Optimality of uncertainty principles for joint time-frequency representations

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Degree project

Optimality of uncertainty principles for joint time- frequency representations

Author: Leonie Callies

Supervisor: Patrik Wahlberg

Examiner: Joachim Toft

Date: 2014-06-10

Course Code: 4MA11E

Subject: Mathematics

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joint time-frequen y representations

Leonie Callies

June 17, 2014

Abstra t

Thestudyofjointtime-frequen y representationsis alargeeld ofmathemati s

andphysi s,espe iallysignalanalysis. BasedonHeisenberg's lassi alun ertainty

prin iplevariousinequalitiesforsu htime-frequen ydistributions havebeenstud-

ied. The obje tive of this thesis is to examine the role that Gaussian fun tions,

in luding those with a hirp ontribution, play in inequalities for the Short-Time

Fourier transformand the Wignerdistribution. We show that Grö henig'sun er-

taintyprin iplesfortheShort-TimeFouriertransformarenotoptimalwithregardto

thesefun tions. AsfortheWignerdistributionweshowhowanexistingun ertainty

prin iplebyJanssen anbemodiedtorea hoptimalityforChirpGaussians.

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1 Introdu tion 3

2 Time-Frequen y representations 4

2.1 TheFourier Transform . . . 4

2.2 TheShort-Time Fourier Transform (STFT) . . . 6

2.3 Time-Frequen y representations ofCohen's lass . . . 7

3 Un ertainty prin iplesfor the pair

(f, ˆ f )

¹⁰ 3.1 Heisenberg'sun ertainty prin iple . . . 10

3.2 Un ertaintyprin iple byCohen . . . 10

3.3 Un ertaintyprin iple byDonohoand Stark . . . 11

4 Un ertainty prin iplesfor the Short-Time Fourier Transform 12 4.1 Grö henig's inequalities . . . 12

4.1.1 Originalversion . . . 12

4.1.2 Adaptation to thealternative denitionof theSTFT . . . 13

4.2 Non-optimalityof thestrong un ertainty prin iple . . . 15

4.2.1 Gaussian fun tions . . . 15

4.2.2 ChirpGaussians . . . 19

5 Un ertainty prin iplesfor the Wignerdistribution 22 5.1 Janssen'sun ertaintyprin iple . . . 22

5.2 Equality inJanssen's prin iple. . . 23

5.3 Astronger lowerboundfor the ase

M = I

^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ²⁷

5.4 Astronger lowerboundfor a general matrix

M

^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ²⁸

6 Con lusion 31

Referen es 32

Appendix 33

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In the early 19 th

entury Jean-Baptiste Joseph Fourier laid the foundations for what is

now alled Fourier Analysis. He dis overed that periodi fun tions an be expressedas

seriesofsineand osinefun tionsofdis retefrequen ies,theso- alledFourierseries. The

equivalentexpressionfornon-periodi fun tionsistheFouriertransformationwhi hhasa

ontinuousratherthandis retefrequen yspe trum. Theappli ationsofFourieranalysis

are ountless, both inphysi s andmathemati s. Inthis work we willbe on erned with

theeldof signalanalysis. Hen e,fun tionswilloftenbe alledsignalsand the ommon

variablesaretime andfrequen y.

When a fun tion des ribes the behavior ofa signal in time, thenthe Fourier transform

des ribesits behaviorinthe frequen y domain. Both fun tionsdes ribe thesignal om-

pletely be ause ea h of the representations an be omputed from the other, but when

dealing with only one of them it la ks an important part of the information. It would

be desirable to simultaneouslydes ribe thesignal'sbehavior inthetime and inthefre-

quen y domain, even determine the frequen y at a spe i point in time, a so- alled

instantaneous frequen y. However, a ording to the un ertainty prin iple dis overed by

WernerKarlHeisenberginthemiddleofthe20 th

enturyitisimpossibletoexpresssu h

an instantaneous frequen y. His inequality was originally formulated for quantities in

quantum me hani s. Yet, it an be arriedoverto signalanalysis.

The issue of joint time-frequen y representation is a question in ongoing resear h and

various ways have been found. Among them are the Short-Time Fourier Transform

(STFT),distributionsofCohen's lassliketheWignerdistribution,Gaboranalysisorthe

Wavelettransformtonameonlyafew. InequalitiesanalogoustoHeisenberg'sun ertainty

prin iple arevalidfor su h joint representations.

Chapter two and three serve as an introdu tion to time-frequen y representations and

un ertainty prin iples. Subsequently, we fo us on inequalities for theSTFT derived by

Karlheinz Grö henig and show that they are not optimal for Gaussian fun tions. The

last hapterisdevotedtoanun ertaintyprin iple fortheWignerdistributionthatisdue

to A.J.E.M. Janssenand theroleof Gaussian fun tionsinhisinequality.

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This hapter serves as an introdu tion to the basi s of Fourier and Time-Frequen y

Analysis. In general analysis is done on

L ^p

^, ^the ^Bana
h ^spa
e ^of ^measurable ^fun
tions

on themeasurespa e

S

^where ^the ^norm^is ^hosen ^to ^be

kf k p := Z

S |f (x)| ^p dx ¹ _p

with

1 ≤ p < ∞

^. ^Mostôften^we^willôn
ernôurselves^with

L ²

^whi
h^turns^into^a^Hilbert

spa eafter introdu ing the innerprodu tof

f, g ∈ L ²

^as

hf, gi =

Z

S

f (x)g(x) dx.

Two fundamental inequalities ne essary for the study of un ertainty prin iples are due

to Hölder and Cau hy. They aretaken for granted and used without proof throughout

this work.

Theorem 2.1 (Hölder's inequality). Let

S

^be ^a ^measure ^spa
e ^and

1 ≤ p, q ≤ ∞

^with

1 p + ¹ _q = 1

^. ^Let^further

f ∈ L ^p (S)

^and

g ∈ L ^q (S)

^. ^Then

f g ∈ L ¹ (S)

^and

kf gk 1 ≤ kf k p kgk q .

⁽¹⁾

Cau hy'sinequality isa spe ial aseof Hölder'sstatement:

Theorem 2.2 (Cau hy-S hwarz inequality).

|hf, gi| ≤ kf k ² kgk ² .

⁽²⁾

Thisisobtained bysetting

p = q = 2

ⁱⁿ^Hölder's inequality:

|hf, gi| = | Z

S f (x)g(x)dx| ≤ Z

S |f (x)g(x)| dx = kf gk 1 ≤ kf k 2 kgk 2 .

Themeasurespa e

S

ⁱⁿ^this^dis
ussion^is^usually^the^spa
e

R

,equippedwiththeLebesgue measure, andthe fun tions

f

^and

g

^are^real- ^or omplex-valued. Mostresults an easily be transferred to higher dimensionsbut this is not done hereto not ompli ate matters

unne essarily. The quantity

|f (x)| ²

îs ôften âlled ^the ênergy ^density ôf ^the ^fun
tion

f

^. Âsêxplained ^by ^Cohenⁱⁿ ^{[Coh95 ℄} ^the ^name^has îts ôrigin ⁱⁿ ^physi
s. Îf

f

^des
ribes

the ele tri eld, then the quantity

|f (x)| ²

orresponds to the ele tri energy density. Analogies an be foundinother areasof physi s, aswell.

