Solving the GPS Problem in Almost Linear Complexityshamgar/SETA12-June5-2012.pdfOded Schwartz (EE &...

Solving the GPS Problem in Almost Linear Complexity

Shamgar Gurevich

Madison

SETA2012

Shamgar Gurevich (Madison) GPS in Linear Complexity SETA2012 1 / 21

CONGRATULATIONS

MAZEL TOV SOL

Joint work with:

Alexander Fish (Math, Sydney)

Ronny Hadani (Math, Austin)

Akbar Sayeed (EE & CE, Madison)

Oded Schwartz (EE & CS, Berkeley)

(0) MOTIVATION - GPS

CLIENT WANT: Coordinates of satellite and time delay (enables tocalculate distance to a satellite)?

Motivation - GPS

S , R ∈ H = CN —Hilbert space of digital sequences, N � 1000.

S , R : ZN = {0, ....,N − 1} → C.

Satellite transmits b · S , b ∈ {1,−1} coordinates.

FactClient receives

R [n] = b · α0 · e2πiN ω0 ·n · S [n− τ0] +W [n], n ∈ ZN ,

α0 ∈ C attenuation, ω0 ∈ ZN Doppler, τ0 ∈ ZN delay, W ∈ H randomwhite noise.

Problem (The GPS Problem)

Design S ∈ H, and effective method to extract (b, τ0), using R and S .

Motivation - GPS

S , R : ZN = {0, ....,N − 1} → C.

FactClient receives

Motivation - GPS

S , R : ZN = {0, ....,N − 1} → C.

FactClient receives

Motivation - GPS

S , R : ZN = {0, ....,N − 1} → C.

FactClient receives

Motivation - GPS

S , R : ZN = {0, ....,N − 1} → C.

FactClient receives

Motivation - TIME-FREQUENCY SHIFT

Simpler scenario

R [n] = e2πiN ω0 ·n · S [n− τ0] + W [n].

Problem (Time-Frequency Shift)

Design S ∈ H, and method of extracting (τ0,ω0), using R and S .

Motivation - TIME-FREQUENCY SHIFT

Simpler scenario

R [n] = e2πiN ω0 ·n · S [n− τ0] + W [n].

Problem (Time-Frequency Shift)

Design S ∈ H, and method of extracting (τ0,ω0), using R and S .

(I) SOLUTION - Matched Filter

DefinitionMatched filter M(R, S) :

Time-Frequency︷︸︸︷ZN ×ZN → C,

M(R,S)[τ,ω] =⟨R [n] , e

2πiN ω·n · S [n− τ]

Question: What S to use for extracting (τ0,ω0) fromM(R,S) ?

(I) SOLUTION - Matched Filter

DefinitionMatched filter M(R, S) :

Time-Frequency︷︸︸︷ZN ×ZN → C,

M(R,S)[τ,ω] =⟨R [n] , e

2πiN ω·n · S [n− τ]

Question: What S to use for extracting (τ0,ω0) fromM(R,S) ?

Solution - MATCHED FILTER

Typical solution: S = pseudo-random.

Figure: |M(R, S)|, (τ0,ω0) = (50, 50).

Using FFT computeM(R,S) in O(N2 · log(N)) operations.

Question (Goresky—Klapper, SETA08): Can you design S andmethod to make almost linear number of operations?

Figure: |M(R, S)|, (τ0,ω0) = (50, 50).

(II) FLAG ALGORITHM - Idea

For every line L ⊂ ZN ×ZN we will construct a sequence SL ∈ H withM(R,SL) of the form

Figure: |M(R, SL)|, L = {(0,ω)}, (τ0,ω0) = (50, 50).

Then we have algorithm of complexity O(N · log(N)) !!Shamgar Gurevich (Madison) GPS in Linear Complexity SETA2012 9 / 21

(III) SEQUENCE DESIGN - Heisenberg (Line) Sequences

Heisenberg operators on H = CN

π(τ,ω)f [n] = e2πiN ω·n · f [n+ τ], τ,ω ∈ ZN .

Property:

π(τ,ω) ◦ π(τ′,ω′) = e2πiN

det︷︸︸︷(τω′ −ωτ′) · π(τ′,ω′) ◦ π(τ,ω).

