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 ∈ PW then, with convergence in L (R),
∞ 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 ∈ PW then, with convergence in L (R),
∞ 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 ∈ PW then, with convergence in L (R),
∞ 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
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The problem: sinc not well localized
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Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting The problem: sinc not well localized
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Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting The problem: sinc not well localized
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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 ∈PW, 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 ∈PW, 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 ∈PW, 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 ∈PW, 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 ∈PW, 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 ∈PW, 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 ∈PW, 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) ϕn has n zeros in [−1, 1] (Kellogg, 1916 for general principle)
Eigenfunction properties I
Solutions: Prolate spheroidal wave functions ϕ0, ϕ1,...
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 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) ϕn has n zeros in [−1, 1] (Kellogg, 1916 for general principle)
Eigenfunction properties I
Solutions: Prolate spheroidal wave functions ϕ0, ϕ1,... Graphs ... coming later
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Relationships to special functions (Legendre, Bessel) Stratton, 1935 Tridiagonal method: expands PSFWs locally in Legendre polynomials (Bouwkamp, 1946, tables) ϕn has n zeros in [−1, 1] (Kellogg, 1916 for general principle)
Eigenfunction properties I
Solutions: Prolate spheroidal wave functions ϕ0, ϕ1,... Graphs ... coming later Study of heat flow (C. Niven, 1879)
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Tridiagonal method: expands PSFWs locally in Legendre polynomials (Bouwkamp, 1946, tables) ϕn has n zeros in [−1, 1] (Kellogg, 1916 for general principle)
Eigenfunction properties I
Solutions: Prolate spheroidal wave functions ϕ0, ϕ1,... Graphs ... coming later Study of heat flow (C. Niven, 1879) Relationships to special functions (Legendre, Bessel) Stratton, 1935
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting ϕn has n zeros in [−1, 1] (Kellogg, 1916 for general principle)
Eigenfunction properties I
Solutions: Prolate spheroidal wave functions ϕ0, ϕ1,... 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)
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Eigenfunction properties I
Solutions: Prolate spheroidal wave functions ϕ0, ϕ1,... 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) ϕn has n zeros in [−1, 1] (Kellogg, 1916 for general principle)
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 Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 1 1 − α N(α) = 2ΩT + log log ΩT + O(log ΩT ). π2 α
Eigenvalue properties I: the 2ΩT Theorem
Theorem For any T > 0 and Ω > 0 and 0 < α < 1 the number N(α) of eigenvalues of PΩQT greater than α satisfies
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 1 1 − α + log log ΩT + O(log ΩT ). π2 α
Eigenvalue properties I: the 2ΩT Theorem
Theorem For any T > 0 and Ω > 0 and 0 < α < 1 the number N(α) of eigenvalues of PΩQT greater than α satisfies
N(α) = 2ΩT
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting + O(log ΩT ).
Eigenvalue properties I: the 2ΩT Theorem
Theorem For any T > 0 and Ω > 0 and 0 < α < 1 the number N(α) of eigenvalues of PΩQT greater than α satisfies 1 1 − α N(α) = 2ΩT + log log ΩT π2 α
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Eigenvalue properties I: the 2ΩT Theorem
Theorem For any T > 0 and Ω > 0 and 0 < α < 1 the number N(α) of eigenvalues of PΩQT greater than α satisfies 1 1 − α N(α) = 2ΩT + log log ΩT + O(log ΩT ). π2 α
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting DFT illustration
Eigenvalues for 1025 points, normalized area of 64 1
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Figure: Eigenvalues for one frequency interval. N = 1025 point centered DFT. S ∼ 513 + [−128, 128]; Σ ∼ 513 + [−128, 128]. c =#T × #Σ/N≈ 64. Plunge region ∼ 61 ≤ n ≤ 69.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Eigenvector # 0 Eigenvector # 2 Eigenvector # 4 Eigenvector # 6 0 0.2 0.3 0.2 −0.05 0.15 0.25 0.15 0.1 0.2 −0.1 0.1 0.05 0.15 0.05 0 0.1 −0.15 0 −0.05 0.05 −0.2 −0.05 −0.1 −0.1 0 −0.25 −0.15 −0.15 −0.05 −0.2 −0.2 −0.1 −0.3 −0.25 −0.25 −0.15 −0.35 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120
Eigenvector # 8 Eigenvector # 10 Eigenvector # 12 Eigenvector # 14 0.3 0.15 0.15 0.1 0.25 0.1 0.1 0.05 0.2 0.05 0.05 0 0.15 0 0 −0.05 0.1 −0.05 −0.05 −0.1 0.05 −0.1 −0.15 0 −0.1
−0.2 −0.05 −0.15 −0.15 −0.2 −0.25 −0.1 −0.2 −0.3 −0.15 −0.25 −0.25 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120
Figure: Even eigenvectors for one frequency interval. N = 129 point centered DFT. S ∼ 65 + [−16, 16]; Σ ∼ 65 + [−16, 16]. c = #T × #Σ/N = 8.44. Plunge region ∼ 7 ≤ n ≤ 12.
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 Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Energy concentration: ϕ0 maximizes kQT f k/kf k over PWΩ Fourier covariance: r Ωξ n T ϕbn = ±i QT ϕn(ξ). T Ωλn
(Double orthogonality)
hQT ϕn, QT ϕmi = hϕn, PΩQT ϕmi = λnhϕn, ϕmi = λnδnm.
2 {ϕn} forms a complete, orthogonal family in L [−T , T ].
Eigenfunction properties II
Properties established by Landau, Slepian and Pollak in the first 3 “Bell Labs” papers (1960s)
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Fourier covariance: r Ωξ n T ϕbn = ±i QT ϕn(ξ). T Ωλn
(Double orthogonality)
hQT ϕn, QT ϕmi = hϕn, PΩQT ϕmi = λnhϕn, ϕmi = λnδnm.
2 {ϕn} forms a complete, orthogonal family in L [−T , T ].
Eigenfunction properties II
Properties established by Landau, Slepian and Pollak in the first 3 “Bell Labs” papers (1960s) Energy concentration: ϕ0 maximizes kQT f k/kf k over PWΩ
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting (Double orthogonality)
hQT ϕn, QT ϕmi = hϕn, PΩQT ϕmi = λnhϕn, ϕmi = λnδnm.
