Random Signals and Noise

ELEMENTARY PROBABILITY THEORY The Probability Function A Bit of Philosophy The One-Dimensional Random Variable The Discrete Random Variable and the PMF A Bit of Combinatorics The Binomial Distribution The Continuous Random Variable, the CDF, and the PDF The Expected Value Two Dimensional Random Variables The Characteristic Function Gaussian Random Variables Exercises AN INTRODUCTION TO STOCHASTIC PROCESSES What Is a Stochastic Process? The Autocorrelation Function What Does the Autocorrelation Function Tell Us? The Evenness of the Autocorrelation Function Two Proofs that Rxx(0) => |Rxx(t)| Some Examples Exercises THE WEAK LAW OF LARGE NUMBERS The Markov Inequality Chebyshev's Inequality A Simple Example The Weak Law of Large Numbers Correlated Random Variables Detecting a Constant Signal in the Presence of Additive Noise A Method for Determining the CDF of a Random Variable Exercises THE CENTRAL LIMIT THEOREM Introduction The Proof of the Central Limit Theorem Detecting a Constant Signal in the Presence of Additive Noise Detecting a (Particular) Non-Constant Signal in the Presence of Additive Noise The Monte Carlo Method Poisson Convergence Exercises EXTREMA AND THE METHOD OF LAGRANGE MULTIPLIERS The Directional Derivative and the Gradient Over-Determined Systems The Method of Lagrange Multipliers The Cauchy-Schwarz Inequality Under-Determined Systems Exercises THE MATCHED FILTER FOR STATIONARY NOISE White Noise Colored Noise The Autocorrelation Matrix The Effect of Sampling Many Times in a Fixed Interval More about the Signal to Noise Ratio Choosing the Optimal Signal for a Given Noise Type Exercises FOURIER SERIES AND TRANSFORMS The Fourier Series The Functions en(t) Span-a Plausibility Argument The Fourier Transform Some Properties of the Fourier Transform Some Fourier Transforms A Connection between the Time and Frequency Domains Preservation of the Inner Product Exercises THE WIENER-KHINCHIN THEOREM AND APPLICATIONS The Periodic Case The Aperiodic Case The Effect of Filtering The Significance of the Power Spectral Density White Noise Low-Pass Noise Low-Pass Filtered Low-Pass Noise The Schottky Formula for Shot Noise A Semi-Practical Example Johnson Noise and the Nyquist Formula Why Use RMS Measurements The Practical Resistor as a Circuit Element The Random Telegraph Signal-Another Low-Pass Signal Exercises SPREAD SPECTRUM Introduction The Probabilistic Approach A Spread Spectrum Signal with Narrow Band Noise The Effect of Multiple Transmitters Spread Spectrum-The Deterministic Approach Finite State Machines Modulo Two Recurrence Relations A Simple Example Maximal Length Sequences Determining the Period An Example Some Conditions for Maximality What We Have Not Discussed Exercises MORE ABOUT THE AUTOCORRELATION AND THE PSD The "Positivity" of the Autocorrelation Another Proof that Rxx(0) => |Rxx(t)| Estimating the PSD The Properties of the Periodogram Exercises WIENER FILTERS A Non-Causal Solution White Noise and a Low-Pass Signal Causality, Anti-Causality and the Fourier Transform The Optimal Causal Filter Two Examples Exercises APPENDIX: A BRIEF OVERVIEW OF LINEAR ALGEBRA The Space CN Linear Independence and Bases A Preliminary Result The Dimension of CN Linear Mappings Matrices Sums of Mappings and Sums of Matrices The Composition of Linear Mappings-Matrix Multiplication A Very Special Matrix Solving Simultaneous Linear Equations The Inverse of a Linear Mapping Invertibility The Determinant-A Test for Invertibility Eigenvectors and Eigenvalues The Inner Product A Simple Proof of the Cauchy-Schwarz Inequality The Hermitian Transpose of a Matrix Some Important Properties of Self-Adjoint Matrices Exercises BIBLIOGRAPHY INDEX