Gab or and chirplet transforms

in the study of pseudo dierential

yz

op erators

Ryuichi ASHINO Michihiro NAGASE

Rmi VAILLANCOURT

CRM

January

In memoriam NobuhisaIwasaki

y

To app ear in RIMS Symp osium on the Structure of Solutons for Partial Dierential Equations

z

This work was partially supp orted through aGrantinAid of Japan NSERCofCanada and the Centre de recherches

mathmatiques of the Universit de Montral

Abstract

Gab or transforms and Wilson bases are reviewed in the study of the b oundedness of pseudo dif

ferential op erators in mo dulation spaces and of the decay of the singular values of compact

pseudo dierential op erators Similarly the is reviewed in the study of the

L b oundedness of pseudo dierential op erators The parameter Gab or transform and the

parameter wavelet transforms are generalized to parameter chirplet transforms in the context

of signal pro cessing Op en questions concerning chirplets are addressed for p ossible applications

to pseudo dierential op erators and Fourier integral op erators

Keywords and phrases Pseudo dierential op erators Gab or transform wavelet transform chirplet

transform

Rsum

On fait la synthse de lutilisation des transformations de Gab or et des bases de Wilson dans

ltude de la continuit des op rateurs pseudodirentiels sur les espaces de mo dulation et de la

mme on dcroissance des valeurs singulires des op rateurs pseudodirentiels compacts De

fait la synthse de lutilisation des transformations en ondelettes dans ltude de la continuit

L des op rateurs pseudodirentiels Les transformations deux paramtres de Gab or et en

ondelettes se gnralisent aux transformation en p pielettes huit paramtres dans le contexte du

traitement du signal On p ose des questions sur lutilisation p ossible des p pielettes dans ltude

des op rateurs pseudodirentiels et Fourier intgraux

Intro duction

n n

Given a symbol S R R the Weyl corresp ondence denes a pseudo dierential

op erator

n n

L D X S R S R

by the formula

ZZ

ixy

L f x e x y f y dy d

The Weyl representation is common in harmonic analysis and its relation with quantum theory The repre

sentation of pseudo dierential op erators as a generalization of partial dierential op erators is

ZZ

ixy

x D f x e x f y dy d

In fact the name pseudodierential operator comes from its use in the study of parametrices of elliptic

op erators A further generalization to Fourier integral op erators

ZZ

ixy

Af x e ax y f y dy d

was needed to obtain an asymptotic representation of the solutions of hyp erb olic partial dierential equations

see also

To establish the continuity of the op erator L on certain spaces of functions with symbol in certain

classes of functions one often attempts to express the op erator as a sum of simpler almost orthogonal

op erators or use other metho ds see for example In the use of the

es the b est p ossible result whichismuch more precise than the results obtained bythe PaleyLittlewo o d giv

use Cotlars lemma under the hyp otheses of the following theorem

Theorem Let n and Nn be two integers Suppose that the symbol x and al l its derivatives

n n

x for jjN and j jN arecontinuous on R R and verify

x

j jjj

j x jC j j jjN j jN

x

n

are two constants Then the operator x D is boundedonL R where and C

In the context of op erator theoryinYves Meyer has used Littlewo o dPaley which provide

p n

unconditional bases for the spaces L R toprovethe T theorem of David and Journ and thus to derive

the L continuity of nonclassic pseudo dierential op erators in a very simple way and study their symb olic

p

calculus For the L theory of pseudo dierential op erators see for example

Wavelets have also found applications in new trends in microlo cal analysis In several

microlo cal spaces are considered A windo wed of hyp erb olic chirps is used as a digression

to intro duce a more involved hyp erb olic paving of the timefrequency plane pro duced by the unfolding of an

orthonormal wavelet basis called an adapted basis

In the context of signal pro cessing in and Gab or transforms are used to establish the continuity

of pseudo dierential op erators on mo dulation spaces and to study the decayofthesingularvalues of compact

pseudo dierential op erators

n

In and KazuyaTachizawa has used Wilsons bases which are unconditional bases for L R

p n

but not for L R with p to establish the b oundedness of pseudo dierential op erators on mo dulation

spaces and to study the sp ectrum of these op erators

The Gab or transform or shorttime STFT pro duces a shift in time and in frequency