2 {ϕn} forms a complete, orthogonal family in L [−T , T ].
Eigenfunction properties II
Properties established by Landau, Slepian and Pollak in the first 3 “Bell Labs” papers (1960s) Energy concentration: ϕ0 maximizes kQT f k/kf k over PWΩ Fourier covariance: r Ωξ n T ϕbn = ±i QT ϕn(ξ). T Ωλn
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 2 {ϕn} forms a complete, orthogonal family in L [−T , T ].
Eigenfunction properties II
Properties established by Landau, Slepian and Pollak in the first 3 “Bell Labs” papers (1960s) Energy concentration: ϕ0 maximizes kQT f k/kf k over PWΩ Fourier covariance: r Ωξ n T ϕbn = ±i QT ϕn(ξ). T Ωλn
(Double orthogonality)
hQT ϕn, QT ϕmi = hϕn, PΩQT ϕmi = λnhϕn, ϕmi = λnδnm.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Eigenfunction properties II
Properties established by Landau, Slepian and Pollak in the first 3 “Bell Labs” papers (1960s) Energy concentration: ϕ0 maximizes kQT f k/kf k over PWΩ Fourier covariance: r Ωξ n T ϕbn = ±i QT ϕn(ξ). T Ωλn
(Double orthogonality)
hQT ϕn, QT ϕmi = hϕn, PΩQT ϕmi = λnhϕn, ϕmi = λnδnm.
2 {ϕn} forms a complete, orthogonal family in L [−T , T ].
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Thomson 1982 Multitaper method (see also Walden et al) Growth of numerical methods since 2000 (Beylkin and Monzon, Xiao Yarvin and Rokhlin, Boyd, Walter and Shen, Karoui and Moumni, Moore and Cada, others Growth of methods related to wireless communications: channel modeling (Zemen et al., Lyman and Sikora, etc) and pulse shaping (Hua and Sarker (1988); Chen et al (2007))
Applications and extensions I
Slepian (1978): index-limited DPSS sequences : main observation: find the correct commuting operator
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Growth of numerical methods since 2000 (Beylkin and Monzon, Xiao Yarvin and Rokhlin, Boyd, Walter and Shen, Karoui and Moumni, Moore and Cada, others Growth of methods related to wireless communications: channel modeling (Zemen et al., Lyman and Sikora, etc) and pulse shaping (Hua and Sarker (1988); Chen et al (2007))
Applications and extensions I
Slepian (1978): index-limited DPSS sequences : main observation: find the correct commuting operator Thomson 1982 Multitaper method (see also Walden et al)
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Growth of methods related to wireless communications: channel modeling (Zemen et al., Lyman and Sikora, etc) and pulse shaping (Hua and Sarker (1988); Chen et al (2007))
Applications and extensions I
Slepian (1978): index-limited DPSS sequences : main observation: find the correct commuting operator Thomson 1982 Multitaper method (see also Walden et al) Growth of numerical methods since 2000 (Beylkin and Monzon, Xiao Yarvin and Rokhlin, Boyd, Walter and Shen, Karoui and Moumni, Moore and Cada, others
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Applications and extensions I
Slepian (1978): index-limited DPSS sequences : main observation: find the correct commuting operator Thomson 1982 Multitaper method (see also Walden et al) Growth of numerical methods since 2000 (Beylkin and Monzon, Xiao Yarvin and Rokhlin, Boyd, Walter and Shen, Karoui and Moumni, Moore and Cada, others Growth of methods related to wireless communications: channel modeling (Zemen et al., Lyman and Sikora, etc) and pulse shaping (Hua and Sarker (1988); Chen et al (2007))
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Sampling and Time-Frequency localization
Problem Can one approximately recover the projection of f ∈ PW onto the space of “approximately time- and bandlimited” functions solely from samples near [−T , T ]?
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Z T If Amk = sinc (t − m) sinc (t − k) dt. −T X then λnϕn(m) = Amk ϕn(k) k
i.e. samples of PSWF ϕn form n-th eigenvector of {Amk }.
Sampling and eigenfunctions: rectangle case
Theorem (Walter and Shen; Khare and George, 2003-2004)
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting X then λnϕn(m) = Amk ϕn(k) k
i.e. samples of PSWF ϕn form n-th eigenvector of {Amk }.
Sampling and eigenfunctions: rectangle case
Theorem (Walter and Shen; Khare and George, 2003-2004)
Z T If Amk = sinc (t − m) sinc (t − k) dt. −T
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting i.e. samples of PSWF ϕn form n-th eigenvector of {Amk }.
Sampling and eigenfunctions: rectangle case
Theorem (Walter and Shen; Khare and George, 2003-2004)
Z T If Amk = sinc (t − m) sinc (t − k) dt. −T X then λnϕn(m) = Amk ϕn(k) k
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Sampling and eigenfunctions: rectangle case
Theorem (Walter and Shen; Khare and George, 2003-2004)
Z T If Amk = sinc (t − m) sinc (t − k) dt. −T X then λnϕn(m) = Amk ϕn(k) k
i.e. samples of PSWF ϕn form n-th eigenvector of {Amk }.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting ∞ X f (k) ϕn(k) ϕn(t). k=−∞
Corollary (Second Shen-Walter formula) If f ∈ PW then
∞ X PQT f (t) = λn n=0
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Corollary (Second Shen-Walter formula) If f ∈ PW then
∞ ∞ X X PQT f (t) = λn f (k) ϕn(k) ϕn(t). n=0 k=−∞
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Application: approximate time-frequency localization
Problem For suitable N(T ) ≈ T and M(T ) ≈ (T ), in what sense is
trunc span{ϕn : n ≤ N(T )} ≈ trunc span{sinc (t − k): |k| ≤ M(T )}?
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting The decay ought to be “linear” in λn if |k| > M(n, T ) ≈ T
Quadratic decay for PSWF samples
Shen and Walter quadratic√ decay estimate: P 2 |k|>T (ϕn(k)) ≤ CT 1 − λn.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Quadratic decay for PSWF samples
Shen and Walter quadratic√ decay estimate: P 2 |k|>T (ϕn(k)) ≤ CT 1 − λn.
