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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

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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

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(0) Motivation - Diagonalizing DFT

H = C(Fp) — Hilbert space of digital sequences.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 2 / 12

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(0) Motivation - Diagonalizing DFT

H = C(Fp) — Hilbert space of digital sequences.

ψ(t) = exp(2πit/p).

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 2 / 12

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(0) Motivation - Diagonalizing DFT

H = C(Fp) — Hilbert space of digital sequences.

ψ(t) = exp(2πit/p).

DFT : H → H - Discrete Fourier Transform DFT[f ](ω) = 1 √p ∑

t∈Fp

ψ(ωt)f (t).

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 2 / 12

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(0) Motivation - Diagonalizing DFT

H = C(Fp) — Hilbert space of digital sequences.

ψ(t) = exp(2πit/p).

DFT : H → H - Discrete Fourier Transform DFT[f ](ω) = 1 √p ∑

t∈Fp

ψ(ωt)f (t). Fact: DFT 4 = Id = ⇒ λ(DFT) ∈ {±1, ±i}.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 2 / 12

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(0) Motivation - Diagonalizing DFT

H = C(Fp) — Hilbert space of digital sequences.

ψ(t) = exp(2πit/p).

DFT : H → H - Discrete Fourier Transform DFT[f ](ω) = 1 √p ∑

t∈Fp

ψ(ωt)f (t). 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

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Solution - Idea

Find natural Symmetries DFT H C

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 3 / 12

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Solution - Idea

Find natural Symmetries DFT H C

C commutative group.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 3 / 12

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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

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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

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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

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Characterization of DFT

Basic operations

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

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Characterization of DFT

Basic operations

Time shift: τ ∈ Fp,

  • Lτ : H → H,

Lτ[f ](t) = f (t + τ), t ∈ ZN.

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

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Characterization of DFT

Basic operations

Time shift: τ ∈ Fp,

  • Lτ : H → H,

Lτ[f ](t) = f (t + τ), t ∈ ZN. Frequency shift: ω ∈ Fp,

  • Mω : H → H,

Mω[f ](t) = ψ(ωt)f (t).

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

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Characterization of DFT

Basic operations

Time shift: τ ∈ Fp,

  • Lτ : H → H,

Lτ[f ](t) = f (t + τ), t ∈ ZN. Frequency shift: ω ∈ Fp,

  • Mω : H → H,

Mω[f ](t) = ψ(ωt)f (t).

Intertwining relations

  • DFT ◦ Lτ = Mτ ◦ DFT,

DFT ◦ Mω = L−ω ◦ DFT.

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

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Characterization of DFT - Cont.

Combine

  • π : Fp × Fp → U(H),

π(τ, ω) = ψ(− 1

2τω) · Mω ◦ Lτ

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 5 / 12

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Characterization of DFT - Cont.

Combine

  • π : Fp × Fp → U(H),

π(τ, ω) = ψ(− 1

2τω) · Mω ◦ Lτ

Intertwining relations ΣW : DFT ◦ π τ ω

  • = π(

W

  • −1

1 τ ω

  • ) ◦ DFT.

System of p2 linear equations.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 5 / 12

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Characterization of DFT - Cont.

Combine

  • π : Fp × Fp → U(H),

π(τ, ω) = ψ(− 1

2τω) · Mω ◦ Lτ

Intertwining relations ΣW : DFT ◦ π τ ω

  • = π(

W

  • −1

1 τ ω

  • ) ◦ DFT.

System of p2 linear equations.

Theorem (Stone - von Neumann)

dim Sol(ΣW ) = 1.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 5 / 12

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Characterization of DFT - Cont.

Combine

  • π : Fp × Fp → U(H),

π(τ, ω) = ψ(− 1

2τω) · Mω ◦ Lτ

Intertwining relations ΣW : DFT ◦ π τ ω

  • = π(

W

  • −1

1 τ ω

  • ) ◦ DFT.

