Course Review for Final ECE460 Spring, 2012

Common Fourier Transform Pairs 2

Fourier Transform Properties 3

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 4

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 5

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: 6

Angular Modulation Angle Modulation Definitions: FM (sinusoidal signal) 7

Combinatorics 1.Sampling with replacement and ordering 2.Sampling without replacement and with ordering 3.Sampling without replacement and without ordering 4.Sampling with replacement and without ordering Bernoulli Trials Conditional Probabilities 8

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 9

Functions of a Random Variable General: Statistical Averages Mean Variance 10

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 ρ X,Y ? Independence of X and Y 11

Jointly Gaussian R.V.’s X and Y are jointly Gaussian if Matrix Form: Function: 12

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 13

Transfer Through a Linear System Mean of Y(t) where X(t) is wss Cross-correlation function R XY (t 1,t 2 ) Autocorrelation function R Y (t 1,t 2 ) Spectral Analysis 14

Energy & Power Processes For a sample function For Random Variables we have Then the energy and power content of the random process is 15

Zero-Mean White Gaussian Noise A zero mean white Gaussian noise, W ( t ), is a random process with 4.For any n and any sequence t 1, t 2, …, t n the random variables W (t 1 ), W( t 2 ), …, W( t n ), are jointly Gaussian with zero mean and covariances 16

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 X c ( t ) and X s ( t ) will be zero-mean jointly stationary processes: Giving 17

Performance on an Analog System in Noise Metric: SNR Message Signal Power m(t): Noise: 18

SNR for Amplitude Modulated Systems 19

Digital Systems Discrete Memoryless Source (DMS) completely defined by: Alphabet: Probability Mass Function: Self-Information Log 2 -bits (b) Log e - nats Entropy - measure of the average information content per source symbol and is measured in b/symbol Discrete System : Bounded: – Joint entropy of two discrete random variables ( X, Y ) – Conditional entropy of the random variable X given Y – Relationships 20

Mutual Information Mutual Information denotes the amount of uncertainty of X that has been removed by revealing random variable Y. If H(X) is the uncertainty of channel input before channel output is observed and H(X|Y) is the uncertainty of channel input after channel output is observed, then I(X;Y) is the uncertainty about the channel input that is resolved by observing channel output 21

Source Coding Viable Source Codes Uniquely decodable properties Prefix-free instantaneously decodable Theorem: A source with entropy H can be encoded with arbitrarily small error probability at any rate R (bits/source output)as long as R > H. Conversely if R < H, the error probability will be bounded away from zero, independent of the complexity of the encoder and the decoder employed. : the average code word length per source symbol Huffman Coding 22

Quantization Quantization Function: Squared-error distortion for a single measurement: Distortion D for the source since X is a random variable In general, a distortion measure is a distance between X and its reproduction. Hamming distortion: 23

Rate Distortion Minimum number of bits/source output required to reproduce a memoryless source with distortion less than or equal to D is call the rate-distortion function, denoted by R ( D ): For a binary memoryless source And with Hamming distortion, the rate-distortion function is For a zero-mean, Gaussian Source with variance σ 2 24

Geometric Representation Gram-Schmidt Orthogonalization 1.Begin with first waveform, s 1 ( t ) with energy ξ 1: 2.Second waveform a.Determine projection, c 21, onto ψ 1 b.Subtract projection from s 2 (t) c.Normalize 3.Repeat 25

Pulse Amplitude Modulation Bandpass Signals What type of Amplitude Modulation signal does this appear to be? 26 X

PAM Signals Geometric Representation M-ary PAM waveforms are one-dimensional where For Bandpass: 27 d d d d d 0 d = Euclidean distance between two points

Optimum Receivers Start with the transmission of any one of the M-ary signal waveforms: 1.Demodulators a.Correlation-Type b.Matched-Filter-Type 2.Optimum Detector 3.Special Cases (Demodulation and Detection) a.Carrier-Amplitude Modulated Signals b.Carrier-Phase Modulation Signals c.Quadrature Amplitude Modulated Signals d.Frequency-Modulated Signals 28 DemodulatorDetector Sampler Output Decision

Demodulators Correlation-Type 29 Next, obtain the joint conditional PDF

Demodulators Matched-Filter Type Instead of using a bank of correlators to generate { r k }, use a bank of N linear filters. The Matched Filter 30 Demodulator Key Property: if a signal s(t) is corrupted by AGWN, the filter with impulse response matched to s(t) maximizes the output SNR

Optimum Detector Maximum a Posterior Probabilities (MAP) If equal a priori probabilities, i.e., for all M and the denominator is a constant for all M, this reduces to maximizing called maximum-likelihood (ML) criterion. 31

Probability of Error Binary PAM Baseband Signals Consider binary PAM baseband signals where is an arbitrary pulse which is nonzero in the interval and zero elsewhere. This can be pictured geometrically as Assumption: signals are equally likely and that s 1 was transmitted. Then the received signal is Decision Rule: The two conditional PDFs for r are 32 0

Probability of Error M-ary PAM Baseband Signals Recall baseband M-ary PAM are geometrically represented in 1- D with signal point values of And, for symmetric signals about the origin, where the distance between adjacent signal points is. Each signal has a different energies. The average is 33