The decay ought to be “linear” in λn if |k| > M(n, T ) ≈ T
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Hardy-Littlewood implies ... Lemma
X 2 (ψ ∗ (QeT ϕn)(k)) ≤ C(1 − λn).
ϕ = QT ϕ + QeT ϕ = ψ ∗ (QT ϕ) + ψ ∗ (QeT ϕ) ψ: Fourier bump;
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting ϕ = QT ϕ + QeT ϕ = ψ ∗ (QT ϕ) + ψ ∗ (QeT ϕ) ψ: Fourier bump; Hardy-Littlewood implies ... Lemma
X 2 (ψ ∗ (QeT ϕn)(k)) ≤ C(1 − λn).
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting ϕn: n-th eigenfunction of PQT ψ: Fourier bump ; Ψ: majorant M(n, T ) :
Z r1 − λ Ψ(t) dt ≤ n . |t|>M(n,T )−T −1/2 λn
Then X 2 ((ψ ∗ (QT ϕn))(k)) ≤ (1 − λn). |k|>M(n,T ) provided
M(n, T ) = (π2(T − n/2) + T )(1 + logγ T ).
Lemma
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting ϕn: n-th eigenfunction of PQT ψ: Fourier bump ; Ψ: majorant M(n, T ) :
Z r1 − λ Ψ(t) dt ≤ n . |t|>M(n,T )−T −1/2 λn
Then X 2 ((ψ ∗ (QT ϕn))(k)) ≤ (1 − λn). |k|>M(n,T ) provided
M(n, T ) = (π2(T − n/2) + T )(1 + logγ T ).
Lemma
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting ψ: Fourier bump ; Ψ: majorant M(n, T ) :
Z r1 − λ Ψ(t) dt ≤ n . |t|>M(n,T )−T −1/2 λn
Then X 2 ((ψ ∗ (QT ϕn))(k)) ≤ (1 − λn). |k|>M(n,T ) provided
M(n, T ) = (π2(T − n/2) + T )(1 + logγ T ).
Lemma ϕn: n-th eigenfunction of PQT
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting M(n, T ) :
Z r1 − λ Ψ(t) dt ≤ n . |t|>M(n,T )−T −1/2 λn
Then X 2 ((ψ ∗ (QT ϕn))(k)) ≤ (1 − λn). |k|>M(n,T ) provided
M(n, T ) = (π2(T − n/2) + T )(1 + logγ T ).
Lemma ϕn: n-th eigenfunction of PQT ψ: Fourier bump ; Ψ: majorant
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Then X 2 ((ψ ∗ (QT ϕn))(k)) ≤ (1 − λn). |k|>M(n,T ) provided
M(n, T ) = (π2(T − n/2) + T )(1 + logγ T ).
Lemma ϕn: n-th eigenfunction of PQT ψ: Fourier bump ; Ψ: majorant M(n, T ) :
Z r1 − λ Ψ(t) dt ≤ n . |t|>M(n,T )−T −1/2 λn
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting provided
M(n, T ) = (π2(T − n/2) + T )(1 + logγ T ).
Lemma ϕn: n-th eigenfunction of PQT ψ: Fourier bump ; Ψ: majorant M(n, T ) :
Z r1 − λ Ψ(t) dt ≤ n . |t|>M(n,T )−T −1/2 λn
Then X 2 ((ψ ∗ (QT ϕn))(k)) ≤ (1 − λn). |k|>M(n,T )
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Lemma ϕn: n-th eigenfunction of PQT ψ: Fourier bump ; Ψ: majorant M(n, T ) :
Z r1 − λ Ψ(t) dt ≤ n . |t|>M(n,T )−T −1/2 λn
Then X 2 ((ψ ∗ (QT ϕn))(k)) ≤ (1 − λn). |k|>M(n,T ) provided
M(n, T ) = (π2(T − n/2) + T )(1 + logγ T ).
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Corollary
Let ϕn be the n-th eigenfunction of PQT . Then there is a C > 0 independent of n such that
X 2 ϕn(k) ≤ C(1 − λn). |k|>M(n,T )
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting N X X fN,T = f (k) ϕn(k) ϕn(t) n=0 |k|≤M(T )
Proposition
N N 2 X 2 2X kQT fN − fN,T k ≤ λn|h(fN −fN,T ), ϕni| ≤ Ckf k λn(1 − λn). n=0 n=0
Approximate time-localized projections: the solution
For f ∈ PW ...
N X X fN (t) = f (k) ϕn(k) ϕn(t) n=0 k
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Proposition
N N 2 X 2 2X kQT fN − fN,T k ≤ λn|h(fN −fN,T ), ϕni| ≤ Ckf k λn(1 − λn). n=0 n=0
Approximate time-localized projections: the solution
For f ∈ PW ...
N X X fN (t) = f (k) ϕn(k) ϕn(t) n=0 k N X X fN,T = f (k) ϕn(k) ϕn(t) n=0 |k|≤M(T )
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Approximate time-localized projections: the solution
For f ∈ PW ...
N X X fN (t) = f (k) ϕn(k) ϕn(t) n=0 k N X X fN,T = f (k) ϕn(k) ϕn(t) n=0 |k|≤M(T )
Proposition
N N 2 X 2 2X kQT fN − fN,T k ≤ λn|h(fN −fN,T ), ϕni| ≤ Ckf k λn(1 − λn). n=0 n=0
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Standard: tridiagonal method produces Legendre coefficients of PSWFs on [−1, 1], Want: integer samples of ϕn (T ) = ψn (k/T ): dilate of ϕn (T ) (T ) ψn (k/T ): Fourier coefficients of ψbn on [−1, 1] Karoui and Moumni, (ACHA, 2008): Legendre expansion