System of p2 linear equations.

Theorem (Stone - von Neumann)

dim Sol(ΣW ) = 1. = ⇒ DFT is characterized by ΣW .

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 5 / 12

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(II) The Weil Representation

Note W ∈ SL2(Fp) = a b c d

  • ; a, b, c, d ∈ Fp,

ad − bc = 1

  • .

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 6 / 12

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(II) The Weil Representation

Note W ∈ SL2(Fp) = a b c d

  • ; a, b, c, d ∈ Fp,

ad − bc = 1

  • .

Generalization: g ∈ SL2(Fp) Σg : ρ(g) ◦ π τ ω

  • = π(g ·

τ ω

  • ) ◦ ρ(g).

System of p2 linear equations.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 6 / 12

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(II) The Weil Representation

Note W ∈ SL2(Fp) = a b c d

  • ; a, b, c, d ∈ Fp,

ad − bc = 1

  • .

Generalization: g ∈ SL2(Fp) Σg : ρ(g) ◦ π τ ω

  • = π(g ·

τ ω

  • ) ◦ ρ(g).

System of p2 linear equations.

Theorem (Stone - von Neumann)

dim Sol(Σg ) = 1.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 6 / 12

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(II) The Weil Representation

Note W ∈ SL2(Fp) = a b c d

  • ; a, b, c, d ∈ Fp,

ad − bc = 1

  • .

Generalization: g ∈ SL2(Fp) Σg : ρ(g) ◦ π τ ω

  • = π(g ·

τ ω

  • ) ◦ ρ(g).

System of p2 linear equations.

Theorem (Stone - von Neumann)

dim Sol(Σg ) = 1. = ⇒ ρ(g) is characterized by Σg.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 6 / 12

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Weil Representation

Theorem

∃! collection of operators ρ(g) ∈ Sol(Σg ), g ∈ SL2(Fp), such that ρ(gh) = ρ(g) ◦ ρ(h).

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 7 / 12

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Weil Representation

Theorem

∃! collection of operators ρ(g) ∈ Sol(Σg ), g ∈ SL2(Fp), such that ρ(gh) = ρ(g) ◦ ρ(h). The homomorphism ρ : SL2(Fp) → U(H), H = C(Fp), is called the Weil Representation.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 7 / 12

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(III) Diagonalizing the DFT

We have ρ : SL2(Fp) → U(H) ⊃ C =? W → ρ(W ) = DFT;

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 8 / 12

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(III) Diagonalizing the DFT

We have ρ : SL2(Fp) → U(H) ⊃ C =? W → ρ(W ) = DFT; Consider symmetries of W : TW = {g ∈ SL2(Fp); gW = Wg} =

  • g ∈ SL2(Fp); ggt = I
  • =

SO2(Fp) - finite rotations.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 8 / 12

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(III) Diagonalizing the DFT

We have ρ : SL2(Fp) → U(H) ⊃ C =? W → ρ(W ) = DFT; Consider symmetries of W : TW = {g ∈ SL2(Fp); gW = Wg} =

  • g ∈ SL2(Fp); ggt = I
  • =

SO2(Fp) - finite rotations.

Lemma

TW is a maximal commutative subgroup (torus) of SL2(Fp).

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 8 / 12

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(III) Diagonalizing the DFT

We have ρ : SL2(Fp) → U(H) ⊃ C =? W → ρ(W ) = DFT; Consider symmetries of W : TW = {g ∈ SL2(Fp); gW = Wg} =

  • g ∈ SL2(Fp); ggt = I
  • =

SO2(Fp) - finite rotations.

Lemma

TW is a maximal commutative subgroup (torus) of SL2(Fp).

Proof.

PW (x) = det(xI − W ) = x2 + 1. Hence λ(W ) = ±√−1.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 8 / 12

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Diagonalizing the DFT

Symmetries of DFT C = Im(TW ) = {ρ(g) ; g ∈ TW } .

