Diagonalization of the Discrete Fourier Transform using Weil - PowerPoint PPT Presentation

Diagonalization of the Discrete Fourier Transform using Weil Representation Shamgar Gurevich Madison August 3, 2014 Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 1 / 12

(0) Motivation - Diagonalizing DFT H = C ( F p ) — Hilbert space of digital sequences. Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 2 / 12

(0) Motivation - Diagonalizing DFT H = C ( F p ) — Hilbert space of digital sequences. ψ ( t ) = exp ( 2 π it / p ) . Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 2 / 12

(0) Motivation - Diagonalizing DFT H = C ( F p ) — Hilbert space of digital sequences. ψ ( t ) = exp ( 2 π it / p ) . DFT : H → H - Discrete Fourier Transform 1 DFT [ f ]( ω ) = ψ ( ω t ) f ( t ) . √ p ∑ t ∈ F p Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 2 / 12

(0) Motivation - Diagonalizing DFT H = C ( F p ) — Hilbert space of digital sequences. ψ ( t ) = exp ( 2 π it / p ) . DFT : H → H - Discrete Fourier Transform 1 DFT [ f ]( ω ) = ψ ( ω t ) f ( t ) . √ p ∑ t ∈ F p Fact: DFT 4 = Id = ⇒ λ ( DFT ) ∈ {± 1 , ± i } . Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 2 / 12

(0) Motivation - Diagonalizing DFT H = C ( F p ) — Hilbert space of digital sequences. ψ ( t ) = exp ( 2 π it / p ) . DFT : H → H - Discrete Fourier Transform 1 DFT [ f ]( ω ) = ψ ( ω t ) f ( t ) . √ p ∑ t ∈ F p Fact: DFT 4 = Id = ⇒ λ ( DFT ) ∈ {± 1 , ± i } . Problem ( Diagonalization ) Find natural basis of eigenfunctions for DFT . Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 2 / 12

Solution - Idea Find natural Symmetries DFT � H � C Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 3 / 12

Solution - Idea Find natural Symmetries DFT � H � C C commutative group. Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 3 / 12

Solution - Idea Find natural Symmetries DFT � H � C C commutative group. Take common eigenfunctions! Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 3 / 12

Solution - Idea Find natural Symmetries DFT � H � C C commutative group. Take common eigenfunctions! Question: C = ? . Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 3 / 12

Solution - Idea Find natural Symmetries DFT � H � C C commutative group. Take common eigenfunctions! Question: C = ? . Answer: Characterization of DFT . Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 3 / 12

Characterization of DFT Basic operations Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 4 / 12

Characterization of DFT Basic operations Time shift: τ ∈ F p , � L τ : H → H , L τ [ f ]( t ) = f ( t + τ ) , t ∈ Z N . Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 4 / 12

Characterization of DFT Basic operations Time shift: τ ∈ F p , � L τ : H → H , L τ [ f ]( t ) = f ( t + τ ) , t ∈ Z N . Frequency shift: ω ∈ F p , � M ω : H → H , M ω [ f ]( t ) = ψ ( ω t ) f ( t ) . Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 4 / 12

Characterization of DFT Basic operations Time shift: τ ∈ F p , � L τ : H → H , L τ [ f ]( t ) = f ( t + τ ) , t ∈ Z N . Frequency shift: ω ∈ F p , � M ω : H → H , M ω [ f ]( t ) = ψ ( ω t ) f ( t ) . Intertwining relations � DFT ◦ L τ = M τ ◦ DFT , DFT ◦ M ω = L − ω ◦ DFT . Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 4 / 12

Characterization of DFT - Cont. Combine � π : F p × F p → U ( H ) , π ( τ , ω ) = ψ ( − 1 2 τω ) · M ω ◦ L τ Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 5 / 12

Characterization of DFT - Cont. Combine � π : F p × F p → U ( H ) , π ( τ , ω ) = ψ ( − 1 2 τω ) · M ω ◦ L τ Intertwining relations W � �� � � τ � � 0 � � τ � − 1 Σ W : DFT ◦ π = π ( ) ◦ DFT . ω 1 0 ω System of p 2 linear equations. Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 5 / 12

Characterization of DFT - Cont. Combine � π : F p × F p → U ( H ) , π ( τ , ω ) = ψ ( − 1 2 τω ) · M ω ◦ L τ Intertwining relations W � �� � � τ � � 0 � � τ � − 1 Σ W : DFT ◦ π = π ( ) ◦ DFT . ω 1 0 ω System of p 2 linear equations. Theorem (Stone - von Neumann) dim Sol ( Σ W ) = 1 . Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 5 / 12