2.1 The Fourier Transform

All time-frequen y representations onsidered inthe following are basedon theFourier

transform. A little in onvenien e of time-frequen y analysisis theexisten e of multiple

versions of the denition of the Fourier transform. Consequently this arries over to

the denitions of the time-frequen y representations and makes omparisons of works

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dieronlyinamultipli ativefa torandafa torintheexponential. CohenandDanguse

equation(3)fromdenition2.3intheirwork([Coh95 ℄and[PGT13℄),whereasGrö henig,

JanssenandCarypisuseequation(4)intheirwork([Grö01 ℄,[FS03 ℄,[Jan89 ℄and[Car14 ℄).

Denition 2.3. TheFourier transform of afun tion

f ∈ L ¹ (R)

^is^dened ^as

f (ω) = ˆ 1

√ 2π Z

R

f (x)e ^−ixω dx,

⁽³⁾

alternatively:

f (ω) = ˆ Z

R

f (x)e ^−2πixω dx.

⁽⁴⁾

Asnotation for theFourier transform of

f

^both

f ˆ

^and

F[f ]

^will^be ^used.

The rst denitionis theone to be usedin this report. The results found by thelatter

three authors need therefore be adaptedto thedierent denitionof the Fourier trans-

form.

Gaussian fun tions play an important role in this report. Hen e, it is worth to note

Hörmander's theorem on the Fourier transform of a omplex-valued Gaussian fun tion

(see [Hör85 ℄)

Theorem 2.4 (Fourier transform of a omplex-valued Gaussian). Let

f : R → C, x 7→

e ^−dx ² ^/2

^with

d ∈ C

^and

Re(d) ≥ 0

^. ^Then^the^Fourier^transform^of

f

^is^another^Gaussian

fun tion

f : R ∋ ω 7→ d ˆ ⁻ ¹ ² e ⁻ ^ω2 ^2d .

⁽⁵⁾

If

Re(d) = 0

^then

f : R ∋ ω 7→ |d| ˆ ⁻ ¹ ² e ^πi ⁴ ⁺ⁱ ^2Im(d) ^ω2 .

⁽⁶⁾

Remark 2.5. For

Im(d) = 0

^and

Re(d) > 0

^denition ^{(5 )}^redu
es ^to

f (ω) = ˆ √ ¹

2π

R

R f (x)e ^−iωx dx

^asⁱⁿ^equation ^{(3) .}

Afewpropertiesareessentialfor thestudy ofFourier analysis. TheFourier transformis

invertible andtheinversionisgiven by

Theorem 2.6 (InversionFormula). Let

f ∈ L ¹ (R)

^and

f ∈ L ˆ ¹ (R)

^, ^then

f (x) = 1

√ 2π Z

R

f (x)e ˆ ^ixω dω.

Remark2.7. Ingeneralthe Fourier transformis amapfrom

L ¹

^to ^the^spa
e ^of^ontin-

uousfun tionsthatvanishat innity. Inthat aseadditional requirementsfor

f

^have^to

be setsothatthe inversionformulaholds. However,thisisnot the aseforthefun tions

thatareof relevan efor thisreport.

Ifin addition

f ∈ L ² (R)

^, ^then^the ^Fourier ^transform ^has^the^property^of ^preserving^the

energy ofa signal. Thisstatement is dueto Plan herel.

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Theorem 2.8 (Plan herel). Let

f ∈ L ¹ ∩ L ² (R)

^. ^Then

kf k 2 = k ˆ f k 2

^.

An important operator for thefollowing studyis the onvolutionof fun tions.

Denition 2.9 (Convolution). Let

f, g ∈ L ¹ (R)

^. ^Then

(f ∗ g)(x) =

Z

R f (t)g(x − t) dt

is alledthe onvolution of

f

^and

g

^.

Theorem 2.10 (Convolutiontheorem). Let

f, g ∈ L ¹ (R)

^. ^The ^F^ourier ^transform^of ^the

onvolved fun tions is

F[f ∗ g] = √

2πF[f ]F[g].

Thestatement isproven bysimple al ulation usingFubini's theorem.

2.2 The Short-Time Fourier Transform (STFT)

In signal analysis the fun tion

f

^represents ^for înstan
e ân â
ousti ^signal. Ît ôntains

informationaboutthetemporalbehavior,inthe aseofthea ousti signaltheamplitude

overtime, whereas the Fourier transform ontains information about the frequen y be-

havior, thatis, whi hfrequen ieswere present. Unfortunately,neitherfun tionprovides

any information whatsoever about the hange of the spe trum over time. However, in

many ases this is very relevant. An analogy that Grö henig introdu es motivates the

sear hforsimultaneoustime-frequen y representations. Composershavefound awayto

express timeand frequen y behavior at the same time sothat musi ians areable to re-

onstru tthe omposer'sideasfromasheetofmusi . Foramusi iantheverti alposition

of the note in the staves provides information about the frequen y and the horizontal

arrangement ofthenotesinformationaboutthedurationofea hnoteandthetimewhen

thefrequen y isto beplayed.

Arstandratherintuitiveattempttoa hievethesameintime-frequen yrepresentation

is the Short-Time Fourier transform (STFT). The idea behind it is to emphasize the

fun tion of interest

f

^on ^smaller ^subsets ^of ^the ^domain ^before determining the Fourier transform. TheSTFT anberegardedasalo alversionofthe lassi alFouriertransform.

Toemphasize thefun tion,

f

^is^multiplied^by^a^so-
alled ^window ^fun
tion

g

^. ^This^might

be a fun tion

g(x) ≡ 1

ôn â ômpa
t ^set ^that ^goes ^to ^zero ^smoothly ôutside ^this ^set.

Other frequent window fun tions are Gaussian fun tions. The hoi e of the window is

ru ialand highlydependsonthe respe tive obje tive. It isimportant thatthewindow

fun tion emphasizes

f

^but ât ^the^same ^time ^doesn't âlter ît signi antly on thesubset.

An illustrationofthis method an beseen ingure1.

Denition 2.11. The Short-Time Fourier transform of

f ∈ L ² (R)

^with ^respe
t ^to

g ∈ L ² (R)

^is ^dened ^as

V g f (x, ω) = 1

√ 2π Z

R f (t)g(t − x)e ^−itω dt,

⁽⁷⁾

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−1.5 0 −1 −0.5 0 0.5 1 1.5 2 2.5 3 0.2

0.4 0.6 0.8 1 1.2 1.4

t

g(t−x) f(t)

g(t)

f(t)g(t−x)

Figure1:Visualizationofanexemplary fun tion

f

^,^a^window^fun
tion

g

^and^their ^prod-

u twhi h isneededto determine theSTFT

or alternatively Grö henig'sversion:

V _g f (x, ω) = Z

R f (t)g(t − x)e ^−2πitω dt

⁽⁸⁾

with

x, ω ∈ R

^.

