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ECE310 – Lecture 23

Random Signal Analysis04/27/01

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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

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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

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Mean Sample mean

Expected value/Population mean

Sample mean is an estimation of population mean Example (brighter/darker) MATLAB: mean()

N

iixN

x1

1

N

ii

Nx x

NxE

1

1lim

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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()

222

1

21lim XEXEXEXEx

N

N

ixi

Nx

Square of mean

Meansquare

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Covariance (*) A measure of how much two

random variables vary together

YEXEXYEYEYEXEXEXY

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Histogram A graph indicating what percentage of

the time a random variable spends in various ranges of values

Example: x=[2 3 4 5 4 3 2 1 6 7 4 5 3 2 3 4]hist(x)

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Probability Density Function Raw histogram 1st normalization

Divide each frequency with total number of occurrence – relative frequency

It’s the probability!!! 2nd normalization

The width of the bin is approaching to zero

It’s the pdf

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% 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');

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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) xN

nxp

xN

X

0

lim

1Pr

dxxpX X

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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

zpzpzpYXZ YXZ ,

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Gaussian pdf x: mean x: standard deviation Normal distribution

mean = 0 std = 1

2

2

2

2

1x

Xx

X

X exp

2

2

2

1 x

X exp

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Error Function Gaussian pdf: N~(, 2) erf(x): error function

Pr(X<=3)

22

1

2

1Pr

aerfaX

2

3

2

1

2

1

2

13Pr

2

2

2

33

erfdxedxxpXx

X

x

u duexerf22

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

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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?

fGfHfG

tRththtR

XY

XY

2

0

22

2

2

1lim0

dffGdffG

XEdttXtXT

R

XX

T

TT

X

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Summary Probability

Prob. of anevent Prob. of

independent event

Prob. of disjoint event

Statistics mean: = E(x) variance: 2 = E(x2)-[E(x)]2

std: Histogram pdf Gaussian pdf Error function Autocorrelation and PSD

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Test 3 Statistics

ECE310 Test 3 Statistics

0123456

90-100

80-89 70-79 60-69 <60

Avg. 77.9, [49.5, 99]

Nr.

of S

tude

nts

Series1