Mechanism to generate ’good sequences’:

Take any line L ⊂ ZN ×ZN ;

Restrict π to L get commutative collection of Heisenberg operators

π(`) : H → H, ` ∈ L;

Property:

det︷︸︸︷(τω′ −ωτ′) · π(τ′,ω′) ◦ π(τ,ω).

π(`) : H → H, ` ∈ L;

Property:

det︷︸︸︷(τω′ −ωτ′) · π(τ′,ω′) ◦ π(τ,ω).

π(`) : H → H, ` ∈ L;

Property:

det︷︸︸︷(τω′ −ωτ′) · π(τ′,ω′) ◦ π(τ,ω).

π(`) : H → H, ` ∈ L;

Property:

det︷︸︸︷(τω′ −ωτ′) · π(τ′,ω′) ◦ π(τ,ω).

π(`) : H → H, ` ∈ L;

Sequence Design - HEISENBERG (LINE) SEQUENCES

Theorem (Linear Algebra —Simultaneous Diagonalization)We have a basis for H of common eigen-sequences

BL = {fLψ ; π(`)[ fLψ ] =

e.v.︷︸︸︷ψ(`) · fLψ , for every ` ∈ L}.

Theorem (Support, [Calderbank—Howard—Moran, Howe])For fL ∈ BL

|M(fL, fL)[τ,ω]| ={1, (τ,ω) ∈ L;0, (τ,ω) /∈ L.

Sequence Design - HEISENBERG (LINE) SEQUENCES

Theorem (Linear Algebra —Simultaneous Diagonalization)We have a basis for H of common eigen-sequences

BL = {fLψ ; π(`)[ fLψ ] =

e.v.︷︸︸︷ψ(`) · fLψ , for every ` ∈ L}.

Theorem (Support, [Calderbank—Howard—Moran, Howe])For fL ∈ BL

|M(fL, fL)[τ,ω]| ={1, (τ,ω) ∈ L;0, (τ,ω) /∈ L.

Heisenberg (Line) Sequences —NUMERICS

Figure: |M(fL, fL)| , L = {(τ, τ)}.

What is next?

Just add any pseudo-random sequence.

Heisenberg (Line) Sequences —NUMERICS

Figure: |M(fL, fL)| , L = {(τ, τ)}.

What is next?

Just add any pseudo-random sequence.

Sequence Design - WEIL (SPIKE) SEQUENCES

Group SL2(ZN ) = {g =(a bc d

); a, b, c, d ∈ ZN , ad − bc = 1}.

Theorem (Fourier—Weil representation)

There exists a collection of unitary operators ρ(g) : H → H,g ∈ SL2(ZN ), with

1 ρ(g · h) = ρ(g) ◦ ρ(h);

(0 −11 0

)=Fourier Transform.

); a, b, c, d ∈ ZN , ad − bc = 1}.

1 ρ(g · h) = ρ(g) ◦ ρ(h);

(0 −11 0

); a, b, c, d ∈ ZN , ad − bc = 1}.

1 ρ(g · h) = ρ(g) ◦ ρ(h);

(0 −11 0

); a, b, c, d ∈ ZN , ad − bc = 1}.

1 ρ(g · h) = ρ(g) ◦ ρ(h);

(0 −11 0

Take maximal commutative subgroup T ⊂ SL2(ZN ), i.e., gh = hg forevery g , h ∈ T .Get a commutative collection of Weil operators

ρ(g) : H → H, g ∈ T .

Get a basis for H of common eigen-sequences

BT = {ϕTχ; ρ(g)[ ϕTχ

e.v.︷︸︸︷χ(g) · ϕTχ

, for every g ∈ T}.

Take maximal commutative subgroup T ⊂ SL2(ZN ), i.e., gh = hg forevery g , h ∈ T .

