Provide an introduction to the theory of Detection and Estimation, with applications mainly in the area of digital communications.
Syllabus (2 hours each class)
First hour: Course organization, objectives, textbooks, exam details. Sneaky preview of the course, motivations, applications. Second hour: basic probability theory refresher: total probability, Bayes rule in discrete/continuous/mixed versions, double conditioning. A first elementary exercise on binary hypothesis testing.
First hour: completion of proposed exercise. Second hour: Bayes Tests.
First hour: exercise on Bayes Test (Laplacian distributions) Second hour: MiniMax Test.
First hour: esercise on Minimax. Second hour: Neyman Pearson Test with example.
First hour: ROC properties. NP test with distrete RVs: randomization. Second hour: Exercise on Bayes, Minimax, Neyman-Pearson tests.
First hour: Multiple hypothesis testing, Bayesian approach. MAP and ML tests. Decision regions, boundaries among regions: examples in R^1 and R^2. Second hour: exercise: 3 equally-likely signal "hypotheses" -A,0,A in AWGN noise: Bayes rule (ML) based on the sample-mean (sufficient statistic).
First hour: Minimax in multiple hypotheses. Sufficient statistics: introduction. Second hour: Factorization theorem, irrelevance theorem. Reversibility theorem. Gaussian vectors refresher: joint PDF, MGF/CF.
First hour: Summary of known main results on Gaussian random vectors: Gaussian MGF, 4th order statistics from moment theorem, MGF-based proof of Gaussianity of linear transformations. Examples of Gaussian vectors: Fading Channel. Second hour: A: MAP Test with Gaussian signals. B: Additive Gaussian noise channel. Decision regions are hyperplanes.
First hour: examples of decision regions. Optimal detection of continuous-time signals: motivation for their discrete representation. Second hour: Discrete signal representation: definitions. Inner product, norm, distance, linear independence. Orthonormal bases and signal coordinates.
Gram-Schmidt orthonormalization. Detailed example. Operations on signals, and dual operations on signal images.
Unitary matrices in change of basis. Orthorgonal matrices: rotations and reflections. Orthogonality principle. Projection theorem. Interpretation of Gram-Schmidt procedure as repeated projections. Complete ON bases: motivations and definition.
First hour: exercises: 1. product of unitary matrices is unitary. 2. unitary matrix preserves norm of vectors. Projection matrices, eigenvectors, eigenvalues, spectral decomposition. Properties. Second hour: examples of complete bases in L2: the space of band-limited functions, evaluation of series coefficients, sampling theorem, ON check. More examples of complete bases: Legendre, Hermite, Laguerre.
Discrete representation of a stochastic process. Mean and covariance of process coefficients. Properties of covariance matrices for finite random vectors: Hermitianity and related properties. Whitening. Karhunen Leove (KL) theorem for whitening of discrete process representation (hint to proof). Statement of Mercer theorem. KL bases.
Summary of useful matrices: Normal and their subclasses: unitary, hermitian, skew-hermitian. If noise process is white, any ON complete basis is KL. Digital modulation. Example: QPSK. Digital demodulation with correlators bank or matched-filter bank.
First hour: Matched filter properties. Max SNR, physical reason of peak at T. Second hour: back to M-ary hypothesis testing with time-continuous signals: receiver structure. With white noise, irrelevance of noise components outside signal basis. Optimal MAP receiver in AWGN. Basis detector. Signal detector.
Examples of MAP RX and evaluation of symbol error probability Pe. First hour: MAP RX for QPSK signals and its Pe. Second hour: MAP RX for generic binary signals, basis detector, reduced complexity signal detector. Evaluation of Pe.
First hour: Techniques to evaluate Pe: rotational invariance in AWGN and signal image shifts. Center of gravity for minimum energy. Second hour: Pe evaluation for binary signaling. Comparisons between antipodal and orthogonal signals. Calculation of Pe for 16-QAM (begin).
First hour: Calculation of Pe for 16-QAM (end). Second hour: Calculation of Pe for M-ary orthogonal signaling. Begin calculation of Bit error rate (BER).
