Sound in the Time Domain 1st Edition by Mikio Tohyama ISBN 9789811058899 981105889X

1 Signal Dynamics as Superpositions of Sinusoidal Waves in the Time Domain

1.1 Sinusoidal Waves

1.1.1 Representation of Sinusoidal Waves

1.1.2 Complex Exponential Function

1.1.3 Complex Variables and Logarithmic Functions

1.2 Temporal Fluctuations of Sinusoidal Waves in the Time Domain

1.2.1 Analytic Signals and Envelopes

1.2.2 Beats

1.2.3 Modulation

1.2.4 Phase and Group Speed

1.2.5 Period and Fundamental Frequency

1.2.6 Missing Fundamental

1.2.7 Harmonic, Nonharmonic, and Almost Periodic Functions

1.3 Spectral Modification by the Superposition of Sinusoidal Waves with Different Phases

1.3.1 Interference of Sinusoidal Waves

1.3.2 Superposition with Different Phases in the Time Domain

1.3.3 Cross-Correlation of Sinusoidal Waves

1.3.4 Correlation Functions

1.4 Exercises

References

2 Sinusoidal Waves as Random Variables

2.1 Random Variables

2.1.1 Probability Distribution and Expectation

2.1.2 Sum of Independent Random Variables and Convolution

2.2 Correlations

2.2.1 Correlation Functions for Random Variables

2.2.2 Correlation and Square Correlation

2.3 Probability Distribution for Sinusoidal Waves

2.3.1 Probability Density Function

2.3.2 Probability Density Function for Sinusoidal Wave

2.3.3 Uncorrelation and Independence of Random Variables

2.3.4 Binaural Merit in Listening to Pairs of Signals

2.4 Exercises

References

3 Fourier Transform and Superposition of Sinusoidal Functions

3.1 Convolution, Generating Functions, and Fourier Transform

3.1.1 Generating Function and Combination

3.1.2 Fourier Transform and Convolution

3.1.3 Periodicity of the Fourier Transform

3.1.4 Inverse Fourier Transform

3.1.5 Auto-Correlation and Auto-Convolution for Signal Dynamics in Time Domain

3.1.6 Decomposition of Sequence into Even and Odd Sequences

3.1.7 Analytic Sequence and Envelope

3.2 Symbolic Expression of Time Delay and Convolution

3.2.1 Magnitude Spectral Modification by Superposition of Direct and Delayed Sound

3.2.2 Effect of Phase on the Direct Sound from the Time Delay

3.2.3 Clustered Time Sequence and Its Spectral Effect

3.2.4 Sinc Function and Auto-Correlation

3.3 Fourier Transform of Functions

3.3.1 Fourier Transform

3.3.2 Sinc Function

3.3.3 Spectral Weighting and Auto-Correlation Function

3.3.4 Windowing in the Time Domain

3.4 Triangular Windowing and Group Delay

3.4.1 Phase and Group Delay Functions

3.4.2 Group Delay Function for N-Sample Delay of Reflection

3.4.3 Group Delay Functions and Locations of Zeros

3.5 Fourier Series Expansion of a Periodic Function

3.5.1 Fourier Series

3.5.2 Auto-Convolution and Auto-Correlation of Spectral Sequence

3.5.3 Dirichlet Function

3.6 Exercises

References

4 Differential Equations for Sinusoidal Waves

4.1 Differential Equations and Linear Systems

4.1.1 Single-Degree-of-Freedom System

4.1.2 Sinusoidal Function and General Solution of Single-Degree-of-Freedom System

4.1.3 Impulse Response

4.1.4 Response to an External Force

4.1.5 Transient and Resonant Responses to a Sinusoidal Function

4.2 Source Waveform Estimation Through Differential Equation

4.2.1 Observed Response from a Single-Degree-of-Freedom System

4.2.2 Source Waveform Recovery for a Single-Degree-of-Freedom System

4.3 Wave Equations for Sinusoidal Traveling Waves

4.3.1 Equation of Waves

4.3.2 Propagation of Sinusoidal Waves

4.4 Exercises

