- Source: Modulation space
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- Modulation space
- Modulation
- Space vector modulation
- Space modulation
- Amplitude modulation
- Quadrature amplitude modulation
- Pulse-amplitude modulation
- Single-sideband modulation
- Frequency modulation
- Pulse-width modulation
Modulation spaces are a family of Banach spaces defined by the behavior of the short-time Fourier transform with
respect to a test function from the Schwartz space. They were originally proposed by Hans Georg Feichtinger and are recognized to be the right kind of function spaces for time-frequency analysis. Feichtinger's algebra, while originally introduced as a new Segal algebra, is identical to a certain modulation space and has become a widely used space of test functions for time-frequency analysis.
Modulation spaces are defined as follows. For
1
≤
p
,
q
≤
∞
{\displaystyle 1\leq p,q\leq \infty }
, a non-negative function
m
(
x
,
ω
)
{\displaystyle m(x,\omega )}
on
R
2
d
{\displaystyle \mathbb {R} ^{2d}}
and a test function
g
∈
S
(
R
d
)
{\displaystyle g\in {\mathcal {S}}(\mathbb {R} ^{d})}
, the modulation space
M
m
p
,
q
(
R
d
)
{\displaystyle M_{m}^{p,q}(\mathbb {R} ^{d})}
is defined by
M
m
p
,
q
(
R
d
)
=
{
f
∈
S
′
(
R
d
)
:
(
∫
R
d
(
∫
R
d
|
V
g
f
(
x
,
ω
)
|
p
m
(
x
,
ω
)
p
d
x
)
q
/
p
d
ω
)
1
/
q
<
∞
}
.
{\displaystyle M_{m}^{p,q}(\mathbb {R} ^{d})=\left\{f\in {\mathcal {S}}'(\mathbb {R} ^{d})\ :\ \left(\int _{\mathbb {R} ^{d}}\left(\int _{\mathbb {R} ^{d}}|V_{g}f(x,\omega )|^{p}m(x,\omega )^{p}dx\right)^{q/p}d\omega \right)^{1/q}<\infty \right\}.}
In the above equation,
V
g
f
{\displaystyle V_{g}f}
denotes the short-time Fourier transform of
f
{\displaystyle f}
with respect to
g
{\displaystyle g}
evaluated at
(
x
,
ω
)
{\displaystyle (x,\omega )}
, namely
V
g
f
(
x
,
ω
)
=
∫
R
d
f
(
t
)
g
(
t
−
x
)
¯
e
−
2
π
i
t
⋅
ω
d
t
=
F
ξ
−
1
(
g
^
(
ξ
)
¯
f
^
(
ξ
+
ω
)
)
(
x
)
.
{\displaystyle V_{g}f(x,\omega )=\int _{\mathbb {R} ^{d}}f(t){\overline {g(t-x)}}e^{-2\pi it\cdot \omega }dt={\mathcal {F}}_{\xi }^{-1}({\overline {{\hat {g}}(\xi )}}{\hat {f}}(\xi +\omega ))(x).}
In other words,
f
∈
M
m
p
,
q
(
R
d
)
{\displaystyle f\in M_{m}^{p,q}(\mathbb {R} ^{d})}
is equivalent to
V
g
f
∈
L
m
p
,
q
(
R
2
d
)
{\displaystyle V_{g}f\in L_{m}^{p,q}(\mathbb {R} ^{2d})}
. The space
M
m
p
,
q
(
R
d
)
{\displaystyle M_{m}^{p,q}(\mathbb {R} ^{d})}
is the same, independent of the test function
g
∈
S
(
R
d
)
{\displaystyle g\in {\mathcal {S}}(\mathbb {R} ^{d})}
chosen. The canonical choice is a Gaussian.
We also have a Besov-type definition of modulation spaces as follows.
M
p
,
q
s
(
R
d
)
=
{
f
∈
S
′
(
R
d
)
:
(
∑
k
∈
Z
d
⟨
k
⟩
s
q
‖
ψ
k
(
D
)
f
‖
p
q
)
1
/
q
<
∞
}
,
⟨
x
⟩
:=
|
x
|
+
1
{\displaystyle M_{p,q}^{s}(\mathbb {R} ^{d})=\left\{f\in {\mathcal {S}}'(\mathbb {R} ^{d})\ :\ \left(\sum _{k\in \mathbb {Z} ^{d}}\langle k\rangle ^{sq}\|\psi _{k}(D)f\|_{p}^{q}\right)^{1/q}<\infty \right\},\langle x\rangle :=|x|+1}
,
where
{
ψ
k
}
{\displaystyle \{\psi _{k}\}}
is a suitable unity partition. If
m
(
x
,
ω
)
=
⟨
ω
⟩
s
{\displaystyle m(x,\omega )=\langle \omega \rangle ^{s}}
, then
M
p
,
q
s
=
M
m
p
,
q
{\displaystyle M_{p,q}^{s}=M_{m}^{p,q}}
.
Feichtinger's algebra
For
p
=
q
=
1
{\displaystyle p=q=1}
and
m
(
x
,
ω
)
=
1
{\displaystyle m(x,\omega )=1}
, the modulation space
M
m
1
,
1
(
R
d
)
=
M
1
(
R
d
)
{\displaystyle M_{m}^{1,1}(\mathbb {R} ^{d})=M^{1}(\mathbb {R} ^{d})}
is known by the name Feichtinger's algebra and often denoted by
S
0
{\displaystyle S_{0}}
for being the minimal Segal algebra invariant under time-frequency shifts, i.e. combined translation and modulation operators.
M
1
(
R
d
)
{\displaystyle M^{1}(\mathbb {R} ^{d})}
is a Banach space embedded in
L
1
(
R
d
)
∩
C
0
(
R
d
)
{\displaystyle L^{1}(\mathbb {R} ^{d})\cap C_{0}(\mathbb {R} ^{d})}
, and is invariant under the Fourier transform. It is for these and more properties that
M
1
(
R
d
)
{\displaystyle M^{1}(\mathbb {R} ^{d})}
is a natural choice of test function space for time-frequency analysis. Fourier transform
F
{\displaystyle {\mathcal {F}}}
is an automorphism on
M
1
,
1
{\displaystyle M^{1,1}}
.