Uncertainty Principles in Signal Processing

Aline Bonami,Universite d’Orleans

Buea, May 2, 2011

Uncertainty principle in Signal Processing

Assume that φ is a signal (a function depending of the time, forinstance). Doing its spectrum analysis means : take its Fouriertransform

φ(ω) :=

∫R

e−2iπωtφ(t) dt.

The signal is given by φ in the frequency domain.

One can reconstruct f from its “spectrum”(under appropriateassumptions) :

φ(t) =

∫ ∞−∞

φ(ω)e2iπωtdω.

The Uncertainty principle : a signal cannot be localized both intime and frequency.

Uncertainty principle in Quantum Mechanics

The position and the momentum (velocity) of a sub-atomicparticle cannot both be known with arbitrary precision. The moreprecisely one is known, the less precisely the other can be known.

To quote Heisenberg : Any use of the words “position” and“velocity” with an accuracy exceeding that given by theuncertainty equation is just as meaningless as the use of wordswhose sense is not defined.” (from Physical Principles of theQuantum Theory, 1930)

I it is impossible to determine simultaneously both the positionand velocity of a particle with any great degree of accuracy orcertainty.

I it is a statement about the nature of the system itself asdescribed by the equations of quantum mechanics.

The link : Uncertainty principle in Fourier Analysis

Let f integrable in Rd . Let its Fourier transform be

f (ξ) :=

∫Rd

f (x)e−ix ·ξ dx

well defined for f ∈ L1(Rd), then, using Plancherel Identity, forf ∈ L2(Rd), then for tempered distributions.

Uncertainty Principle. A function f and its Fourier transform fcannot be simultaneously well localized.

In quantum mechanics, the distribution of the position (resp. themomentum) of a particle is given by its wave function (resp. theFourier transform of its wave function).

Uncertainty principle : examples of statementsLet δa be the Dirac mass at a Rd , which is highly localized. Thenits Fourier transform

δa(ξ) :=

∫Rd

f (x)e−ix ·ξ dδa(x) = e−iaξ

has modulus 1 everywhere.Let f be compactly supported. Then its Fourier transform vanishesonly at isolated points.Proof : f extends into a holomorphic function on the complexplane.

Remark : one can exchange the role of f and f .

Hardy’s Theorem. If

|f (x)| ≤ Ce−a|x |2 |f (ξ)| ≤ Ce−b|ξ|2 ,

thenI if ab > 1/4, the function f is 0 ;I if ab = 1/4, the function f is, up to a constant, the Gaussian

function e−a|x |2 .

Heisenberg Uncertainty InequalityHeisenberg Inequality (in Dimension one) is

2∆x∆ξ ≥ 1.

Mathematically, this means the inequality

2

(∫R

x2|f (x)|2 dx

)1/2

×(

1

2π

∫Rξ2|f (ξ)|2 dξ

)1/2

≥∫

R|f (x)|2 dx ,

with equality for Gaussian functions, that is, functions e−a|x−x0|2 .

f (ξ) :=

∫ +∞

−∞e−ixξf (x)dx

and Plancherel Identity reads∫ +∞

−∞|f (x)|2dx =

1

2π

∫ +∞

−∞|f (x)|2dx .

The proof ! (for physicists)Want to prove that

2

(∫R

x2|f (x)|2 dx

)1/2

×(

1

2π

∫Rξ2|f (ξ)|2 dξ

)1/2

≥ 1,

for∫

R |f (x)|2 dx = 1.

Use of Plancherel Identity

1

2π

∫Rξ2|f (ξ)|2 dξ =

∫| ∂∂x

f (x)|2dx .

1 =

∫|f |2dx = −2

∫ (∂

∂xf

)(xf )dx .

Then use of Cauchy Schwarz Inequality.Define the operators Q : multiplication by x , and P := −i ∂∂x .Heisenberg commutation properties :

[P,Q] := PQ − QP = −iId.

The spectral gap of the Harmonic oscillator

2

(∫R

x2|f (x)|2 dx

)1/2

×(∫

Rξ2|f (ξ)2 dξ

)1/2

≥∫

R|f (x)|2 dx

leads to ∫R

x2|f (x)|2 dx +

∫Rξ2|f (ξ)|2 dξ ≥

∫R|f (x)|2 dx ,

or, which is equivalent,⟨x2f −

(d

dx

)2

f , f

⟩≥ ‖f ‖22.

1 is the smallest eigenvalue of the harmonic oscillator

H := x2 −(

ddx

)2and e−x2/2 is the corresponding eigenfunction.

The eigenvalues of the Harmonic oscillator

2n + 1 is the nth eigenvalue of the harmonic oscillator

H := x2 −(

ddx

)2and the Hermite function Hn(x)e−x2/2 is the

corresponding eigenfunction. Here Hn is the polynomial

Hn(x) := (−1)nex2

(d

dx

)n

(e−x2).

Moreover, by the CourantFischerWeyl min-max principle forcompact operators (here the inverse of H), then 2n + 1 is equal to

maxf1,···fn

min{〈Hf , f 〉 ; f ∈ (f1, · · · , fn)⊥, ‖f ‖2 = 1}.

Extrema are given by Hermite functions.

Slepian Extremal problemHow large on [−1,+1] are functions in L2(R) with spectrum in[−c ,+c] ? Let Bc be the space of such functions.

I Then

λ0(c) = max

{∫ +1

−1|f |2dx ; ‖f ‖2 = 1, f ∈ Bc

}.

I

λn(c) = minf1,···fn

max{∫ +1

−1|f |2dx ; f ∈ (f1, · · · , fn)⊥, ‖f ‖2 = 1, f ∈ Bc}.

Consider the operator on L2([−1,+1])

Fc(f )(x) :=( c

2π

)1/2∫ +1

−1e icxξf (ξ)dξ.

Then λn(c) are the eigenvalues of F∗cFc .Key point : F∗cFc(f ) = f ∗ χ[−c,+c] is compact.

The asymptotic behavior of λn(c).

(Landau and Widom 1980)Asymptotically for c tending to ∞, the sequence λn(c) stays closeto 1 for n ≤ 2

π c , then decreases exponentially rapidly.

For c large, and looking at the restrictions of Bc on the interval[−1,+1], roughly speaking one may do as if this space was offinite dimension N ≈ 2

π c .

May be extremely useful for compression of data in SignalProcessing.

The eigenfunctions.

Let ψn,c be the eigenfunction corresponding to λn,c .

The main tool : (Slepian) The eigenfunctions λn are alsoeigenfunctions of an explicit Sturm-Liouville operator on (−1,+1)

Lcφ := − d

dx

[(1− x2)φ′

]+ c2x2φ.

Arises in finding particular “radial” solutions of the Helmotzequation ∆Φ + k2Φ = 0 in three dimensions in the “ prolatespheroidal coordinates”. Explains the name : Prolate SpheroidalWave Functions PSWF.C. Niven : On the Conduction of Heat in Ellipsoids ofRevolution, Philosophical Trans. R. Soc. Lond. 1880 171,117-151. In this remarkable piece of work, Niven has developed adetailed computational and asymptotic methods for the PSWFsand the eigenvalues χn(c).Justifies to see PSWF appear in many scientific fields.