1. Review for Exam I
ECE460
Spring, 2012

2. 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

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

4. Common Fourier Transform Pairs

5. Fourier Transform Properties

6. 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

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

8. Narrowband Signals
Given:

9. 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

10. 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

11. Analog Modulation
Angle Modulation Definitions: FM (sinusoidal signal)

12. 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

13. 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

14. Functions of a Random Variable
General:
Statistical Averages
Mean
Variance

15. 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

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

17. 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

18. 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

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

20. 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

21. 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