Remark 2.12. The STFT is not limited to fun tions in

L ² (R)

^. ^Depending ^on ^the

respe tive ontext other spa es su h as

L ¹ (R)

^,^the ^S
hwartz ^spa
e

S(R)

^or ^the^spa
e ^of

distributions

S ^′ (R)

ân ^beûsed, âs^well.

To simplifyexpressions like (7)the two operators

T _x

^and

M _ω

^are^introdu
ed ^su
h ^that

T _x f (t) = f (t − x),

M _ω f (t) = e ^iω·t f (t).

The operator

T _x

îs âlledâ ^time-shift ând

M _ω

^a frequen y-shift. Equation (7 ) an then be written as

V _g f (x, ω) = 1

√ 2π hf, M ω T _x gi,

provided thatthe hosenfun tion spa eis aHilbertspa e withan innerprodu t.

2.3 Time-Frequen y representations of Cohen's lass

Starting from the 1940s onsiderable progress was made in the study of joint time-

frequen y distributions. In the1960sLeon Cohendis overed thatmany ofthedistribu-

tions that hadbeen found ould be derived byone and thesame method and unitedin

a lass that is now alled Cohen's lass. Cohen presents the method, the lass and an

overviewofthe most important distributions in[Coh95 ℄.

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C _f (x, ω) = 1 4π ²

Z Z Z

f (u + 1

2 τ )f (u − 1

2 τ )Φ(θ, τ )e −iθx−iτ ω+iθu du dτ dθ.

Thefun tion

Φ(θ, τ )

îs âlled^the^kernel ôf^thedistribution.

Distributions of Cohen's lass are often alled quadrati or bilinear be ause the signal

is bilinearly involved. The properties of the distribution are determined by the kernel.

Dependingontheappli ationdierentpropertiesmightbedesirable. Alldistributionsof

Cohen's lasshave ertainshift-propertiesin ommon,thatis, translationsand modula-

tions of the signalimply orresponding shiftsof thedistribution [MH97℄. Janssenoers

a omprehensivedis ussionof furtherpropertiesin[MH97℄. A ompressed versionofhis

listisgiven inthe following. For details see, for instan e,Janssen'sor Cohen's work.

It isdesirable thatthefollowing onditions holdfor all

f

^and^for ^all

x, ω ∈ R

^:

•

^Corre
t ^marginal^onditions

Z

R

C _f (x, ω) dω = |f (x)| ² Z

R

C _f (x, ω) dx = | ˆ f (ω)| ²

•

^Corre
t ^total ^energy

Z Z

R ²

C _f (x, ω) dx dω = kf k ²

•

^W^eak ^support^property

f (x) = 0, |x| > a ⇔ C _f (x, ω) = 0, |x| > a, ω ∈ R, a > 0 f (ω) = 0, ˆ |ω| > b ⇔ C _f (x, ω) = 0, |ω| > b, x ∈ R, b > 0

•

^Strong^support^property

f (x) = 0 ⇔ C _f (x, ω) = 0, ω ∈ R f (ω) = 0 ˆ ⇔ C _f (x, ω) = 0, x ∈ R

•

Real-valuedness

C _f (x, ω) ∈ R, x, ω ∈ R

•

^Moyal's ^ondition

Z Z

R 2

C _f (x, ω)C _g (x, ω) dx dω = |hf, gi| ²

•

Non-negativity ondition

C _f (x, ω) ≥ 0, x, ω ∈ R

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dilation, onvolution and the Fourier transform might be required for a time-frequen y

distribution. The best-known time-frequen y distribution of this lass is the Wigner

distribution. It was introdu ed by Eugene Wigner in the 1930s and has been of great

importan e eversin e.

Denition 2.14. The Wignerdistribution ofa fun tion

f ∈ L ² (R)

^is ^dened ^as

W _f (x, ω) = 1

2π Z

R

f (x + y

2 )f (x − y

2 )e ^−iωy dy.

⁽⁹⁾

This an be obtained from the general time-frequen y distribution of theorem 2.13 by

settingthe kernel

Φ(θ, τ ) = 1

^.

An alternative,widely useddenitionof theWignerdistribution is

W _f (x, ω) = Z

R

f (x + y

2 )f (x − y

2 )e ^−2πiωy dy.

⁽¹⁰⁾

ThisversionisbasedontheFouriertransformasinequation(4 )andusedbyGrö henig,

Janssenand Carypis.

There is no perfe t time-frequen y representation be ause no distribution of Cohen's

lass an fulll all the requirements mentioned above asexplained indetail in [MH97℄.

Infa t,foreverydistributionofCohen's lassthatsatisesthemarginal onditionsthere

is at least one fun tion

f

^su
h ^that ^the distribution takes negative values. However, among Cohen's distributions the Wigner distribution is losest to being non-negative.

Moreover, it an be shown that the Wigner distribution satises ten out of the twelve

onditions in Janssen's list. The fa t that it an take on negative values ould be seen

asitsmost signi ant aw be ause thisleads tointerpretation di ultiesinphysi s. As

explainedbyJanssenin[Jan89℄this an be over ome byweightingthedistributionwith

ertain square-integrable fun tions. The se ond property that the Wigner distribution

doesn't satisfy is the strong support ondition. A ommon inherent attribute of all

bilineardistributions areso- alled ross-terms. Theseareterms thatareex lusively due

to the bilinear involvement of the fun tion but la ka physi al meaning and hen e lead

to interpretationdi ulties.

The STFT isnot a dire tmember of Cohen's lassbut thesquare of its absolutevalue

is. Thisdistributionis alledtheSpe trogram.

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3 Un ertainty prin iples for the pair

(f, ˆ f )

3.1 Heisenberg's un ertainty prin iple

In1927 thephysi istWernerHeisenbergformulated hisfamousun ertaintyprin iple for

quantumme hani s thatstatesthat ertainpairsof quantitiessu haspositionand mo-

mentum ofa parti le or energy andtime annot both bedetermined arbitrarilypre ise.

There are numerous un ertainty prin iples in the literature, not only in quantum me-

hani sbut alsoin otherelds ofphysi sand mathemati s,manyof whi h arebasedon

Heisenberg'sun ertainty prin iple. The versionof interest of Heisenberg's prin iple for

this study isaninequalityfor afun tion and itsFourier transform.

Theorem 3.1 (Heisenberg's un ertainty prin iple). Let

f ∈ L ² (R)

^and

a, b ∈ R

^arbi-

trary. Then

Z

R (x − a) ² |f (x)| ² dx 1/2 Z

R (ω − b) ² | ˆ f (ω)| ² dω 1/2

≥ 1

2 kf k ² 2

⁽¹¹⁾

Equalityholdsifandonlyif

f (x) = De ^ib(x−a) · e ⁻ ^(x−a)2 ^2c

^for^some

D ∈ C

^and

c ∈ R, c > 0

^.