in the timefrequency plane of a given analyzing compactlysupp orted window g On the other hand the

wavelet transform WT pro duces a shift and a scaling in time in the time domain of a given analyzing

mother wavelet

In and the STFT and WT are generalized to the by considering an

parameter transformation of the timefrequency plane By restricting this generalization to planar subspaces

one obtains useful transformations such as bowtie disp ersion warblet and p ersp ective transforms It is

an op en question whether results for Fourier integral op erators would follow simply bytheuseofchirplet

expansions

The mo dulation space approachis not new in the sense that it is a typ e of coherent state expansion

and coherent states have b een around a long time The new asp ect is that Feichtinger and Gr chenig

haveshown that coherent states are concrete bases or basislike systems for the mo dulation spaces

and as a result one has convenient equivalent norms for these spaces in terms of the basis co ecients and

this fact can be applied to derive results on pseudo dierential op erators In one sense this is similar to

howwavelets have simplied some asp ects of the LittlewoodPaley theory Wavelets nowprovide concrete

unconditional bases for a large class of function spaces In this sense Besov spaces are the wavelet analogue

of the mo dulation spaces And Yves Meyer has used wavelets to go o d eect in the study of integral op er

ators Wavelets are simpler and easier to use than say functiondep endent atomic decomp ositions The

Gr chenigHeil approach is complementary it uses uniform decomp osition of phase space instead of dyadic

decomp ositions but it is the basislike asp ect that makes things simple

Preliminaries

Weintro duce general notation

Inner pro duct

Z

n

fg f x g x dx f g L R

Fourier and inverse Fourier transforms

Z

ix

f F f e f x dx

Z

ix

f xF f xf x e f d

Translation T and mo dulation M

ix n

T f xf x y M f xe f x x R

y

Dilation

n

D f xjaj f xa a R nfg

a

We notice that

iy

F T f M f F M f T f M T e T M

y y y y

and

F D f D f

a

a

n n

The shorttime Fourier transform STFT of a function f S R with resp ect to a window g SR

Z

iy

S x fM T g e g y x f y dy

g x

n n

The wavelet transform WT of a function f S R with resp ect to a window g SR is

Z

x y

g y dy x ajaj S g

g

a

n n n n

The Schr dinger representation of the Heisenb erg group H R R T on L R

ix iy

x f y e e f y x

If the toral variable is unimp ortant with x the reduced Schr dinger representation is

f y x f y

ix iy

e e f y x

ix

T f y e M

x

The radar ambiguity function Af goff and g

Af gx x f g

ix

e S f x

g

n n

The Wigner distribution W f g of f and g is the R R Fourier transform of the ambiguity

function

W f gF Af g

n

The mappings A and W are sesquiunitary on L R

Af g Af g f f g g

Wf g Wf g

The last equalityisMoyals identity

n n

Using the linear transformation M R R

M

where each blo c kisamultiple of the n n identity matrix wehave

W f g M g W f g

n n

where and are the Schr dinger representations on L R and L R resp ectively

The pseudo dierential op erator L can b e expressed in the following forms

L f g b Ag f

n

Wg f f g SR

Mo dulation spaces

n n

R Aweight function on R Mo dulation spaces measure the joint timefrequency distribution of f S

n

w R satises the inequality

s

w x y C jxj w y for some s

n n

Denition Given a weight function w on R a window g S R and two numbers p q such that

w n

p q we dene the mo dulation space M R to b e the Banach space of all distributions of slow

pq

n

growth f S R for which the following norm is nite

Z Z

q pq

p p

w

d kf k jS f x j w x dx

M g

pq

with obvious mo dications if p or q

n s s n

If w we write M R and if w x jxj j j wewriteM R If w x w x

pq

pq

w n

then M R is invariant under Fourier transformation since

pp

S f x S f x

g g

The following wellknown function spaces are mo dulation spaces

n n

R L R M

s

The weighted L spaces with weight w x w x jxj

s

w n n s n

M R L R ff f x jxj L R g

s

s

The Sob olev spaces with weight w x w j j

s

s n s n w n

R ff f j j L R g M R H

s

The space with weight w x jxj j j

s n n s n

M R L R H R

s

n

The Feichtinger algebra M S R

p

is a very nice space For It is to b e noted that M is not equal to L The Feichtinger algebra M

pp

example it is an algebra under b oth p ointwise multiplication and convolution It is invariant under Fourier

transform It is the minimal Banach space that is isometrically invariant using its own norm under b oth

p

translations and mo dulations It is contained in every L spaces in the Fourier algebra consisting of Fourier

transforms of L functions and in the Wiener algebra C l of continuous functions with norm