Characterization of DFT - Cont. Combine � π : F p × F p → U ( H ) , π ( τ , ω ) = ψ ( − 1 2 τω ) · M ω ◦ L τ Intertwining relations W � �� � � τ � � 0 � � τ � − 1 Σ W : DFT ◦ π = π ( ) ◦ DFT . ω 1 0 ω System of p 2 linear equations. Theorem (Stone - von Neumann) dim Sol ( Σ W ) = 1 . ⇒ DFT is characterized by Σ W . = Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 5 / 12

(II) The Weil Representation Note �� a � � b W ∈ SL 2 ( F p ) = ; a , b , c , d ∈ F p , ad − bc = 1 . c d Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 6 / 12

(II) The Weil Representation Note �� a � � b W ∈ SL 2 ( F p ) = ; a , b , c , d ∈ F p , ad − bc = 1 . c d Generalization: g ∈ SL 2 ( F p ) � τ � � τ � Σ g : ρ ( g ) ◦ π = π ( g · ) ◦ ρ ( g ) . ω ω System of p 2 linear equations. Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 6 / 12

(II) The Weil Representation Note �� a � � b W ∈ SL 2 ( F p ) = ; a , b , c , d ∈ F p , ad − bc = 1 . c d Generalization: g ∈ SL 2 ( F p ) � τ � � τ � Σ g : ρ ( g ) ◦ π = π ( g · ) ◦ ρ ( g ) . ω ω System of p 2 linear equations. Theorem (Stone - von Neumann) dim Sol ( Σ g ) = 1 . Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 6 / 12

(II) The Weil Representation Note �� a � � b W ∈ SL 2 ( F p ) = ; a , b , c , d ∈ F p , ad − bc = 1 . c d Generalization: g ∈ SL 2 ( F p ) � τ � � τ � Σ g : ρ ( g ) ◦ π = π ( g · ) ◦ ρ ( g ) . ω ω System of p 2 linear equations. Theorem (Stone - von Neumann) dim Sol ( Σ g ) = 1 . = ⇒ ρ ( g ) is characterized by Σ g . Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 6 / 12

Weil Representation Theorem ∃ ! collection of operators ρ ( g ) ∈ Sol ( Σ g ) , g ∈ SL 2 ( F p ) , such that ρ ( gh ) = ρ ( g ) ◦ ρ ( h ) . Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 7 / 12

Weil Representation Theorem ∃ ! collection of operators ρ ( g ) ∈ Sol ( Σ g ) , g ∈ SL 2 ( F p ) , such that ρ ( gh ) = ρ ( g ) ◦ ρ ( h ) . The homomorphism ρ : SL 2 ( F p ) → U ( H ) , H = C ( F p ) , is called the Weil Representation. Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 7 / 12

(III) Diagonalizing the DFT We have � ρ : SL 2 ( F p ) → U ( H ) ⊃ C = ? W �→ ρ ( W ) = DFT ; Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 8 / 12

(III) Diagonalizing the DFT We have � ρ : SL 2 ( F p ) → U ( H ) ⊃ C = ? W �→ ρ ( W ) = DFT ; Consider symmetries of W : T W = { g ∈ SL 2 ( F p ) ; gW = Wg } � � g ∈ SL 2 ( F p ) ; gg t = I = = SO 2 ( F p ) - finite rotations. Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 8 / 12

(III) Diagonalizing the DFT We have � ρ : SL 2 ( F p ) → U ( H ) ⊃ C = ? W �→ ρ ( W ) = DFT ; Consider symmetries of W : T W = { g ∈ SL 2 ( F p ) ; gW = Wg } � � g ∈ SL 2 ( F p ) ; gg t = I = = SO 2 ( F p ) - finite rotations. Lemma T W is a maximal commutative subgroup (torus) of SL 2 ( F p ) . Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 8 / 12

(III) Diagonalizing the DFT We have � ρ : SL 2 ( F p ) → U ( H ) ⊃ C = ? W �→ ρ ( W ) = DFT ; Consider symmetries of W : T W = { g ∈ SL 2 ( F p ) ; gW = Wg } � � g ∈ SL 2 ( F p ) ; gg t = I = = SO 2 ( F p ) - finite rotations. Lemma T W is a maximal commutative subgroup (torus) of SL 2 ( F p ) . Proof. P W ( x ) = det ( xI − W ) = x 2 + 1 . Hence λ ( W ) = ±√− 1 . Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 8 / 12

Diagonalizing the DFT Symmetries of DFT C = Im ( T W ) = { ρ ( g ) ; g ∈ T W } . Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 9 / 12

Diagonalizing the DFT Symmetries of DFT C = Im ( T W ) = { ρ ( g ) ; g ∈ T W } . C commutative group of unitary operators commuting with DFT . Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 9 / 12

Diagonalizing the DFT Symmetries of DFT C = Im ( T W ) = { ρ ( g ) ; g ∈ T W } . C commutative group of unitary operators commuting with DFT . Can diagonalize C simultaneously. Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 9 / 12