222 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 57, NO. 2, FEBRUARY 2008 Filtering of Randomly Sampled Time-Stamped Measurements Lee Barford, Member, IEEE Abstract—This paper concerns the filtering of measurements that are taken by networked sensors at nonuniform intervals but that are accurately time stamped. Traditional digital filtering methods are difficult or impossible to use due to nonuniform sampling. Two filtering methods are described. Both are based on making an assumption about the signal behavior between measurements, such as the signal being constant between measure- ments. In the first method, a filter is formulated as an ordinary differential equation that is incrementally solved as measurements arrive. Such filtering is general; nonlinear and nontime invariant filters may be constructed. In the second method, signal con- volution with a continuous-time finite impulse response filter is efficiently performed using a spline representation for the filter response. Such filters are “FIR like” in the sense that they have frequency-domain performance similar to FIR filters and have only slightly worse asymptotic computation time and memory requirements compared to FIR filters, yet have the advantage of being able to deal with nonuniformly sampled measurements. Examples of the operation of both sorts of filters are shown on actual measured data. Index Terms—Continuous-time filters, digital filters, intelligent sensors, measurement system data handling. I. I NTRODUCTION T HERE HAS been considerable research and engineering in defining the hardware, architectural, and middleware interfaces needed to realize the vision of measurement systems interconnected by IP networks. However, IP networks that are commonly available today make no guarantees concerning the time that data will be delivered. One way to work around such lack of temporal guarantees in the communications network used as the control and data communications means in a measurement system is to take a so-called time-stamped approach. In such an approach, the times—often nonuniformly spaced—that the samples are taken are accurately recorded. These records are the so-called “time stamps,” after which the approach is named. It is desirable to stabilize inexpensive real-time clocks in networked sensors and to synchronize the clocks in the various sensors and instruments in a measurement system, so that the relative timing of stimuli and responses may be estab- lished. Network Time Protocol [1] or the IEEE 1588 Precision Clock Synchronization Protocol [2], [3] may be used for those purposes. In many measurement applications, it is desirable to digitally filter the measurements. Reasons for doing so include filtering Manuscript received July 15, 2006; revised May 9, 2007. The author is with Agilent Laboratories, Santa Clara, CA 95051 USA (e-mail: lee.barford@agilent.com). Digital Object Identifier 10.1109/TIM.2007.909616 out interfering signals with known frequency content (e.g., interference from 50- or 60-Hz ac power lines) and using low- pass filters to average a number of measurements to improve measurement accuracy. For most previous applications, the designers of the measurement systems have taken care to ensure that samples are obtained at precisely regular intervals so that efficient and well-understood methods, particularly digital finite impulse response (FIR) and infinite impulse response (IIR) filters, may be used. However, in the setting under consideration in this paper, conventional FIR and IIR filters cannot be used because of nonuniform sampling. In many measurement systems, FIR filters are used because of the following reasons: 1) they parsimoniously use memory; 2) they require a small amount of computation (linear in the number of sample rate); 3) they require only a few different mathematical operations (+, ×); and 4) they can be extremely efficiently implemented on inexpensive less capable microprocessors, digital signal processors (DSPs), and field- programmable gate arrays (FPGAs). This paper explores two different approaches to filter design and implementation for such nonuniformly sampled but time- stamped measurements. One approach emphasizes flexibility, permitting nonlinear and time-varying filters. The second ap- proach emphasizes achieving “FIR-like” performance in senses that will be made precise later in this paper. This paper is an extension of the work reported in [4]. II. PROPOSED APPROACHES Both methods are based on the same basic idea but differ greatly in the tradeoff between generality and implementation efficiency. That idea is stated as follows: when the precision and accuracy of the clocks at the sensors and instruments are high and sampling intervals are far from uniform, each measured signal can be treated as a continuous-time signal. A reasonable assumption is made about the behavior of the signal between sampled instants. Typically, the signal is assumed to be constant up until the moment each measurement is made (the zeroth-order hold assumption), changes linearly between measurements (the first-order hold assumption), or is band limited. Once such an assumption is made, an algorithm is developed that simulates the behavior of a continuous system that has the desired filtering characteristics when operating on a piecewise-constant, a piecewise-linear, or a band-limited signal, as the case may be. That algorithm is the desired filter. A straightforward method of filtering time-stamped data would be to convert it to a nontime-stamped uniformly sampled time series by resampling and then applying traditional digital 0018-9456/$25.00 © 2008 IEEE