
  • $\begingroup$ I think you have selected a book that is not suitable for your interests. You might want to start with very simple basics of sound and spectrum analysis of sound first. That would give a handle on the topics : the scales, octaves and timbre of instruments. Many music theory sites would give a musician's point of view of scales and notes. Then audio engineering sites will give you the effect of certain operations such as filtering etc. Then you can back these intuitions up by mathematical gibberish :) $\endgroup$ – user13838 Sep 24 '11 at 22:47
  • $\begingroup$ Also, if you can edit your question and include more specific details I might give some suggestions. $\endgroup$ – user13838 Sep 24 '11 at 22:49
  • $\begingroup$ You are looking for articles on Fourier analysis. There is a tutorial at sunlightd.com/Fourier and a discussion that avoids complex numbers at colorado.edu/MCEN/Measlab/backgroundfourier.pdf $\endgroup$ – Ross Millikan Sep 24 '11 at 22:50
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    $\begingroup$ It might interest you to know that there is also a dsp.SE. $\endgroup$ – J. M. is a poor mathematician Sep 25 '11 at 1:51

This is probably heretical for math.SE, but you don't need to understand that equation. Just skim over it. You aren't going to use it for anything anyway.

Signal processing isn't mathematically rigorous (see the intro of Dirac delta "function", for instance). You don't actually work out integrals to find Fourier transforms. Instead, you memorize the most common Fourier transform pairs, and learn how mathematical operations in the time domain translate to the frequency domain (multiplication ⇔ convolution, for instance), so you can represent complicated signals as a combination of simple signals that you can work with easily.

Engineering is all about applying mathematics to build practical things, and taking lots of shortcuts and simplifications in the process. We transform to the Laplace domain and use phasors to avoid doing differential equations, converting them into polynomials and algebra. We memorize tables of common Fourier transforms to avoid doing the integrals, etc.

Fourier transform pairs: Fourier transform pairs

For instance, say you have a recording of a tuning fork at 440 Hz (a sine wave), and you want to send it over the radio at 1 MHz. To do this, you multiply the 440 Hz sine wave with another sine wave at 1 MHz. This is amplitude modulation.

$x(t) = \cos(2 \pi 440 t) \cdot \cos(2 \pi 1000000 t)$

You know the Fourier transform of each sinusoid is a Dirac spike (as in the above graphic), and you know that multiplication in the time domain is equivalent to convolution in the frequency domain, so you can convolve the spectra of the two sine waves to get the spectrum of the result. Once you learn convolution, you'll know that this is just two spikes at the sum and difference frequencies: 1000000-440 and 1000000+440. You don't actually go through the trouble of solving the integral

$X(\Omega) = \int_{-\infty}^\infty \cos(2 \pi 440 t) \cdot \cos(2 \pi 1000000 t)e^{-i\Omega t} dt$

Solving this is not trivial, but applying transform tables is. It's more important to see in your head what's happening.

To demodulate at the other end, you multiply by 1 MHz again, producing frequency components at the sum and difference frequencies again, which are now 440 Hz, 2000440 Hz, and 1999560 Hz. The latter two can be thrown away by filtering, which just means multiplying by 0 in the frequency domain using a rectangle function, and you're left with the original recording. (And again, this is not mathematically rigorous; real filters are not rectangular, and calculating real filters' actual effects mathematically can be very difficult.)

For the stuff you want to know about audio signal processing, this is sufficient. When you get into more advanced stuff and need to know the details, you can go back and learn it in more depth.

The relationship of formal mathematics to the real world is ambiguous. Apparently, in the early history of mathematics the mathematical abstractions of integers, fractions, points, lines, and planes were fairly directly based on experience in the physical world. However, much of modern mathematics seems to have its sources more in the internal needs of mathematics and in esthetics, rather than in the needs of the physical world. 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. (Hamming, Digital Filters, 1998 Dover edition, page 72.)

