| Abstract: |
| Most wireless communications (e.g., WiFi, cellular networks, or modern mobile phone protocols like 5G) rely on a fixed time band and frequency-limited signals, despite the uncertainty principle showing they are technically incompatible. However, frequency-limited signals can be created that are almost limited to a given time band. In this talk, we will investigate how to determine the number of orthogonal functions that are almost limited to specific time and frequency regions.
This question can be investigated using the asymptotic and clustering behaviour of eigenvalues of time-frequency limiting operators. These operators are compact, self-adjoint and positive semi-definite, and understanding their eigenvalue distribution is crucial for improving wireless communication protocols through multiplexing. While the one-dimensional setting is well explored by a series of Bell Labs papers by H. Landau, H. Pollak, D. Slepian, H. Widom and I. Daubechies between 1960-1980, the clustering behaviour of eigenvalues in higher dimensions is a hard and open problem in general.
The higher-dimensional case is a crucial aspect in many applications, particularly in scientific imaging problems such as cryoelectron microscopy (cryo-EM) and MRI, and certain optimal orthogonal systems that are approximately space-limited and bandlimited functions in two or more variables play a vital role in achieving accurate and efficient representation of complex multi-dimensional data.
Although the question is well studied in dimension d=1 for the signals of one-variable, the situation becomes much more complex in higher dimensions, and numerous questions still need to be addressed in this regard, as we aim to address some of them in this talk. This is a joint work with Arie Israel. |
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