Introduction TheIdentificationProblem SufficientConditions NecessaryConditions Epilogue References Identification of Operators on Elementary Locally Compact Abelian Groups Go¨khan Civan [email protected] NorbertWienerCenterforHarmonicAnalysisandApplications DepartmentofMathematics UniversityofMaryland,CollegePark July 23, 2015 IdentificationofOperatorsonElementaryLocallyCompactAbelianGroups G¨okhanCivan(UMD) (cid:73) Time-invariant operators = convolution operators: (cid:90) g → τ(·−y)g(y)dy (cid:73) Time-variant operators: (cid:90) g → τ(·,·−y)g(y)dy (cid:73) Set κ(x,y)=τ(x,x−y): (cid:90) g → κ(·,y)g(y)dy Introduction TheIdentificationProblem SufficientConditions NecessaryConditions Epilogue References Time-Variant Linear Channels (cid:73) Time-variant channels arise in mobile communications [Str06] and super-resolution radar [BGE11]. IdentificationofOperatorsonElementaryLocallyCompactAbelianGroups G¨okhanCivan(UMD) (cid:73) Time-variant operators: (cid:90) g → τ(·,·−y)g(y)dy (cid:73) Set κ(x,y)=τ(x,x−y): (cid:90) g → κ(·,y)g(y)dy Introduction TheIdentificationProblem SufficientConditions NecessaryConditions Epilogue References Time-Variant Linear Channels (cid:73) Time-variant channels arise in mobile communications [Str06] and super-resolution radar [BGE11]. (cid:73) Time-invariant operators = convolution operators: (cid:90) g → τ(·−y)g(y)dy IdentificationofOperatorsonElementaryLocallyCompactAbelianGroups G¨okhanCivan(UMD) (cid:73) Set κ(x,y)=τ(x,x−y): (cid:90) g → κ(·,y)g(y)dy Introduction TheIdentificationProblem SufficientConditions NecessaryConditions Epilogue References Time-Variant Linear Channels (cid:73) Time-variant channels arise in mobile communications [Str06] and super-resolution radar [BGE11]. (cid:73) Time-invariant operators = convolution operators: (cid:90) g → τ(·−y)g(y)dy (cid:73) Time-variant operators: (cid:90) g → τ(·,·−y)g(y)dy IdentificationofOperatorsonElementaryLocallyCompactAbelianGroups G¨okhanCivan(UMD) Introduction TheIdentificationProblem SufficientConditions NecessaryConditions Epilogue References Time-Variant Linear Channels (cid:73) Time-variant channels arise in mobile communications [Str06] and super-resolution radar [BGE11]. (cid:73) Time-invariant operators = convolution operators: (cid:90) g → τ(·−y)g(y)dy (cid:73) Time-variant operators: (cid:90) g → τ(·,·−y)g(y)dy (cid:73) Set κ(x,y)=τ(x,x−y): (cid:90) g → κ(·,y)g(y)dy IdentificationofOperatorsonElementaryLocallyCompactAbelianGroups G¨okhanCivan(UMD) (cid:73) (cid:90) g → η(x,ω)M T gdxdω ω x T : translation (time delay) x M : modulation (Doppler shift) ω (cid:73) g is transformed into a weighted sum of time-frequency shifts of itself. Introduction TheIdentificationProblem SufficientConditions NecessaryConditions Epilogue References Time-Variant Linear Channels (cont.) (cid:73) The spreading function: (cid:90) η(x,ω)= κ(y,y −x)e−2πyωdy η is a function of time and frequency. IdentificationofOperatorsonElementaryLocallyCompactAbelianGroups G¨okhanCivan(UMD) (cid:73) g is transformed into a weighted sum of time-frequency shifts of itself. Introduction TheIdentificationProblem SufficientConditions NecessaryConditions Epilogue References Time-Variant Linear Channels (cont.) (cid:73) The spreading function: (cid:90) η(x,ω)= κ(y,y −x)e−2πyωdy η is a function of time and frequency. (cid:73) (cid:90) g → η(x,ω)M T gdxdω ω x T : translation (time delay) x M : modulation (Doppler shift) ω IdentificationofOperatorsonElementaryLocallyCompactAbelianGroups G¨okhanCivan(UMD) Introduction TheIdentificationProblem SufficientConditions NecessaryConditions Epilogue References Time-Variant Linear Channels (cont.) (cid:73) The spreading function: (cid:90) η(x,ω)= κ(y,y −x)e−2πyωdy η is a function of time and frequency. (cid:73) (cid:90) g → η(x,ω)M T gdxdω ω x T : translation (time delay) x M : modulation (Doppler shift) ω (cid:73) g is transformed into a weighted sum of time-frequency shifts of itself. IdentificationofOperatorsonElementaryLocallyCompactAbelianGroups G¨okhanCivan(UMD) (cid:73) Is it possible to identify this family by a single probing signal? (cid:73) Kailtah’s conjecture: Yes if µ(R)≤1 No if µ(R)>1 (cid:73) Bello [Bel69] removed the restriction that R should be a rectangle. Introduction TheIdentificationProblem SufficientConditions NecessaryConditions Epilogue References Work of Kailath and Bello (cid:73) Kailath [Kai62] considered a family of operators where η is supported in a fixed rectangle R in the time-frequency plane. IdentificationofOperatorsonElementaryLocallyCompactAbelianGroups G¨okhanCivan(UMD) (cid:73) Kailtah’s conjecture: Yes if µ(R)≤1 No if µ(R)>1 (cid:73) Bello [Bel69] removed the restriction that R should be a rectangle. Introduction TheIdentificationProblem SufficientConditions NecessaryConditions Epilogue References Work of Kailath and Bello (cid:73) Kailath [Kai62] considered a family of operators where η is supported in a fixed rectangle R in the time-frequency plane. (cid:73) Is it possible to identify this family by a single probing signal? IdentificationofOperatorsonElementaryLocallyCompactAbelianGroups G¨okhanCivan(UMD)