introduction to sound processinga davide rocchesso p 248 d 2003n3983_h

1、Introduction toSound ProcessingDavide Rocchesso11Universitdi Verona Dipartimento di Informaticaemail: D.Rwww: http:/www.scienze.univr.it/rocchessCopyright c 2003 Davide Rocchesso. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Fre

2、e Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no In- variant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the appendix entitled “GNU Free Documentation License”.The book is accessible from th

3、e authors web site: http:/www.scienze.univr.it/rocchess. The book is listed in , where reviews can beposted.ISBN 88-901126-1-1Cover Design: Claudia Calvaresi.Editorial Production Staff: Federico Fontana, Alessandra Ceccherelli, Nicola Giosmin, Anna Meo.Produced from LATEX tex

4、t sources and PostScript and TIFF images. Compiled with VTEX/free.Online distributed in Portable Document Format.Printed and bound in Italy by PHASAR Srl, Firenze.Contents1Systems, Sampling and Quantization11361112131.4Continuous-Time Systems . . . . . . . . . . . . . . . . . . .The Sampl

5、ing Theorem . . . . . . . . . . . . . . . . . . . . .Discrete-Time Spectral Representations . . . . . . . . . . . .Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . ..21.4.3The Impulse Response . . . . . . . . . . . . . . . . .The Shift Theorem . . . . . . . . . . . . . . . . . .

6、 .Stability and Causality . . . . . . . . . . . . . . . . .1.5Continuous-time to discrete-time systemconversion . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.5.1Impulse Invariance . . . . . . . . . . . . . . . . . . .1.5.2Bilinear Transformation . . . . . . . . . . . . . . . .Quantization

7、 . . . . . . . . . . . . . . . . . . . . . . . . . . .151517191.62Digital Filters232424293240434447555862642.1FIR Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . ...4The Simplest FIR Filter. . . . . . . . . . . . . . . .The Phase Response . . . . . . . . . . . . . . .

8、.Higher-Order FIR Filters. . . . . . . . . . . . . . .Realizations of FIR Filters . . . . . . . . . . . . . . . .2.2IIR Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . ...4The Simplest IIR Filter. . . . . . . . . . . . . . . .Higher-Order IIR Filters . . . . . . . . .

9、. . . . . . .Allpass Filters. . . . . . . . . . . . . . . . . . . .Realizations of IIR Filters . . . . . . . . . . . . . . .2.32.4Complementary filters and filterbanks . . . . . . . . . . . . .Frequency warping . . . . . . . . . . . . . . . . . . . . . . .iiiD. Rocchesso: Sound Processing3Delays and

10、 Effects6767686972747678798181893.13.2The Circular Buffer . . . . . . . . . . . . . . . . . . . . . . .Fractional-Length Delay Lines . . . . . . . . . . . . . . . . ..2FIR Interpolation Filters . . . . . . . . . . . . . . . .Allpass Interpolation Filters . . . . . . . . . . . . . .3.33.4The

11、Non-Recursive Comb Filter . . . . . . . . . . . . . . . .The Recursive Comb Filter . . . . . . . . . . . . . . . . . . .3.4.1The Comb-Allpass Filter. . . . . . . . . . . . . . .3.53.6Sound Effects Based on Delay Lines . . . . . . . . . . . . . .Spatial sound processing . . . . . . . . . . . . . . .

12、. . . . .3.6.1Spatialization . . . . . . . . . . . . . . . . . . . . . .3.6.2Reverberation . . . . . . . . . . . . . . . . . . . . . .4Sound Analysis4.1Short-Time Fourier Transform . . . . . . . . . . . . . . . . .9999991001031081101..44.1.5The Filterbank View . . . . . . . . . .

13、. . . . . . . .The DFT View . . . . . . . . . . . . . . . . . . . . .Windowing . . . . . . . . . . . . . . . . . . . . . . .Representations . . . . . . . . . . . . . . . . . . . . .Accurate partial estimation . . . . . . . . . . . . . .4.2Linear predictive coding (with Federico Fontana) . . . . . .