X



kf k kf k

L

C l kk

The notation C l is meant to imply that this space is an amalgam space of functions that are lo cally

continuous and globally l

nn

Lemma Let the matrix A R be symmetric and the convolution T and multiplication U with the

ixAx

chirp e be denedby

iA

F Tf e f U f

s n

Then T leaves M R invariant for each s and p q

pq

Proof See p and

Boundedness of pseudo dierential op erators on mo dulation spaces

We represent a symbol as a sup erp osition of timefrequency shifts by means of the following inversion

formula

n



Theorem If SR and kk then

L

ZZ

S M T dd

w n

If M R with p q the integral converges in the norm of this space If p or q

pq

n

the integral converges weakly or if S R

As a consequence of this result wehave the following representation of a pseudo dierential op erator

ZZ

L S L d d

M T

Thus arbitrary op erators L can b e studied in terms of the elementary pseudo dierential op erators L

M T

and the STFT S of the symbol

In K Gr chenig and C Heil haveproved the following b oundedness theorem for pseudo dierential

op erators on mo dulation spaces

n

Theorem If the symbol M R then the pseudodierential operator L is a bounded mapping

n

of M R onto itself for each p p and there exists a constant C such that

pp p

kL k C k k

M M p M

pp pp 

Proof The pro of of the theorem follows from several lemmas to b e found in and included here without

pro ofs for the convenience of the reader

n

Wecho ose a xed convenient windowon R

 

n x

x e

n

W x x R

where is the Gaussian function



n x n

x e x R

For this window the elementary op erators L are rankone pro jections as follows from next Lemma

M T

For this purp ose we use the n n real matrix

N diag M

and note that

N M

Lemma Using the notation N and noting that where

n

for R

we have the fol lowing three formulae

i

M T e W

i

f L f e

M T

ZZ

i

L f S N e f dd

n

Lemma If M R then

Z

jS N j d k k sup

M



n

R

and

Z

jS N j d k k sup

M



n

R

Hence the mapping T denedby

Z

TG S N G d

n

R

p n

is a bounded mapping of L R into itself for each p and

p p

kT k k k

L L M



Finally we need a result of H G Feichtinger and K Gr chenig which is stated in terms of the

w n

weighted mixednorm space L R

pq

Z Z

pq q

w p p

kF k jF x y j w x y dx dy

pq

n n n w

R then there exists Prop osition Let w be a weight function on R and x SR If F L

pq

aconstant C C such that

ZZ

w n

f F x y x y dxdy M R

pq

w n

where the integral converges in M R for p q and

pq

w w

kf k C kF k

M L

pq pq

These results combine to give the pro of of Theorem

n n n

that L is b ounded on L R M R when M R In contrast to the We remark

traditional classes memb ership in M do es not imply smo othness of and Theorem applies to cer

s n

tain nonsmo oth symb ols On the other hand since the Lipschitz classes R for s n and hence

n n n

C R in particular are emb edded in M R Theorem is an improvement in a certain direction

of the classical CaldernVaillancourt theorem

Asymptotic behavior of singular values of compact pseudo dier

ential op erators

In signal pro cessing the decay prop erties of singular values of pseudo dierential op erators L may b e a to ol

to determine the quality of timefrequency lters

Let H b e a separable Hilb ert space and B H b e the Banach space of b ounded op erators mapping H to

H with op erator norm kk

B H

The singular values s L of a compact op erator L B H are dened via the sp ectral theory

k

k

That is since L is compact L L has a discrete sp ectrum

as k

k

p

Since H is a Hilb ert space wehave The singular values of L are s

k k

s L inf kL T k rankT k

k

B H

Denition The Schatten class I is the set of compact op erators L BH such that

p

p

s L l

k

k

It is a Banach space under the norm

P

p

p

s L p

k

k

p

kLk kfs Lgk

I k l

p

sup s Ls LkLk p

k

B H

k

Wehave the following wellknown Schatten classes

I is the space of compact op erators on H

I is the space of traceclass op erators on H

I is the space of Hilb ertSchmidt op erators on H

The Schatten quasiideal I consists of compact op erators whose singular values lie in the Lorentz space

pq

pq

l dened by

q

X

q

pq

pq

k c kfc gk

k k l

k

The traceclass statement in Theorem improves analogous results of Ingrid Daub echies and Karlheinz