References

5 Discrete Signals and Linear Systems

5.1 Linear Equations

5.1.1 Linear Combination of Vectors

5.1.2 Linear Equation and Linear Combination

5.1.3 Minimum-Norm Solution

5.1.4 Minimum-Norm Solution by Least-Squares Error Criterion

5.2 Linear Regression

5.2.1 Least-Squares Error Solution

5.2.2 Example of Linear Regression Analysis

5.3 Orthogonal Regression Analysis

5.3.1 Orthogonal Decomposition and Regression

5.3.2 Quadratic Form

5.3.3 Representation of a Matrix by Its Diagonal Matrix of Eigenvalues

5.3.4 Quadratic Form and Orthogonal Eigenvectors

5.3.5 Binaural Merit and Principal Component Analysis

5.3.6 Source Separation by Decorrelation of Pair of Signals

5.3.7 Decorrelation of Signals and Estimation of Impulse Response

5.4 Discrete Fourier Transform and Spectral Representation of Sequences

5.4.1 Fourier and Discrete Fourier Transforms of a Sequence

5.4.2 Examples of Spectral Sequences by Discrete Fourier Transform

5.4.3 Violation of Sampling Condition

5.5 Exercises

References

6 Transfer Functions and Sequences in the Time Domain

6.1 z-Transforms

6.1.1 Traveling Waves and the Pure Delay System

6.1.2 Discrete System and z-Transform

6.2 Phase Response and Symmetric Zeros

6.2.1 Accumulation Phase and Group Delay

6.2.2 Linear Phase and Pair of Symmetric Zeros with Respect to the Unit Circle

6.2.3 Power Spectral Functions for Pair of Symmetric Zeros

6.3 Poles for Resonance Systems and Transient Time Responses

6.3.1 Cumulative Spectral Analysis of the Transient Response for a Resonance System

6.4 Unstable Pole for Causal System

6.4.1 Poles and Howling

6.4.2 Detection of Unstable Time Responses

6.5 Poles and Filtering

6.5.1 Notch Filtering

6.5.2 Inverse and All-Pass Filtering for Predicted Pole

6.5.3 Inverse Filter for the Minimum Phase System

6.6 Exercises

References

7 Signal Dynamics for Time and Frequency Domains

7.1 Enhancement of Signal Dynamics for Noisy Waveforms

7.1.1 Superposition of Noisy Signals

7.1.2 Multiplication of Noisy Signals

7.2 Estimation of Path Information

7.3 Spectral Peak Selection and Representation of Time Waveforms

7.3.1 Estimation of Spectral Sequence of Sinusoidal Function

7.3.2 Spectral Peak Estimation for Single Sinusoidal Sequence

7.3.3 Time-Variant Spectral Peaks in Time Waveforms

7.3.4 Fundamental Frequency Tracking of Time Waveform

7.4 Modal Overlap and Representation of Envelopes

7.4.1 Separation of Auto-correlation Functions

7.4.2 Estimating Envelopes of Sound in the Time Domain

7.4.3 Examples of Signal Decomposition into Clustered Sinusoids

7.4.4 Separation of Direct Wave

7.5 Exercises

References

8 Time and Frequency Responses of Spherical Source

8.1 Impulse Responses of a Point Source

8.1.1 Pressure Response of a Point Source

8.1.2 Steady-State Response of a Point Source

8.1.3 Acoustic Transfer Impedance and Impulse Response of a Point Source

8.2 Transient Response of a Spherical Source

8.2.1 Steady-State Response

8.2.2 Acoustic Transfer Impedance of Spherical Source

8.3 Acoustic Radiation Impedance and Pressure Response in the Time Domain

8.3.1 Acoustic Radiation Impedance

8.3.2 Radiation Impedance in the Time Domain

8.3.3 Pressure Response in Time Domain

8.4 Exercises

References

9 Wave Equation and Its General Solution in the Time Domain

9.1 Initial Disturbance and One-Dimensional Waves

9.1.1 Initial Condition and Impulse Response

9.1.2 Response to Initial Disturbance of One-Dimensional Wave Equation