X Z 1 k βn e2πikξ P (ξ)dξ = µ ψ . ` ` n n T `≥0: `=n mod 2 −1
2 P`: `-th Legendre polynomial, L -norm one over [−1, 1].
Accurate estimates of integer samples of PSWFs
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Standard: tridiagonal method produces Legendre coefficients of PSWFs on [−1, 1], Want: integer samples of ϕn (T ) = ψn (k/T ): dilate of ϕn (T ) (T ) ψn (k/T ): Fourier coefficients of ψbn on [−1, 1] Karoui and Moumni, (ACHA, 2008): Legendre expansion
X Z 1 k βn e2πikξ P (ξ)dξ = µ ψ . ` ` n n T `≥0: `=n mod 2 −1
2 P`: `-th Legendre polynomial, L -norm one over [−1, 1].
Accurate estimates of integer samples of PSWFs
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Want: integer samples of ϕn (T ) = ψn (k/T ): dilate of ϕn (T ) (T ) ψn (k/T ): Fourier coefficients of ψbn on [−1, 1] Karoui and Moumni, (ACHA, 2008): Legendre expansion
X Z 1 k βn e2πikξ P (ξ)dξ = µ ψ . ` ` n n T `≥0: `=n mod 2 −1
2 P`: `-th Legendre polynomial, L -norm one over [−1, 1].
Accurate estimates of integer samples of PSWFs
Standard: tridiagonal method produces Legendre coefficients of PSWFs on [−1, 1],
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting (T ) = ψn (k/T ): dilate of ϕn (T ) (T ) ψn (k/T ): Fourier coefficients of ψbn on [−1, 1] Karoui and Moumni, (ACHA, 2008): Legendre expansion
X Z 1 k βn e2πikξ P (ξ)dξ = µ ψ . ` ` n n T `≥0: `=n mod 2 −1
2 P`: `-th Legendre polynomial, L -norm one over [−1, 1].
Accurate estimates of integer samples of PSWFs
Standard: tridiagonal method produces Legendre coefficients of PSWFs on [−1, 1], Want: integer samples of ϕn
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting (T ) (T ) ψn (k/T ): Fourier coefficients of ψbn on [−1, 1] Karoui and Moumni, (ACHA, 2008): Legendre expansion
X Z 1 k βn e2πikξ P (ξ)dξ = µ ψ . ` ` n n T `≥0: `=n mod 2 −1
2 P`: `-th Legendre polynomial, L -norm one over [−1, 1].
Accurate estimates of integer samples of PSWFs
Standard: tridiagonal method produces Legendre coefficients of PSWFs on [−1, 1], Want: integer samples of ϕn (T ) = ψn (k/T ): dilate of ϕn
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Karoui and Moumni, (ACHA, 2008): Legendre expansion
X Z 1 k βn e2πikξ P (ξ)dξ = µ ψ . ` ` n n T `≥0: `=n mod 2 −1
2 P`: `-th Legendre polynomial, L -norm one over [−1, 1].
Accurate estimates of integer samples of PSWFs
Standard: tridiagonal method produces Legendre coefficients of PSWFs on [−1, 1], Want: integer samples of ϕn (T ) = ψn (k/T ): dilate of ϕn (T ) (T ) ψn (k/T ): Fourier coefficients of ψbn on [−1, 1]
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Legendre expansion
X Z 1 k βn e2πikξ P (ξ)dξ = µ ψ . ` ` n n T `≥0: `=n mod 2 −1
2 P`: `-th Legendre polynomial, L -norm one over [−1, 1].
Accurate estimates of integer samples of PSWFs
Standard: tridiagonal method produces Legendre coefficients of PSWFs on [−1, 1], Want: integer samples of ϕn (T ) = ψn (k/T ): dilate of ϕn (T ) (T ) ψn (k/T ): Fourier coefficients of ψbn on [−1, 1] Karoui and Moumni, (ACHA, 2008):
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 2 P`: `-th Legendre polynomial, L -norm one over [−1, 1].
Accurate estimates of integer samples of PSWFs
Standard: tridiagonal method produces Legendre coefficients of PSWFs on [−1, 1], Want: integer samples of ϕn (T ) = ψn (k/T ): dilate of ϕn (T ) (T ) ψn (k/T ): Fourier coefficients of ψbn on [−1, 1] Karoui and Moumni, (ACHA, 2008): Legendre expansion
X Z 1 k βn e2πikξ P (ξ)dξ = µ ψ . ` ` n n T `≥0: `=n mod 2 −1
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Accurate estimates of integer samples of PSWFs
Standard: tridiagonal method produces Legendre coefficients of PSWFs on [−1, 1], Want: integer samples of ϕn (T ) = ψn (k/T ): dilate of ϕn (T ) (T ) ψn (k/T ): Fourier coefficients of ψbn on [−1, 1] Karoui and Moumni, (ACHA, 2008): Legendre expansion
X Z 1 k βn e2πikξ P (ξ)dξ = µ ψ . ` ` n n T `≥0: `=n mod 2 −1
2 P`: `-th Legendre polynomial, L -norm one over [−1, 1].
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Legendre Fourier coefficients
( νq 2ν+1 (−i) 2k Jν+1/2(πk), k 6= 0 Ik,ν = δ0,ν, k = 0. The values can be calculated using the matlab builtin function besselj. Approximate by finite sum
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting ( νq 2ν+1 (−i) 2k Jν+1/2(πk), k 6= 0 Ik,ν = δ0,ν, k = 0. The values can be calculated using the matlab builtin function besselj. Approximate by finite sum
Legendre Fourier coefficients
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting The values can be calculated using the matlab builtin function besselj. Approximate by finite sum
Legendre Fourier coefficients
( νq 2ν+1 (−i) 2k Jν+1/2(πk), k 6= 0 Ik,ν = δ0,ν, k = 0.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Approximate by finite sum
Legendre Fourier coefficients
( νq 2ν+1 (−i) 2k Jν+1/2(πk), k 6= 0 Ik,ν = δ0,ν, k = 0. The values can be calculated using the matlab builtin function besselj.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Legendre Fourier coefficients
( νq 2ν+1 (−i) 2k Jν+1/2(πk), k 6= 0 Ik,ν = δ0,ν, k = 0. The values can be calculated using the matlab builtin function besselj. Approximate by finite sum
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Proposition (Karoui and Moumni)
For k 6= 0 and even `0 ≥ 2([2T e] + 1), k 1 1 X n 2 ψn − β` Ik,` ≤ ` |µn| . T µn 2 0 `≤`0: `=n mod 2
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 0.8 1 1
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KM loc Figure: Plots of ϕn (solid) and ϕn for n = 0, 4, 8, 12, 16 and T = 5.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 0.16
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KM loc Figure: Plots of ϕn (solid) and ϕn on [−5, 5] for n = 12.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting −3 x 10
5
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−5 −5 −4 −3 −2 −1 0 1 2 3 4 5 KM WS Figure: Plots of ϕ15 (solid) and ϕ15 (dashed) for T = 5. One the left, the time interval is [−2T , 2T ]. On the right it is [−T , T ].