Gr chenig which require sn

The continuoustyp e expansion used in Theorem leads to the following prop osition proved in

n

Prop osition If the symbol S R thenL I

q

p

This result is interpreted by aspecialcaseofemb edding of L FL into S proved in

a

b

Prop osition

n s n n

R H R S R If snthen L

s

n s n n

If s nthen L R H R S R

s

The main result of this section is the next theorem whichisproved in

n s n

Theorem If L R H R with s then the singular values fs L satisfy the

k

s

k

asymptotic estimate

sn

s L O k as k

k

Thus

L I I for each pnn s

p

nns

In particular L is traceclass if sd

It is convenient to use a discrete series expansion of the symb ol b ecause singular values can b e estimated

via niterank approximations The following lemma found in will b e used

Lemma If L I then

X

s L s L inf kL T k rankT K

K k

I



K K

kK

The series expansions are based on the theory of Gab or frames Sp ecically the following results are

used

  

n x y n x

A If e W wherex e is the Gaussian function and a b are

chosen so that ab then there exist constants A B suchthat

X

n

jS f na mbj B kf k for all f L R Akf k

 

L L

n

mnZ

n

n

The collection fM T g is called a Gab or frame for L R

mb na

mnZ

n

B There exists a dual window SR suchthat

X

S f na mbM T f

mb na

n

mnZ

X

n

S f na mbM T for all f L R

mb na

n

mnZ

with unconditional convergence of the series in the L norm

w n

C The frame expansion in B also holds for all f M R with unconditional convergence in

pq

w

the norm of M if p q and weak convergence if p or q Furthermore the following is an

pq

w n

equivalent norm for M R for each p q

pq

qp q

X X

p p

w

jS f na mbj w na mb kf k

M

pq

n n

nZ mZ

which holds for arbitrary windows provided they are in the Finally one uses the following prop erty

Wiener algebra

X X

fc g l B jc j c M T

mn mn mn ma na



L

mn mn

Putting all this together wehave a pro of of Theorem

n s n

Corollary If L R H R with s then

s

L I

nns

Pseudo dierential op erators and Wilson bases

L R generate a complete orthonormal basis of The BalianLow theorem states that if the function

R of the form L

ilx

e x n l n Z

b

then either x x L R or L R

By replacing the complex exp onentials by sines and cosines one obtains Wilsons bases as follows

One constructs a function R R satisfying the inequalities

ajxj bj j

b

j xjCe and j jCe a b

and such that the shifted and realmo dulated functions

x x n

n

p

x x n cosln l l n Z

ln

p

x n sinln l l n Z x

ln

form an orthonormal basis for L R

In and Kazuya Tachizawa uses an ndimensional unconditional Wilson basis on mo dulation

w n

spaces M R to analyze global pseudo dierential op erators whose Weyl symbol x is such that

pq

x as jxj j j

An example is the symbol j j jxj of the Schr dinger op erator jxj A second example is the op erator

with symb ol of the form

x z z sin z

n n

where z x R R and z jxj j j The metho d based on the existence of a

parametrix is not applicable to the op erator with this symbol

n

The results in are based on the following observation If f x L R is lo calized around x

n n

R and its Fourier transform f is lo calized around R then the function x D f x can be

n

approximated by the function x f x Moreover if ff g is an orthonormal basis of L R and each

i

f is lo calized in phase space then one can get an approximate diagonalization of x D by ff g

i i

n

n

of L R is used to give a new sucient condition on the In a tight Gab or frame fg g

mn mnZ

symb ols to ensure that the singular values of the corresp onding compact pseudo dierential op erators b elong

to the Schatten quasiideal I Lo cal orthonormal trigonometric bases of the form

pq

n

Y

n n n

x g x m cos n x m m n Z x R

mn i i i

i

are used to obtain an approximate diagonalization of elliptic pseudo dierential op erators similar to the

diagonalization obtained in by means of Wilson bases

Pseudo dierential op erators and wavelets

The reader is referred to for denitions and references on wavelets In this section we only mention one

n

indirect use of wavelets in the study of the continuity of pseudo dierential op erators in L R