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 9 / 12

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Diagonalizing the DFT

Symmetries of DFT C = Im(TW ) = {ρ(g) ; g ∈ TW } .

C commutative group of unitary operators commuting with DFT.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 9 / 12

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Diagonalizing the DFT

Symmetries of DFT C = Im(TW ) = {ρ(g) ; g ∈ TW } .

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

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Diagonalizing the DFT

Symmetries of DFT C = Im(TW ) = {ρ(g) ; g ∈ TW } .

C commutative group of unitary operators commuting with DFT. Can diagonalize C simultaneously.

We have orthogonal decomposition H = ⊕

χ:TW →C∗ Hχ,

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 9 / 12

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Diagonalizing the DFT

Symmetries of DFT C = Im(TW ) = {ρ(g) ; g ∈ TW } .

C commutative group of unitary operators commuting with DFT. Can diagonalize C simultaneously.

We have orthogonal decomposition H = ⊕

χ:TW →C∗ Hχ,

ϕχ ∈ Hχ iff ρ(g)ϕχ = χ(g)ϕχ for every g ∈ TW .

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 9 / 12

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Diagonalizing the DFT

Symmetries of DFT C = Im(TW ) = {ρ(g) ; g ∈ TW } .

C commutative group of unitary operators commuting with DFT. Can diagonalize C simultaneously.

We have orthogonal decomposition H = ⊕

χ:TW →C∗ Hχ,

ϕχ ∈ Hχ iff ρ(g)ϕχ = χ(g)ϕχ for every g ∈ TW .

Theorem

dim Hχ = 1.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 9 / 12

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Diagonalizing the DFT

Symmetries of DFT C = Im(TW ) = {ρ(g) ; g ∈ TW } .

C commutative group of unitary operators commuting with DFT. Can diagonalize C simultaneously.

We have orthogonal decomposition H = ⊕

χ:TW →C∗ Hχ,

ϕχ ∈ Hχ iff ρ(g)ϕχ = χ(g)ϕχ for every g ∈ TW .

Theorem

dim Hχ = 1. Obtained the Canonical basis of eigenfunctions of DFT BTW = {ϕχ ∈ Hχ}.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 9 / 12

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(IV) The Oscillator Dictionary

Generalization

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 10 / 12

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(IV) The Oscillator Dictionary

Generalization Each Ti, i = 1, ..., p2, torus in SL2(Fp).

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 10 / 12

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(IV) The Oscillator Dictionary

Generalization Each Ti, i = 1, ..., p2, torus in SL2(Fp). Oscillator Dictionary D = ∐

T ⊂SL2(Fp)

BT .

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 10 / 12

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(IV) The Oscillator Dictionary

Generalization Each Ti, i = 1, ..., p2, torus in SL2(Fp). Oscillator Dictionary D = ∐

T ⊂SL2(Fp)

BT .

#D ≈ p3.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 10 / 12

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Oscillator Dictionary - Properties

Theorem (Pseudo-Randomness)

We have

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 11 / 12

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Oscillator Dictionary - Properties

Theorem (Pseudo-Randomness)

We have

1

Auto-correlations. For every ϕ ∈ D | ϕ, π(τ, ω)ϕ| =

  • 1 if (τ, ω) = (0, 0),

≤ 2/√p other.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 11 / 12

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Oscillator Dictionary - Properties

Theorem (Pseudo-Randomness)

We have

1

Auto-correlations. For every ϕ ∈ D | ϕ, π(τ, ω)ϕ| =

  • 1 if (τ, ω) = (0, 0),

≤ 2/√p other.

2

Cross-correlation. For every ϕ = φ ∈ D | ϕ, π(τ, ω)φ| ≤ 4/√p. Figure: ϕ, π(τ, ω)ϕ , p = 199.

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 11 / 12

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THANK YOU

Thank You!

Shamgar Gurevich (Madison) Diagonalizing DFT using Weil Repn August 3, 2014 12 / 12