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SPECTRAL AUDIO
SIGNAL PROCESSING
JULIUS O. SMITH III
Center for Computer Research in Music and Acoustics (CCRMA)
Department of Music, Stanford University, Stanford, California 94305 USA
October 2008 DRAFT
Preface ............................................................................................................................................ 17
Acknowledgments ................................................................................................................... 17
Book Series Overview ............................................................................................................ 18
Chapter 1 Introduction and Overview ..................................................................................... 20
1.1 Organization ............................................................................................................ 20
1.2 Overview ................................................................................................................. 21
1.2.1 Elementary Spectrum Analysis ....................................................................... 21
1.2.2 The Short-Time Fourier Transform (STFT) and Time-Frequency Displays ... 22
1.2.3 Short-Time Analysis, Modification, and Resynthesis ..................................... 22
1.2.4 Applications .................................................................................................... 22
1.2.5 Multirate Polyphase Filter and Wavelet Banks ............................................... 23
1.2.6 Appendices ...................................................................................................... 23
Chapter 2 Fourier Transforms for Continuous/Discrete Time/Frequency ............................... 24
2.1 Discrete Time Fourier Transform (DTFT) .............................................................. 25
2.2 Fourier Transform (FT) and Inverse ........................................................................ 26
2.2.1 Existence of the Fourier Transform ................................................................. 26
2.3 Fourier Theorems for the DTFT .............................................................................. 27
2.3.1 Linearity of the DTFT ..................................................................................... 27
2.3.2 Time Reversal ................................................................................................. 28
2.3.3 Symmetry of the DTFT for Real Signals ........................................................ 28
2.3.3.1 Real Even (or Odd) Signals ..................................................................... 29
2.3.4 Shift Theorem for the DTFT ........................................................................... 30
2.3.5 Convolution Theorem for the DTFT ............................................................... 31
2.3.6 Correlation Theorem for the DTFT ................................................................. 32
2.3.7 Autocorrelation ............................................................................................... 33
2.3.8 Power Theorem for the DTFT ......................................................................... 33
2.3.9 Stretch Operator .............................................................................................. 34
2.3.10 Repeat (Scaling) Operator ............................................................................... 35
2.3.11 Stretch/Repeat (Scaling) Theorem .................................................................. 36
2.3.12 Downsampling and Aliasing ........................................................................... 37
2.3.12.1 Proof of Aliasing Theorem .............................................................. 38
2.3.13 Differentiation Theorem Dual ......................................................................... 39
2.4 Continuous-Time Fourier Theorems ....................................................................... 40
2.4.1 Scaling Theorem ............................................................................................. 40
2.4.2 Spectral Roll-Off ............................................................................................. 41
2.5 Spectral Interpolation .............................................................................................. 42
2.5.1 Ideal Spectral Interpolation ............................................................................. 42
2.5.2 Interpolating a DFT ......................................................................................... 43
2.5.3 Zero Padding in the Time Domain .................................................................. 43
2.5.3.1 Practical Zero Padding ............................................................................ 44
2.5.4 Zero-Phase Zero Padding ................................................................................ 47
2.5.4.1 Matlab/Octave fftshift utility ................................................................... 49
Chapter 3 Spectrum Analysis Windows ................................................................................... 52
3.1 Rectangular Window ............................................................................................... 52
3
3.1.1 Definition ( odd): ...................................................................................... 53
3.1.2 Transform: ....................................................................................................... 53
3.1.3 Properties: ....................................................................................................... 53
3.2 Generalized Hamming Window Family .................................................................. 54
3.2.1 Hann or Hanning or Raised Cosine ................................................................. 55
3.2.2 Matlab for the Hann Window .......................................................................... 56
3.2.3 Summary of Hann window properties: ........................................................... 58
3.2.4 Hamming Window .......................................................................................... 58
3.2.5 Matlab for the Hamming Window .................................................................. 61
3.2.6 Summary of Generalized Hamming Windows ................................................ 62
3.2.7 Definition: ....................................................................................................... 62
3.2.8 Transform: ....................................................................................................... 62
