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. For , a non-negative function on and a test function , the modulation space
is defined by
In the above equation, denotes the short-time Fourier transform of with respect to evaluated at , namely
In other words, is equivalent to . The space 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.[3]
- ,
where is a suitable unity partition. If , then .