Chapter
1.4.3 Complex demodulation
1.4.4 Bedrosian's theorem: the Hilbert transform of a product
1.4.5 Instantaneous amplitude, frequency, and phase
1.4.6 Hilbert transform and SSB modulation
1.4.7 Passband filtering at baseband
1.5 Complex signals for the efficient use of the FFT
1.5.2 Twofer: two real DFTs from one complex DFT
1.5.3 Twofer: one real 2N-DFT from one complex N-DFT
1.6 The bivariate Gaussian distribution and its complex representation
1.6.1 Bivariate Gaussian distribution
1.6.2 Complex representation of the bivariate Gaussian distribution
1.6.3 Polar coordinates and marginal pdfs
1.7 Second-order analysis of the polarization ellipse
1.8 Mathematical framework
1.9 A brief survey of applications
2 Introduction to complex random vectors and processes
2.1 Connection between real and complex descriptions
2.1.1 Widely linear transformations
2.1.2 Inner products and quadratic forms
2.2 Second-order statistical properties
2.2.1 Extending definitions from the real to the complex domain
2.2.2 Characterization of augmented covariance matrices
2.3 Probability distributions and densities
2.3.1 Complex Gaussian distribution
2.3.2 Conditional complex Gaussian distribution
2.3.3 Scalar complex Gaussian distribution
2.3.4 Complex elliptical distribution
2.4 Sufficient statistics and ML estimators for covariances: complex Wishart distribution
Complex Wishart distribution
2.5 Characteristic function and higher-order statistical description
2.5.1 Characteristic functions of Gaussian and elliptical distributions
2.5.2 Higher-order moments
2.5.3 Cumulant-generating function
Do circular random vectors have spherical pdf contours?
2.6 Complex random processes
2.6.1 Wide-sense stationary processes
2.6.2 Widely linear shift-invariant filtering
Part II Complex random vectors
3 Second-order description of complex random vectors
3.1 Eigenvalue decomposition
3.1.1 Principal components
3.1.2 Rank reduction and transform coding
3.2 Circularity coefficients
3.2.2 Strong uncorrelating transform (SUT)
3.2.3 Characterization of complementary covariance matrices
3.3 Degree of impropriety
3.3.1 Upper and lower bounds
3.3.2 Eigenvalue spread of the augmented covariance matrix
3.3.3 Maximally improper vectors
3.4 Testing for impropriety
3.5 Independent component analysis
4.1 Foundations for measuring multivariate association between two complex random vectors
4.1.1 Rotational, reflectional, and total correlations for complex scalars
4.1.2 Principle of multivariate correlation analysis
4.1.3 Rotational, reflectional, and total correlations for complex vectors
4.1.4 Transformations into latent variables
4.2 Invariance properties
4.2.1 Canonical correlations
4.2.2 Multivariate linear regression (half-canonical correlations)
4.2.3 Partial least squares
4.3 Correlation coefficients for complex vectors
4.3.1 Canonical correlations
4.3.2 Multivariate linear regression (half-canonical correlations)
4.3.3 Partial least squares
4.5 Testing for correlation structure
4.5.2 Independence within one data set
4.5.3 Independence between two data sets
5.1 Hilbert-space geometry of second-order random variables
5.2 Minimum mean-squared error estimation
5.3 Linear MMSE estimation
5.3.1 The signal-plus-noise channel model
5.3.2 The measurement-plus-error channel model
5.3.5 Concentration ellipsoids
5.4 Widely linear MMSE estimation
5.4.2 Performance comparison between LMMSE and WLMMSE estimation
5.5 Reduced-rank widely linear estimation
5.5.1 Minimize mean-squared error (min-trace problem)
5.5.2 Maximize mutual information (min-det problem)
5.6 Linear and widely linear minimum-variance distortionless response estimators
5.6.1 Rank-one LMVDR receiver
Relation to LMMSE estimator
5.6.2 Generalized sidelobe canceler
5.6.3 Multi-rank LMVDR receiver
Generalized sidelobe canceler
5.6.4 Subspace identification for beamforming and spectrum analysis
5.6.5 Extension to WLMVDR receiver
5.7 Widely linear-quadratic estimation
5.7.1 Connection between real and complex quadratic forms
6 Performance bounds for parameter estimation
6.1 Frequentists and Bayesians
6.1.1 Bias, error covariance, and mean-squared error