14、. .5Sound Modelling1171171171221231241241251281291301301351361371381405.1Spectral modelling . . . . . . . . . . . . . . . . . . . . . . ..25.1.3The sinusoidal model . . . . . . . . . . . . . . . . .Sines + Noise + Transients . . . . . . . . . . . . . . .LPC Modelling . . . . . . . . . . . .

15、. . . . . . . . .5.2Time-domain models . . . . . . . . . . . . . . . . . . . . . ...4The Digital Oscillator. . . . . . . . . . . . . . . . .The Wavetable Oscillator . . . . . . . . . . . . . . . .Wavetable sampling synthesis . . . . . . . . . . . . .Granular synthesis (with Giovanni

16、 De Poli) . . . . . . .5.3Nonlinear models . . . . . . . . . . . . . . . . . . . . . . . ..2Frequency and phase modulation . . . . . . . . . . . .Nonlinear distortion . . . . . . . . . . . . . . . . . .5.4Physical models . . . . . . . . . . . . . . . . . . . . . . . . ..25.4.3A physi

17、cal oscillator . . . . . . . . . . . . . . . . . .Coupled oscillators . . . . . . . . . . . . . . . . . . .One-dimensional distributed resonators . . . . . . . .iiiAMathematical Fundamentals145145145146147148152154158158161164164168169170171172173173175A.1Classes of Numbers . . . . . . . . . . . . .

18、 . . . . . . . . . .A.1.1A.1.2A.1.3Fields . . . . . . . . . . . . . . . . . . . . . . . . . .Rings . . . . . . . . . . . . . . . . . . . . . . . . . .Complex Numbers . . . . . . . . . . . . . . . . . . .A.2A.3A.4Variables and Functions . . . . . . . . . . . . . . . . . . . . .Polynomials . . . . . .

19、 . . . . . . . . . . . . . . . . . . . . .Vectors and Matrices . . . . . . . . . . . . . . . . . . . . . .A.4.1Square Matrices . . . . . . . . . . . . . . . . . . . .Exponentials and Logarithms. . . . . . . . . . . . . . . . .Trigonometric Functions . . . . . . . . . . . . . . . . . . . .Derivatives

20、 and Integrals . . . . . . . . . . . . . . . . . . . .A.7.1 Derivatives of Functions . . . . . . . . . . . . . . . .A.5A.6A.7A.7.2 Integrals of Functions . . . . . . . . . . . . . . .Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . .A.8A.8.1A.8.2A.8.3The Laplace Transform. . . . . .

21、. . . . . . . . . .The Fourier Transform . . . . . . . . . . . . . . . . .The Z Transform . . . . . . . . . . . . . . . . . . .A.9Computer Arithmetics. . . . . . . . . . . . . . . . . . . . .A.9.1Integer Numbers . . . . . . . . . . . . . . . . . . . .A.9.2Rational Numbers . . . . . . . . . . . . . .

22、 . . . . .BTools for Sound Processing(with Nicola Bernardini)177178179182185186192193195196198200202203205B.1Sounds in Matlab and Octave . . . . . . . . . . . . . . . . . .B.1.1Digression . . . . . . . . . . . . . . . . . . . . . . .Languages for Sound Processing . . . . . . . . . . . . . . . .B.2.1

23、Unit generator. . . . . . . . . . . . . . . . . . . . .B.2.2Examples in Csound, SAOL, and CLM . . . . . . . .Interactive Graphical Building Environments. . . . . . . . .B.3.1Examples in ARES/MARS and pd . . . . . . . . . .Inline sound processing . . . . . . . . . . . . . . . . . . . . .B.4.1 Time-Do

24、main Graphical Editing and Processing . . .B.4.2 Analysis/Resynthesis Packages . . . . . . . . . . . .Structure of a Digital Signal Processor . . . . . . . . . . . . .B.2B.3B.4B.5B.5.1B.5.2B.5.3Memory Management . . . . . . . . . . . . . . . . .Internal Arithmetics. . . . . . . . . . . . . . . . . .