9.2 Wave Propagation of a Spherically Symmetric Wave

9.2.1 Wave Equation for a Spherically Symmetric Wave

9.2.2 General Solution for the Spherically Symmetric Wave

9.2.3 Free Oscillation of a Spherically Symmetric Wave

9.2.4 Propagation of the Initial Disturbance

9.3 General Solution of the Wave Equation

9.3.1 Initial Condition and Wave Propagation in Three-Dimensional Space

9.4 Exercises

References

10 Sound Traveling in One-Dimensional Space

10.1 Time-Domain Approach to Pulse-Like Sound Traveling in Acoustic Tube

10.1.1 Sound Traveling in a Tube with Open Ends

10.1.2 Sound in an Open–Closed Tube

10.1.3 Period and Fundamental Frequency

10.1.4 Magnitude Frequency Characteristics of Pulse Train

10.2 Boundary and Driving Conditions of Sound Traveling in a Tube

10.2.1 Sinusoidal Waves Under Generalized Boundary Conditions

10.2.2 Generalized Boundary Conditions in Time-Delayed Reflection

10.2.3 Driving Condition for the Acoustic Tube

10.3 Phase Trend of a Periodic Wave Along a Finite Length of String

10.3.1 Periodically Repeated Waves Traveling Between both Ends

10.3.2 Poles and Zeros of a Cyclic Wave on a Finite String

10.4 Minimum-Phase Property and Propagation Phase

10.4.1 Poles and Zeros for Minimum Phase

10.4.2 Accumulated Phase for One-Dimensional Systems

10.5 Sound Radiation from a One-Dimensionally Extended Source

10.5.1 Representation of Waves and Radiation Impedance

10.5.2 Geometric Interpretation of Radiation Impedance

10.5.3 Coincidence Frequency and Radiation Condition

10.5.4 Superposition of Vibration

10.5.5 Sound Radiation from an Impulsive Vibration

10.6 Exercises

References

11 Reverberation in Rooms

11.1 Spatial Dynamics for Superposition of Sinusoidal Waves with Random Phase

11.1.1 Statistical Approach to Sound Field in an Enclosed Space

11.1.2 Superposition of Random Sinusoidal Waves

11.1.3 Distribution of Envelopes

11.2 Transmission of Envelopes and Reverberation

11.2.1 Reverberation Response to Wideband Noise Source

11.2.2 Reverberation Response to Modulated Wideband Noise Source

11.2.3 Transmission of Modulated Sinusoidal Signal

11.3 Stochastic Reverberation Process and Reverberation ƒ

11.3.1 Ensemble Average of Reverberation Energy

11.3.2 Binomial Distribution for the Collision Process

11.3.3 Multi-nominal Distribution in the Collision Process

11.3.4 Poisson Distribution as a Limiting Case

11.4 Energy Balance Equation and Poisson Process

11.4.1 Energy Balance Equation

11.4.2 Differential Equation for Reverberation Formula

11.5 Reverberation and the Geometric Dimension of the Space

11.5.1 Mirror Image in Three-Dimensional Space

11.5.2 Mirror-Image Sources in a Two-Dimensional Space

11.5.3 Mirror Image in One-Dimensional Space

11.6 Exercises

References

12 Signal Dynamics and Sound Source Distance

12.1 Direct Sound and Distance from Source

12.1.1 Energy Ratio of Direct Sound and Reverberation

12.1.2 Standard Deviation of Magnitude Frequency Response

12.1.3 Propagation Phase and Distance

12.2 Envelope Spectrum and Sound Source Distance

12.2.1 Random Noise and Modulated Noise

12.2.2 Speech Envelopes and Distance from Source

12.2.3 Intelligibility Test

12.3 Subjective Loudness and Spectral Deformation ƒ

12.3.1 Subjective Loudness by Paired-Comparison Tests

12.3.2 Auto-correlation Functions for Early Echoes

12.4 Exercises

References

Index

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