Paper_BCDO_Vaxjo.dvi


CUBO A Mathematical Journal

Vol.12, No¯ 03, (171–185). October 2010

Generalized Spectrograms

and τ-Wigner Transforms

BOGGIATTO PAOLO, DE DONNO GIUSEPPE, OLIARO ALESSANDRO

Department of Mathematics, University of Turin,

Via Carlo Alberto, 10, 10123 Torino, Italy

email: paolo.boggiatto@unito.it

email: giuseppe.dedonno@unito.it

email: alessandro.oliaro@unito.it

AND

BUI KIEN CUONG

Higher Education Department, Hanoi Pedagogical University 2,

Building G7-144 Xuan Thuy Rd – Hanoi, Vietnam

email: buikiencuong@yahoo.com

ABSTRACT

We consider in this paper Wigner type representations W i gτ depending on a parameter
τ ∈ [0, 1] as defined in [2]. We prove that the Cohen class can be characterized in terms of
the convolution of such W i gτ with a tempered distribution. We introduce furthermore a
class of “quadratic representations” S pτ based on the τ-Wigner, as an extension of the two
window Spectrogram (see [2]). We give basic properties of S pτ as subclasses of the general
Cohen class.



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RESUMEN

Nosotros consideramos en este artículo representaciones de tipo Wigner W i gτ dependiendo
de um parámetro τ ∈ [0, 1] como definido en [2]. Probamos que la clase Cohen puede ser
caracterizada en terminos de la convolución de tales W i gτ con una distribución temper-
ada. Introducimos también la clase de “representaciones cuadraticas” S pτ basado en el
τ-Wigner, como una extensión de dos ventanas espectrograma (ver [2]). Nosotros damos
propiedades básicas de S pτ como subclases de la clase Cohen.

Key words and phrases: Time-Frequency representation, τ-Wigner distribution, generalized

Spectrogram.

Math. Subj. Class.: 42B10, 47A07, 33C05.

1 Introduction

One of the basic problems in time-frequency analysis is the representation of the energy

of a signal simultaneously with respect to time and frequency. Considering for generality

signals as square-integrable functions on Rd , the classical mathematical tool used for this

aim are sesquilinear maps Q : L2(Rd ) × L2(Rd ) → L2(R2d ). For a given signal f , the function

Q( f , f )(x,ω), or for short Q( f )(x,ω), plays a role corresponding to that of density of mass in

classical mechanics or that of probability distribution in statistics. In contrast however to

these situations, in the case of the energy of a signal the time-frequency distribution to be

used is not unique. Many proposals have been presented in the literature, each having ad-

vantages and drawbacks, see [5], [6], [7], [8], [9] for detailed presentations of these topics.

This is due essentially to the presence of the Heisenberg uncertainty principle which

makes some of the natural requirements of a joint time-frequency distribution incompatible

(see [11]).

Two of the most used time-frequency representations are the Wigner distribution:

W i g( f , f )(x,ω) = W i g( f )(x,ω) =

ˆ

Rd

e
−2πitω

f (x + t/2) f (x − t/2) dt (1.1)

and the Spectrogram

S p g( f )(x,ω) = |Vg( f )(x,ω)|
2 (1.2)

where Vg( f ) is the Gabor transform (also known as short-time Fourier transform) and is de-

fined by

Vg( f )(x,ω) =

ˆ

Rd

e
−2πitω

f (t) g(x − t) dt (1.3)



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Generalized Spectrograms ... 173

in dependence on the “window” g(x), which in the most generality can be supposed to be a

tempered distribution.

This paper is based on these two representations of which we present modifications de-

pending on parameters. We shall analyze the properties of these new representations with

respect to classical requirements such as reality of values, marginal distribution conditions,

and their relations with the Cohen class. This is a very general class of time-frequency rep-

resentations, introduced by L. Cohen, see [6], and widely studied since the 1970’s. It can be

defined as the set of representations of the form

C( f ) = σ∗ W i g( f ) (1.4)

where, in our context, σ will be supposed to be a tempered distribution in S ′(R2d ) and will be

called Cohen kernel. The wide possibility of choice of the Cohen kernel permits to cover most

time-frequency representations.

We recall next that some considerations concerning shifts of the ghost frequencies led in

[2] to the introduction of the representations

W i gτ( f , g)(x,ω) =

ˆ

Rd

e
−2πitω

f (x +τt) g(x − (1 −τ)t) dt (1.5)

which are a parameterized version of the Wigner representation in dependence on τ ∈ [0, 1].

It was also showed in [2] that these representations constitute the natural “quadratic form”

counterparts to the τ-pseudo-differential operators which are extensions of the Weyl calculus

on Rd ; classical references on this subject are Shubin [14] and Wong [15], see also [1] for

generalizations concerning global hypo-ellipticity.

In the present paper we analyze at first the role of (1.5) in the definition of the Cohen

class, showing that we can replace W i g( f ) in (1.4) by W i gτ( f ), for an arbitrary fixed τ ∈ [0, 1],

getting equivalent definitions of the Cohen class. In the second part of the paper, we propose

a new form based on the two window spectrogram and the τ-Wigner representation. The

two window spectrogram was studied in [3]-[4] (called there generalized spectrogram) and is

defined by

S pφ,ψ( f , g)(x, w) = Vφ f (x, w)Vψ g(x, w). (1.6)

Using τ-Wigner distribution, we generalize here definition (1.6) by replacing the classical

Wigner distribution with τ-Wigner distributions. We obtain new representations that we

shall call parameterized two window spectrograms and we study some of their basic properties

such as positivity, support properties and boundedness in the L p context. We show that our

definition is motivated by the fact that the parameterized two window spectrograms show in

some basic cases reduced interference phenomena with respect to (1.6) without a loss in the

quality of the time-frequency localization. Finally we prove that among the variety of time-



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frequency representations they constitute a peculiarity as they do not belong to the Cohen

class 1.

2 τ-Wigner Representations and the Cohen Class

In the definition (1.4) of the Cohen class the Wigner representation plays a special role and

one natural question is if it can be replaced by another representation. In general this can be

achieved under some additional conditions. More precisely suppose

C0( f ) = σ0 ∗ W i g( f )

is a fixed representation in the Cohen class; then, as long as Ĉ0( f )/σ̂0 belongs to S
′(R2d) for

every signal f ∈ S (Rd ), we have

W i g( f ) = F −1(Ĉ0( f )/σ̂0).

But even under this somewhat restrictive condition it does not necessarily happen that C0 →

F
−1(Ĉ0( f )/σ̂0) is a convolution. Actually only if this were the case we could write

F
−1(Ĉ0( f )/σ̂0) = σ

′
∗ C0( f )

for a suitable fixed σ′ ∈ S ′(R2d), and then for any generic representation in the Cohen class

C = σ∗ W i g, (with σ ∈ S ′(R2d )), we would obtain

C( f ) = σ∗ W i g( f ) = (σ∗σ′) ∗ C0( f ).

In this case, under the further condition that σ ∗ σ′ ∈ S ′(R2d ), we would have that every

element in the Cohen class could be expressed in terms of C0 instead of W i g.

In view of these observations it is interesting, even if not surprising, that any W i gτ
representation can replace the Wigner representation in the expression of the Cohen class.

In order to prove this assertion we need the explicit expression of W i gτ as a member of

the Cohen class. We recall then from [2] the following result.

Proposition 1. The representation W i gτ( f ) belongs to the Cohen class for every τ ∈ [0, 1], in

particular

W i gτ( f )(x,ω) =
(
στ ∗ W i g( f )

)
(x,ω), (2.1)

for every f ∈ S (Rd ), where

στ =






2d

|2τ−1|d
e2πi

2
2τ−1 xω for τ 6= 12

δ for τ = 12

(2.2)

and δ is the Dirac distribution.

1According to (1.4) we only consider signal independent kernels σ



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Generalized Spectrograms ... 175

We have now the following Proposition:

Proposition 2. Let τ ∈ [0, 1] be fixed, then W i gτ can be used to express the entire Cohen class,

i.e. every representation C in the Cohen class can be written in the form

C( f ) = σ′ ∗ W i gτ( f )

for a suitable σ′ ∈ S ′(R2d).

Proof. Let

C( f ) = σ∗ W i g( f ) (2.3)

with σ ∈ S ′(R2d), be the expression of C( f ) in the Cohen class. From the previous proposition

we have

W i gτ( f ) = στ ∗ W i g( f )

and a straightforward computation yields:

στ ∗σ1−τ = δ.

We have therefore

σ1−τ ∗ W i gτ( f ) = W i g( f )

and substituting in (2.3) we get formally:

C( f ) = (σ∗σ1−τ) ∗ W i gτ( f )

This expression has actually a meaning if we show that σ∗σ1−τ is a well defined tempered

distribution. As σ∗ σ1−τ = F
−1(σ̂σ̂1−τ) and σ ∈ S

′(R2d ), this is equivalent to prove that σ̂1−τ
is a multiplier of S ′(R2d). Since

´

e2πi yρ d y dρ = 1 we have

F σ1−τ(ξ, t) = e
−πi(1−2τ)tξ (2.4)

which is a C∞ function with derivatives with polynomial growth and therefore our assertion

is proved. The thesis is then satisfied with σ′ = σ∗σ1−τ.

We turn now our attention to the spectrograms with the aim of describing how the gen-

eral context above applies to this specific case.

As already pointed out in the Introduction, the classical spectrogram, defined by

S p g ( f )(x, w) = |Vg f (x, w)|
2, (2.5)

is a way to represent the energy of a signal f simultaneously with respect to time and fre-

quency; Vg f is the short-time Fourier transform, or Gabor transform, with window g, see



176 Boggiatto Paolo, et. al CUBO
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for reference [13], [16], [10]. In [3], the two window spectrogram has been introduced and

studied: it depends on two windows and it is defined by the skew-linear form

S pφ,ψ( f , g)(x, w) = Vφ( f )Vψ( g)(x, w); (2.6)

when φ = ψ, f = g, formula (2.6) becomes the classical spectrogram.

The following relationship between Wigner distribution and two window spectrogram

holds (see [3]):

S pφ,ψ( f , g)(x, w) = W i g(ψ̃,φ̃) ∗ W i g( f , g)(x, w), (2.7)

where φ̃(s) := φ(−s) and ψ̃(s) := ψ(−s). Relation (2.7), valid in suitable functional settings,

for example when f , g,φ,ψ ∈ S (Rd), gives us the expression of the two window spectrogram

as an element of the Cohen class, where σ in (1.4) is given now by W i g(ψ̃,φ̃). As proved

in Proposition 2, we can re-write S pφ,ψ( f , g) through the τ-Wigner transform. In the special

case of the two window spectrogram this can be made more explicit as showed by the following

result.

Proposition 3. For every f , g,φ,ψ ∈ S (Rd) and for every τ ∈ [0, 1], we have

S pφ,ψ( f , g) = W i g1−τ(ψ̃,φ̃) ∗ W i gτ( f , g)(x, w).

Proof. Since

W i g1−τ(ψ̃,φ̃) = W i gτ(φ̃,ψ̃), (2.8)

we have to prove that

S pφ,ψ( f , g) = W i gτ(φ̃,ψ̃) ∗ W i gτ( f , g)(x, w). (2.9)

Let us observe that, by a simple change of variables, we can write

W i gτ( f , g)(x − y, w −η) = Ft→η
(
e

2πiωt
f (x − y −τt) g(x − y + (1 −τ)t)

)
.

Since

W i gτ(φ̃,ψ̃)( y,η) = Ft→η
(
φ̃( y +τt)ψ̃( y − (1 −τ)t)

)
,

by the standard properties of the Fourier transform we get

W i gτ(φ̃,ψ̃) ∗ W i gτ( f , g)(x, w)

=

(
φ̃( y +τt)ψ̃( y − (1 −τ)t), e2πiωt f (x − y −τt) g(x − y + (1 −τ)t)

)

L2 (R2d
y,t )

.

Finally, by the change of variables

{
y +τt = Y

y − (1 −τ)t = T



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Generalized Spectrograms ... 177

in the L2-product, we have

W i gτ(φ̃,ψ̃) ∗ W i gτ( f , g)(x, w) =
(
φ̃(Y )ψ̃(T), e2πiω(Y −T) f (x − Y ) g(x − T)

)

L2(R2d
Y ,T

)
.

This shows that W i gτ(φ̃,ψ̃) ∗ W i gτ( f , g)(x, w) is independent of τ ∈ [0, 1], and so for every

τ ∈ [0, 1],

W i gτ(φ̃,ψ̃) ∗ W i gτ( f , g)(x, w) = W i g(φ̃,ψ̃) ∗ W i g( f , g)(x, w).

From (2.8), (2.7) and this last identity, we get (2.9).

3 The Parameterized Two Window Spectrogram: Definition and Mo-

tivations

So far we have been concerned with relationships between τ−Wigner and spectrograms rep-

resentations within the frame of the Cohen class. In this section we want to consider relation-

ships between these two types of representations under another point of view which will bring

us to the definition of a further representation. We start with some preliminary remarks. It

is well-known that the Wigner transform can be expressed in function of the spectrogram by

the following equality

W i g( f , g)(x, w) = 2d e4πixwVg̃ f (2x, 2w), (3.1)

and viceversa we have

Vg f (x, w) = 2
−d

e
−πixw

W i g( f , g̃)(
x

2
,
w

2
). (3.2)

From (2.6) it is then clear that we can then rewrite the two window spectrogram as

S pφ,ψ( f , g)(x, w) = 4
−d

W i g( f ,φ̃)(
x

2
,
w

2
)W i g( g,ψ̃)(

x

2
,
w

2
). (3.3)

In view of this equality it is natural to introduce the following generalization of the spectro-

gram:

Definiton 4. Let τ1,τ2 ∈ [0, 1] be two parameters, the parameterized two window spectro-

gram, denoted S p
(τ1 ,τ2)
φ,ψ ( f , g), is defined by

S p
(τ1 ,τ2 )
φ,ψ ( f , g)(x, w) = 4

−d
W i gτ1 ( f ,φ̃)(

x

2
,
w

2
)W i gτ2 ( g,ψ̃)(

x

2
,
w

2
), (3.4)

where φ,ψ are window functions and f , g are signals in suitable functional or distributional

spaces.



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Remark 5. When τ1 = τ2 = 1/2, the parameterized two window spectrogram becomes the two

window spectrogram

S p
(τ1 ,τ2 )
φ,ψ ( f , g)(x, w) = S pφ,ψ( f , g)(x, w).

The introduction of this new family of parameterized representations is not due to pure search

of mathematical generality. Actually, as we describe next, the form S p
(τ1 ,τ2)
φ,ψ ( f , g) shows an

interesting behavior for what concerns localization properties and reduction of interference

disturbances in particular in the cases where frequencies occur in time intervals very close

to one another. To this aim let us consider a signal f containing the frequency ω = 2 in the

time interval [−4, 0] and the frequency ω = 3 in the time interval [0, 4]; we fix the window

functions φ = χ[−10,10] and ψ = χ[− 110 ,
1

10 ]
, where χ[a,b] denotes the characteristic function of the

interval [a, b] and we compare the pictures of the parameterized two window spectrograms

S p
(τ1 ,τ2 )
φ,ψ ( f , g) for different values of τ1 and τ2. The two window spectrogram S pφ,ψ( f , f ),

corresponding to case τ1 = τ2 =
1
2 , is visualized in Figure 1:

Figure 1: S p
( 12 ,

1
2 )

φ,ψ ( f , f ) = S pφ,ψ( f , f )

As we can see, although the localization is good both in time and in frequency, the picture

presents disturbing interference patterns. The explanation of this fact is the following. The

Gabor transform Vφ f with a large window φ gives better information regarding frequencies,

and the Gabor transform Vψ f with a narrow window ψ gives better information concerning

time. When we consider the two window spectrogram

S pφ,ψ( f , g) = Vφ f Vψ g

we take a product of one Gabor transform well localized in time and another one well localized

in frequency, and so the reciprocal cut-off effect yields good localization both in time and



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Generalized Spectrograms ... 179

frequency, see [4] for a detailed discussion on this subject. It could seem therefore that we

have overcome the Heisenberg uncertainty principle but of course it is not so. Actually what

is obtained in good localization, is “paid” terms of interference. More precisely, the fact that

each Gabor transform is well localized in one variable and, consequently, badly localized in

the other, implies that the supports of the two Gabor transforms also intersects in places

where no frequency is present. This is what is observed in Figure 1 and clearly represents a

considerable drawback in the use of the classical two window spectrogram.

Let us consider now the parameterized two window spectrogram, with the same win-

dows and signal as above. In Picture 2 we have a representation of S p
(0.3,0.3)
φ,ψ ( f , f ) and

S p
(0.2,0.2)
φ,ψ ( f , f ) (for simplicity we take here τ1 = τ2).

S p
(0.3,0.3)
φ,ψ ( f , f ) S p

(0.2,0.2)
φ,ψ ( f , f )

Figure 2: Parameterized two window spectrogram for different values of τ1,τ2.

As we observe from the pictures, although the windows φ and ψ are kept fixed, the in-

terference between the two frequencies is considerably reduced when the parameter τ in

S p
(τ,τ)
φ,ψ ( f , f ) becomes small, keeping on the other hand the good level of localization. Inci-

dentally we also remark that the improvement of frequency localization is only apparent as

it is essentially the consequence of an effect of vertical contraction and horizontal dilation

compensated in the picture by a relabeling of the axis.

4 Properties of the Parameterized Two Window Spectrogram

In this section we analyze some properties of the representation S p
(τ1 ,τ2 )
φ,ψ ( f , g) with τ1,τ2 ∈

[0, 1]. More precisely we consider positivity, L p−boundedness and support property, we con-



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clude then our investigations by showing that the parameterized two window spectrogram

does not belong to the Cohen class.

For what positivity is concerned we limit ourself to the following basic fact, we have

S p
(τ)
φ

( f )(x, w) := S p
(τ,τ)
φ,φ ( f , f )(x, w) = 4

−d
|W i gτ( f ,φ̃)(x, w)|

2
≥ 0.

and therefore the following property holds:

Proposition 6. For τ1 = τ2, f = g and φ = ψ the parameterized two window spectrogram is a

positive time-frequency representation.

We consider next the parameterized two window spectrogram in the context of the L p spaces.

For this purpose we shall need the following Proposition, which is proved in [2].

Proposition 7. Let us fix q and p satisfying q ≥ 2 and q′ ≤ p ≤ q, ( 1
q
+

1
q′
= 1). Then:

i) For each τ ∈ (0, 1), W i gτ : L
p′ (R) × L p(R) → Lq(R2d ) is continuous, in particular:

‖W i gτ( g, f )‖Lq ≤
1

|1 −τ|
d( 1

p
−

1
q

)

1

|τ|
d(1− 1

p
−

1
q

)
‖g‖

L p
′ ‖f ‖L p . (4.1)

ii) For τ = 0, W i g0( g, f )(x, w) = R( g, f )(x, w) and W i g0 : L
q(R) × Lq

′

(R) → Lq(R2d) is contin-

uous, in particular

‖W i g0( g, f )‖Lq ≤ ‖g‖Lq′ ‖f ‖Lq . (4.2)

iii) For τ = 1, W i g1( g, f )(x, w) = R( g, f )(x, w) and W i g1 : L
q′ (R) × Lq(R) → Lq(R2d) is contin-

uous, in particular

‖W i g1( g, f )‖Lq ≤ ‖g‖Lq ‖f ‖Lq′ . (4.3)

Furthermore for p, q in the remaining cases the τ-Wigner transform is not bounded as sesquilin-

ear map: L p
′

(R) × L p(R) → Lq(R2d).

The L p behavior of the parameterized two window spectrogram is specified by the follow-

ing proposition.

Theorem 8. Let q ≥ 1, q j ≥ 2, p j ≥ 1, ( j = 1, 2) satisfy the following conditions:
1
q1

+
1
q2

=
1
q

; q
′

j
≤

p j ≤ q j , ( j = 1, 2), where
1
q j

+
1
q′

j

= 1. Then

i) The parameterized two window spectrogram S p(τ1 ,τ2 ) : L p
′

1 × L p1 × L p
′

2 × L p2 → Lq is con-

tinuous (0 < τ1,τ2 < 1), in particular

‖S p
(τ1 ,τ2)
φ,ψ ( f , g)‖Lq ≤ C‖f ‖

L
p
′

1
‖φ‖L p1 ‖g‖

L
p
′

2
‖ψ‖L p2 , (4.4)

where C = C1C2 with C j =
1

|1−τ|
d( 1p j

−
1

q j
)

1

|τ|
d(1− 1p j

−
1

q j
)
, j = 1, 2.



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Generalized Spectrograms ... 181

ii) When τ1 = 1,τ2 = 0 then S p
(1,0) : Lq1 × Lq

′

1 × Lq
′

2 × Lq2 → Lq is continuous, in particular

‖S p
(1,0)
φ,ψ ( f , g)‖Lq ≤ ‖f ‖Lq1 ‖φ‖

L
q
′

1
‖g‖

L
q
′

2
‖ψ‖Lq2 . (4.5)

iii) When τ1 = 0,τ2 = 1 then S p
(0,1) : Lq

′

1 × Lq1 × Lq2 × Lq
′

2 → Lq is continuous, in particular

‖S p
(0,1)
φ,ψ ( f , g)‖Lq ≤ ‖f ‖

L
q
′

1
‖φ‖Lq1 ‖g‖Lq2 ‖ψ‖

L
q
′

2
. (4.6)

iv) When τ1 = τ2 = 1 then S p
(1,1) : Lq1 × Lq

′

1 × Lq2 × Lq
′

2 → Lq is continuous, in particular

‖S p
(1,1)
φ,ψ ( f , g)‖Lq ≤ ‖f ‖Lq1 ‖φ‖

L
q
′

1
‖g‖Lq2 ‖ψ‖

L
q
′

2
. (4.7)

v) When τ1 = τ2 = 0 then S p
(0,0) : Lq

′

1 × Lq1 × Lq
′

2 × Lq2 → Lq is continuous, in particular

‖S p
(0,0)
φ,ψ ( f , g)‖Lq ≤ ‖f ‖

L
q
′

1
‖φ‖Lq1 ‖g‖

L
q
′

2
‖ψ‖Lq2 . (4.8)

Proof. It is an easy consequence of Proposition 7 and the generalized Hölder’s inequality

‖f g‖Lq ≤ ‖f ‖Lq1 ‖f ‖Lq2 for
1

q1
+

1

q2
=

1

q
, q1 ≥ q,

We recall now some notations. We indicate with H(supp f ) the convex hull of supp f and

with Πx,Πw the orthogonal projections on the first and the second factor in R
d
x ×R

d
w respec-

tively. Properties on the support of time-frequency representations is a widely studied subject

because too large projections Πx and Πw of the support of a representation in comparison

with the supports of the signal itself and its Fourier transform respectively would indicate a

“spreading” of the energy that is seen as disturbance in the applications, see for instance [12].

We have the following basic results.

Lemma 9. Let W i gτ( f , g) be the τ-Wigner representation defined by (1.5); then

Πx(suppW i gτ( f , g)) ⊂ H(supp f + supp g). (4.9)

and

Πw(suppW i gτ( f , g)) ⊂ H(supp f̂ + supp ĝ). (4.10)

Proof. Suppose that W i gτ( f , g)(x,ω) 6= 0, then there exists t ∈ R
d such that f ( y1) 6= 0 and

g( y2) 6= 0 with y1 = x +τt and y2 = x − (1 −τ)t. On the other hand x = λy1 +µy2 with λ = 1 −τ

and µ = τ, i.e. x can be written as convex linear combination of y1 and y2. We have therefore



182 Boggiatto Paolo, et. al CUBO
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that x belongs to the segment [ y1, y2] and (4.9) follows then immediately. To obtain (4.10) we

just need to recall that

W i gτ( f , g)(x, w) = W i gτ( f̂ , ĝ)(w,−x).

and repeat the argument above with x replaced by ω.

From (4.9), (4.10), and the equality supp( f g) = supp f ∩ supp g, we obtain the “support”

property of the parameterized two window spectrogram.

Proposition 10. The support of the parameterized two window spectrogram satisfies the fol-

lowing properties:

Πx(suppS p
(τ1 ,τ2 )
φ,ψ ( f , g)) ⊂ H(supp f + suppφ̃) ∩ H(supp g + suppψ̃) (4.11)

and

Πw(suppS p
(τ1 ,τ2 )
φ,ψ ( f , g)) ⊂ H(supp f̂ + supp

ˆ̃φ) ∩ H(supp ĝ + supp ˆ̃ψ). (4.12)

Remark 11. The meaning of the Proposition 10 becomes even more evident if we consider the

case where f = g is a signal and we suppose that one window is well localized in time and the

other one in frequency. Assume for example that supp φ ⊂ Bδ and supp ψ̂ ⊂ Bδ, with Bδ ball of

radius δ > 0, then Proposition 10 implies that

supp S p
(τ1 ,τ2 )
φ,ψ ( f , f ) ⊂ H(supp f + B

δ) × H(supp f̂ + Bδ),

i.e. we have good localization both in time and in frequency, having reduced the spread of the

energy to a ball of radius δ with respect to each variable.

Finally we prove that the parameterized two window spectrogram, in general, does not

belong to the Cohen class. Let us consider for simplicity the case τ1 = τ2 := τ in Definition

4, with τ 6= 12 ( actually for τ =
1
2 , the representation S p

( 12 ,
1
2 )

φ,ψ ( f , g) belongs to the Cohen class,

since, as proved in [3], it coincides with S pφ,ψ( f , g) ). We denote for shortness S p
τ
φ,ψ( f , g) :=

S p
(τ,τ)
φ,ψ ( f , g); the following proposition holds.

Proposition 12. For τ 6= 12 there does not exist a tempered distribution σ = στ,φ,ψ ∈ S
′(R2d)

such that

S p
τ
φ,ψ = σ∗ W i g, (4.13)

i.e. S pτ
φ,ψ( f , g) = σ∗ W i g( f , g) for every f , g ∈ S (R

d).



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Generalized Spectrograms ... 183

Proof. By Definition 4 and simple changes of variables we have:

S p
τ
φ,ψ( f , g) = 4

−d

ˆ

e
−2πit ω2 f

( x
2
+τt

)
φ̃

( x
2
− (1 −τ)t

)
dt

ˆ

e
2πit ω2 g

( x
2
+τt

)
ψ̃

( x
2
− (1 −τ)t

)
dt

=

ˆ

e
−2πisω

f (2τs)φ
(
2(1 −τ)s −

x

2τ

)
ds

ˆ

e
−2πisω

g(−2τs)ψ
(
−2(1 −τ)s −

x

2τ

)
ds.

By standard properties of the Fourier transform we can write the inverse Fourier transform

of S pτ
φ,ψ( f , g)(x,ω) in the following way:

F
−1
x→t
ω→ξ

(
S p

τ
φ,ψ( f , g)(x,ω)

)
=

= F
−1
x→t

[
f (2τξ)φ

(
2(1 −τ)ξ−

x

2τ

)]
∗ F

−1
x→t

[
g(−2τξ)ψ

(
−2(1 −τ)ξ−

x

2τ

)]

= (2τ)2d
[
e

2πi(4τ(1−τ))tξ
f (2τξ)φ̂(2τt)

]
∗

[
e
−2πi(4τ(1−τ))tξ

g(−2τξ) ψ̂(2τt)
]
,

where the convolution is performed in both the variables (t,ξ). Finally, writing explicitly the

convolution, we obtain

F
−1
x→t
ω→ξ

(
S p

τ
φ,ψ( f , g)(x,ω)

)
= (2τ)2d e2πi(4τ(1−τ))tξ

ˆ

e
−2πi(4τ(1−τ))tx

f (2τ(ξ− x)) g(−2τx) dx
ˆ

e
−2πi(4τ(1−τ))ξs

φ̂(2τ(t − s))ψ̂(2τs) ds.