25、The Pipeline . . . . . . . . . . . . . . . . . . . . . .ivD. Rocchesso: Sound ProcessingCFundamentals of psychoacoustics207207209213215217217219C.1C.2The ear . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Sound Intensity . . . . . . . . . . . . . . . . . . . . . . . . .C.2.1Psychophysics.

26、. . . . . . . . . . . . . . . . . . . .Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Critical Band . . . . . . . . . . . . . . . . . . . . . . . . . .Masking . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Spatial sound perception . . . . . . . . . . . . . . . . . . . .

27、C.3C.4C.5C.6DGNU Free Documentation License223224226226227229229D.1D.2D.3D.4D.5D.6D.7APPLICABILITY AND DEFINITIONS . . . . . . . . . . . .VERBATIM COPYING. . . . . . . . . . . . . . . . . . . .COPYING IN QUANTITY . . . . . . . . . . . . . . . . . . .MODIFICATIONS . . . . . . . . . . . . . . . . . .

28、. . . . .COMBINING DOCUMENTS . . . . . . . . . . . . . . . . .COLLECTIONS OF DOCUMENTS . . . . . . . . . . . . . .AGGREGATION WITH INDEPENDENTWORKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .TRANSLATION . . . . . . . . . . . . . . . . . . . . . . . .TERMINATION . . . . . . . . . . . .

29、. . . . . . . . . . . .230230231231D.8D.9D.10 FUTURE REVISIONS OF THIS LICENSE . . . . . . . . . .Index233References241PrefaceWhat you have in your hands, or on your screen, is an introductory book on sound processing. By reading this book, you may expect to acquire some knowledge on the mathematica

30、l, algorithmic, and computational tools that I consider to be important in order to become proficient sound designers or ma- nipulators.The book is targeted at both science- and art-oriented readers, even though the latter may find it hard if they are not familiar with calculus. For this purposean a

31、ppendix of mathematical fundamentals has been prepareduch a waythat the book becomes self contained. Of course, the mathematical appendixis not intended to be a substitute of a thorough mathematical preparation, but rather as a shortcut for those readers that are more eager to understand the applica

32、tions.Indeed, this book was conceived in 1997, when I was called to teach in- troductory audio signal processing in the course “Specialisti in Informatica Musicale” organized by the Centro Tempo Reale in Firenze. In that class, the majority of the students were excellent (no kidding, really superb!)

33、 music com- posers. Only two students had a scientific background (indeed, a really strong scientific background!). The task of introducing this audience to filters and trasforms was so challenging for me that I started planning the lectures and laboratory material much earlier and in a structured f

34、orm. This was the ini- tial form of this book. The course turned out to be an exciting experience for me and, based on the music and the research material that I heard from them afterward, I have the impression that the students also made good use of it.After the course in Firenze, I expanded and im

35、proved the book during four editions of my course on sound processing for computer science students at the University of Verona. The mathematical background of these students is different from that of typical electrical engineering students, as it is stronger in discrete mathematics and algebra, and

36、 with not much familiarity with advancedvviD. Rocchesso: Elaborazione del Suonoand applied calculus. Therefore, the books presents the basics of signals, sys- tems, and transforms in a way that can be immediately used in applications and experienced in computer laboratory sessions.This is a free boo

37、k, thus meaning that it was written using free software tools, and it is freely downloadable, modifiable, and distributable in electronic or printed form, provided that the enclosed license and link to its original web location are included in any derivative distribution. The book web site also con-

