# Advanced signal processing Dr. Mohamad KAHLIL Islamic University of Lebanon - PowerPoint PPT Presentation

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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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#### Presentation Transcript

Islamic University of Lebanon

### 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

• 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

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

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

Hypotheses :

estimated

Known or unknown

### Parameters definition

• False alarm

• Detect H1, H0 is correct

• Detection

• Detect H1, H1 is correct

• Miss detection

• Detect H0, H1 is correct

### Likelihood ratio

• Detection in signals

### 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

### Parameters

• False Alarm probability

• Detection probability

### Neymen pearson method

• Fix the probability of false alarm

• Estimate the threshold

### Detection: multidimensional case

• Multidimensional case

Distribution de Fisher-Snedecora = 0,05

DISTRIBUTION DU KHI-DEUX

DISTRIBUTION DU KHI-DEUX (suite)

LOI NORMALE CENTRÉE RÉDUITE