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1
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
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()
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
7
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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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');
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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