Sound in the Time Domain

334 Pages·2018·19.882 MB·English

by  Mikio Tohyama (auth.)

Signals and Communication Technology
Mikio Tohyama
Sound in
the Time
Domain
Signals and Communication Technology
More information about this series at http://www.springer.com/series/4748
Mikio Tohyama
Sound in the Time Domain
123
MikioTohyama
Research Consulting Firm
WaveScience Study
Kanagawa
Japan
ISSN 1860-4862 ISSN 1860-4870 (electronic)
Signals andCommunication Technology
ISBN978-981-10-5887-5 ISBN978-981-10-5889-9 (eBook)
https://doi.org/10.1007/978-981-10-5889-9
LibraryofCongressControlNumber:2017952503
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Preface
Sounds create an evolving signal of the events occurring around us in our daily
lives. The field of acoustics is that branch of natural science concerned with the
physicalandpsychologicalpropertiesofsuchsounds.Aswithotherareasofnatural
sciences,thepropertiesofsoundcanbedescribedbymathematics.Asmathematics
isanabstractscience,soundisalsoanalyzedintermsofabstractspaces.However,
theprimitivestructurescontainedwiththesoundsencountered inourreallivesare
left without one’snotice. The motivation of this book isthe return to the nature of
sound from the perspective of time structures.
Fouriertransformingafunctionintoitsimagefunctionhasbeenthefundamental
and historical approach to the science of waves and particularly the field of
acoustics. Given a function that depends on the time variable, the Fourier trans-
formation provides a means to extract the spectral properties of the function. The
spectral properties are normally represented by the frequency variable. Thus, the
spectral function can be characterized by relating each spectral component to a
sinusoidal function. Any function of the time variable or waveform can be repre-
sented as a superposition of sinusoidal functions. This approach by decomposition
in treating the waveform as a sum of sinusoidal functions has been heavily
exploited in signal analysis of waves and particularly sound in the frequency
domain using Fourier transformation theory. Consequently, the behavior of sound
wavesinthetimedomainhashadlessattentionalthoughsoundwaveformscanbe
normally recorded in the time domain.
Thisbookfocusesonthecharacteristicsofsoundwavesinthetimedomain.This
time domain approach provides an informative and intuitively understandable
descriptionofvariousacoustictopicssuchassoundwavestravelinginanacoustic
tube or in other media where spectral or modal analysis can be intensively per-
formed. The effects of reflection from the boundary on propagating waves are
explicitly manifest, for example.
Theenvelopeorthesignaldynamicsisrepresentativeofsoundsignaturesinthe
time domain. The global behavior of the waveform defined by the envelope is
determined by local structures related to the spectral characteristics including the
phase of the waveform. The signal dynamics as represented by envelopes provide
v
vi Preface
themaincuesinspeechintelligibility.Thisfacthasorientedtheauthor'sresearchto
the time domain approach and reestablished the importance of the phase of the
wave, which has been given less attention in the frequency domain.
Theperiodicnatureofsoundisanothertypicalsignatureassociatedwiththetime
domain that relates to the harmonic and nonharmonic structures in the global
spectralpropertiesthatareindependentofthephase.Sensationofpitchthatmostly
relates to the period of sound is a historical topic in sound perception; however, it
seems still a very attractive topic to the author. Sound generated by a musical
instrument such as a piano looks nonharmonic. Such sound waveforms may be
interpretedthroughperiodanalysesinthetimedomaintobesubjecttothemissing
fundamental condition where the fundamental is out of auditory range.
Alternatively, sound waves travel in media basically governed by the theory of
linear system as described by linear equations. Here the impulse responses and
convolution are key concepts. The solutions of the linear equations are superpo-
sition of the particular and general solutions. This decomposition reminds the
author of the decomposition of sound waves into direct and reflected (or rever-
beration) sound waves in the time domain. This type of decomposition would be
informative not only for room acoustics but the study of musical sounds such as
those from piano string vibrations, where the waveform can be understood as a
superposition of the direct wave resulting from the excitation by the hammer and
the resonant modal waves generated in the string vibration.
