The Heisenberg Representation and the Fast Fourier Transform - PowerPoint PPT Presentation

The Heisenberg Representation and the Fast Fourier Transform Shamgar Gurevich UW Madison July 31, 2014 Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 1 / 25

Motivation: Discrete Fourier Transform � N τ · ω � 1 2 π i DFT = √ e N 0 ≤ τ , ω ≤ N − 1 Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 2 / 25

Motivation: Discrete Fourier Transform � N τ · ω � 1 2 π i DFT = √ e N 0 ≤ τ , ω ≤ N − 1 Compute:   f ( 0 )   .   �   f = DFT [ f ] ; f = . ; Fast!!     . f ( N − 1 ) Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 2 / 25

Motivation: Discrete Fourier Transform � N τ · ω � 1 2 π i DFT = √ e N 0 ≤ τ , ω ≤ N − 1 Compute:   f ( 0 )   .   �   f = DFT [ f ] ; f = . ; Fast!!     . f ( N − 1 ) Cooley—Tukey (1965): O ( N · log ( N )) operations! Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 2 / 25

Solution (Auslander—Tolimieri) (I) Heisenberg Group Representation H = C ( Z N ) – Hilbert space of digital signals. Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 3 / 25

Solution (Auslander—Tolimieri) (I) Heisenberg Group Representation H = C ( Z N ) – Hilbert space of digital signals. f : { 0 , ...., N − 1 } → C . Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 3 / 25

Solution (Auslander—Tolimieri) (I) Heisenberg Group Representation H = C ( Z N ) – Hilbert space of digital signals. f : { 0 , ...., N − 1 } → C . Basic operations Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 3 / 25

Solution (Auslander—Tolimieri) (I) Heisenberg Group Representation H = C ( Z N ) – Hilbert space of digital signals. f : { 0 , ...., N − 1 } → C . Basic operations Time shift: τ ∈ Z N , L τ : H → H , L τ [ f ]( t ) = f ( t + τ ) , t ∈ Z N . Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 3 / 25

Solution (Auslander—Tolimieri) (I) Heisenberg Group Representation H = C ( Z N ) – Hilbert space of digital signals. f : { 0 , ...., N − 1 } → C . Basic operations Time shift: τ ∈ Z N , L τ : H → H , L τ [ f ]( t ) = f ( t + τ ) , t ∈ Z N . Frequency shift: ω ∈ Z N , M ω : H → H , 2 π i N ω t f ( t ) . M ω [ f ]( t ) = e Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 3 / 25

Solution (Auslander—Tolimieri) (I) Heisenberg Group Representation H = C ( Z N ) – Hilbert space of digital signals. f : { 0 , ...., N − 1 } → C . Basic operations Time shift: τ ∈ Z N , L τ : H → H , L τ [ f ]( t ) = f ( t + τ ) , t ∈ Z N . Frequency shift: ω ∈ Z N , M ω : H → H , 2 π i N ω t f ( t ) . M ω [ f ]( t ) = e Note: 2 π i N ωτ · L τ ◦ M ω M ω ◦ L τ = e – Heisenberg commutation relations Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 3 / 25

Heisenberg Rep’n, cont. τ , ω , z ∈ Z N Combine: 2 ωτ + z } · M ω ◦ L τ . 2 π i N { 1 π ( τ , ω , z ) = e Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 4 / 25

Heisenberg Rep’n, cont. τ , ω , z ∈ Z N Combine: 2 ωτ + z } · M ω ◦ L τ . 2 π i N { 1 π ( τ , ω , z ) = e Identity: � � � � π ( τ , ω , z ) ◦ π ( τ � , ω � , z � ) = π ( τ + τ � , ω + ω � , z + z � + 1 τ ω � � � ) . � τ � ω � 2 Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 4 / 25

Heisenberg Rep’n, cont. τ , ω , z ∈ Z N Combine: 2 ωτ + z } · M ω ◦ L τ . 2 π i N { 1 π ( τ , ω , z ) = e Identity: � � � � π ( τ , ω , z ) ◦ π ( τ � , ω � , z � ) = π ( τ + τ � , ω + ω � , z + z � + 1 τ ω � � � ) . � τ � ω � 2 Question. How to think on this? Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 4 / 25

Heisenberg Rep’n, cont. τ , ω , z ∈ Z N Combine: 2 ωτ + z } · M ω ◦ L τ . 2 π i N { 1 π ( τ , ω , z ) = e Identity: � � � � π ( τ , ω , z ) ◦ π ( τ � , ω � , z � ) = π ( τ + τ � , ω + ω � , z + z � + 1 τ ω � � � ) . � τ � ω � 2 Question. How to think on this? Answer. Heisenberg group: Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 4 / 25

Heisenberg Rep’n, cont. τ , ω , z ∈ Z N Combine: 2 ωτ + z } · M ω ◦ L τ . 2 π i N { 1 π ( τ , ω , z ) = e Identity: � � � � π ( τ , ω , z ) ◦ π ( τ � , ω � , z � ) = π ( τ + τ � , ω + ω � , z + z � + 1 τ ω � � � ) . � τ � ω � 2 Question. How to think on this? Answer. Heisenberg group: H = Z N × Z N × Z N ; � �� � ���� V Z Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 4 / 25

