Time- and bandlimiting IMAHA, November 14, 2009, 11:00{11:40 Joe Lakey (w Scott Izu and Jeff Hogan)1 November 16, 2009 Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting sampling theory and history Time- and bandlimiting, history, definitions and basic properties connecting sampling and time- and bandlimiting The multiband case Outline Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Time- and bandlimiting, history, definitions and basic properties connecting sampling and time- and bandlimiting The multiband case Outline sampling theory and history Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting connecting sampling and time- and bandlimiting The multiband case Outline sampling theory and history Time- and bandlimiting, history, definitions and basic properties Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting The multiband case Outline sampling theory and history Time- and bandlimiting, history, definitions and basic properties connecting sampling and time- and bandlimiting Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Outline sampling theory and history Time- and bandlimiting, history, definitions and basic properties connecting sampling and time- and bandlimiting The multiband case Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting R −2πitξ Fourier transform: bf (ξ) = f (t) e dt R _ Bandlimiting: PΣf (x) = (bf 1Σ) (x) Paley-Wiener space : 2 PWΣ = PΣ(L (R)) Bandlimiting Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting _ Bandlimiting: PΣf (x) = (bf 1Σ) (x) Paley-Wiener space : 2 PWΣ = PΣ(L (R)) Bandlimiting R −2πitξ Fourier transform: bf (ξ) = f (t) e dt R Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Paley-Wiener space : 2 PWΣ = PΣ(L (R)) Bandlimiting R −2πitξ Fourier transform: bf (ξ) = f (t) e dt R _ Bandlimiting: PΣf (x) = (bf 1Σ) (x) Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Bandlimiting R −2πitξ Fourier transform: bf (ξ) = f (t) e dt R _ Bandlimiting: PΣf (x) = (bf 1Σ) (x) Paley-Wiener space : 2 PWΣ = PΣ(L (R)) Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting _ Bandlimiting: Pf (x) = (bf 1[−1=2; 1=2]) (x) 2 Paley-Wiener space : PW = P(L (R)) 2 Sampling theorem: If f 2 PW then, with convergence in L (R), 1 X sin π(t − k) f (t) = f (k) π(t − k) k=−∞ The Shannon sampling theorem Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 2 Paley-Wiener space : PW = P(L (R)) 2 Sampling theorem: If f 2 PW then, with convergence in L (R), 1 X sin π(t − k) f (t) = f (k) π(t − k) k=−∞ The Shannon sampling theorem _ Bandlimiting: Pf (x) = (bf 1[−1=2; 1=2]) (x) Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting The Shannon sampling theorem _ Bandlimiting: Pf (x) = (bf 1[−1=2; 1=2]) (x) 2 Paley-Wiener space : PW = P(L (R)) 2 Sampling theorem: If f 2 PW then, with convergence in L (R), 1 X sin π(t − k) f (t) = f (k) π(t − k) k=−∞ Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 0.04 0.03 0.02 0.01 0 −0.01 −0.02 −0.03 10 20 30 40 50 60 70 80 90 100 The problem: sinc not well localized 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −10 −8 −6 −4 −2 0 2 4 6 8 10 Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 0.04 0.03 0.02 0.01 0 −0.01 −0.02 −0.03 10 20 30 40 50 60 70 80 90 100 The problem: sinc not well localized 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −10 −8 −6 −4 −2 0 2 4 6 8 10 Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 0.04 0.03 0.02 0.01 0 −0.01 −0.02 −0.03 10 20 30 40 50 60 70 80 90 100 The problem: sinc not well localized 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −10 −8 −6 −4 −2 0 2 4 6 8 10 Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Bandlimited functions can be recovered from samples Sinc function not well localized Find a subspace of PW whose elements can be approximated locally by finite sinc series The problem: how many samples Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Sinc function not well localized Find a subspace of PW whose elements can be approximated locally by finite sinc series The problem: how many samples Bandlimited functions