Gravitational Wave Data Analysis

7.1 Random Processes for Probability Theory 269

• 7.1.1 Power Spectrum
• 7.1.2 Gaussian Noise 273

7.2 Optimal Detection Statistic 275

• 7.2.1 Bayes’s Theorem 275
• 7.2.2 Matched Filter 276
• 7.2.3 Unknown Matched Filter Parameters 277
• 7.2.4 Statistical Properties of the Matched Filter 279
• 7.2.5 Matched Filter with Unknown Arrival Time 281
• 7.2.6 Template Banks of Matched Filters 282

7.3 Parameter Estimation 286

• 7.3.1 Measurement Accuracy 286
• 7.3.2 Systematic Errors in Parameter Estimation 289
• 7.3.3 Confidence Intervals 291

7.4 Detection Statistics for Poorly Modelled Signals 293

• 7.4.1 Excess-Power Method 293

7.5 Detection in Non-Gaussian Noise 295

7.6 Networks of Gravitational-Wave Detectors 298

• 7.6.1 Co-located and Co-aligned Detectors 298
• 7.6.2 General Detector Networks 300
• 7.6.3 Time-Frequency Excess-Power Method for a Network of Detectors
• 7.6.4 Sky Position Localization for Gravitational-Wave Bursts 305

7.7 Data Analysis Methods for Continuous-Wave Sources 307

• 7.7.1 Search for Gravitational Waves from a Known, Isolated Pulsar 309
• 7.7.2 All-Sky Searches for Gravitational Waves from Unknown Pulsars 316

7.8 Data Analysis Methods for Gravitational-Wave Bursts 317

• 7.8.1 Searches for Coalescing Compact Binary Sources 318
• 7.8.2 Searches for Poorly Modelled Burst Sources 332

7.9 Data Analysis Methods for Stochastic Sources 333

• 7.9.1 Stochastic Gravitational-Wave Point Sources 344

Basic Signal Processing

• Signal
  • Analog vs. Digital
  • Sampling Frequency
• Fourier Transform
• Convolution
  • Windowing
• Filtering
  • LTI System
  • Transfor Function
• Discrete Time Filtering
• Nyquist Sampling Theorem
• Discrete Fourier Transform
  • Fast Fourier Transform
• DFT & Discrete Time Convolution
• Digital Filtering
  • Digital Filter Design
• Linear Algebra Approach to Signal Processing
  • Signal and Vector Space
  • Fourier Basis
  • Wavelet Basis
• Time-frequency Analysis

Appendix Introduction to Signal Processing

• Sampling and Reconstruction
• Discrete-Time Systems
• FIR Filtering and Convolution
• z-Transforms
• Transfer Functions
• Digital Filter Realizations
• Signal Processing Applications
• DFT/FFT Algorithms
• FIR Digital Filter Design
• IIR Digital Filter Design

Appendix Bayesian Inference and Computation

• Markov Chain Monte Carlo
• Model Selection and Nested Sampling
• Assigning Prior Distributions

Appendix Gravitational-Wave Detector Data

• Gravitational-Wave Detector Site Data
• Idealized Initial LIGO Model