6.1.2 Connection between frequentist and Bayesian approaches
6.1.3 Extension to augmented errors
6.2 Quadratic frequentist bounds
6.2.1 The virtual two-channel experiment and the quadratic frequentist bound
6.2.2 Projection-operator and integral-operator representations of quadratic frequentist bounds
6.2.3 Extension of the quadratic frequentist bound to improper errors and scores
6.3 Fisher score and the Cramer-Rao bound
6.3.1 Nuisance parameters
6.3.2 The Cramer-Rao bound in the proper multivariate Gaussian model
6.3.3 The separable linear statistical model and the geometry of the Cramer-Rao bound
6.3.4 Extension of Fisher score and the Cramer-Rao bound to improper errors and scores
6.3.5 The Cramer-Rao bound in the improper multivariate Gaussian model
6.3.6 Fisher score and Cramer-Rao bounds for functions of parameters
6.4 Quadratic Bayesian bounds
6.5 Fisher--Bayes score and Fisher-Bayes bound
6.5.1 Fisher-Bayes score and information
6.6 Connections and orderings among bounds
7.1 Binary hypothesis testing
7.1.1 The Neyman-Pearson lemma
7.1.3 Adaptive Neyman-Pearson and empirical Bayes detectors
7.2 Sufficiency and invariance
7.3 Receiver operating characteristic
7.4 Simple hypothesis testing in the improper Gaussian model
7.4.1 Uncommon means and common covariance
7.4.2 Common mean and uncommon covariances
7.4.3 Comparison between linear and widely linear detection
7.5 Composite hypothesis testing and the Karlin-Rubin theorem
7.6 Invariance in hypothesis testing
7.6.1 Matched subspace detector
7.6.2 CFAR matched subspace detector
Part III Complex random processes
8 Wide-sense stationary processes
8.1 Spectral representation and power spectral density
8.2.1 Analytic and complex baseband signals
8.2.2 Noncausal Wiener filter
8.3.1 Spectral factorization
8.3.2 Causal synthesis, analysis, and Wiener filters
8.4 Rotary-component and polarization analysis
8.4.2 Rotary components of random signals
Interpretation of the random ellipse
Statistical properties of the random ellipse
8.4.3 Polarization and coherence
8.4.4 Stokes and Jones vectors
8.4.5 Joint analysis of two signals
8.5.1 Moment spectra and principal domains
9 Nonstationary processes
9.1 Karhunen-Loeve expansion
9.2 Cramer-Loeve spectral representation
9.2.1 Four-corners diagram
9.2.2 Energy and power spectral densities
Wide-sense stationary signals
9.2.4 Discrete-time signals
9.3 Rihaczek time-frequency representation
9.4 Rotary-component and polarization analysis
9.5 Higher-order statistics
10 Cyclostationary processes
10.1 Characterization and spectral properties
10.1.1 Cyclic power spectral density
10.1.2 Cyclic spectral coherence
10.1.3 Estimating the cyclic power-spectral density
10.2 Linearly modulated communication signals
10.2.1 Symbol-rate-related cyclostationarity
10.2.2 Carrier-frequency-related cyclostationarity
10.2.3 Cyclostationarity as frequency diversity
10.3 Cyclic Wiener filter
10.4 Causal filter-bank implementation of the cyclic Wiener filter
10.4.1 Connection between scalar CS and vector WSS processes
10.4.2 Sliding-window filter bank
10.4.3 Equivalence to FRESH filtering
10.4.4 Causal approximation
Appendix 1: Rudiments of matrix analysis
A1.1 Matrix factorizations
A1.1.1 Partitioned matrices
A1.1.2 Eigenvalue decomposition
A1.1.3 Singular value decomposition
A1.2 Positive definite matrices
A1.2.1 Matrix square root and Cholesky decomposition
A1.2.2 Updating the Cholesky factors of a Grammian matrix
A1.3.1 Partitioned matrices
A1.3.2 Moore-Penrose pseudo-inverse
Appendix 2: Complex differential calculus (Wirtinger calculus)
A2.1.1 Holomorphic functions
A2.1.2 Complex gradients and Jacobians
A2.1.3 Properties of Wirtinger derivatives
A2.3.2 Extension to complex-valued functions
Appendix 3: Introduction to majorization
A3.1.2 Schur-convex functions
A3.2 Tests for Schur-convexity
A3.2.2 Functions defined on D
A3.3 Eigenvalues and singular values
A3.3.1 Diagonal elements and eigenvalues
A3.3.2 Diagonal elements and singular values
A3.3.3 Partitioned matrices
Statistical Signal Processing of Complex-Valued Data
：The Theory of Improper and Noncircular Signals
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