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Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting −3 x 10 0.5 5 0.4 4 0.3 3
0.2 2
0.1 1
0 0
−0.1 −1
−2 −0.2 −3 −0.3 −4 −0.4 −5 −0.5 −5 −4 −3 −2 −1 0 1 2 3 4 5 −10 −8 −6 −4 −2 0 2 4 6 8 10 KM WS Figure: Plots of ϕ15 (solid) and ϕ15 (dashed) for T = 5. One the left, the time interval is [−2T , 2T ]. On the right it is [−T , T ].
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Potential in communications such as “cognitive radio” Potential use of GDPS eigenvectors of “PΣQT ” as modulation waveforms
A more general special case: multiband
Orthogonal eigenfunctions for symmetric bands
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Potential use of GDPS eigenvectors of “PΣQT ” as modulation waveforms
A more general special case: multiband
Orthogonal eigenfunctions for symmetric bands Potential in communications such as “cognitive radio”
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting A more general special case: multiband
Orthogonal eigenfunctions for symmetric bands Potential in communications such as “cognitive radio” Potential use of GDPS eigenvectors of “PΣQT ” as modulation waveforms
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Ac = PcΣQS PcΣ N(Ac , α) = #{λ(Ac ) > α} Landau and Widom (1980)
N N 1 − α N(A , α) = c|S||Σ| + S Σ log log c + o(log c) c π2 α
NS NΣ: width of “plunge region”
The “Σ T ”-theorem, N(α), Multiple intervals
S, Σ: finite unions of intervals
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting N(Ac , α) = #{λ(Ac ) > α} Landau and Widom (1980)
N N 1 − α N(A , α) = c|S||Σ| + S Σ log log c + o(log c) c π2 α
NS NΣ: width of “plunge region”
The “Σ T ”-theorem, N(α), Multiple intervals
S, Σ: finite unions of intervals Ac = PcΣQS PcΣ
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Landau and Widom (1980)
N N 1 − α N(A , α) = c|S||Σ| + S Σ log log c + o(log c) c π2 α
NS NΣ: width of “plunge region”
The “Σ T ”-theorem, N(α), Multiple intervals
S, Σ: finite unions of intervals Ac = PcΣQS PcΣ N(Ac , α) = #{λ(Ac ) > α}
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting N N 1 − α + S Σ log log c + o(log c) π2 α
NS NΣ: width of “plunge region”
The “Σ T ”-theorem, N(α), Multiple intervals
S, Σ: finite unions of intervals Ac = PcΣQS PcΣ N(Ac , α) = #{λ(Ac ) > α} Landau and Widom (1980)
N(Ac , α) = c|S||Σ|
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting + o(log c)
NS NΣ: width of “plunge region”
The “Σ T ”-theorem, N(α), Multiple intervals
S, Σ: finite unions of intervals Ac = PcΣQS PcΣ N(Ac , α) = #{λ(Ac ) > α} Landau and Widom (1980)
N N 1 − α N(A , α) = c|S||Σ| + S Σ log log c c π2 α
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting NS NΣ: width of “plunge region”
The “Σ T ”-theorem, N(α), Multiple intervals
S, Σ: finite unions of intervals Ac = PcΣQS PcΣ N(Ac , α) = #{λ(Ac ) > α} Landau and Widom (1980)
N N 1 − α N(A , α) = c|S||Σ| + S Σ log log c + o(log c) c π2 α
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting The “Σ T ”-theorem, N(α), Multiple intervals
S, Σ: finite unions of intervals Ac = PcΣQS PcΣ N(Ac , α) = #{λ(Ac ) > α} Landau and Widom (1980)
N N 1 − α N(A , α) = c|S||Σ| + S Σ log log c + o(log c) c π2 α
NS NΣ: width of “plunge region”
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting φj frequency concentrated on Ij , |Ij | = 1 2πim t φj (t) = e j ϕj (t) mj = Ij . ϕj frequency concentrated on [−1/2, 1/2].
Z 1/2 2πi(mj −mk )t hQφj , Qφk i = e ϕj (t)ϕk (t) dt −1/2 = ϕc1 ∗ ϕc2 ∗ sinc (m1 − m2) = O(1/|m1 − m2|)
Each Ij gives one eigenvalue ≈ 1/2 If each Ij were very short: no large eigenvalues.
Plunge width and ∼ NS NΣ
“Separated at infinity”
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 2πim t φj (t) = e j ϕj (t) mj = Ij . ϕj frequency concentrated on [−1/2, 1/2].
Z 1/2 2πi(mj −mk )t hQφj , Qφk i = e ϕj (t)ϕk (t) dt −1/2 = ϕc1 ∗ ϕc2 ∗ sinc (m1 − m2) = O(1/|m1 − m2|)
Each Ij gives one eigenvalue ≈ 1/2 If each Ij were very short: no large eigenvalues.
Plunge width and ∼ NS NΣ
“Separated at infinity” φj frequency concentrated on Ij , |Ij | = 1
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting ϕj frequency concentrated on [−1/2, 1/2].
Z 1/2 2πi(mj −mk )t hQφj , Qφk i = e ϕj (t)ϕk (t) dt −1/2 = ϕc1 ∗ ϕc2 ∗ sinc (m1 − m2) = O(1/|m1 − m2|)
Each Ij gives one eigenvalue ≈ 1/2 If each Ij were very short: no large eigenvalues.
Plunge width and ∼ NS NΣ
“Separated at infinity” φj frequency concentrated on Ij , |Ij | = 1 2πim t φj (t) = e j ϕj (t) mj = Ij .