Completion of BER evaluation in M-ary orthogonal signaling. Example: M-FSK. Occupied bandwidth. Limit as M->infinity and connection with Shannon channel capacity. Notes on Simplex constellation. BER evaluation for QPSK: natural vs. Gray mapping.
Further notes on Gray mapping. Approximate BER calculation: union upper bound, minimum distance bound, nearest-neighbor bound. Lower bounds. Example: M-PSK. Review of cartesian(X,Y)-to-polar(R,Q) probability transformation. For zero-mean normal (X,Y), (R,Q) are independent with Rayleigh and Uniform marginals.
For non-zero-mean normal (X,Y), (R,Q) are dependent, with Rice and Bennet marginals. Properties of Rayleigh, Rice, Bennet PDFs. Use of Bennet PDF in the exact evaluation of Pe in M-PSK.
Composite hypothesis testing: introduction. Bayesian approach: Example of partially known signals in AWGN.
Partially known signals in AWGN: Bayesian MAP decision rule. Application to incoherent reception of passband signals. Optimal incoherent MAP receiver structure.
Alternative more compact derivation of incoherent MAP receiver for passband signals using complex envelopes. Incoherent OOK receiver and its BER evaluation.
Detection in additive colored Gaussian noise. Karhunen-Loeve formulation. Hints about the analog whitening filter. Reversibility theorem and whitening of the discretized signal sample. Example 1: whitening by unitary transformation that alignes the orthonormal eigenvectors of the noise covariance matrix to the canonical basis. Example 2: Cholesky decomposition of covariance matrix and noise whitening. Example of calculation of Cholesky decomposition.
Exercise: whitening and Pe evaluation for sampled signals in colored Gaussian noise.
Detection with stochastic signals: the case of Gaussian signals. Binary hypothesis testing: Radiometer. BER evaluation.
Estimation theory: introduction. Classical (Fisherian) estimation. MSE cost. The bias-variance tradeoff. Example and motivation for unbiased estimators.
Asymptotically unbiased and consistent estimators. MVUE. Cramer Rao Lower Bound: motivazion, theorem statement, example: signals in AWGN (both discrete and continuous-time). Amplitude estimation.
Phase estimation. Proof of CRLB. Extension of CRLB to vector parameters: theorem statement and examples. ML estimation, introduction. If an efficient estimator exists, it is ML.
ML: asymptotic properties and invariance. Examples: 1) Gaussian observations with unknown (constant) mean and variance. 2) Linear Gaussian model and comparison with least-squares solution. 3) Phase estimation of passband signals (begin)
ML: Phase estimation of passband signals (end). Bayesian Estimation: 1) MMSE estimator and minimum error. Orthogonality principle. Unbiasedness. Note on regression curve. Gaussian example. Exercise: both observations and parameter are negative exponentials.
Bayesian estimation: MAP estimator. Example. ML Criterion as a paticular MAP case. Ex: linear Gaussian model (homework, with solution). Extension to vector parameters. Gaussian multivariate regression. MMSE linear Bayesian estimates. Optimal filter coefficients through orthogonality principle. Yule-Walker equations. LMMSE optimal estimator and minimal MSE.
Review of optimal scalar LMMSE estimator and minimum MSE. Extension to vector estimator. Wiener Filter: problem statement, objectives. A) Smoothing, optimal non-causal filter, MMSE error, case of additive noise channel. Alternative evaluation of MMSE with error filter.
B) Causal Wiener filter: problem setting in 2 steps: whitening and innovations estimation. Whitening: 1) review of two-sided Z-transform and its ROC. 2) review: Z-transform of PSD of the output of a linear system. 3) statement of Spectral Factorization (SF) theorem.
SF theorem: key to proof. Calculation of innovations filter L(z) for real processes through the SF. Regular processes classification with L(z) a rational fraction. AR, MA, ARMA processes. Example: AR(1).
Wiener causal filter, formula in z. Example. r-step predictor: form of filter in z. Error Formula.
Predictor: example: prediction of AR(p) processes. r-step filtering and prediction: formula in z. General error formula for additive noise channels. Example.