Matched Subspace Detectors

1
Matched Subspace Detectors

• L. L. Scharf and B. Friedlander
• Selami Ciftci
• 11/26/2007

2
Motivation

• Detection of subspace signals in subspace
  interference and broadband noise
• Matched subspace detector Matched signal
  detector (matched filter)
• Generalized Likelihood Ratio Test (GLRT) and its
  natural invariances
• Uniformly Most Powerful Test (UMP)
• Narrowband Array Processing Problem

3
Data Model

• Structured Noise Model

4
Detection Problem

• Assumptions
• H, S and R0 known and R0I
• H and S full-rank matrices ltHgt andltSgt
  full-rank subspaces
• H and S are not orthogonal
• Detection or
• Rank of ltHgt is 1 or p ? Noise level is known or
  unknown ?
• Rank-1 known noise level
• Rank-p known noise level
• Rank-1 unknown noise level
• Rank-p unknown noise level

5
Signal Subspace and Projections

• Orthogonal projection onto signal subspace
• Oblique projections
• Orthogonal projections

6
Detection Tools

• Projections
• Generalized Likelihood Ratio (GLR)
• Known noise level
• Unknown noise level
• Generalized Likelihood Ratio Test (GLRT)
• A function to be T-invariant

7
Rank-1 Signal Subspace, Known Noise Level

• Problem
• Maximum Likelihood Estimates of ,
  Logarithmic GLR Monotonic
• GLRT
• Detector
• Invariance Rotations and translations in

8
Receiver Operating Characteristic

• Performance

9
Rank-1 Signal Subspace, Unknown Noise Level

• Problem unknown
• Maximum Likelihood Estimates of ,
  , (N/2)-root GLR Monotonic
• GLRT
• Detector
• Invariance Rotations and scaling in

10
Receiver Operating Characteristic

• Performance

11
Rank-p Signal Subspace, Known Noise Level

• Problem
• Maximum Likelihood Estimates of ,
  Logarithmic GLR
• GLRT
• Detector
• Invariance rotations and translations around the
  space

12
Receiver Operating Characteristic

• Performance

13
Rank-p Signal Subspace, Unknown Noise Level

• Problem unknown
• Maximum Likelihood Estimates of ,
  , (N/2)-root GLR
• GLRT
• Detector
• Invariance non-negative scalings around the
  space

14
Receiver Operating Characteristic

• Performance

15
Conclusion

• GLRTs UMP invariant characteristic optimality
• Optimum Detectors Interference Rejecting
  Matched Subspace Detectors
• Energy Detectors