On a Novel Time-Varying Up-Sampling Rate (TVUSR) Structure and Its Statistical Properties

Several real-world signals exhibit semi-periodicity in that the period of repetition varies from pulse to pulse about a mean value instead of being constant. Some examples, including among others, are ECG signals, voiced phonemes in speech, carrier jitter in communication etc. In order to model/generate such signals, one can pass a train of discrete-time delta functions, having zeros between two ones in a semi-periodic manner, through an LTI system with the impulse response matched to the pulse. Strictly speaking, the number of zeros between two ones should be a random variable, with the mean value being an average period of repetition. As a first step towards this objective, we propose a time-varying up-sampling rate
$\left(\text{TVUSR}-Q\right)$
structure, where
$Q$
up-samplings, namely
$\uparrow\left(L_{1}+1\right),\cdots,\uparrow\left(L_{Q}+1\right)$
are introduced periodically. We demonstrate that the proposed
$\text{TVUSR}-Q$
, followed by the LTI system, results in the DT-LTI system/
$\text{DT-TVUSR}-Q$
system under distinct up-sampling rate conditions. Since modeling of any real-life signal mentioned earlier would be effected by passing a stochastic input through such a system, we investigate the spectral properties of the outputs under various input conditions. We demonstrate the proposed structure's utility by synthesizing ECG signals. In order to demonstrate the utility of the proposed structure we take a segment of ECG and, after extracting some required information from it, synthesize the same using the proposed
$\text{DT-TVUSR}-15$
system. Simulation findings validate the proposed theory.