The celebrated T theorem of David and Journ has b een used to prove b oundedness of pseudo dif

ferential op erators To state this theorem Yves Meyer denes a sp ecial class of singular integrals on an

n n

intermediary top ological vector space V suchthat D R V L R

Denition A continuous linear op erator T V V is asso ciated to a singular integral op erator if there

exist an exp onent two constants C and C and a function K C suchthat

n

jK x y jC jx y j

n

y K x y jC jx xj jx y j jx y j jK x if jx xj

n

jx y j jK x y K x y jC jy y j jx y j if jy y j

and

Z

Tfx K x y f y dy

for every function f V and every x supp f

Theorem DavidJourn Let T be as in Denition Then a necessary and sucient condition for

n

T to extend by continuity to L R is that the fol lowing threeconditions be satised

n

T belongs to BM OR

t n

T belongs to BM OR

n

T is weakly continuous on L R

A pro of of the T theorem based on wavelets is found in It is enough to establish the L continuity

of T under the hyp otheses and We mayassumethat V is the top ological vector space of C

n

functions with compact supp ort Wexamultiresolution analysis V j Zof L R leading to a scaling

j

function x and wavelets

nj j j j n

x x k k f g nfg

n

of class C with compact supp ort We recall that the family fx k g k Z forms an orthonormal basis



of V The functions x are real valued and the idea of the pro of is to estimate the mo duli jt j of the

matrix entries

t

  

t T T

of the op erator T in the Hilb ert wavelet basis These matrices are characterized by the decrease of their

yof such matrices follows from a lemma of entries as one moves away form the diagonal The l continuit

Schur

A second pro of of the T theorem based on the lemma of Cotlar and Stein is given in whichwe

quote for the convenience of the reader

Lemma Let H be a Hilbert spaceandT H H for j Z be continuous linear operators and denote

j

their adjoints by T Suppose that there exists a sequence j for j Z such that

j

X

p

j

j

and the fol lowing inequalities hold

kT T k j k kT T k j k for al l j k Z

k j

j k

Then pour al l x H the series

X

T x

j

j

converges in the H norm Putting

X

T x T x

j

j

we obtain

X

p

j kT k

j

Consider pseudo dierential op erators T of the form

Z

ix

x D f x e x f d

ZZ

ixy

e x f y dy d

Z

K x y f y dy

with kernel

Z

iy

K x y e x d

If b oth the symbol x of T and the symb ol of the adjoint T of T satisfy the inequalities

j jjj

j x jC j j

x

it follows by an application of the T theorem that T b elongs to a subalgebra A of b ounded linear

n n

op erators from L R to L R

An op erator in A can b e describ ed byits kernel its symboloritsmatrixin an orthonormal wavelet

basis of LittlewoodPaley and this last prop ertyallows a symb olic calculus in A



The standard matrix realization t of the op erator T in a wavelet basis has the obvious advantages

of supplying a mechanism for the compression of op erators that is simply a representation of the op erator

in as suitably chosen basis In the nonstandard form of a kernel K x y is obtained byevaluating the

expressions

ZZ ZZ

   

y dx dy K x y x y dx dy K x y x

I I II I I II

ZZ

 

K x y x y dx dy

II I

j j

where the notation I k k and

jk

j

x x x k

jk I I

jk

is used and similarly for x The nonstandard form gives exact estimates for pseudo dierential op erators

and CaldernZygmund op erators and other translation invariant op erators The nonstandard form has

the numerical advantage of pro ducing a much sparser matrix which requires only O N op erations to apply

suchanN N matrix to a vector dep ending on the particular op erator as opp osed to O N log N with the

standard realization of the op erator

Chirplets

The chirplet transform rst prop osed in and develop ed in dierent directions is an extension of the

shorttime Fourier transform and wavelet transform Informally sp eaking a chirplet is a piece of chirp see

Figure in the same manner a wavelet could b e lo osely regarded as a piece of wave Yves Meyer rst coined

the term ondelette from the French word onde English wave and the diminutive ette The closest

English translation wavelet b ecame the accepted word in pap ers written in the English language Steve

Mann and Simon Haykin have coined the French term ppielette combining the diminutive with the French

word p pier or p piement to designate a piece of chirp and similarly in English they have coined the

term chirplet

In this and the next sections we shall restrict attention to functions dened on R Small Latin letters

will refer to time or physical domain and capital letters will refer to frequency or Fourier domain

A general class of chirplets has b een prop osed by S Mann and S Haykin by means of the two

transformations

q t

g w tg pt e

wpq

and

Qf

G W f G P f e

WPQ

where w and W are the temp oral and sp ectral windows resp ectively p and P are the physical and Fourier

p ersp ective resampling or warping functions and q and Q if pure imaginary are the mo dulation

functions

Physicaldomain windows are wellknown to p eople working in the signal pro cessing eld It has been

transformation also proves to be very useful Alternately found that windowing the data after Fourier

one may think of just convolving in the physical domain with the inverse Fourier transform of W Thus Wave signal Wavelet signal 0 0 Real Real

0 0 Imag Imag

Time-frequency for wave Time-frequency for wavelet

0 0 Frequency Frequency

Time Time Chirp signal Chirplet signal 0 0 Real Real

0 0 Imag Imag

Time-frequency for chirp Time-frequency for chirplet

0 0 Frequency Frequency

Time Time

Figure Wave wavelet chirp and chirplet

conceptually b oth w and W are implementedinthephysical domain one windows with w and convolves

with F W Windowing by means of partitions of unity in the time and Fourier domains has b een used in

the study of pseudo dierential op erators

Wenow fo cus attention on an imp ortant distinction b etween timefrequency and timescale metho ds

Mo dulates

The onedimensional timefrequency metho d may b e regarded as expansions into usually overdetermined

bases which may be regarded as translates and mo dulates of one primitive In terms of the STFT this

primitive is the data window Translates are time shifts and mo dulates are frequency shifts A frequency

shift by the amount f may b e accomplished through multiplication by a complex exp onential of frequency

c

f The Wigner distribution is an expansion of a signal into translates and mo dulates of itself reversed

c

Hence the Wigner transform emb o dies this mo dulate class

Scaling

The timescale metho d suchasthewavelet transform may b e regarded as expansions into usually overde

ariable may b e regarded as translates and dilates of one termined bases which for functions of one real v

mother wavelet Intro ducing the camera metaphor where the D mother wavelet is reimaged we see that

the D camera can b e shifted along translated and zo omed in within a D space This combination of

translation and zo om generates the family of wavelets as a collection of D images of one mother wavelet

The dierence b etween the two phenomena may b e b est appreciated by a couple of examples To those

familiar with amateur ham radio a Typ eJ mo dulated signal singlesideband suppressed carrier often

has asso ciated with it a Donald Duck voice eect the voice is p erhaps higher or lower in pitchthanthe

original But this pitch shift results in a very dissonantsounding voice The voice jiggles and the sound

is very dierent from that of a tap e recording b eing played at the wrong sp eed The dierence is that the

jiggle is the result of a frequencyshift while the change in sp eed gives rise to a scale shift

The Doppler eect is a pro cess which b elongs to scalings For example an automobile coming toward

you with horns sounding an Fma jor rd interval will sound a higher pitch but the harmonic structure of

the sound is preserved so that it still sounds a ma jor rd interval On the other hand if one listens to

a meter band with a singlesideband receiver a horn would likely sound very dissonant b ecause each

frequency comp onent would be shifted by a certain number of herz and the frequencies of the harmonic

comp onents would not likely end up b eing multiples of each other

The chirplet is a generalization of this dichotomy which includes the mo dulates and the scalings as sp ecial

cases Thus the chirplet can b e broken down into the same dichotomy evolutive mo dulates and evolutive

scalings

Evolutive mo dulates

The family of q chirplets is generated by a translation in frequency mo dulation which is allowed to vary in

time acting on a particular mother chirplet

chirplets are generated bymodulationintheFourier domain the Fourier transform of the mother The Q

chirplet is computed and then acted on by the timevarying FM pro cess and then the result is brought

backtothephysical domain byinverse Fourier transformation The Qchirplets represent time shifts which