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    $\begingroup$ This "practical"attitude towards our most technical disciplines is why America is between 25th and 37th in the world-depending on who does the evaluating-in mathematics and physical science training among produced scientists and why most of our top professionals are born in other countries. I'm sure you meant well,but encouraging this isn't helping ANYONE. $\endgroup$ – Mathemagician1234 Sep 25 '11 at 17:25
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    $\begingroup$ @Mathemagician1234: Engineering is practical. I'm encouraging him to focus on learning signal processing and not worry about the more difficult math used to derive it, since it's not needed anyway. (You don't need to learn about Fourier's habit of wearing blankets, either.) Engineers apply math to get things done, not to dive deeper into math. It's not hurting anything to encourage shortcuts for small stuff so that you can focus on bigger-picture stuff. Anyone who wants to dive into the details can, but it's not necessary for engineering. $\endgroup$ – endolith Sep 25 '11 at 22:25
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    $\begingroup$ I've gotten in trouble on Math Overflow for getting too passionate and opinionated in the past.So I'm going to merely say we'll have to agree to disagree on this one. $\endgroup$ – Mathemagician1234 Sep 27 '11 at 4:07

You first need to have a fairly good background in undergraduate real variables. That's really going to be essential regardless of how deeply into the theoretical aspects you want to do. That means working your way through Rudin's Principles Of Mathematical Analysis or Charles Pugh's Real Mathematical Analysis. You need to do that before tackling anything specific in signal processing. (Of course, it goes without saying you need to be very good at basic calculus first!) You also need to be pretty good at linear algebra since so much of signal processing theory is decomposition by linear basis function approximation. An applied book like Strang will do very nicely for your purposes since it does everything carefully,but with an applied flavor.

The great book serious applied mathematicians learned Fourier analysis and integrals from for a generation was Dym and McKean's Fourier Series And Integrals. This is absolute must read if you're interested in the theory behind signal processing-it presents the basic theory very rigorously,but very concretely with a lot of applications. Another excellent book at the same level,but with more of an applied bent for physicists and engineers, is James Walker's Fourier Analysis. It's another terrific text with everything done carefully,but with lots of applications.

Lastly,one of the great unmentioned books on Fourier Analysis is the 2 book text by Tom Korner. Loads of applications,beautifully written and with lots of mathematical insight you won't find in other texts.

All those books will give you a great place to start and from there, you can begin reading texts specifically on signal processing. Good luck!

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    $\begingroup$ I somewhat strongly disagree here. Suggesting "baby" Rudin or Pugh is very much overkill here and likely to lead to more frustration, not less, for the OP. At least in North America, I would guess that the signal-processing engineers that have ever had a course at the level of Rudin or Pugh are in the very minute minority. Much like the physicists, they seem to get along fine with a mostly "mechanical" knowledge of the associated math. Oftentimes, their intuition in that area will be quite strong, even if they don't have a strong grasp of the underlying details. $\endgroup$ – cardinal Sep 24 '11 at 23:24
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    $\begingroup$ As a mathematician, please be assured that I am completely sympathetic to this viewpoint and agree that the more math the better. However, I do not feel that it addresses the OP's intent or concerns and I do feel the suggestions here, at least in the short term, are likely to be counterproductive. $\endgroup$ – cardinal Sep 25 '11 at 0:01
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    $\begingroup$ As a pithy rejoinder to your adviser, R. W. Hamming was known to have said: "Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane." Feynman had an equally snarky (or perhaps moreso) comment to this effect. :) At any rate, the empirical evidence seems to suggest that such a background, while desirable, is not a prerequisite. $\endgroup$ – cardinal Sep 25 '11 at 0:06
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    $\begingroup$ @cardinal There's a little known exchange between Alan Turing and Ludwig Wittengenstien in the early 1930's which had the German master chastising Turing with nearly the same comment. But Turing stuck to his guns because he understood rigorous mathematics can lead to breakthroughs in physical theory and vice versa-and that kind of creativity doesn't occur if one treats mathematics as a blunt toolbox only. $\endgroup$ – Mathemagician1234 Sep 25 '11 at 1:46
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    $\begingroup$ @Mathemagician1234 The OP wants learn signal processing but he does not have access to the mathematical knowledge (yet!). Wouldn't it be first more proper to get him in the direction of signal processing rather than Rudin and other hardcore-for-beginners mathematics? The required mathematics level to get you going in DSP is much simpler and less rigorous than you might imagine(sad but true you might say). So I have to agree with cardinal here. $\endgroup$ – user13838 Sep 25 '11 at 10:53

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