An overcomplete stitching algorithm for time decoding machines

Abstract

We investigate a class of finite-dimensional time decoding algorithms that: 1) is insensitive with respect to the time-encoding parameters; 2) is highly efficient and stable; and 3) can be implemented in real time. These algorithms are based on the observation that the recovery of time encoded signals given a finite number of observations has the property that the quality of signal recovery is very high in a reduced time range. We show how to obtain a local representation of the time encoded signal in an efficient and stable manner using a Vandermonde formulation of the recovery algorithm. Once the signal values are obtained from a finite number of possibly overlapping observations, the reduced-range segments are stitched together. The signal obtained by segment stitching is subsequently filtered for improved performance in recovery. Finally, we evaluate the complexity of the algorithms and their computational requirements for real-time implementation.

Error analysis and implementation considerations of decoding algorithms for time-encoding machine

“…Using the same basis, the reconstruction process can also be formulated by a Vandermonde system as in [3]:…”

“…For all these methods, we make the bases finite by applying a window function w(t) to cut the signal into clips as in [3]:…”

“…In contrast, if we set the tolerance level of the other two methods to a low value to boost the low noise performance, their performance would be much worse at high noise level, as shown in Figure 5c (for example the performance of the sinc basis-blue curve-is 7.7 dB worse than in Figure 5a when input ENOB is 6). An interesting observation from Figure 5c is that even though the Toeplitz matrix also has a large condition number, it is not sensitive to the tolerance level until a critical level because of its robustness against small noise [3,7]. When the tolerance is below 2.5e-13, its output ENOB cannot pass 10.5 bits.…”

“…This phenomenon is known as the Runge phenomenon. Employing 2M time points outside the reconstruction window is suggested in [3]. Setting M to a large value reduces the boundary effect and improves the reconstruction result, but the improvement levels off quickly.…”

“…However, the reconstruction algorithm requires the inversion of an infinite matrix. This problem can be solved by reconstructing small intervals of the signal ("clips") and stitching these "clips" together [3]. A Toeplitz formulation of the reconstruction problem was proposed by Lazar and co-workers [4] to increase the speed of the reconstruction algorithm.…”