The fun tions

|f | ²

^and

| ˆ f | ²

^an ^be ^regarded ^asprobability densityfun tions (aslongas

f ∈ L ² (R)

^and

kf k ² = 1

^). ^The întegrals ⁱⁿ^{(11 )}ân ^then^be^seen âs^the^vârian
e ôf

|f | ²

and

| ˆ f | ²

^. ^The ^prin
iple ^states ^that ^if ^either

f

^or

f ˆ

^is ^strongly on entrated around a point,inotherwords,its varian e issmall,theotherone annotbe on entrated aswell,

inotherwordsits varian ehasto be large.

Thisinterpretation suggeststhe widelyusedand insignalanalysispopularnotation

σ _x σ _ω ≥ 1

2 ,

⁽¹²⁾

provided

kf k 2 = 1

^. ^The ^quantities

σ _x ²

^and

σ ² _ω

âre ^the^varian
e ôf ^the ^fun
tion ând ^the

Fouriertransform,respe tively. They analsobeseenasthedurationandthebandwidth

ofthe signal.

3.2 Un ertainty prin iple by Cohen

StartingfromHeisenberg's lassi alun ertaintyprin ipleCohenderivesastrongerbound

in[Coh95 ℄. Inhis onsiderations hexes thearbitraryparameters

a

^and

b

^as

a = hxi =

Z

R x|f (x)| ² dx b = hωi =

Z

R ω| ˆ f (ω)| ² dω.

Thevarian es intime andfrequen y (orduration and bandwidth)thenbe ome

σ _x ² = Z

R (x − hxi) ² |f (x)| ² dx σ _ω ² =

Z

R (ω − hωi) ² | ˆ f (ω)| ² dω.

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Inhisworkhe onsiderssignals

f (x) = ρ(x)e ^iϕ(x)

^where

ρ(x)

îs^theâmplitudeôf^the^signal

overtime and

ϕ(x)

^a real-valuedfun tion des ribing thefrequen y. The ovarian e of a signalis dened as

Cov = Z

R

xϕ ^′ (x)|f (x)| ² dx − hxihωi.

Using thesedenitions Cohen statesand provesthefollowing un ertainty prin iple:

Theorem 3.2 (Cohen's un ertainty prin iple). Let

f ∈ L ² (R)

^,

kf k 2 = 1

^. ^Then

σ _x σ _ω ≥ 1

2 p 1 + 4Cov ² .

⁽¹³⁾

Equalityisattainedonlyforfun tionsoftheform

f (x) = Ce ^{−a(x−hxi)} ² e −i(b(x−hxi) ² +hωix)

,

where

a, b ∈ R, a > 0

^and

C ∈ C

^su
h ^that

kf k 2 = 1

^.

3.3 Un ertainty prin iple by Donoho and Stark

Another instrument to express theun ertainty dis overed by Heisenberg is thesupport

of afun tion. A qualitative versionofthe un ertainty prin iple is given bythetheorem

ofBenedi ks([FS03 ℄):

Theorem 3.3. Assume

f ∈ L ¹ (R)

^(or

f ∈ L ^p (R)

^). ^If

|suppf | < ∞

^and

|supp ˆ f | < ∞

^,

then

f ≡ 0

^.

Sin eafun tionanditsFourierTransform annotbothhave ompa t support,asstated

intheorem3.3, itis onvenient to dene theessential support of afun tion asin[FS03℄

whi histhesubsetofthedomainthat oversmostofthefun tion. Thishelpstoformulate

quantitative un ertainty prin iples.

Denition 3.4 (essential support). A fun tion

f ∈ L ² (R)

^is

ε

- on entrated on a mea- surableset

T ⊂ R

^,^if

Z

T ^c |f (x)| ² dx 1/2

≤ εkf k ²

with

0 ≤ ε ≤ 1

^.

T

îsâlled^the êssential ^support ôf

f

^.

DonohoandStark introdu ed anun ertaintyprin iple basedontheideaoftheessential

support[Grö01℄.

Theorem 3.5. Suppose that

f ∈ L ² (R)

^,

f 6= 0

^is

ε _T

- on entrated on

T ⊂ R

^and

f ˆ

^is

ε _Ω

- on entrated on

Ω ⊂ R

^. ^Then

|T ||Ω| ≥ (1 − ε T − ε Ω ) ² .

Thisrelation istherefore alledan essential support ondition.

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Transform

Asall time-frequen y representations are basedon theFourier transform itis only nat-

ural that un ertainty prin iples for the pair

(f, ˆ f )

^arry ^over ^to ^joint time-frequen y distributions. This hapterisdevotedto inequalitiesfor theSTFT thatwerederived by

Karlheinz Grö henig.

4.1 Grö henig's inequalities

4.1.1 Original version

Theweak un ertainty prin ipleasproposedbyGrö henigin[Grö01 ℄isanother example

oftheuseoftheessentialsupportintrodu edindenition3.4 . Inthefollowingtheresults

byGrö heningwillbepresentedusinghisalternativedenitionsoftheFouriertransform

and theSTFT(equation (4)and (8 )).

Theorem 4.1 (Weak un ertainty prin iple). Suppose that

kf k 2 = kgk 2 = 1

^and ^that

U ⊂ R ²

^and

0 ≤ ε ≤ 1

^are ^su
h ^that

Z Z

U |V g f (x, ω)| ² dx dω ≥ 1 − ε.

Then

|U| ≥ 1 − ε

^where

|U|

^is^the ^measure ^of

U

^.

A stronger estimate for the essential support of

|V g f | ²

^an ^be ^derived ^by ^using ^Lieb's

inequalities as they are stated and proven in [Grö01 ℄. The proof is in luded in the

appendix.

Theorem 4.2 (Lieb). Let

f, g ∈ L ² (R)

^, ^then

kV g f k ^p p =

Z Z

R ² |V g f (x, ω)| ^p dx dω

( ≤ ² _p

(kf k 2 kgk 2 ) ^p

^if

2 ≤ p < ∞,

≥ ² _p

(kf k 2 kgk 2 ) ^p

^if

1 ≤ p ≤ 2.

The rstinequalityallowed Grö henigto establishthefollowing stronger versionof the

weak un ertainty prin iple 4.1 :

Theorem 4.3(Strongun ertaintyprin iple). Suppose that

kf k 2 = kgk 2 = 1

^. ^If

U ⊂ R ²

and

0 ≤ ε ≤ 1

^are ^su
h^that

Z Z

U |V g f (x, ω)| ² dx dω ≥ 1 − ε,

then

|U| ≥ (1 − ε) ^p−2 ^p p 2

_p−2 ²

for all p > 2.

⁽¹⁴⁾

In parti ular,

|U| ≥ sup

p>2 (1 − ε) ^p−2 ^p p 2

_p−2 ²

≥ 2(1 − ε) ² .

