1 ECE310 – Lecture 23 Random Signal Analysis 04/27/01

2 Review The concept of randomness Random signal analysis needs Probability Probability of an event Probability of disjoint events (summation) Probability of independent events (multiplication) Statistics

3 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

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

5 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() Square of mean Mean square

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

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

8 Probability Density Function Raw histogram 1 st normalization Divide each frequency with total number of occurrence – relative frequency It’s the probability!!! 2 nd normalization The width of the bin is approaching to zero It’s the pdf

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10 % demonstrate histogram, normalized histogram, and pdf rx = randn(1,1000); % 1000 random numbers subplot(221); plot(rx); title('the random numbers'); [M, X] = hist(rx); subplot(222); bar(X, M); title('histogram with 10 bins'); % 10 bins subplot(223); bar(X, M/1000); % first normalization title('normalized histogram'); [M, X] = hist(rx, 200); % 200 bins, bin width getting smaller subplot(224); bar(X, M/1000); title('histogram with 200 bins');

11 Properties of pdf X: the random variable x: value of the random variable  x: the bin width N: total number of random numbers n: the random numbers fall within each bin pdf definition: pdf is a function of x Area under pdf function: x pX(x)pX(x)

12 Central Limit Theorem The pdf of a sum of two random variables is the convolution of their individual pdf’s The shape of the convolution over many independent random variable pdf’s approaches a limiting shape called the “Gaussian” shape

13 Gaussian pdf  x : mean  x : standard deviation Normal distribution mean = 0 std = 1

14 Error Function Gaussian pdf: N~( ,  2 ) erf(x): error function Pr(X<=3)

15 Example Pr(X>3) Pr(3<X<5)

16 Power Spectral Density Problem with pdf? psd and autocorrelation frequency behavior In case of random signal, the only thing we know is the average power of the signal. How to calculate?

17 Summary Probability Prob. of anevent Prob. of independent event Prob. of disjoint event Statistics mean:  = E(x) variance:  2 = E(x 2 )-[E(x)] 2 std:  Histogram pdf Gaussian pdf Error function Autocorrelation and PSD

18 Test 3 Statistics