Get a commutative collection of Weil operators

ρ(g) : H → H, g ∈ T .

e.v.︷︸︸︷χ(g) · ϕTχ

ρ(g) : H → H, g ∈ T .

e.v.︷︸︸︷χ(g) · ϕTχ

ρ(g) : H → H, g ∈ T .

e.v.︷︸︸︷χ(g) · ϕTχ

Theorem (Pseudo-Randomness [G—Hadani—Sochen, 2008])For ϕT ∈ BT we have

|M(ϕT , ϕT )[τ,ω]| ={1, (τ,ω) = (0, 0);≤ 2√

N, (τ,ω) 6= (0, 0).

Figure: M(ϕT , ϕT ), T = {(a 00 a−1

); 0 6= a ∈ ZN }.

HEISENBERG+WEIL (FLAG) SEQUENCES

Theorem ([Fish—G—Hadani—Sayeed—Schwartz, 2012])Take SL = fL︸︷︷︸

+ ϕT︸︷︷︸∈BT

. Then

1 Flag. We have

|M(SL,SL)[τ,ω]| ≈

2, if (τ,ω) = (0, 0);1, if (τ,ω) ∈ Lr (0, 0);≤ 7√

N, otherwise.

2 Almost orthogonality. For L 6= M we have|M(SL,SM )[τ,ω]| ≤ 7√

N, for every (τ,ω) ∈ ZN× ZN .

HEISENBERG+WEIL (FLAG) SEQUENCES

Theorem ([Fish—G—Hadani—Sayeed—Schwartz, 2012])Take SL = fL︸︷︷︸

+ ϕT︸︷︷︸∈BT

. Then

1 Flag. We have

|M(SL,SL)[τ,ω]| ≈

2, if (τ,ω) = (0, 0);1, if (τ,ω) ∈ Lr (0, 0);≤ 7√

N, otherwise.

2 Almost orthogonality. For L 6= M we have|M(SL,SM )[τ,ω]| ≤ 7√

N, for every (τ,ω) ∈ ZN× ZN .

Heisenberg+Weil (Flag) Sequences - NUMERICS

Figure: |M(SL, SL)| for Heisenberg—Weil flag with L = {(τ, τ)}.

(IV) APPLICATION - GPS

Satellite transmits waveform SL + b · SM .

Client receives

R [n] = α0 · e2πiN ω0 ·n︸︷︷︸Doppler

·SL[n− τ0︸︷︷︸delay

]+b · α0 · e2πiN ω0 ·n ·SM [n− τ0]+W [n].

Goal: Extract (b, τ0), using R, SL, and SM .

Satellite transmits waveform SL + b · SM .Client receives

]+b · α0 · e2πiN ω0 ·n ·SM [n− τ0]+W [n].

Satellite transmits waveform SL + b · SM .Client receives

]+b · α0 · e2πiN ω0 ·n ·SM [n− τ0]+W [n].

Application - GPS

We have

R [n] = α0 · e2πiN ω0 ·n · SL[n− τ0] + b · α0 · e

2πiN ω0 ·n · SM [n− τ0] +W [n].

Compute

M(R, SL)[τ,ω] ≈

2 · α0, if (τ,ω) = (τ0,ω0);1 · α0, if (τ,ω) ∈ L+ (τ0,ω0)r (τ0,ω0);≤ O( 1√

N), otherwise,

=⇒ get (α0, τ0,ω0).

ComputeM(R, SM )[τ,ω],=⇒ get b · α0;=⇒ get b.

Flag method solves The GPS Problem in O(N log(N)) !!

Application - GPS

We have

2πiN ω0 ·n · SM [n− τ0] +W [n].

Compute

M(R, SL)[τ,ω] ≈

N), otherwise,

=⇒ get (α0, τ0,ω0).

Application - GPS

We have

2πiN ω0 ·n · SM [n− τ0] +W [n].

Compute

M(R, SL)[τ,ω] ≈

N), otherwise,

=⇒ get (α0, τ0,ω0).

Application - GPS

We have

2πiN ω0 ·n · SM [n− τ0] +W [n].

Compute

M(R, SL)[τ,ω] ≈

N), otherwise,

=⇒ get (α0, τ0,ω0).

THANK YOU

Sasha Ronny

Akbar Oded

CONGRATULATIONS

MAZEL TOV SOL

Solving the GPS Problem in Almost Linear Complexityshamgar/SETA12-June5-2012.pdfOded Schwartz (EE &...

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