⁽¹⁵⁾

(14)

Let

f ˆ ^C

^be^the ^Fôurier ^transformâ

ording^to êquation⁽³⁾ând

f ˆ ^G

^the^F^ourier ^transform

asdenedinequation(4)where

C

abbreviatesCohenand

G

^Grö
henig. ^Let^the^respe
-

tiveShort-timeFourier transformsbe

V _g ^C f

^and

V _g ^G f

^. ^The^relation^between ^the^dierent

denitions aregiven by:

f ˆ ^C (ω) = 1

√ 2π f ˆ ^G ( ω

2π )

⁽¹⁶⁾

V _g ^C f (x, ω) = 1

√ 2π V _g ^G f (x, ω

2π ).

⁽¹⁷⁾

Inthefollowingtheindex

”C”

^will^beômitted^butîtîsûnderstood^that^Cohen's^denition

isused. Theweak un ertainty an berestated as

Theorem4.4(Theweakun ertaintyprin iple-adapted). Supposethat

kf k 2 = kgk 2 = 1

andthat

U ⊂ R ²

^and

0 ≤ ε ≤ 1

^are ^su
h ^that

Z Z

U |V g f (x, ω)| ² dx dω ≥ 1 − ε.

Then

|U| ≥ 2π(1 − ε)

^where

|U|

^is ^the ^measure ^of

U

^.

Proof. The proof is essentially the same as given by Grö henig in [Grö01 ℄. Using the

denitionoftheSTFT(2.11 ),Cau hy'sinequality2.2andtheisometryoftime-frequen y

shiftsthefollowing holds:

|V g f (x, ω)| = | 1

√ 2π Z

R f (t)g(t − x)e ^−itω dt|

= 1

√ 2π |hf, M ω T _x gi|

≤ 1

√ 2π kf k 2 kM ω T _x gk 2

≤ 1

√ 2π kf k 2 kgk 2 = 1

√ 2π .

Thelastequalityresults fromtheassumption thatbothfun tionsarenormalized. Thus,

itfollows immediatelythat

1 − ε ≤ Z Z

U |V ^g f (x, ω)| ² dx dω ≤ Z Z

U

1 2π dx dω = 1 2π |U|

whi h anbe rewrittenas

|U| ≥ 2π(1 − ε)

^.

Lieb'sinequalities intheir adaptedversionaregiven by

Theorem 4.5 (Lieb - adapted). Let

f, g ∈ L ² (R)

^, ^then

kV g f k ^p p =

Z Z

R ² |V g f (x, ω)| ^p dx dω







≤ _2π ¹ ^p−2 _p ₂

p

(kf k 2 kgk 2 ) ^p

^if

2 ≤ p < ∞,

≥ _2π ¹ ^p−2 _p ₂

p

(kf k ² kgk ² ) ^p

^if

1 ≤ p ≤ 2.

(18)

(15)

kV g ^G f (x, ω)k ^p p = Z Z

R ² |V g ^G f (x, ω)| ^p dx dω

= Z Z

R ²

1 √ 2π V _g ^G f (x, ω 2π )

^p (2π) ^p ² 1

2π dx dω

= (2π) ^p−2 ² Z Z

R 2 |V g ^C f (x, ω)| ^p dx dω

= (2π) ^p−2 ² kV g ^C f (x, ω)k ^p p

and theestimation

V _g ^G f (x, ω) ^p

p ≤ 2 p

kf k 2 kgk 2

p

for theSTFTinGrö henig'sdenition, the

p−

^norm ôf^the^STFT ân^be êstimatedâs

V _g ^C f (x, ω)

^p

p = 1 2π

^p−2 _p

V _g ^G f (x, ω) ^p

p ≤ 1 2π

^p−2 _p 2 p

kf k 2 kgk 2

p

.

Theseinequalities allow tostate the adapted strongun ertainty prin iple:

Theorem4.6(Thestrongun ertaintyprin iple-adapted). Supposethat

kf k ² = kgk ² = 1

^. ^If

U ⊂ R ²

^and

0 ≤ ε ≤ 1

^are ^su
h ^that

Z Z

U |V ^g f (x, ω)| ² dx dω ≥ 1 − ε,

then

|U| ≥ 2π(1 − ε) ^p−2 ^p p 2

_p−2 ²

for all p > 2.

⁽¹⁹⁾

In parti ular,

|U| ≥ sup

p>2 2π(1 − ε) ^p−2 ^p p 2

_p−2 ²

≥ 4π(1 − ε) ² .

⁽²⁰⁾

Proof. This proof is the adaptation of the proof given by Grö henig in [Grö01 ℄. From

theassumtionit followsthat:

1 − ε ≤ Z Z

U |V g f (x, ω)| ² dx dω

= Z Z

R ²

χ _U (x, ω)|V g f (x, ω)| ² dx dω

= kχ U (x, ω)|V g f (x, ω)| ² k 1 .

⁽²¹⁾

Now, Hölder's inequality 2.1 an be applied to (21 ). The hoi e of parameters is

a = ^p ₂

and

b = _p−2 ^p

^. ^These ^parameters ^fulll^therequirements:

1 a + 1

b = 2

p + p − 2

p = 1.

(16)

Moreover, ithasto beshown that

|V ^g f (x, ω)| ² ∈ L ² ^p (R ² )

^and

χ U (x, ω) ∈ L ^p−2 ^p (R ² )

^:

Z Z

R ² |V g f (x, ω)| ^2· ^p ² dx dω 2/p

= kV g f (x, ω)k ² p .

Hen e it follows from (18 ) that

|V g f (x, ω)| ² ∈ L ^p ² (R ² )

^. ^The ^fun
tion

χ _U (x, ω)

^belongs

to

L ^p−2 ^p

^as^long^as

U

^has^nite^measure. ^F^or ^any

U

^with ^innite^measure^the^inequality

(19 )is trivial.

In the ase of

|U| < ∞

âpplying ^Hölder's înequality ^to ⁽²¹⁾ ând ^then ^Lieb's înequality

(4.5 ) to (22)gives:

1 − ε ≤ kχ U (x, ω)|V g f (x, ω)| ² k 1

≤ kχ U (x, ω)k _p/(p−2) k|V g f (x, ω)| ² k p/2

= Z Z

R ² |χ U (x, ω)| ^p−2 ^p ^p−2 _p Z Z

R ² |V g f (x, ω)| ^2· ^p ² dx dω ² _p

= |U| ^p−2 ^p Z Z

R 2 |V ^g f (x, ω)| ^p dx dω ² _p

(22)

≤ |U| ^p−2 ^p 1 2π

^p−2 ₂ 2 p

kf k 2 kgk 2 p ² _p

= 1

2π |U| ^p−2 _p 2 p

² _p

.

⁽²³⁾

For

p > 2

^the ^inequality⁽²³⁾ ^an ^be reformulated as:

|U| ≥ 2π(1 − ε) ^p−2 ^p p 2

_p−2 ²

and inparti ular for

p = 4

|U| ≥ 4π(1 − ε) ²

whi h proves the theorem.

