Identification of Operators on Elementary Locally Compact Abelian Groups

109 Pages·2015·1.09 MB·English

by  Gökhan Civan` `%%%`#`&12_`__~~~alse

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)