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Z 1/2 2πi(mj −mk )t hQφj , Qφk i = e ϕj (t)ϕk (t) dt −1/2 = ϕc1 ∗ ϕc2 ∗ sinc (m1 − m2) = O(1/|m1 − m2|)
Each Ij gives one eigenvalue ≈ 1/2 If each Ij were very short: no large eigenvalues.
Plunge width and ∼ NS NΣ
“Separated at infinity” φj frequency concentrated on Ij , |Ij | = 1 2πim t φj (t) = e j ϕj (t) mj = Ij . ϕj frequency concentrated on [−1/2, 1/2].
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Each Ij gives one eigenvalue ≈ 1/2 If each Ij were very short: no large eigenvalues.
Plunge width and ∼ NS NΣ
“Separated at infinity” φj frequency concentrated on Ij , |Ij | = 1 2πim t φj (t) = e j ϕj (t) mj = Ij . ϕj frequency concentrated on [−1/2, 1/2].
Z 1/2 2πi(mj −mk )t hQφj , Qφk i = e ϕj (t)ϕk (t) dt −1/2 = ϕc1 ∗ ϕc2 ∗ sinc (m1 − m2) = O(1/|m1 − m2|)
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting If each Ij were very short: no large eigenvalues.
Plunge width and ∼ NS NΣ
“Separated at infinity” φj frequency concentrated on Ij , |Ij | = 1 2πim t φj (t) = e j ϕj (t) mj = Ij . ϕj frequency concentrated on [−1/2, 1/2].
Z 1/2 2πi(mj −mk )t hQφj , Qφk i = e ϕj (t)ϕk (t) dt −1/2 = ϕc1 ∗ ϕc2 ∗ sinc (m1 − m2) = O(1/|m1 − m2|)
Each Ij gives one eigenvalue ≈ 1/2
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Plunge width and ∼ NS NΣ
“Separated at infinity” φj frequency concentrated on Ij , |Ij | = 1 2πim t φj (t) = e j ϕj (t) mj = Ij . ϕj frequency concentrated on [−1/2, 1/2].
Z 1/2 2πi(mj −mk )t hQφj , Qφk i = e ϕj (t)ϕk (t) dt −1/2 = ϕc1 ∗ ϕc2 ∗ sinc (m1 − m2) = O(1/|m1 − m2|)
Each Ij gives one eigenvalue ≈ 1/2 If each Ij were very short: no large eigenvalues.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting What about N(PΣQ, 1/2)?
Proposition (Landau (1993), Izu (2009)) Let Σ = [−1/2, 1/2] and let S be a union of m pairwise disjoint intervals of total length c. Set
ν = max #{k ∈ :(k, k + 1) ⊂ S + α}, α Z
µ = min #{` ∈ Z :(`, ` + 1) ∩ S + β 6= ∅}. β
Then the eigenvalues λk of QS P satisfy
λν−1 ≥ 1/2 ≥ λµ.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting DFT illustration
Eigenvalues for 1025 points, normalized area of 64 1
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Figure: Eigenvalues for one frequency interval. N = 1025 point centered DFT. S ∼ 513 + [−128, 128]; Σ ∼ 513 + [−128, 128]. c =#T × #Σ/N≈ 64. Plunge region ∼ 61 ≤ n ≤ 69.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Eigenvector # 0 Eigenvector # 2 Eigenvector # 4 Eigenvector # 6 0 0.2 0.3 0.2 −0.05 0.15 0.25 0.15 0.1 0.2 −0.1 0.1 0.05 0.15 0.05 0 0.1 −0.15 0 −0.05 0.05 −0.2 −0.05 −0.1 −0.1 0 −0.25 −0.15 −0.15 −0.05 −0.2 −0.2 −0.1 −0.3 −0.25 −0.25 −0.15 −0.35 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120
Eigenvector # 8 Eigenvector # 10 Eigenvector # 12 Eigenvector # 14 0.3 0.15 0.15 0.1 0.25 0.1 0.1 0.05 0.2 0.05 0.05 0 0.15 0 0 −0.05 0.1 −0.05 −0.05 −0.1 0.05 −0.1 −0.15 0 −0.1
−0.2 −0.05 −0.15 −0.15 −0.2 −0.25 −0.1 −0.2 −0.3 −0.15 −0.25 −0.25 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120
Figure: Even eigenvectors for one frequency interval. N = 129 point centered DFT. S ∼ 65 + [−16, 16]; Σ ∼ 65 + [−16, 16]. c = #T × #Σ/N = 8.44. Plunge region ∼ 7 ≤ n ≤ 12.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Plunge width ∼ NS NΣ
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Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Grid spectrum
Corollary When T = 1 and Σ is a union of integer intervals [k, k + 1], λc = 1/2. Conjecture When Σ is a symmetric union of “grid intervals” of length 1/T (so c ∈ N) one has λc−k + λc+k = 1, k = 1,..., [c].
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 1
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Figure: Σ: 10 symmetrized length 16 “grid intervals” c = 40 (real part)
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 1
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Figure: Eigenvalues for DFT localization, N = 1024, T = 128, 10 symmetrized length 16 intervals c = 40 (real part), Note symmetry
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting I Z T /2 Z T /2 |f (t)|2 dt ≤ |(bf ?)∨(t)|2 dt. −T /2 −T /2
I optimal concentration: Σ is an interval if T is small enough.
I Rearrangement inequality fails for large measure.
Largest energy concentration for a given area?
I Donoho and Stark (1993): if |Σ| = 1 and T ≤ 0.8 then ...
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting I optimal concentration: Σ is an interval if T is small enough.
I Rearrangement inequality fails for large measure.
Largest energy concentration for a given area?
I Donoho and Stark (1993): if |Σ| = 1 and T ≤ 0.8 then ...
I Z T /2 Z T /2 |f (t)|2 dt ≤ |(bf ?)∨(t)|2 dt. −T /2 −T /2
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting I Rearrangement inequality fails for large measure.
Largest energy concentration for a given area?
I Donoho and Stark (1993): if |Σ| = 1 and T ≤ 0.8 then ...
I Z T /2 Z T /2 |f (t)|2 dt ≤ |(bf ?)∨(t)|2 dt. −T /2 −T /2
I optimal concentration: Σ is an interval if T is small enough.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Largest energy concentration for a given area?
