|Ph.D Student||Levinson Yaron|
|Subject||Signal Reconstruction with Finite Impulse Response|
Constraints: A System Theoretic Approach
|Department||Department of Mechanical Engineering||Supervisor||Professor Leonid Mirkin|
|Full Thesis text|
Interpolation (or signal reconstruction) is a fundamental operation in digital signal processing. Among its many applications are sampling rate conversion (mostly in audio applications), pulse shaping in communication, geometric transformation of images, etc.
This work explores a new method for enforcing finite support constraint in the design of signal reconstructors, which might be beneficial in many applications. The first part is dedicated to the development of a solution to a pure discrete FIR estimation problem. Formalizing the problem in state-space terms enabled operations over large matrices be carried by manipulation of systems, which leads to computationally efficient and numerically stable solution. The
use of a broader family of state-space systems, namely descriptor implicit systems, alleviated the need for restrictive assumptions that are common in the literature. This approach can also meet asymptotic constraints via the use of unstable weighing functions. As a result, necessary and sufficient conditions for the stabilization by an FIR system in one-sided model matching problem was derived; a result that stand on its own.
The second part of the thesis extends the discrete solution to hybrid systems, which leads to L2-optimal FIR interpolators (reconstructors). To this end, the lifting transformation is used to reduce the problem to a pure discrete estimation problem in the lifted domain. The main technical challenge in this part is to handle infinite-dimensional parameters of the lifted problem. This is successfully carried out and results in closed-form formulae. The solution comprises an FIR (discrete) filter and a generalized zero-order (i.e., having finite support) hold. It requires two discrete Riccati equations and two matrix recursions.
In the last part of the thesis, two applications are studied. The first one demonstrates advantages of the derived theoretical results in (2D) image processing tasks. It is shown that the proposed algorithm outperforms available (ad hoc) FIR interpolators and is about 20% faster than the (compatible performancewise) IIR cardinal splines. The second application is a real world visual tracking problem, which involves velocity estimation from position measurements and a subsequent resampling of this signal to adjust measurements to the sampling rate of the associated controller .