Review for Exam I ECE460 Spring, 2012

Dirichlet Conditions Fourier Transform Fourier Series x(t) has a finite number of minima and maxima in any interval on the real line x(t) has a finite number of discontinuities over any interval on the real line Fourier Series x(t) has a finite number of minima and maxima over one period x(t) has a finite number of discontinuities over one period

Fourier Series (Periodic Functions) Exponential Form Real Coefficient Trigonometric Form Complex Coefficient Trigonometric Form

Common Fourier Transform Pairs

Fourier Transform Properties

Sampling Theorem Able to reconstruct any bandlimited signal from its samples if we sample fast enough. If X(f) is band limited with bandwidth W then it is possible to reconstruct x(t) from samples

Example Linear Time-Invariant Causality Stability Filter Properties of a System: Linear Time-Invariant Causality Stability

Narrowband Signals Given:

Bandpass Signals & Systems Frequency Domain: Low-pass Equivalents: Let Giving To solve, work with low-pass parameters (easier mathematically), then switch back to bandpass via

Analog Modulation Amplitude Modulation (AM) Message Signal: Sinusoidal Carrier: AM (DSB) DSB – SC SSB Started with DSB-SC signal and filtered to one sideband Used ideal filter: Vestigial

Analog Modulation Angle Modulation Definitions: FM (sinusoidal signal)

Combinatorics Sampling with replacement and ordering Sampling without replacement and with ordering Sampling without replacement and without ordering Sampling with replacement and without ordering Bernouli Trials Conditional Probabilities

Random Variables Cumulative Distribution Function (CDF) Probability Distribution Function (PDF) Probability Mass Function (PMF) Key Distributions Bernoulli Random Variable Uniform Random Variable Gaussian (Normal) Random Variable

Functions of a Random Variable General: Statistical Averages Mean Variance

Multiple Random Variables Joint CDF of X and Y Joint PDF of X and Y Conditional PDF of X Expected Values Correlation of X and Y Covariance of X and Y - what is ρX,Y

Jointly Gaussian R.V.’s X and Y are jointly Gaussian if Matrix Form: Function:

Random Processes Notation: Understand integration across time or ensembles Mean Autocorrelation Auto-covariance Power Spectral Density Stationary Processes Strict Sense Stationary Wide-Sense Stationary (WSS) Cyclostationary Ergodic

Transfer Through a Linear System Mean of Y(t) where X(t) is wss Cross-correlation function RXY(t1,t2) Autocorrelation function RY(t1,t2) Spectral Analysis

Energy & Power Processes For a sample function For Random Variables we have Then the energy and power content of the random process is

Zero-Mean White Gaussian Noise A zero mean white Gaussian noise, W(t), is a random process with For any n and any sequence t1, t2, …, tn the random variables W(t1), W(t2), …, W(tn), are jointly Gaussian with zero mean and covariances

Bandpass Processes X(t) is a bandpass process Filter X(t) using a Hilbert Transform: and define If X(t) is a zero-mean stationary bandpass process, then Xc(t) and Xs(t) will be zero-mean jointly stationary processes: Giving