can be recovered from samples Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Find a subspace of PW whose elements can be approximated locally by finite sinc series The problem: how many samples Bandlimited functions can be recovered from samples Sinc function not well localized Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting The problem: how many samples Bandlimited functions can be recovered from samples Sinc function not well localized Find a subspace of PW whose elements can be approximated locally by finite sinc series Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting I (QT f )(x) = 1[−T ;T ](x) f (x) I PQT P: self-adjoint, I 2 λmax = λ0 = kPQT k = sup kQT (f )k f 2PW; kf k=1 I Uncertainty principle: λmax < 1 I PΩQT commutes with certain differential operator ::: d2 d (4T 2 − t2) − 2t − Ω2t2 dt2 dt I Eigenfunctions: Prolate Spheroidal Wave Functions Time- and bandlimiting: Bell Labs Theory _ I Pf (x) = (bf 1[−1=2; 1=2]) (x) Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting I PQT P: self-adjoint, I 2 λmax = λ0 = kPQT k = sup kQT (f )k f 2PW; kf k=1 I Uncertainty principle: λmax < 1 I PΩQT commutes with certain differential operator ::: d2 d (4T 2 − t2) − 2t − Ω2t2 dt2 dt I Eigenfunctions: Prolate Spheroidal Wave Functions Time- and bandlimiting: Bell Labs Theory _ I Pf (x) = (bf 1[−1=2; 1=2]) (x) I (QT f )(x) = 1[−T ;T ](x) f (x) Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting I 2 λmax = λ0 = kPQT k = sup kQT (f )k f 2PW; kf k=1 I Uncertainty principle: λmax < 1 I PΩQT commutes with certain differential operator ::: d2 d (4T 2 − t2) − 2t − Ω2t2 dt2 dt I Eigenfunctions: Prolate Spheroidal Wave Functions Time- and bandlimiting: Bell Labs Theory _ I Pf (x) = (bf 1[−1=2; 1=2]) (x) I (QT f )(x) = 1[−T ;T ](x) f (x) I PQT P: self-adjoint, Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting I Uncertainty principle: λmax < 1 I PΩQT commutes with certain differential operator ::: d2 d (4T 2 − t2) − 2t − Ω2t2 dt2 dt I Eigenfunctions: Prolate Spheroidal Wave Functions Time- and bandlimiting: Bell Labs Theory _ I Pf (x) = (bf 1[−1=2; 1=2]) (x) I (QT f )(x) = 1[−T ;T ](x) f (x) I PQT P: self-adjoint, I 2 λmax = λ0 = kPQT k = sup kQT (f )k f 2PW; kf k=1 Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting I PΩQT commutes with certain differential operator ::: d2 d (4T 2 − t2) − 2t − Ω2t2 dt2 dt I Eigenfunctions: Prolate Spheroidal Wave Functions Time- and bandlimiting: Bell Labs Theory _ I Pf (x) = (bf 1[−1=2; 1=2]) (x) I (QT f )(x) = 1[−T ;T ](x) f (x) I PQT P: self-adjoint, I 2 λmax = λ0 = kPQT k = sup kQT (f )k f 2PW; kf k=1 I Uncertainty principle: λmax < 1 Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting I Eigenfunctions: Prolate Spheroidal Wave Functions Time- and bandlimiting: Bell Labs Theory _ I Pf (x) = (bf 1[−1=2; 1=2]) (x) I (QT f )(x) = 1[−T ;T ](x) f (x) I PQT P: self-adjoint, I 2 λmax = λ0 = kPQT k = sup kQT (f )k f 2PW; kf k=1 I Uncertainty principle: λmax < 1 I PΩQT commutes with certain differential operator ::: d2 d (4T 2 − t2) − 2t − Ω2t2 dt2 dt Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Time- and bandlimiting: Bell Labs Theory _ I Pf (x) = (bf 1[−1=2; 1=2]) (x) I (QT f )(x) = 1[−T ;T ](x) f (x) I PQT P: self-adjoint, I 2 λmax = λ0 = kPQT k = sup kQT (f )k f 2PW; kf k=1 I Uncertainty principle: λmax < 1 I PΩQT commutes with certain differential operator ::: d2 d (4T 2 − t2) − 2t − Ω2t2 dt2 dt I Eigenfunctions: Prolate Spheroidal Wave Functions Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Eigenvalue properties Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Graphs ::: coming later Study of heat flow (C. Niven, 1879) Relationships to special functions (Legendre, Bessel) Stratton, 1935 Tridiagonal method: expands PSFWs locally in Legendre polynomials (Bouwkamp, 1946, tables) 'n has n zeros in [−1; 1] (Kellogg, 1916 for general principle) Eigenfunction properties I Solutions: Prolate spheroidal wave functions '0;'1;::: Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Study of heat flow (C.