# RIT SIMG 713 - Study Guide (5 pages)

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## Study Guide

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- School:
- Rochester Institute of Technology
- Course:
- Simg 713 - Noise and Random Processes

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SIMG 713 SPRING 2002 REVIEW TOPICS FOR FINAL EXAM Outline of Topics 1 Concept of an experiment that produces outcomes that can be di erent on each trial 2 Concept of events as sets of possible experiment outcomes 3 Probability as a number associated with events a Properties of probability b Joint probabilities c Conditional probabilities d Statistical independence e Mutually exclusive events 4 Random variables 5 Averages over random variables a Computation of averages and moments b Physical interpretation of mean and variance c Correlation and covariance 6 Averages over functions of a random variable 7 Modeling of discrete events and random arrivals a Binomial distribution b Poisson distribution c Normal distribution d Error function and z score e Normal approximation to binomial 8 Application to detector arrays 9 Transfer function modeling a Mean value of output as a function of q b Variance of output as a function of q c Detective Quantum E ciency d Detection of input level change SNR 10 Random Process a Ensemble of sample functions b Ensemble averages 1 2 3 4 Mean value Autocorrelation and crosscorrelation Variance Function of random variable 1 c Stationary vs nonstationary d Wide sense stationary 11 Parameter Estimates a Why is an estimate based on a sample function always a random variable b What is an estimator as opposed to an estimate c What is bias as it relates to an estimator d How does the mean and variance of an estimator of the mean change with the number of samples used 12 Correlation Covariance and Crosscorrelation a De nition of covariance b De nition of correlation c De nition of crosscorrelation d Correlation coe cient e Bivariate normal distribution 13 Power spectrum a De nition as transform of autocorrelation function Wiener Khintchine Theorem b Relationship to Fourier transform of a sample function c Describe the periodogram algorithm for spectral estimation d Describe the correlogram algorithm for spectral estimation 14 Digital Filter Models a Di erence equation b Block diagram c Impulse response d System function e Relationships between models 15 Special Topics Not for nal exam a Z Transform 1 De nition of z transform of a sequence 2 Interpretation of z as a delay operator 3 Relationship between input output and system function in z domain b What information does the substitution X z with z ei provide 1 What range of values of is available 2 How do we relate to time or frequency for a time sampled application 3 How does X for a function x t relate to X ei for a discrete function c Generation of random sequences d How to simulate a bandlimited random process 2 Questions from old exams 1 Random variables X and Y have a joint discrete the table below X Y 1 2 a b c d Find Find Find Find the the the the probability function PXY X m Y n shown in 1 2 0 1 0 2 0 3 0 4 mean value of X variance of X correlation coe cient XY expected value of the function g X Y X Y X Y 2 A random number generator can produce a sequence of samples Xn n 1 2 N a How would you determine an approximation to the probability density function fX x Describe an algorithm that you would use to construct the approximation List any parameters associated with the algorithm b Provide an estimator for the mean value of X and determine the variance of the estimate it provides under the assumption that the samples are statistically independent 3 A physial analysis of the e ciency of a photon detector has shown that 80 of the incoming photons cause an electron to be produced inside the detector Construct a model for the detector output Y in terms of the detector input X and derive a formula for the DQE based on this model List any assumptions you use in constructing the model and deriving the DQE formula 4 Random variables X and Y take on the values x1 x2 x3 x4 and y1 y2 y3 The values for P X xi and P Y yj X xi are given in the table below Note that some of the information in the positions labeled A B C D is missing X P X xi P Y y1 X xi P Y y2 X xi P Y y3 X xi x1 0 1 B 0 35 0 4 x2 0 2 0 25 0 45 0 3 x3 A 0 25 0 25 D x4 0 3 0 25 C 0 25 a Determine the values that should go into locations A B C D in the table b Compute P X x2 Y y3 5 We have four boxes Box 1 contains 2000 components of which 5 are defective Box 2 contains 500 components of which 40 are defective Boxes 3 and 4 contain 1000 each with 10 defective We select at random one of the boxes and remove at random a single component a What is the probability that the selected component is defective b We examine the defective component and nd it defective On the basis of this evidence what is the probability it was drawn from Box 2 6 Two random processes x n and y n are related by y n n X k where P k 0 h2 k is bounded 3 x k h n k a Find an expression for the cross correlation between the x and y sequences in terms of the impulse response function h n under the condition that x is a white noise sequence with x 0 and variance x2 b Find an expression for the variance of the y sequence in terms of the impulse response function h n under the condition that x is a white noise sequence with x 0 and variance x2 7 Let X n be a discrete random process that can assume two values 0 and 1 with probabilities 0 2 and 0 8 respectively The process is ergodic The values X n and X m are statistically independent if m 6 n This random process is the input to a digital lter with impulse response h n an step n where a is a real number with a 1 The output of the digital lter is a random process Y n a Determine whether Y n is a wide sense stationary random process b Find the expected value of Y n in terms of a c Find the variance of Y n in terms of a 8 A calibrated photon source is designed to produce photons at an average rate of 0 1000 photons per second The sources are tested after they have been manufactured to see if they are acceptable A source is acceptable if the actual rate satis es 0 5 photons per second The acceptance test is to count the number N of photons in T 40 seconds and accept the source if N 40 000 200 a What is the probability that a source will be rejected by the test if it has 0 You may want to use the …

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