AN INTRODUCTION TO STATISTICAL SIGNAL PROCESSING
CONTENTS INTRODUCTION PROBABILITY SPINNING POINTERS AND FLIPPING COINS PROBABILITY SPACES SAMPLE SPACES EVENT SPACES PROBABILITY MEASURES DISCRETE PROBABILITY SPACES CONTINUOUS PROBABILITY SPACES INDEPENDENCE ELEMENTARY CONDITIONAL PROBABILITY PROBLEMS RANDOM OBJECTS RANDOM VARIABLES RANDOM VECTORS RANDOM PROCESSES DISTRIBUTIONS OF RANDOM VARIABLES DISTRIBUTIONS MIXTURE DISTRIBUTIONS DERIVED DISTRIBUTIONS RANDOM VECTORS AND RANDOM PROCESSES DISTRIBUTIONS OF RANDOM VECTORS ?MULTIDIMENSIONAL EVENTS MULTIDIMENSIONAL PROBABILITY FUNCTIONS CONSISTENCY OF JOINT AND MARGINAL DISTRIBUTIONS INDEPENDENT RANDOM VARIABLES IID RANDOM VECTORS CONDITIONAL DISTRIBUTIONS DISCRETE CONDITIONAL DISTRIBUTIONS CONTINUOUS CONDITIONAL DISTRIBUTIONS STATISTICAL DETECTION AND CLASSI?CATION ADDITIVE NOISE 3 10 BINARY DETECTION IN GAUSSIAN NOISE 3 11 STATISTICAL ESTIMATION 3 12 CHARACTERISTIC FUNCTIONS 3 13 GAUSSIAN RANDOM VECTORS 3 14 EXAMPLES: SIMPLE RANDOM PROCESSES 3 15 DIRECTLY GIVEN RANDOM PROCESSES 3 15 1 THE KOLMOGOROV EXTENSION THEOREM 3 15 2 IID RANDOM PROCESSES 3 15 3 GAUSSIAN RANDOM PROCESSES 3 16 DISCRETE TIME MARKOV PROCESSES 3 16 1 A BINARY MARKOV PROCESS 3 16 2 THE BINOMIAL COUNTING PROCESS 3 16 3 ?DISCRETE RANDOM WALK 3 16 4 THE DISCRETE TIME WIENER PROCESS 3 16 5 HIDDEN MARKOV MODELS 3 17 ?NONELEMENTARY CONDITIONAL PROBABILITY 3 18 PROBLEMS EXPECTATION AND AVERAGES AVERAGES EXPECTATION EXAMPLES: EXPECTATION FUNCTIONS OF SEVERAL RANDOM VARIABLES PROPERTIES OF EXPECTATION EXAMPLES: FUNCTIONS OF SEVERAL RANDOM VARIABLES CORRELATION COVARIANCE COVARIANCE MATRICES MULTIVARIABLE CHARACTERISTIC FUNCTIONS EXAMPLE: DI?ERENTIAL ENTROPY OF A GAUSSIAN VECTOR CONDITIONAL EXPECTATION ?JOINTLY GAUSSIAN VECTORS EXPECTATION AS ESTIMATION ?IMPLICATIONS FOR LINEAR ESTIMATION 4 10 CORRELATION AND LINEAR ESTIMATION 4 11 CORRELATION AND COVARIANCE FUNCTIONS 4 12 ?THE CENTRAL LIMIT THEOREM 4 13 SAMPLE AVERAGES 4 14 CONVERGENCE OF RANDOM VARIABLES 4 15 WEAK LAW OF LARGE NUMBERS 4 16 ?STRONG LAW OF LARGE NUMBERS 4 17 STATIONARITY 4 18 ASYMPTOTICALLY UNCORRELATED PROCESSES 4 19 PROBLEMS SECOND-ORDER MOMENTS LINEAR FILTERING OF RANDOM PROCESSES SECOND-ORDER LINEAR SYSTEMS I/O RELATIONS POWER SPECTRAL DENSITIES LINEARLY FILTERED UNCORRELATED PROCESSES LINEAR MODULATION WHITE NOISE ?TIME-AVERAGES ?DI?ERENTIATING RANDOM PROCESSES ?LINEAR ESTIMATION AND FILTERING 5 10 PROBLEMS A MENAGERIE OF PROCESSES DISCRETE TIME LINEAR MODELS SUMS OF IID RANDOM VARIABLES INDEPENDENT STATIONARY INCREMENTS ?SECOND-ORDER MOMENTS OF ISI PROCESSES SPECI?CATION OF CONTINUOUS TIME ISI PROCESSES MOVING-AVERAGE AND AUTOREGRESSIVE PROCESSES THE DISCRETE TIME GAUSS-MARKOV PROCESS GAUSSIAN RANDOM PROCESSES ?THE POISSON COUNTING PROCESS 6 10 COMPOUND PROCESSES 6 11 ?EXPONENTIAL MODULATION 6 12 ?THERMAL NOISE 6 13 ERGODICITY AND STRONG LAWS OF LARGE NUMBERS 6 14 PROBLEMS