I Donoho and Stark (1993): if |Σ| = 1 and T ≤ 0.8 then ...
I Z T /2 Z T /2 |f (t)|2 dt ≤ |(bf ?)∨(t)|2 dt. −T /2 −T /2
I optimal concentration: Σ is an interval if T is small enough.
I Rearrangement inequality fails for large measure.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting I ... at rate N: N eigenvalues ≥ 1/2
I Rationale: basis functions ∼ codes
I Which pairs support information?
Information problem
I (S, Σ) supports information if kPΣQS PΣk ≥ 1/2.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting I Rationale: basis functions ∼ codes
I Which pairs support information?
Information problem
I (S, Σ) supports information if kPΣQS PΣk ≥ 1/2.
I ... at rate N: N eigenvalues ≥ 1/2
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting I Which pairs support information?
Information problem
I (S, Σ) supports information if kPΣQS PΣk ≥ 1/2.
I ... at rate N: N eigenvalues ≥ 1/2
I Rationale: basis functions ∼ codes
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Information problem
I (S, Σ) supports information if kPΣQS PΣk ≥ 1/2.
I ... at rate N: N eigenvalues ≥ 1/2
I Rationale: basis functions ∼ codes
I Which pairs support information?
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Fix S ⊂ ZN and generate Σ ⊂ ZN randomly from the uniform distribution of subsets of ZN of size |Σ|, subject to N 1 |S| + |Σ| ≤ √ + o(1) . p(β + 1) log N 6
Then with probability at least 1 − O((log N)1/2/Nβ), every signal x frequency supported in Σ satisfies 1 kx1 k2 ≤ kxk2 or N(P Q , 1/2) = 0 S 2 Σ S
Probabilistic bounds: finite case
Theorem (Cand`es,Romberg, Tao) Fix N ≥ 512 and β such that 1 ≤ β ≤ (3/8) log N.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Then with probability at least 1 − O((log N)1/2/Nβ), every signal x frequency supported in Σ satisfies 1 kx1 k2 ≤ kxk2 or N(P Q , 1/2) = 0 S 2 Σ S
Probabilistic bounds: finite case
Theorem (Cand`es,Romberg, Tao)
Fix N ≥ 512 and β such that 1 ≤ β ≤ (3/8) log N. Fix S ⊂ ZN and generate Σ ⊂ ZN randomly from the uniform distribution of subsets of ZN of size |Σ|, subject to N 1 |S| + |Σ| ≤ √ + o(1) . p(β + 1) log N 6
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting or N(PΣQS , 1/2) = 0
Probabilistic bounds: finite case
Theorem (Cand`es,Romberg, Tao)
Fix N ≥ 512 and β such that 1 ≤ β ≤ (3/8) log N. Fix S ⊂ ZN and generate Σ ⊂ ZN randomly from the uniform distribution of subsets of ZN of size |Σ|, subject to N 1 |S| + |Σ| ≤ √ + o(1) . p(β + 1) log N 6
Then with probability at least 1 − O((log N)1/2/Nβ), every signal x frequency supported in Σ satisfies 1 kx1 k2 ≤ kxk2 S 2
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Probabilistic bounds: finite case
Theorem (Cand`es,Romberg, Tao)
Fix N ≥ 512 and β such that 1 ≤ β ≤ (3/8) log N. Fix S ⊂ ZN and generate Σ ⊂ ZN randomly from the uniform distribution of subsets of ZN of size |Σ|, subject to N 1 |S| + |Σ| ≤ √ + o(1) . p(β + 1) log N 6
Then with probability at least 1 − O((log N)1/2/Nβ), every signal x frequency supported in Σ satisfies 1 kx1 k2 ≤ kxk2 or N(P Q , 1/2) = 0 S 2 Σ S
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55 1 1.5 2 2.5
Figure: Norm of PΣQT versus entropy, N = 512, c = 8
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Approximation and Sampling in the Multiband case
Problem Describe projection onto localized eigenspaces of PΣQT PΣ in terms of samples of eigenvectors in the multiband case.
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 2 V ⊂ L (R): RKHS 2 PK : L (R) → V: Z ∞ PK f (t) = K(t, s) f (s) ds −∞
QS f (t) = f (t)1S (t). Sk , k ∈ Z, interpolates V sampled at {tk }: ∞ X f (t) = f (tk )Sk (t). k=−∞
PK QS compact, self-adjoint operator, Z Amk = K(tm, s)Sk (s) ds. S
Eigenvectors of Amk are samples of eigenvectors of PK QS
Reproducing kernel Hilbert space version
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting 2 PK : L (R) → V: Z ∞ PK f (t) = K(t, s) f (s) ds −∞
QS f (t) = f (t)1S (t). Sk , k ∈ Z, interpolates V sampled at {tk }: ∞ X f (t) = f (tk )Sk (t). k=−∞
PK QS compact, self-adjoint operator, Z Amk = K(tm, s)Sk (s) ds. S
Eigenvectors of Amk are samples of eigenvectors of PK QS
Reproducing kernel Hilbert space version
2 V ⊂ L (R): RKHS
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting QS f (t) = f (t)1S (t). Sk , k ∈ Z, interpolates V sampled at {tk }: ∞ X f (t) = f (tk )Sk (t). k=−∞
PK QS compact, self-adjoint operator, Z Amk = K(tm, s)Sk (s) ds. S
Eigenvectors of Amk are samples of eigenvectors of PK QS
Reproducing kernel Hilbert space version
2 V ⊂ L (R): RKHS 2 PK : L (R) → V: Z ∞ PK f (t) = K(t, s) f (s) ds −∞
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Sk , k ∈ Z, interpolates V sampled at {tk }: ∞ X f (t) = f (tk )Sk (t). k=−∞
PK QS compact, self-adjoint operator, Z Amk = K(tm, s)Sk (s) ds. S
Eigenvectors of Amk are samples of eigenvectors of PK QS
Reproducing kernel Hilbert space version
2 V ⊂ L (R): RKHS 2 PK : L (R) → V: Z ∞ PK f (t) = K(t, s) f (s) ds −∞
QS f (t) = f (t)1S (t).