Sound propagation in a medium or radiation from a sound source can also be
formulated in accordance with convolution theory. This formulation has also
motivated the author to take up the time domain approach. In addition to that, the
analysisofroomreverberationinthetimedomainhashistoricalrootsinthefieldof
acoustics.
Thisbookdealswith thesignature analysis ofsoundfrom theviewpoint ofthe
timedomain.Theauthorhastriedtomakethisbookself-containedandusefulasa
textbook for graduate and undergraduate courses offered at universities. The book
starts from the very introductory topic of sinusoidal waves basing the material on
theformalrelationshipbetweenthetime andfrequencydomains.Forthatpurpose,
the fundamental notions of Fourier or z-transformations and linear systems theory
aresummarizedalongwithinterestingexamplesfromacousticalresearch.Research
engineers and scientists might also have interest in this book for its different
approach. In particular, the expressions concerning waveforms including the
impulse responses might be informative to audio engineers who are familiar with
digital signal analysis.
Exerciseshavebeenpreparedforeverychapter.Theseexercisesareverysimple
and designed to be solved straightforwardly by hand without the need for a com-
puter.Thus,theyassistinreconfirmingthefundamentalideasandnotionspresentin
every chapter. However, the author has had to assume some prior knowledge on
acoustics and signal processing so that from a practical point of view the finished
book remained concise. For readers who are not familiar with acoustics in general
including signal analysis, the references below have been found to be useful.
Preface vii
W.M.Hartmann, Signals, sound, and sensation, Springer (1997)
J. Blauert and N. Xiang, Acoustics for engineers, Springer (2008)
M. Tohyama, Sound and signals, Springer (2011)
M. Tohyama, Waveform analysis of sound, Springer (2015)
TheauthorthanksTomomiHasegawa,MiyabiKonishi,YoshifumiHara,Satoru
Gotoh, Yoshinori Takahashi, Takatoshi Okuno, Mitsuo Matsumoto, and Michiko
Kazama (Waseda University, Kogakuin University), Prof. Manabu Fukushima
(Nippon Bunri University), Prof. Hirofumi Nakajima and Prof. Kazunori Miyoshi
(Kogakuin University), Prof. Yoshio Yamasaki, and Prof. Katsuhiko Shirai
(Waseda University) for their research collaborations that have oriented the author
towards the issues of sound synthesis in the time domain described in this book.
The author also thanks Hirofumi Onitsuka and his research colleagues (Yamaha
Corporation), Hiroyuki Satoh and his group members (Ono Sokki Co. Ltd), and
YouskeTanabeandhiscolleagues(HitachiResearchCorporation)fortheirfruitful
and intensive discussions. The author also thanks Edanz Group Japan Co. Ltd for
checking the author’s written English.
The author acknowledges that this book is motivated by the very kind and
responsive editorship of Dr. Christoph Baumann (Springer). The author also
acknowledges thevery kindguidance provided by Dr. Kouji Maruyama (Wolfram
Research Asia Ltd) in using Mathematica. The author expresses his great appre-
ciationtoProf.TammoHoutgast(AmsterdamFreeUniversity)andDr.Yoshimutsu
Hirata (SV Research Associates) for the long-term research cooperation spanning
many years and theveryfruitful suggestionsand discussions.The author sincerely
thanks Prof. Yoichi Ando (Kobe University) for the inspiration and motivation to
takeupthetimedomainapproach.Finally,theauthorextendshisappreciationtoall
authors of research articles referred to in this book.