3.2.9 Common Properties ......................................................................................... 62
3.2.10 Rectangular window properties: ...................................................................... 63
3.2.11 Hann window properties: ................................................................................ 63
3.2.12 Hamming window properties: ......................................................................... 64
3.2.13 The MLT Sine Window ................................................................................... 64
3.2.13.1 Properties: ....................................................................................... 64
3.3 Blackman-Harris Window Family .......................................................................... 65
3.3.1 Window Definition: ......................................................................................... 65
3.3.2 Window Transform: ........................................................................................ 65
3.3.3 Blackman Window Family .............................................................................. 65
3.3.4 Classic Blackman ............................................................................................ 66
3.3.5 Matlab for the Classic Blackman Window ...................................................... 66
3.3.6 Three-Term Blackman-Harris Window ........................................................... 67
3.3.7 Frequency-Domain Implementation of the Blackman-Harris Family .......... 69
3.3.8 Power-of-Cosine Window Family ................................................................... 69
3.3.8.1 Definition: ............................................................................................... 69
3.3.8.2 Properties: ............................................................................................... 69
3.3.8.3 Special Cases: .......................................................................................... 69
3.4 Example: Spectrum Analysis of an Oboe Tone ....................................................... 70
3.4.1 Rectangular-Windowed Oboe Recording ........................................................ 70
3.4.2 Hamming-Windowed Oboe Recording ........................................................... 71
3.4.3 Blackman-Windowed Oboe Recording ........................................................... 71
3.4.4 Conclusions ..................................................................................................... 72
3.5 Bartlett (``Triangular'') Window .............................................................................. 72
3.5.1 Definition: ....................................................................................................... 72
3.5.2 Transform: ....................................................................................................... 73
3.5.3 Properties: ....................................................................................................... 73
3.5.4 Matlab for the Bartlett Window: ..................................................................... 73
3.6 Poisson Window ...................................................................................................... 74
3.6.1 Definition: ....................................................................................................... 74
3.7 Hann-Poisson Window ............................................................................................ 76
3.7.1 Definition: ....................................................................................................... 76
4
3.7.2 Matlab for the Hann-Poisson Window ............................................................ 78
3.8 Slepian or DPSS Window ....................................................................................... 78
3.8.1 Matlab for the DPSS Window ......................................................................... 80
3.9 Kaiser Window ........................................................................................................ 81
3.9.1 Definition: ....................................................................................................... 81
3.9.2 Window transform: ......................................................................................... 81
3.9.3 Kaiser Window Beta Parameter ...................................................................... 82
3.9.4 Kaiser Windows and Transforms .................................................................... 82
3.9.5 Minimum Frequency Separation vs. Window Length ..................................... 87
3.9.6 Kaiser and DPSS Windows Compared ........................................................... 88
3.10 Dolph-Chebyshev Window ..................................................................................... 90
3.10.1 Matlab for the Dolph-Chebyshev Window ..................................................... 91
3.10.2 Example Chebyshev Windows and Transforms .............................................. 91
3.10.3 Dolph-Chebyshev and Hamming Windows Compared .................................. 94
3.10.4 Dolph-Chebyshev Window Theory ................................................................. 94
3.10.4.1 Chebyshev Polynomials .................................................................. 94
3.10.4.2 Dolph-Chebyshev Window Definition ............................................ 95
3.10.4.3 Dolph-Chebyshev Window Main-Lobe Width ................................ 96
3.10.4.4 Dolph-Chebyshev Window Length Computation ........................... 96
3.11 Gaussian Window and Transform ........................................................................... 97
3.11.1 Matlab for the Gaussian Window .................................................................... 97
3.11.2 Gaussian Window and Transform ................................................................... 98
3.11.3 Exact Discrete Gaussian Window ................................................................... 98
3.12 Optimized Windows ................................................................................................ 99
3.12.1 Optimal Windows for Audio Coding............................................................... 99
3.12.2 General Rule.................................................................................................. 100
3.13 Optimal Window Design by Linear Programming ................................................ 100
3.13.1 Linear Programming (LP) ............................................................................. 100