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting PK QS compact, self-adjoint operator, Z Amk = K(tm, s)Sk (s) ds. S
Eigenvectors of Amk are samples of eigenvectors of PK QS
Reproducing kernel Hilbert space version
2 V ⊂ L (R): RKHS 2 PK : L (R) → V: Z ∞ PK f (t) = K(t, s) f (s) ds −∞
QS f (t) = f (t)1S (t). Sk , k ∈ Z, interpolates V sampled at {tk }: ∞ X f (t) = f (tk )Sk (t). k=−∞
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Eigenvectors of Amk are samples of eigenvectors of PK QS
Reproducing kernel Hilbert space version
2 V ⊂ L (R): RKHS 2 PK : L (R) → V: Z ∞ PK f (t) = K(t, s) f (s) ds −∞
QS f (t) = f (t)1S (t). Sk , k ∈ Z, interpolates V sampled at {tk }: ∞ X f (t) = f (tk )Sk (t). k=−∞
PK QS compact, self-adjoint operator, Z Amk = K(tm, s)Sk (s) ds. S
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Reproducing kernel Hilbert space version
2 V ⊂ L (R): RKHS 2 PK : L (R) → V: Z ∞ PK f (t) = K(t, s) f (s) ds −∞
QS f (t) = f (t)1S (t). Sk , k ∈ Z, interpolates V sampled at {tk }: ∞ X f (t) = f (tk )Sk (t). k=−∞
PK QS compact, self-adjoint operator, Z Amk = K(tm, s)Sk (s) ds. S
Eigenvectors of Amk are samples of eigenvectors of PK QS Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Lemma (Multiband spectrum) M Σ = ∪m=1Im bounded, pairwise disjoint Im ϕn : n-th eigenfunction of PIm QT Then, for any f ∈ PWΣ
M ∞ X X Im Im f (t) = hf ϕn iϕn m=1 n=0 M ∞ ∞ X 1 X X k Im k Im = (PIm f ) ϕn ϕn (t). |Im| |Im| |Im| m=1 n=0 k=−∞
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Numerous aproaches: Venkataramani and Bresler,... also Herley and Wong, Behmard Faridani and Walnut, Avdonin and Moran ... Bunched sampling
Sampling of multiband signals
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting ... also Herley and Wong, Behmard Faridani and Walnut, Avdonin and Moran ... Bunched sampling
Sampling of multiband signals
Numerous aproaches: Venkataramani and Bresler,
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Bunched sampling
Sampling of multiband signals
Numerous aproaches: Venkataramani and Bresler,... also Herley and Wong, Behmard Faridani and Walnut, Avdonin and Moran ...
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Sampling of multiband signals
Numerous aproaches: Venkataramani and Bresler,... also Herley and Wong, Behmard Faridani and Walnut, Avdonin and Moran ... Bunched sampling
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting J−1 Σ: L-subtiling set, Σj : 1-tiling, Σ = ∪j=0 Σj PM−1 Ijm = Im + βjm; Σj = m=1 Ijm; ∪mIjm = [0, 1); βjm ∈ Z −2πijβνm/L Wm(j, ν) = e (J × J) −1 Cm(j, ν) = Wm (ν, j)Wm(j, ν). Then ∀ f ∈ PWΣ ...
J−1 ∞ X X j j f , ϕIνm = C (j, ν)f k + ϕIνm k + n m L n L j=0 k=−∞
Theorem (Izu)
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting PM−1 Ijm = Im + βjm; Σj = m=1 Ijm; ∪mIjm = [0, 1); βjm ∈ Z −2πijβνm/L Wm(j, ν) = e (J × J) −1 Cm(j, ν) = Wm (ν, j)Wm(j, ν). Then ∀ f ∈ PWΣ ...
J−1 ∞ X X j j f , ϕIνm = C (j, ν)f k + ϕIνm k + n m L n L j=0 k=−∞
Theorem (Izu) J−1 Σ: L-subtiling set, Σj : 1-tiling, Σ = ∪j=0 Σj
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting −2πijβνm/L Wm(j, ν) = e (J × J) −1 Cm(j, ν) = Wm (ν, j)Wm(j, ν). Then ∀ f ∈ PWΣ ...
J−1 ∞ X X j j f , ϕIνm = C (j, ν)f k + ϕIνm k + n m L n L j=0 k=−∞
Theorem (Izu) J−1 Σ: L-subtiling set, Σj : 1-tiling, Σ = ∪j=0 Σj PM−1 Ijm = Im + βjm; Σj = m=1 Ijm; ∪mIjm = [0, 1); βjm ∈ Z
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting −1 Cm(j, ν) = Wm (ν, j)Wm(j, ν). Then ∀ f ∈ PWΣ ...
J−1 ∞ X X j j f , ϕIνm = C (j, ν)f k + ϕIνm k + n m L n L j=0 k=−∞
Theorem (Izu) J−1 Σ: L-subtiling set, Σj : 1-tiling, Σ = ∪j=0 Σj PM−1 Ijm = Im + βjm; Σj = m=1 Ijm; ∪mIjm = [0, 1); βjm ∈ Z −2πijβνm/L Wm(j, ν) = e (J × J)
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Theorem (Izu) J−1 Σ: L-subtiling set, Σj : 1-tiling, Σ = ∪j=0 Σj PM−1 Ijm = Im + βjm; Σj = m=1 Ijm; ∪mIjm = [0, 1); βjm ∈ Z −2πijβνm/L Wm(j, ν) = e (J × J) −1 Cm(j, ν) = Wm (ν, j)Wm(j, ν). Then ∀ f ∈ PWΣ ...
J−1 ∞ X X j j f , ϕIνm = C (j, ν)f k + ϕIνm k + n m L n L j=0 k=−∞
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting With the sample coefficients one obtains:
M−1 J−1 ∞ X X X Iνm Iνm f (t) = f , ϕn ϕn (t) m=0 ν=0 n=0 M−1 J−1 ∞ ∞ X X X X Iνm Iνm Σ Σ = f , ϕn ϕn , ϕ` ϕ` (t). m=0 ν=0 n=0 `=0
Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting