Modulation spaces[1] 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,[2] 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.
, a non-negative function
m ( x , ω )
and a test function
, the modulation space
is defined by In the above equation,
denotes the short-time Fourier transform of
evaluated at
( x , ω )
, namely In other words,
is the same, independent of the test function
chosen.
The canonical choice is a Gaussian.
We also have a Besov-type definition of modulation spaces as follows.
is a suitable unity partition.
m ( x , ω ) = ⟨ ω
m ( x , ω ) = 1
, the modulation space
is known by the name Feichtinger's algebra and often denoted by
for being the minimal Segal algebra invariant under time-frequency shifts, i.e. combined translation and modulation operators.
is a Banach space embedded in
, and is invariant under the Fourier transform.
It is for these and more properties that
is a natural choice of test function space for time-frequency analysis.
Fourier transform