are allowed to have a frequency dep endency in the same sense that the q chirplets represent frequency shifts

which are allowed to have a time dep endency

Evolutive scalings

The physical p ersp ectives p represent pro jections Wemay think of the mother chirplet as b eing pro jected

onto a surface or as existing on that surface and b eing imaged bya hyp othetical camera Now the surface

orientation and camera angle giverisetoachange in scale which is dierent for dierent parts of the original

time axis In terms of the tap e recorder metaphor the physical p ersp ectives pchirplets are the result of

varying the tap e sp eed while one mother chirplet is b eing played Wemay view the mother chirplet as b eing

sub jected to arbitrary Doppler shifts to generate the family of p chirplets

The Fourier p ersp ective P chirplets are generated by arbitrary pro jections in the Fourier domain

Starting with a mother chirplet in the physical domain we compute its Fourier transform then p erform a

pro jective co ordinate transformation in Fourier space and then compute the inverse Fourier transform

Two classes of chirplet transforms are intro duced in and are called metaplectic transforms based on

q chirplets and pro jectivechirplet transforms based on pchirplets Both are generalizations of the wavelet

transform

Metaplectic chirplet transforms

Metaplectic chirplet transform MCT is an inner pro duct representation into the space of chirplets which

may b e generated from wavelets using multiplication by a complex exp onential with a quadratic time term

for q and convolution by a complex exp onential with another quadratic in time for Q It can b e shown that

quadratic exp onentials corresp ond resp ectively to shear in frequency and to shear in time in the these

timefrequency plane The MCT of a signal s generalizes the wavelet jaj g ta as follows

 

iq t iQ f

t f a q Qjaj e g t e js ba

s

where the wavelet parameter b is redundantandmay b e set to zero Here and later we use the Dirac notation

Z

fjgf f t tjg t g t dt

for the inner pro duct of two functions f g L R in order to indicate dummyvariables

The algebraic structure of the metaplectic chirplet transform is that of the symplectic group In

it is applied to signal pro cessing

Pro jective chirplet transforms

The pro jective chirplet transform PCT is an inner pro duct representation on pro jections of a wavelet

family The physical pro jectivechirplet transform of s is the representation

a t

c a t ps jg

s

p

The Fourier pro jectivechirplet of a signal s may b e written

a f

C a f r s jg

s

r

where the Fourier p ersp ective dilation parameter a is recipro cally related to the physical p ersp ective dilation

Note that the physical and Fourier pro jectivechirplet representations together also embody ve parameters

t f a p r

Co ordinate transforms of the timefrequency plane

In previous sections we have identied eight primitive co ordinate transforms We list the corresp onding

rst six chirplet op erators and their eect on a function g t and on its Wigner timefrequency distribution

W f The last two transformations are given in the form of a scalar pro duct

g

Translation in time

T g g t W t

t g

Translation in frequency

if

M g e g W f

f g

Dilation in time

D g jaj g a W a a

a g

Dilation in frequency

D g jaj gaf W a a

g

a

Shear in time



i Q

Q e g W

g

Shear in frequency



i q

e g W q

g

Persp ective pro jection in the time domain

a t

c a t ps jg

s

p

Persp ective pro jection in the frequency domain

a f

a f r s jg C

s

r

The six ane transformations of the timefrequency plane and the two p ersp ective pro jections are illus

trated in Figure

In practice from a computational data storage and display point of view the parameter chirplet

transform is dicult to handle Therefore one considers subspaces of the entire parameter space Planes

are particularly attractive choices b ecause of the ease with which they may be printed or displayed on a

computer screen and the fact that they lend themselves to admissible niteenergy signal representations

Known examples are the timefrequency STFT and timescale WT planes discussed already

Bowtie chirplet

The bowtie transform is the threeparameter timefrequencyshear representation obtained by restricting

the computation of the chirplet transform to the surface x t f q in the eightparameter

co ordinate transform The b owtie transform is given by

Z



if iq t

B t f q s g t e e d

s

which is equivalent to a STFT with an added chirp comp onent Thus in addition to measuring the time

frequency content of a signal B t f q also indicates its chirpiness The b owtie chirplet has proven useful

s

for studying the acceleration signature of Dopplershifted radar signals The extra shear parameter q of the

bowtie transform makes it ideally suited to the analysis of signals of arbitrary piecewiseconstantchirp rate

and thus to the analysis of constant acceleration targets Original function Frequency Time Translation in Time Translation in Frequency Frequency Frequency Time Time Dilation in Time Dilation in Frequency Frequency Frequency Time Time Shear in Time Shear in Frequency Frequency Frequency Time Time Perspective in Time Perspective in Frequency Frequency Frequency