Kanagawa, Japan Mikio Tohyama
May 2017
Contents
1 Signal Dynamics as Superpositions of Sinusoidal Waves in the
Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Sinusoidal Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Representation of Sinusoidal Waves . . . . . . . . . . . . . . 1
1.1.2 Complex Exponential Function . . . . . . . . . . . . . . . . . . 2
1.1.3 Complex Variables and Logarithmic Functions . . . . . . 3
1.2 Temporal Fluctuations of Sinusoidal Waves in the Time
Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Analytic Signals and Envelopes . . . . . . . . . . . . . . . . . 5
1.2.2 Beats. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 Phase and Group Speed . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.5 Period and Fundamental Frequency. . . . . . . . . . . . . . . 12
1.2.6 Missing Fundamental . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.7 Harmonic, Nonharmonic, and Almost Periodic
Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Spectral Modification by the Superposition of Sinusoidal
Waves with Different Phases . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.1 Interference of Sinusoidal Waves . . . . . . . . . . . . . . . . 16
1.3.2 Superposition with Different Phases in the Time
Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.3 Cross-Correlation of Sinusoidal Waves . . . . . . . . . . . . 19
1.3.4 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Sinusoidal Waves as Random Variables . . . . . . . . . . . . . . . . . . . . . 31
2.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1.1 Probability Distribution and Expectation . . . . . . . . . . . 31
2.1.2 Sum of Independent Random Variables and
Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
ix
x Contents
2.2 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.1 Correlation Functions for Random Variables . . . . . . . . 35
2.2.2 Correlation and Square Correlation . . . . . . . . . . . . . . . 36
2.3 Probability Distribution for Sinusoidal Waves. . . . . . . . . . . . . . 37
2.3.1 Probability Density Function. . . . . . . . . . . . . . . . . . . . 37
2.3.2 Probability Density Function for Sinusoidal Wave . . . . 39
2.3.3 Uncorrelation and Independence of Random
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.4 Binaural Merit in Listening to Pairs of Signals. . . . . . . 43
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Fourier Transform and Superposition of Sinusoidal Functions. . . . 51
3.1 Convolution, Generating Functions, and Fourier Transform. . . . 52
3.1.1 Generating Function and Combination. . . . . . . . . . . . . 52
3.1.2 Fourier Transform and Convolution. . . . . . . . . . . . . . . 53
3.1.3 Periodicity of the Fourier Transform . . . . . . . . . . . . . . 54
3.1.4 Inverse Fourier Transform. . . . . . . . . . . . . . . . . . . . . . 54
3.1.5 Auto-Correlation and Auto-Convolution for Signal
Dynamics in Time Domain. . . . . . . . . . . . . . . . . . . . . 55
3.1.6 Decomposition of Sequence into Even and Odd
Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1.7 Analytic Sequence and Envelope. . . . . . . . . . . . . . . . . 61
3.2 Symbolic Expression of Time Delay and Convolution . . . . . . . 63
3.2.1 Magnitude Spectral Modification by Superposition of
Direct and Delayed Sound . . . . . . . . . . . . . . . . . . . . . 63
3.2.2 Effect of Phase on the Direct Sound from the Time
Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.3 Clustered Time Sequence and Its Spectral Effect . . . . . 67
3.2.4 Sinc Function and Auto-Correlation. . . . . . . . . . . . . . . 69
3.3 Fourier Transform of Functions . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3.1 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3.2 Sinc Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.3 Spectral Weighting and Auto-Correlation Function. . . . 72
3.3.4 Windowing in the Time Domain. . . . . . . . . . . . . . . . . 76
3.4 Triangular Windowing and Group Delay . . . . . . . . . . . . . . . . . 78
3.4.1 Phase and Group Delay Functions. . . . . . . . . . . . . . . . 78
3.4.2 Group Delay Function for N(cid:1)Sample Delay of
Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.4.3 Group Delay Functions and Locations of Zeros . . . . . . 82
3.5 Fourier Series Expansion of a Periodic Function. . . . . . . . . . . . 83
3.5.1 Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5.2 Auto-Convolution and Auto-Correlation of Spectral
Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84