4.2 Non-optimality of the strong un ertainty prin iple

The role of Gaussian fun tions is oftena spe ial one. They areoptimal withregard to

Heisenberg'sun ertaintyprin iple inthesensethattheyaretheonlyfun tionsforwhi h

equalityin(11) holds. The question ofinterest is now whetheror not this optimality is

preserved for Grö henig's un ertainty prin iples. This se tion is devoted to theanswer

ofthat question.

4.2.1 Gaussian fun tions

Consider theGaussian fun tion

f : R → R

^,

x 7→ ² π

1/4

c ^−1/2 e ^{−((x−b)/c)} ²

^and ^the^window

g : R → R

^,

x 7→ _π ² 1/4

d ^−1/2 e ^{−((x−b)/d)} ²

^with^onstants

b, c, d ∈ R

^where

c

^and

d

^an ^be

assumedpositive. The fun tionsarenormalized sothat

kf k 2 = kgk 2 = 1

^. ^First,^the^set

U ⊂ R ²

^where

RR

U |V g f (x, ω)| ² dx dω ≥ 1 − ε

^holds^for^some

ε ≥ 0

^has^to^bedetermined:

(17)

I :=

Z Z

U |V ^g f (x, ω)| ² dx dω

= 1 π ² (cd) ⁻¹

Z Z

U

Z

R

e ^{− (t−b)/c}

2 e − (t−b−x)/d 2

e ^−itω dt

2 dx dω.

The hange of variables

t − b = cds

^,

dt = cd ds

^yields:

I = 1 π ² cd

Z Z

U

Z

R

e ^−(ds) ² e ^{−(cs−x/d)} ² e ^{−iω(cds+b)} ds

2 dx dω

= 1 π ² cd

Z Z

U

Z

R

e ^{− c} ² ^+d ²

s ² +(cds)(iω−2x/d ² )/(c ² +d ² )

− x/d ² ds

2 dx dω.

Completing the squareintheexponent

I = 1 π ² cd

Z Z

U

e ^c

2 d ² /(4(c ² +d ² ))

iω−2x/d ² 2

− x/d 2

2 × . . .

. . . × Z

R

e ^{− c} ² ^+d ²

s+(cd)(iω−2x/d ² )/(2(c ² +d ² )) 2

ds

2 dx dω

and the hange of variables

s = s + (cd)(iω − 2x/d ˜ ² )/(2(c ² + d ² ))

^gives:

I = 1 π ² cd

Z Z

U

e ^c

2 d ² /(4(c ² +d ² ))

iω−2x/d ² 2

− x/d 2

2 Z

R

e ^{− (c} ² ^+d ² ⁾ ^1/2 ^˜ ^s

2 d˜ s

2 dx dω.

Using

Re[ c ² d ²

2(c ² + d ² ) iω − 2x d ²

2 − 2 x d

2 ]

= − c ² d ² ω ²

2(c ² + d ² ) + 2c ² x ²

d ² (c ² + d ² ) − 2c ² x ²

d ² (c ² + d ² ) − 2x ² c ² + d ²

= − c ² d ² ω ²

2(c ² + d ² ) − 2x ² c ² + d ²

theintegral simplies to

I = 1 π ² cd

Z Z

U

e ^{− (cdω)/} √

2(c ² +d ² ) 2

− ( √ 2x)/( √

c ² +d ² ) 2 √

√ π c ² + d ²

2 dx dω.

A hange of variables shows that hoosing

U ^′

âs â îr
le ⁽

U

^then ^be
omes ^an ^ellipse)

allowsanalyti integration of theGaussian fun tion. Let

u = cdω

p 2(c ² + d ² ) , v =

√ 2x

√ c ² + d ²

withthe orrespondingdeterminant of theJa obian

∂(x, ω)

∂(u, v) =

c ² + d ²

cd .

(18)

I = 1 π

Z Z

U ^′

e ^−(u ² ^+v ² ⁾ du dv = 1 π

Z R 0

Z 2π 0

e ^−r ² r dϕ dr = 1 − e ^−R ²

⁽²⁴⁾

where

U ^′

îs^the^new^domainôfⁱⁿ^tegration. Âs

U ^′

îsâîr
leîts^measureân^beômputed

as

|U ^′ | = πR ²

^.

The measures of the original domain of integration

U

^and

U ^′

^are ^related ^through ^the

determinant ofthe Ja obian:

|U| = c ² + d ²

cd |U ^′ | = c ² + d ² cd πR ² .

Choosing

ε = e ^−R ²

^theônditionsôf^theorem^4.6âre^fullledând^the^STFTîsessentially bounded on

U

^. ^The^strong un ertaintyprin iple (inequality(19)) thus states:

c ² + d ²

cd πR ² ≥ 2π(1 − e ^−R ² ) ^p−2 ^p p 2

²

p−2 .

⁽²⁵⁾

To provethatGaussian fun tionsarenotoptimal inthestrong un ertaintyprin iple for

theSTFT it hasto be shownthat inequality (25 )is a stri t one. First,it an be noted

that

min

c,d

c ² + d ²

2cd = 1

^so^thatîtîs ^su
ient ^to ônsider ^theînequality

R ² ≥ (1 − e ^−R ² ) ^p−2 ^p p

2 ²

p−2 .

⁽²⁶⁾

Denethefun tions

l(R) = R ² and r(R) = (1 − e ^−R ² ) ^p−2 ^p p 2

²

p−2 .

Then

l ^′ (R) = 2R and r ^′ (R) = p 2

²

p−2 p

p − 2 (1 − e ^−R ² ) ^p−2 ² (2Re ^−R ² ).

Sin e

l(0) = r(0) = 0

^showing ^that

l ^′ (R) ≥ r ^′ (R)

^for

R > 0

^is ^su
ient ^to ^prove ^that

(26 )is astri t inequality. Toshow that

l ^′ (R) ≥ r ^′ (R) ⇔ 1 ≥ p 2

²

p−2 p

p − 2 1 − e ^−R ² ²

p−2 e ^−R ²

⁽²⁷⁾

holds for allp onsider thederivative oftheright-handside. Lettheright-handside be

r 2 (R)

^. ^Then

r ₂ ^′ (R) = p 2

_p−2 ² p

p − 2 2Re ^−R ² 1 − e ^−R ² ^4−p _p−2 h p

p − 2 e ^−R ² − 1 i .

Hen e,

r 2 (R)

^is ^maximal ^for

_p

p−2 e ^−R ² − 1

= 0

^, equivalently

e ^−R ² ^M = ^p−2 _p

^where

R M

denotesthe

x

^-value^where^the^maximum^is^taken^on. ^Sin
e

r ₂ (R _M ) = 1

^andfurthermore

r ₂ (0) = lim _R→∞ r ₂ (R) = 0

^, înequality ^{(27 )}îs ^satised ^for âll

R

ând êquality îsâttained

only for

e ^−R ² = ^p−2 _p

^. ^From ^this îtân ^be ôn
luded ^that înequality ⁽²⁵⁾ îs â ^stri
t ône

(19)

for all

R > 0

^. ^The ^bound ^given ^by^the^strong un ertainty prin iple istherefore not the best possible for Gaussian fun tions.

