1 ECE310 – Lecture 22 Random Signal Analysis 04/25/01

2 Random Signals The only way to analyze a random signal is through its Autocorrelation, and Power spectral density

3 PSD ESD: Describes how the signal energy is distributed in frequency ESD: the FT of the autocorrelation for energy signal PSD: Describes how the signal power is distributed in frequency PSD: the FT of the autocorrelation for power signal

4 The Concept of Randomness Random – unpredictable No cause and effect relationship Examples Random signal analysis needs knowledge from two areas Probability Statistics

5 Probability Basics The study of probability is the study of how to quantitatively estimate the likelihood that an event will occur, under certain circumstances Developed in 18 th and 19 th century for estimating the probability of winning at casino games Difficulties in the analysis of random signals in engineered systems No game rules Experimental approach: acquire and analyze the random signal over a long period of time

6 Probability of Event A n A : the number of A events N: the total number of events Probability of event A

7 Disjoint Events Mutually exclusive Example: on page What’s the probability of tossing a 7 with two dice on a single throw?

8 Independent Events The probability that both events occur in independent trials is the product of their probabilities. Example: What is the probability of tossing 3 successive heads with a fair coin?

9 Statistics The study of description and interpretation of data A set of data is a sequence of numerical values Discrete random variables Statistics is to use a few well-chosen descriptors to characterize the random variable Descriptors Mean Variance and standard deviation Covariance Histogram Probability density function Power spectral density

10 Mean Sample mean Expected value/Population mean Sample mean is an estimation of population mean Example (brighter/darker) MATLAB: mean()

11 Variance and STD Mean indicates the center of gravity Standard deviation is the square root of variance, indicating how far away is each value from the center of gravity MATLAB: std(), var() mean(x1) = e-005 mean(x2) = e-006 std(x1) = std(x2) =

12 Covariance (*) A measure of how much two random variables vary together

13 Histogram A graph indicating what percentage of the time a random variable spends in various ranges of values Example: x=[ ] hist(x)

14 Probability Density Function Raw histogram 1 st normalization Divide each frequency with total number of occurrence – relative frequency 2 nd normalization The width of the bin is approaching to zero

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