Error Characterization of Duty Cycle Estimation for Sampled Non-Band-Limited Pulse Signals With Finite Observation Period Hans-Peter Bernhard, Bernhard Etzlinger and Andreas Springer Johannes Kepler University Linz Institute for Communications Engineering and RF-Systems Altenbergerstr. 69, 4040 Linz, Austria, Email: h.p.bernhard@ieee.org Abstract—In many applications the pulse duration of a peri- odic pulse signal is the parameter of interest. Thereby, the non- band-limited pulse signal is sampled during a finite observation period yielding to aliasing and windowing effects, respectively. In this work, the pulse duration estimation based on the mean value of the samples is considered, and an exact expression of the mean squared estimation error (averaged over all possible time shifts) is derived. The resulting mean squared error expression depends on the observation period, the pulse period and the pulse duration. Analyzing the effect of these parameters shows that the mean squared error can be reduced (i) if the observation period is a multiple of the pulse period, (ii) if the pulse period is not a multiple of the sampling period, and (iii) if the total number of samples is a prime number. All results were validated with simulation results. Index Terms—sampling process, band width, signal reconstruc- tion, sampling error, wireless sensor networks (WSN), synchro- nization, localization, ultrasonic. I. I NTRODUCTION Following the established sampling theory, band-limited signals can be perfectly reconstructed from a set of samples that are collected with a sampling frequency larger than the double occupied bandwidth [1], [2]. However, this is only true if the signal is sampled during its infinitely long duration. In practical applications neither an infinite observation time can be achieved, nor the underlying signals are band-limited (e.g., discrete valued and continuous-time signals are not band- limited). Consider time-duration measurements in periodic signals. For example, in ultrasonic reflection measurements for ranging [3], [4], pulses are emitted periodically, and a device counts the number of clock cycles between the emission of the pulse and the reception of the reflection. This setting corresponds to the sampling of a rectangular pulse signal, that is high between emission and reception, and low until the next emission. The pulse duration, i.e., the time that the pulse is on high, is used to determine the range. Another example is round-trip time (RTT) based clock synchronization for which clocks are modeled with discrete events. The corresponding RTT measurements are samples of a pulse function. In that scenario, the pulse duration is related to the propagation delay and the clock offset between two communication nodes [5], [6]. The two examples mentioned use a pulse shaped signal to determine key system parameters. Thereby, both conditions of the sampling theorem are violated, the bandwidth of the signal is infinite and the observation period is limited. Although the signals are considered as noise free, sampling and aliasing distorts the signal reconstruction. In this work the question is raised how accurately signal parameters can be estimated from non band-limited signals with a finite number of samples. Previous contributions have already derived upper bounds on the squared error [7]. Here, for the function space of sampled pulse signals, an exact formulation of the averaged squared error on the estimation of the pulse duration is derived. Moreover, the error analysis gives rise to the following design suggestions in technical systems: • Predicting the estimation accuracy for a given number of samples in a given observation period; • Selecting the number of pulse signal periods (i.e., the total observation period) to achieve a desired estimation accuracy. • Selecting the number of samples to achieve the minimum estimation error for a given sampling period and obser- vation period. II. PRELIMINARIES Sampling is the obvious starting point in all discrete signal processing applications that have a relation to real world processes. In the following, signals x(t) are considered that are integrable in the sense of Lebesgue, i.e., L 1 (R)=  x(t)      ∞ −∞ | x(t) | dt< ∞  . (1) Sampling x(t) with a period T yields the sampled signal x s (n) ∈  x(nT )   n ∈ Z ∧ T = 1 2B ∧ B< ∞  . (2) To reconstruct x(t), the samples x s (n) are interpolated, i.e., x i (t)= ∞  n=−∞ x s (n) sin(2πB(t − nT )) π(t − nT ) , (3) is the interpolated signal. As commonly known [1], [2] x i (t) ≡ x(t) , (4) if and only if x(t) is band-limited by B =[−B,B[. 2016 24th European Signal Processing Conference (EUSIPCO) 978-0-9928-6265-7/16/$31.00 ©2016 IEEE 2136