3.13.2 Matlab's LINPROG ....................................................................................... 100
3.13.3 LP Formulation of Chebyshev Window Design ............................................ 102
3.13.4 Symmetric Window Constraint ..................................................................... 103
3.13.5 Positive Window Sample Constraint ............................................................. 103
3.13.6 DC Constraint................................................................................................ 103
3.13.7 Sidelobe Specification ................................................................................... 104
3.13.8 LP Standard Form ......................................................................................... 105
3.13.8.1 Normal Chebyshev Window ......................................................... 106
3.13.9 Remez Exchange Algorithm .......................................................................... 107
3.13.9.1 Convergence of Remez Exchange ................................................. 107
3.13.10 Monotonicity Constraint ....................................................................... 108
3.13.10.1 Monotonic Chebyshev Window .................................................... 109
3.13.11 L-Infinity Norm of Derivative Objective ...................................................... 110
3.13.12 L-One Norm of Derivative Objective ................................................... 112
Chapter 4 Spectrum Analysis of Sinusoids ............................................................................ 116
4.1 Spectrum of a Sinusoid ......................................................................................... 117
5
4.2 Spectrum of Sampled Complex Sinusoid .............................................................. 119
4.3 Spectrum of a Windowed Sinusoid ....................................................................... 120
4.4 Effect of Windowing ............................................................................................. 122
4.5 The Rectangular Window ...................................................................................... 125
4.5.1 Rectangular Window Side-Lobes .................................................................. 129
4.5.2 Frequency Resolution .................................................................................... 131
4.5.2.1 Two Cosines (``In-Phase'' Case) ............................................................ 132
4.5.2.2 One Sine and One Cosine ``Phase Quadrature'' Case ............................ 133
4.6 Main-Lobe Bandwidth .......................................................................................... 135
4.6.1 Other Definitions of Main Lobe Width ......................................................... 138
4.7 Choosing Window Length to Resolve Sinusoids .................................................. 139
4.7.1 Periodic Signals ............................................................................................. 141
4.7.2 Tighter Bounds for Minimum Window Length ............................................. 143
4.8 Sinusoidal Peak Interpolation ............................................................................... 145
4.8.1 Quadratic Interpolation of Spectral Peaks ..................................................... 146
4.8.1.1 Phase Interpolation at a Peak ................................................................ 148
4.8.1.2 Matlab for Parabolic Peak Interpolation ............................................... 148
4.8.2 Bias of Parabolic Peak Interpolation ............................................................. 149
4.9 Optimal Peak-Finding in the Spectrum ................................................................. 149
4.9.1 Minimum Zero-Padding for High-Frequency Peaks ..................................... 150
4.9.2 Minimum Zero-Padding for Low-Frequency Peaks ...................................... 151
4.9.3 Matlab for Computing Minimum Zero-Padding Factors .............................. 153
4.9.4 Least Squares Sinusoidal Parameter Estimation ........................................... 153
4.9.4.1 Sinusoidal Amplitude Estimation .......................................................... 155
4.9.4.2 Sinusoidal Amplitude and Phase Estimation ......................................... 156
4.9.4.3 Sinusoidal Frequency Estimation .......................................................... 158
4.9.5 Maximum Likelihood Sinusoid Estimation .................................................. 158
4.9.6 Likelihood Function ...................................................................................... 160
4.9.6.1 Multiple Sinusoids in Additive Gaussian White Noise ......................... 161
4.9.6.2 Non-White Noise .................................................................................. 161
4.9.7 Generality of Maximum Likelihood Least Squares ...................................... 161
Chapter 5 Spectrum Analysis of Noise .................................................................................. 163
5.1 Introduction to Noise ............................................................................................. 165
5.1.1 Why Analyze Noise? ..................................................................................... 165
5.1.2 What is Noise? .............................................................................................. 165
5.2 Spectral Characteristics of Noise .......................................................................... 166
5.3 White Noise ........................................................................................................... 166
5.3.1 Testing for White Noise ................................................................................ 167
5.4 Sample Autocorrelation ......................................................................................... 167
5.5 Sample Power Spectral Density ............................................................................ 170
5.6 Biased Sample Autocorrelation ............................................................................. 171
5.7 Smoothed Power Spectral Density ........................................................................ 172
5.8 Cyclic Autocorrelation .......................................................................................... 172
5.9 Practical Bottom Line ........................................................................................... 173
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