Moreover, it an be observed that

lim _R→∞ r(R) = ^p ₂ _p−2 ²

whereas

lim _R→∞ l(R) = ∞

^.

Thismeansthatthelowerboundisagoodestimationwhen onsideringsmallper entages

of the total energy of the STFT (

ε

^lose ^to

1

^). ^However, ît ^looses îts ^meaning âs ân

estimation for the area that overs high per entages of the total energy (

ε

^small). ^F^or

small

ε

^the ^STFT ^is ^mu
h ^less on entrated than the bound inthe strong un ertainty prin iple suggests. Figure2 illustratesthis fa tfor the aseof

p = 4

^.

0 0.5 1 1.5 2 2.5

0 1 2 3 4 5 6 7

R

l(R)=R ²

r(R)=2(1−e ^−R

2 ) ²

Figure2: Qualityof thelowerbound for

p = 4

To investigatethe inuen e ofthedilationfa tors

c

^and

d

ⁱⁿ^the^original^inequality⁽²⁵⁾

itis usefulto rewritethe inequality

c ² + d ²

2cd ≥ 1 ⇔ (c − d) ² ≥ 0.

This expression allows the observation that merely the dieren e of the dilation fa -

tors matters. This orrelates with the expe tations. The STFT of a rather spread out

Gaussian fun tion with respe t to a narrow window (thus,

c

^large ^and

d

^small) ^an ^be

interpreted as the Fourier transform of a fun tion with small essential support. As a

onsequen ethe essentialsupportofthe STFThasto be large. Butfor largedomainsof

integration the lowerboundis not agoodestimation.

The reversed ase (

d

^large ^and

c

^small) orrespondsto the STFT of a narrow fun tion with respe t to a broad window. Su h a window fun tion does not alter the fun tion

signi antly. Thus, like in the rst ase, the STFT an be interpreted as the Fourier

transform of a fun tion withsmall essential support. However, this is not a reasonable

hoi eofthe pair

(f, g)

^as

g

^does^not^emphasize^the^fun
tion

f

^on^subsets^of^the^domain.

(20)

Chirp fun tions are fun tions where the frequen y in reases or de reases with time. A

spe ial ase onsidered in this se tion are Gaussian fun tions with an additional imag-

inary term in the exponent. This term introdu es os illations whose amplitude is en-

velopedbytheGaussianfun tionwithpurelyrealexponent(seegure3). Thefrequen y

ofthe os illationsin reases for

|x| → ∞

^.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

x

Gaussian

Chirp

Figure 3:AGaussian fun tion ompared withthereal partof a hirpfun tion

The obje tive of this se tion is to study the role of hirp fun tions inthe strong un er-

taintyprin ipleintrodu edinse tion3. Theinuen eofthefrequen y

ϕ(x)

^on^the^value

ofthe estimationisto be investigated.

Letthe fun tion be

f : R → C, x 7→ ( π ² ) ¹ ⁴ e ^−x ² ^−iαx ²

^and ^the ^window^be

g : R → R, x 7→

( _π ² ) ¹ ⁴ e ^−x ²

^. Îtîs^su
ient ^toônsider^Gaussian ^fun
tionsêntered ât

x = 0

^be
ause^shifts

alongthe

x

^-axis^don't^haveânyînuen
eôn^the ^result,âs^seenⁱⁿ^subse
tion^4.2.1. ^Sin
e

the inequality (25) was sharpest for the dilation fa tors

c = d = 1

^, ^these ^onstants ^are

usedinthe following omputations, aswell.

(21)

I :=

Z Z

U |V g f (x, ω)| ² dx dω

= 1

π ² Z Z

U

Z

R

e ^−t ² ^−iαt ² e ^−(t−x) ² e ^−itω dt

² dx dω

= 1

π ² Z Z

U

e ^−2x ² Z

R

e ^−t ² (2+iα)−t(iω−2x) dt

² dx dω

= 1

π ² Z Z

U

e ^−2x ²

e ^(iω−2x) ² ^/(2(2+iα))

Z

R

e ^{− 2+iα}

t+(iω−2x)/(2(2+iα)) 2

dt

² dx dω.

Using the hange ofvariable

˜ t = t + (iω − 2x)/(2(2 + iα))

^and ^the^fa
t^that

Re( 1

2(2 + iα) (iω − 2x) ² ) = 1

4 + α ² (−ω ² − 2αxω + 4x ² )

theintegral transformsinto

I = 1 π ²

Z Z

U

e ^−1/(4+α ² ^)(2x ² ^(2+α ² ^)+ω ² ^+2αxω) Z

R

e ^{−(2+iα)˜t} ² d˜ t

² dx dω.

Dening

A := 2(2 + α ² )

4 + α ² , B := 1

4 + α ² , C := 2α 4 + α ²

theintegral an bewritten as

I = 1 π ²

Z Z

U

e ^−(Ax ² ^+Bω ² ^+Cxω) π

(4 + α ² ) ^1/2 dx dω.

Changingvariables to

u = √

Ax + C 2 √

A ω, v = B − C ² 4A

¹ ₂ ω

withthe orrespondingdeterminant of theJa obian

∂(x, ω)

∂(u, v) = (AB − C ²

4 ) ⁻ ¹ ² = (4 + α ² ) ¹ ²

allowsto solve theintegral

I = 1

π(4 + α ² ) ^1/2 Z Z

U ^′

e ^−(u ² ^+v ² ⁾ (4 + α ² ) ¹ ² du dv = 1 π

Z R 0

Z 2π 0

e ^−r ² r dϕ dr = 1 − e ^−R ² .

Themeasure ofthe domain of integration is

|U| = ∂(x, ω)

∂(u, v) |U ^′ | = (4 + α ² ) ¹ ² πR ² .

(22)

(4 + α ² ) ¹ ²

2 R ² ≥ (1 − e ^−R ² ) ^p−2 ^p p 2

²

p−2

(28)

holds.

For

α = 0

^this ^redu
es ^to ^the ^ase ^studied ⁱⁿ ^se
tion ^{4.2.1 .} ^Sin
e

^(4+α ₂ ² ⁾ ^1/2 ≥ 1

^for

all

α

^, înequality ^{(28 )} îs âlways â ^stri
t înequality^. ^Moreover, ît ân ^be ^noted ^that ^the

estimation for thesize ofintegration be omesweaker for largervalues of

α

^. ^The ^STFT

offun tionswithstrongos illationsisthusmu hless on entrated thanthelowerbound

suggests. It is an analogous observation to the one made in the se tion 4.2.1 for the

STFT offun tionswithlarge dilation fa tors.

(23)

The importan e of Cohen's lass of joint time-frequen y distributions naturally implies

thestudy of un ertainty prin iplesfor su hrepresentations. The entral element of this

hapteris aninequalityfor the Wignerdistribution thatisdue to A.J.E.M. Janssen.

