Abstract
As it says in the front of the ComputerMusic Journal number 4, there are many musicians with an interest in musical signal processing with computers, but only a few have much competence in this area. There is of course a huge amount of literature in the field of digital signal processing, including some first-rate textbooks (such as Rabiner and Gold's Theory and Application of Digital Signal Processing, or Oppenheim and Schafer's Digital Signal Processing), but most of the literature assumes that the reader is a graduate student in engineering or computer science (why else would he be interested?), that he wants to know everything about digital signal processing, and that he already knows a great deal about mathematics and computers. Consequently, much of this information is shrouded in mathematical mystery to the musical reader, making it difficult to distinguish the wheat from the chaff, so to speak. Digital signal processing is a very mathematical subject, so to make past articles clearer and future articles possible, the basic mathematical ideas needed are presented in this two-part tutorial. In order to prevent this presentation from turning into several fat books, only the main ideas can be outlined; and mathematical proofs are of course omitted. But keep in mind that learning mathematics is much like learning to play a piano: no amount of reading will suffice -it is necessary to actually practice the techniques described (in this case, by doing the problems) before the concepts become useful in the real world. Therefore some problems are provided (without answers) to give the motivated reader an opportunity both to test his understanding and to acquire some skill. Part I of the tutorial (this part) provides a general review of algebra and trigonometry, including such areas as equations, graphs, polynomials, logarithms, complex numbers, infinite series, radian measures, and the basic trigonometric functions. Part II will discuss the application of these concepts and others in transforms, such as the Fourier and z-transforms, transfer functions, impulse response, convolution, poles and zeroes, and elementary filtering. Insofar as possible, the mathematical treatment always stops just short of using calculus, though a deep understanding of many of the concepts presented requires understanding of calculus. But digital signal processing inherently requires less calculus than analog signal processing, since the integral signs are replaced by the easier-tounderstand discrete summations. It is an experimental goal of this tutorial to see how far into digital signal processing it is possible to explore without calculus.
Published Version
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