Heisenberg Rep’n, cont. τ , ω , z ∈ Z N Combine: 2 ωτ + z } · M ω ◦ L τ . 2 π i N { 1 π ( τ , ω , z ) = e Identity: � � � � π ( τ , ω , z ) ◦ π ( τ � , ω � , z � ) = π ( τ + τ � , ω + ω � , z + z � + 1 τ ω � � � ) . � τ � ω � 2 Question. How to think on this? Answer. Heisenberg group: H = Z N × Z N × Z N ; � �� � ���� V Z � � � � v ( v , z ) · ( v � , z � ) = ( v + v � , z + z � + 1 � � � ) ; � v � 2 Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 4 / 25

Heisenberg Rep’n, cont. τ , ω , z ∈ Z N Combine: 2 ωτ + z } · M ω ◦ L τ . 2 π i N { 1 π ( τ , ω , z ) = e Identity: � � � � π ( τ , ω , z ) ◦ π ( τ � , ω � , z � ) = π ( τ + τ � , ω + ω � , z + z � + 1 τ ω � � � ) . � τ � ω � 2 Question. How to think on this? Answer. Heisenberg group: H = Z N × Z N × Z N ; � �� � ���� V Z � � � � v ( v , z ) · ( v � , z � ) = ( v + v � , z + z � + 1 � � � ) ; � v � 2 ( 0 , 0 ) · ( v , z ) = ( v , z ) · ( 0 , 0 ) = ( v , z ) ; Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 4 / 25

Heisenberg Rep’n, cont. τ , ω , z ∈ Z N Combine: 2 ωτ + z } · M ω ◦ L τ . 2 π i N { 1 π ( τ , ω , z ) = e Identity: � � � � π ( τ , ω , z ) ◦ π ( τ � , ω � , z � ) = π ( τ + τ � , ω + ω � , z + z � + 1 τ ω � � � ) . � τ � ω � 2 Question. How to think on this? Answer. Heisenberg group: H = Z N × Z N × Z N ; � �� � ���� V Z � � � � v ( v , z ) · ( v � , z � ) = ( v + v � , z + z � + 1 � � � ) ; � v � 2 ( 0 , 0 ) · ( v , z ) = ( v , z ) · ( 0 , 0 ) = ( v , z ) ; ( v , z ) · ( − v , − z ) = ( − v , − z ) · ( v , z ) = ( 0 , 0 ) . Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 4 / 25

Heisenberg Rep’n, cont. Summary: Heisenberg Rep’n  π : H → GL ( H ) ;  2 π i N { 1 2 ωτ + z } · M ω ◦ L τ ; π ( τ , ω , z ) = e  π ( h · h � ) = π ( h ) ◦ π ( h � ) – homomorphism. Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 5 / 25

Heisenberg Rep’n, cont. Summary: Heisenberg Rep’n  π : H → GL ( H ) ;  2 π i N { 1 2 ωτ + z } · M ω ◦ L τ ; π ( τ , ω , z ) = e  π ( h · h � ) = π ( h ) ◦ π ( h � ) – homomorphism. Definition A representation of a group H on a complex vector space H is a homomorphism : H → GL ( H ) , π π ( h · h � ) π ( h ) ◦ π ( h � ) . = Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 5 / 25

Heisenberg Rep’n, cont. Summary: Heisenberg Rep’n  π : H → GL ( H ) ;  2 π i N { 1 2 ωτ + z } · M ω ◦ L τ ; π ( τ , ω , z ) = e  π ( h · h � ) = π ( h ) ◦ π ( h � ) – homomorphism. Definition A representation of a group H on a complex vector space H is a homomorphism : H → GL ( H ) , π π ( h · h � ) π ( h ) ◦ π ( h � ) . = Question. DFT ? Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 5 / 25

(II) Representation Theory Definitions We say that ( π 1 , H , H 1 ) , ( π 2 , H , H 2 ) are equivalent, π 1 � π 2 , if ∃ α : H 1 � →H 2 − Intertwiner , such that for every h ∈ H π 1 ( h ) H 1 − − − → H 1     � α � α π 2 ( h ) H 2 − − − → H 2 i.e., α ◦ π 1 ( h ) = π 2 ( h ) ◦ α . Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 6 / 25

(II) Representation Theory Definitions We say that ( π 1 , H , H 1 ) , ( π 2 , H , H 2 ) are equivalent, π 1 � π 2 , if ∃ α : H 1 � →H 2 − Intertwiner , such that for every h ∈ H π 1 ( h ) H 1 − − − → H 1     � α � α π 2 ( h ) H 2 − − − → H 2 i.e., α ◦ π 1 ( h ) = π 2 ( h ) ◦ α . Example: DFT is an intertwiner! Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 6 / 25

Rep’n Theory, cont. H = V × Z = Z N × Z N × Z N ���� ���� ���� T W Z Shamgar Gurevich (UW Madison) Heisenberg Repn and FFT July 31, 2014 7 / 25