38、 ta the source codes listed in the book, and other auxiliary software modules.I encourage additions that may be useful to the reader. Fortance, itwould be nice to have each chapter ended by a section that collects annotations, solutions to the problems that I proposed in footnotes, and other problem

39、s orexercises. Feel free to exploit the open nature of this book to propose your ad- ditional contents.Venezia, 7th October 2003Davide RocchessoChapter 1Systems, Sampling and Quantization1.1Continuous-Time SystemsSound is usually considered as a mono-dimensional signal (i.e., a function of time) rep

40、resenting the air pressure in the ear canal. For the purpose of this book, a Single-Input Single-Output (SISO) System is defined as any algorithm or device that takes a signal in input and produces a signal in output. Most of our discussion will regard linear systems, that can be defined as those sy

41、stems for which the superposition principle holds:Superposition Principle : if y1 and y2 are the responses to the input sequences x1 and x2, respectively, then the input ax1 + bx2 produces the response ay1 + by2.The superposition principle allows us to study the behavior of a linear sys- tem startin

42、g from test signals such as impulses or sinusoids, and obtaining the responses to complicated signals by weighted sums of the basic responses.A linear system is said to be linear time-invariant (LTI), if a time shift in the input results in the same time shift in the output or, in other words, if it

43、 does not change its behavior in time.Any continuous-time LTI system can be described by a differential equa- tion. The Laplace transform, defined in appendix A.8.1 is a mathematical tool that is used to analyze continuous-time LTI systems, since it allows to trans- form complicated differential equ

44、ations into ratios of polynomials of a complex12D. Rocchesso: Sound Processingvariable s. Such ratio of polynomials is called the transfer function of the LTI system.Example 1. Consider the LTI system having as input and output the func- tions of time (i.e., the signals) x(t) and y(t), respectively,

45、 and described by the differential equationdy s0y = x .(1)dtThis equation, transformed into the Laplace domain according to the rules of appendix A.8.1, becomessYL(s) s0YL(s) = XL(s) .(2)Here, as in most of the book, we implicitly assume that the initial conditions are zero, otherwise eq. (2) should

46、 also contain a term in y(0). From the algebraic equation (2) the transfer function is derived as the ratio between the output and input transforms:1s s0H(s) =.(3)#The coefficient s0, root of the denominator polynomial of (3), is called the pole of the transfer function (or pole of the system). Any

47、root of the numerator would be called a zero of the system.The inverse Laplace transform of the transfer function is an equivalent de- scription of the system. In the case of example 1.1, it takes the formes0t0t 0t 0h(t) =,(4)and such function is called a causal exponential.In general, the function

48、h(t), inverse transform of the transfer function, is called the impulse response of the system, since it is the output obtained from the system as a response to an ideal impulse1.The two equivalent descriptions of a linear system in the time domain (im- pulse response) and in the Laplace domain (tra

49、nsfer function) correspond to two alternative ways of expressing the operations that the system performs in order to obtain the output signal from the input signal.1A rigorous definition of the ideal impulse, or Dirac function, is beyond the scope of this book. The reader can think of an ideal impul

50、se as a signal having all its energy lumped at the time tant 0.Systems, Sampling and Quantization3The description in the Laplace domain leads to simple multiplication between the Laplace transform of the input and the system transfer function:Y (s) = H(s)X(s) .(5)This operation can be interpreted as

51、 multiplication in the frequency domain if the complex variable s is replaced by j, being the real variable of the Fourier domain. In other words, the frequency interpretation of (5) is obtained by restricting the variable s from the complex plane to the imaginary axis. The transfer function, whose

52、domain has been restricted to j is called frequency response. The frequency interpretation is particularly intuitive if we imagine the input signal as a complex sinusoid ej0 t, which has all its energy focused on the frequency 0 (in other words, we have a single spectral line at 0). The complex value of the frequency response (magnitude and phase) at the point j0 corresponds to a joint magnitude scaling and phase shift of the sinusoid at that frequency.The description in t