5.1 Janssen's un ertainty prin iple

Theresultsfound byJanssenand Carypiswill rstbepresentedintheir originalversion

using the alternative denition of the Wigner distribution and the Fourier transform.

Let

W _f ^J (x, ω)

^denote ^the ^denition ^a

ording ^to ^{(10 )} ^and

W _f (x, ω)

^the ^version ⁱⁿ ^(9).

The alternative denition of the Fourier transform is denoted by

f ˆ ^J (ω)

^. ^Janssen^states

in[Jan89 ℄

Theorem 5.1(Janssen). Let

M

^be ^a ^positive ^denite

2 × 2

^matrix ^and^denote^its ^deter-

minant by

|M|

^. ^Then

Z Z

R ²

z ^T M zW _f ^J (z) dz ≥ |M| ^1/2

2π kf k ² 2

⁽²⁹⁾

where

z = (x, ω) ^T

^.

Carypis shows in [Car14 ℄ that for

M = I

êquality îs âttained îfând ônly îf

f

^is ^of ^the

form

f (x) = Ce ^−πx ²

^,

C ∈ C

^. ^This ^is ^a ^simple onsequen e of the original version of Heisenberg'sun ertaintyprin iple for

f

^and

f ˆ

^as^shownⁱⁿ^the^following.

Using theelementary inequality

a ² + b ² ≥ 2ab

^with

a = Z

R

x ² |f (x)| ² dx 1/2

and b = Z

R

ω ² | ˆ f ^J (ω)| ² dω 1/2

Heisenberg'sun ertaintyprin iple

Z

R

x ² |f (x)| ² dx 1/2 Z

R

ω ² | ˆ f ^J (ω)| ² dω 1/2

≥ 1 4π kf k ² 2

an be rewritten as

Z

R

x ² |f (x)| ² dx + Z

R

ω ² | ˆ f ^J (ω)| ² dω ≥ 1 2π kf k ² 2 .

Equalityisonlyattainedif

a = b

^. ^This^only^holds^for^those^Gaussians^fun
tions^that^are

invariant underFourier transformation,namelythe fun tions

f (x) = Ce ^−πx ²

^,

C ∈ C

^.

Inserting the marginal onditions of the Wigner distribution gives Janssen's result for

the ase

M = I

^:

Z Z

R ²

(x ² + ω ² )W _f ^J (x, ω) dx dω ≥ 1 2π kf k ² 2 .

Equality isattainedinthe same ases asbefore.

To transform the statement to our denition of theWigner distribution we observe the

relation

(24)

W _f ^J ( x

√ 2π , ω

√ 2π ) = 1

√ 2π Z

R

f 1

√ 2π (x + y 2 )

f 1

√ 2π (x − y 2 )

e ^−iωy dy

= √

2πW _f ₁

√ 2π

(x, ω)

⁽³⁰⁾

with

f √ 1 2π

(x) = f ( √ ¹

2π x)

^. ^A ^hange ^of^variables ⁱⁿ⁽²⁹⁾^gives

Z Z

R ²

√ x 2π , ω

√ 2π

M x

√ 2π , ω

√ 2π

T

W _f ^J ( x

√ 2π , ω

√ 2π ) dx dω ≥ |M| ¹ ² kf k ² 2 .

Using relation(30) this an berewritten as

Z Z

R ²

(x, ω)M (x, ω) ^T W _f ₁

√ 2π

(x, ω) dx dω ≥ √

2π|M| ¹ ² kf k ² 2 .

With

kf √ ¹ 2π k ² 2 = Z

R |f ( x

√ 2π )| ² dx = Z

R |f (x)| ² √

2π dx = √ 2πkf k ² 2

and thesubstitution

g(x) = f √ 1 2π

(x)

^Janssen's un ertaintyprin iple inthetransformed versionreads

Z Z

R ²

(x, ω)M (x, ω) ^T W _g (x, ω) dx dω ≥ |M| ¹ ² kgk ² 2

⁽³¹⁾

withequalityfor

M = I

îfândônly îf

f (x) = Ce ^−πx ²

^or equivalently

g(x) = f ( √ ¹ 2π x) = Ce ⁻ ^x2 ²

^.

5.2 Equality in Janssen's prin iple

In the following the inequality will be onsidered for a general, positive denite

2 × 2

^-

matrix. The question to be answered is whi h fun tions an attain thelower bound in

themore general setting. Itwill beshown that hirpfun tions an yieldequalityunlike

intheun ertaintyprin iples fortheSTFT.

Janssenstatesin[Jan89℄thatWignerdistributions oflinearlytransformedfun tionsare

againWigner distributions:

Theorem5.2 (Janssen). Forany lineartransformation

z ∈ R ² 7→ Az ∈ R ²

^with

|A| = 1

of the phase plane there is a unitary operator

T

^of

L ² (R)

^su
h ^that

W _f (Az) = W _{T f} (z)

^,

f ∈ L ² (R)

^.

The proof requiresWeyl al ulus of pseudodierential operators and group theory and

isbeyond the s ope ofthis report.

DeBruijn derives in[DB73℄the relationbetween thetransformation inthe phaseplane

and theunitaryoperator

T

^.

(25)

Theorem 5.3 (De Bruijn). Consider the transformation

(x, ω) 7→ (x ^′ , ω ^′ ) = (αx + βω, γx + δω)

^with

αδ − βγ = 1

^. ^Then

W _{T f} ^J (x ^′ , ω ^′ ) = W _f ^J (x, ω)

⁽³²⁾

where

T f (x) = α ⁻ ¹ ² f x α

exp iπγx ² α

(33)

for

β = 0

^and

T f (x) = (iβ) ⁻ ¹ ² Z

R exp − π

iβ (δx ² − 2xt + αt ² )

f (t) dt

⁽³⁴⁾

for

β 6= 0

^.

Thisrelation holdsinthe aseofJanssen'sdenitionoftheWignerdistribution. Forthe

fun tion

g

ⁱⁿ^{(31 )}^we ^have ^to ^use^the^relation

g(x) = f ( √ ¹

2π x)

^and^equation^{(30 ).} ^Let

D

be the s aling operator

Df (x) = f ( √ ^x

2π ) = g(x)

^and

D ⁻¹

îts înverse. Âpplying ⁽³⁰⁾^to

both sides ofequation(32) gives

√ 2πW _{DT f} ( √

2πx ^′ , √

2πω ^′ ) = √

2πW _Df ( √ 2πx, √

sπω)

or equivalently

W _{DT f} (x ^′ , ω ^′ ) = W _Df (x, ω).

With

f = D ⁻¹ g

^and

Df = g

^weânôn
lude^that^the^relation^between^theûnitaryôper-

ator

T

ⁱⁿ^De^Bruijn's ^theorem^with^the alternative denitionof theWigner distribution and the operator

Optimality of uncertainty principles for joint time-frequency representations

Degree project