(4.14)

We observe that, by the definition of the Wigner transform,

F
−1
x→t
ω→ξ

(
W i g( f , g)

)
= F

−1
x→t
ω→ξ

[
Fs→ω

(
f
(
x +

s

2

)
g
(
x −

s

2

))]

=

ˆ

e
2πixt

f
(
x +

ξ

2

)
g
(
x −

ξ

2

)
dx.

(4.15)

Now let us suppose that (4.13) holds for some tempered distribution σ; by taking the inverse

Fourier transform and using (4.14) and (4.15), the following should be verified for every f , g ∈

S (Rd ):

(2τ)2d e2πi(4τ(1−τ))tξ
ˆ

e
−2πi(4τ(1−τ))tx

f (2τ(ξ− x)) g(−2τx) dx
ˆ

e
−2πi(4τ(1−τ))ξs

φ̂(2τ(t − s))ψ̂(2τs) ds

= σ̌(t,ξ)

ˆ

e
2πixt

f
(
x +

ξ

2

)
g
(
x −

ξ

2

)
dx,

(4.16)



184 Boggiatto Paolo, et. al CUBO
12, 3 (2010)

where σ̌(t,ξ) is the inverse Fourier transform of σ. In particular, (4.16) should hold for f and

g of the following type:

f (s) = e−πλs
2
, g(s) = e−πµs

2
,

for every λ,µ > 0. In this case we can compute explicitly the integrals involving f and g in

(4.16) and we have:
ˆ

e
−2πi(4τ(1−τ))tx

e
−πλ(2τξ−2τx)2

e
−πµ(−2τx)2

dx =

= e
−4π

λµ
λ+µ

τ2ξ2
ˆ

e
−2πi(4τ(1−τ))tx

e
−π

(
2(λ+µ)1/2τx− 2λτ

(λ+µ)1/2
ξ

)2

dx

= (2τ
√

λ+µ)−d e
−4π

λµ
λ+µ

τ2ξ2
e
−2πi λ

λ+µ
4τ(1−τ)tξ

ˆ

e
−2πi 2(1−τ)

(λ+µ)1/2
t y

e
−πy2

d y

= (2τ
√

λ+µ)−d e
−2πi λ

λ+µ
4τ(1−τ)tξ

e
−4π

λµ
λ+µ

τ2ξ2
e
−π

4(1−τ)2

λ+µ
t2

.

(4.17)

Similarly we obtain that
ˆ

e
2πixt

f
(
x +

ξ

2

)
g
(
x −

ξ

2

)
dx = (

√
λ+µ)−d e

−2πi λ
λ+µ

tξ
e
−π

λµ
λ+µ

ξ2
e
−π 1

λ+µ
t2

. (4.18)

Now, replacing (4.17) and (4.18) in (4.16) we have for σ̌(t,ξ) the following expression

σ̌(t,ξ) = (2τ)d e
2πi(4τ(1−τ))tξ−πitξ−2πi λ

λ+µ
(4τ(1−τ)−1)tξ

e
−π

4λµτ2−λµ
λ+µ

ξ2
e
−π

4(1−τ)2−1
λ+µ

t2
ˆ

e
−2πi(4τ(1−τ))ξs

φ̂
(
2τ(t − s)

)
ψ̂(2τs) ds.

(4.19)

For τ 6= 12 we deduce then that σ̌(t,ξ) necessarily depends on the two parameters λ and µ, and

this is impossible since σ in (4.13) is independent of f and g.

Remark 13. We also observe that in the case τ = 1/2 all terms in (4.19) involving the param-

eters λ and µ cancel, making σ independent of them, and confirming, as expected, that in this

case the representation is in the Cohen class.

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Generalized Spectrograms ... 185

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