Time Time

Figure The six ane transformations and two p ersp ective pro jection of the timefrequency plane

Disp ersion chirplet

Disp ersion artifacts o ccur in signals acquired from media in whichthewave propagation velo cityvaries with

the frequency of the signal The comp onents of the signal are tilted in timefrequency The disp ersion

transform of a signal s is the function on the twoparameter surface x t f Qt f in the

chirplet space

Z

h i

p



i ptf if

D t f ipt f s e g t e d

s

Since Qt f controls the tilt of basis elements called logon at the p oint t f in timefrequency it is called

the tilt function Generally in situations where the disp ersion characteristics of the medium are known or

can b e estimated the disp ersion transform yields a more concentrated timefrequency representation than

either the WT or STFT

Warblet chirplet

The warblet chirplet is used to analyze signal of oscillating frequency Examples of such signal are large

pieces of ice oating in the o cean Such a signal has a frequency that p erio dically rises and falls much like

the vibrato of musical instruments or the wail of a p olice or ambulance siren Although the term chirp

is most often used to describ e a signal with monotonically rising or monotonically falling frequency there is

no requirement that the chirplets in the chirplet transform b e monotone rising or falling Lo oking closely at

the physics of oating ob jects one observes a somewhat sinusoidal evolution of the Doppler radar signals

suggesting that one should use some form of cyclostationary mother chirplet When watching a cork b obbing

up and down at the seaside one notices that it moves around in a circle with a distinct p erio dicity It moves

up and down but it also moves horizontally The instantaneous frequency of the basic function is given by

f cosf t pf

m c

where f is the center carrier frequency p whichvaries on the interval from to is the relative p osition

c

of one of the p eak ep o chs in frequency with resp ect to the origin and f is the mo dulation frequency

m

Integrating one gets the phase

sinf t p

m

f t

c

f

m

whichgives the family of chirplets dened by

i sinf tpf f t

m m c

g Ae

f f

m c

whichmay b e appropriately windowed such as with a Gaussian

Persp ectivefrequency transform

A particular useful plane in image pro cessing applications is one taken b etween the parameters of p chirpi

t

ness due to p ersp ective pro jection in the physical domain and the frequency f Representing a signal

on a set of mo dulated and then pro jected versions of one mother chirplet one nds that the camera ori

entation with resp ect to almostplanar ob jects could be readily determined provided that those ob jects

contained some form of p erio dicity A practical example is given an image photographed with the camera

pointing up at a building one notices that the windows in the building app ear smaller at the top of the

image than those at the b ottom even though they are all the same size in reality

Conclusion and op en questions

The chirplet transform allows for a unied framework for comparison of various timefrequency metho ds

b ecause it embodies many other such metho ds as lowerdimensional subspaces in the chirplet analysis For

example both the Short Time Fourier Transform and the wavelet transform are planar slices through an

parameter signal space

Is there a multiresolution analysis for discrete and continuous chirplets Do chirplets satisfy multiscale

equations Do es one have scaling and waveletlikechirplet functions Do chirplets provide unconditional

bases in some function spaces

In the applications to pseudo dierential and Fourier integral op erators one is interested in compactly

supp orted bases of orthogonal and biorthogonal chirplets in one and higher dimensions Based on the success

of the Gab or and wavelet transforms in the study of some classes of pseudo dierential op erators in natural

spaces one would hop e that new results could b e easily obtained for other until nowuntractable op erators

by using appropriate co ordinate transformations of the timefrequency plane

Oneisalsointerested in the implementation of computationally ecient algorithms for realtime appli

cations

New emb edding results b etween classical function spaces and mo dulation spaces would b e useful for a

b etter description of the latter spaces

Acknowledgment

The authors express their thanks to the organizers of the RIMS Symp osium on Structure of Solutions for

Partial Dierential Equations They also thank Christopher Heil for his numerous emails to answer

many questions and supply many insightful suggestions

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