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Advanced signal processing Dr. Mohamad KAHLIL Islamic University of Lebanon. Outline. Random variables Histogram, Mean, Variances, Moments, Correlation, types, multiple random variables Random functions Correlation, stationarity, spectral density estimation methods

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Advanced signal processing

Dr. Mohamad KAHLIL

Islamic University of Lebanon


Outline
Outline

  • Random variables

    • Histogram, Mean, Variances, Moments, Correlation, types, multiple random variables

  • Random functions

    • Correlation, stationarity, spectral density estimation methods

  • Signal modeling: AR, MA, ARMA,

  • Detection and classification in signals

  • Advanced applications on signal processing:

    • Time frequency and wavelet


Chapter 4 detection and classification in random signals
Chapter 4: detection and classification in random signals

  • Detection

    • Definition

    • Statistical tests for detection

    • Likelihood ratio

    • Example of detection when change in mean

    • Example of detection when change in variances

    • Multidimensional detection


Detection: definition

Hypotheses :

estimated

Known or unknown


Gaussian distributions
Gaussian distributions

Normal distributions


Chi2 distributions

Loi du Chi 2 (Khi-two of Pearson)

10 dof

15 dof

 chi2 with k degree of freedom

E[chi2]=k

Variance of Chi2=2k


Fisher test
Fisher Test

Student distribution

F(6,7)

F(6,10)

Student with k degree of freedom

Fisher-Snédécor Distribution

Fisher with k and l degree of freedom

Example: Detection in signals


Detection definition
Detection: definition

Hypotheses :

estimated

Known or unknown


Parameters definition
Parameters definition

  • False alarm

    • Detect H1, H0 is correct

  • Detection

    • Detect H1, H1 is correct

  • Miss detection

    • Detect H0, H1 is correct


Likelihood ratio
Likelihood ratio

  • Detection in signals


Variation in mean
Variation in mean

  • Detection in mean

    H0: z(t) = 0 + b(t) = b(t)

    H1: z(t) = m + b(t)


Detection in variance
Detection in variance

  • Detection in variance


Parameters
Parameters

  • False Alarm probability

  • Detection probability



Neymen pearson method
Neymen pearson method

  • Fix the probability of false alarm

  • Estimate the threshold


Detection multidimensional case
Detection: multidimensional case

  • Multidimensional case






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