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Page 1: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

Detection and Estimation Theory

Lecture 5

Mojtaba Soltanalian- [email protected]

http://msol.people.uic.edu

Based on ECE 531 Slides- 2011 (Prof. Natasha Devroye)

mailto:[email protected]
http://msol.people.uic.edu/
Page 2: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

The Cramer-Rao Lower Bound

---Last Lecture

• the CRLB give a lower bound on the variance of ANY UNBIASED

estimator

• does NOT guarantee bound can be obtained

• IF find an estimator whose variance = CRLB then it’s MVUE

Page 3: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

The Cramer-Rao Lower Bound

Page 4: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

The Cramer-Rao Lower Bound

Page 5: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

The Cramer-Rao Lower Bound

Basis for CRLB

Page 6: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

The Cramer-Rao Lower Bound

Page 7: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

The Cramer-Rao Lower Bound

Page 8: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

The Cramer-Rao Lower Bound

Page 9: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

The Cramer-Rao Lower Bound

Page 10: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

The Cramer-Rao Lower Bound

Page 11: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

The Cramer-Rao Lower Bound

--- On the Regularity Condition

*

*

Page 12: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

The Cramer-Rao Lower Bound

--- Proof (Scalar Case)

Page 13: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

The Cramer-Rao Lower Bound

--- Proof (Scalar Case)

CRLB

Page 14: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

The Cramer-Rao Lower Bound

--- Proof (Scalar Case)

CRLB

Page 15: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

CRLB examples

See example 3.4

Page 16: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

CRLB and Estimator Efficiency

Page 17: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

Fisher Information

Page 18: Detection and Estimation Theory Lecture 4 5.pdfDetection Taking number used stimuli behavior co vector Hit F stimator Green events random.. appropriate conditions = 3, A) PDF dependence

CRLB Derivation for Signals+WGN


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