The U3 function is a non-linear operator that is a two-sided difference equation. It can be characterized by two parameters: 1) the lateral size window (k)―this is the spacing between the samples whose difference is found, and 2) the size of its central window (N)―which is the spacing between the two side differences that were found from the lateral size window. A diagram of the samples of interest that are retrieved from the input signal to use in the U3 function is shown in Figure 1, where i is the index of the input signal, N = 2m is the size of the central window for given m ≥ 2, and k is the size of the lateral window, where.

After the samples of interest are retrieved from the input signal using the block diagram in Figure 1, the U3 functionat sample i is given in Equation (1) for an input signal x(n).

Geometrically, the U3 function finds the length of the secant line that connects the samples retrieved from lateral window. Using the U3 function, the output signal y(n) evaluated at the i^th sample is given by the recursive function shown in Equation (2), where y(0) = 0.

It can be seen from Equation (1) that so long as the central window size is smaller than the width of a pulse, the rightmost lateral window will not reach the next discontinuity until the left most lateral window has completed its operation on the current discontinuity. In addition, in areas where there is not much transition between the samples, such as the continuous parts of the signal, the U3 function has very little effect on the output signal y(i). The noise of the input signal has also been minimized because of its small transitions.

For a given input signal from one of the aforementioned therapeutic devices, the output signalis a train of pulses of width N whose rising edge occurs when the right lateral window coincides with the discontinuity, and falling edge occurs when the left lateral window coincides with the discontinuity. The number of pulses in the

output signal is equal to the number of discontinuities of the input signal. It should be further noted that if the lateral window size is set so that k = 1, then the U3 function is a backward implementation of squared derivatives, and thus can be thought of as very high-pass filters. These high-pass filters then filter out everything but the discontinuities within the signal, and thus identify the beginning and trailing edges of each pulse.

The U3 algorithm requires less computational recourses than other signal analysis methods such as Fourier transform and wavelet analysis, and reduces the time required for the data analysis and the testing. The algorithm is simple and can be implemented with information of the medical device output signal characteristics including pulse rate, number of outputs, and rise and fall time of the signal and varying k and N parameters of the algorithm. The algorithm runs in linear time, O(N), where N is the number of samples in the signal, rather than the those required by finding the Fourier Transform,. This makes it a great candidate for testing of continuous operation of medical devices under a large number of conditions.

Signals from various therapeutic devices are first analyzed to identify features that could be used to identify a pulse within the input signal. Comparing 30 pulses that will be used as the input to the U3 function from different pacemakers, neurostimulators, ICDs, and TENS devices, it was determined that there are two types: 1) Pulse with two discontinuities and 2) Pulse with three discontinuities, as shown in Figure 2. The discontinuities that are identified within the signal and the known pulse period are used to extract the pulse width, exponential decay, and inter-pulse time.

A complete flowchart of the testing procedure is shown in Figure 3 and was implemented using MATLAB. The sampled signal from the medical device, s(n), is first passed to the recursive function Equation (2) that was described in the previous section.

An adaptive threshold is applied to the output of the recursive function in Equation (2), to find the peaks (left column of Figure 4). The adaptive threshold to identify a peak is to be two standard deviations above the noise floor measured within the signal. This identifies the locations of all of the discontinuities within the signal. In other words, if this was a baseline signal, where there are no expected anomalies and everything is working properly, the locations extracted are the locations of the leading and trailing edges of each pulse. The algorithm assumes the information of the basic characteristics of the output signal such as pulse width, pulse period, and number of the pulses for each delivery of therapy (e.g., device with single vs. three outputs) of the medical devices is available for the baseline analysis. If sampling of the therapeutic device output signal has begun too early or too late, then the beginning and/or the end of the signal, s(n), will contain information from only a portion of a pulse. These portions must be filtered out from the data before checking for anomalies. This is accomplished by checking to see if the spacing between discontinuities is on the order of the pulse period within the first and last sampling periods of the signal. That is, the first and last periods of the output signal from Equation (2) are extracted. Next, the spacing between the discontinuities is identified within the first and last period. If the spacing is on the order of a programmed pulse period, then the pulse is marked as one with missing information and is removed.

The next step is to determine whether the pulse has two or three discontinuities. This can be expressed in terms of f, the operating frequency or number of pulses per second, T, the total time the signal was sampled, and N_peaks, the number of peaks detected and is given in Equation (3).

N_{Discontinuity} is checked to ensure that it is either two or three. If the device is effected by the exposure and part of a pulse is inhibited―for instance, if the third discontinuity of a pulse with three discontinuities is inhibited― then this number will not be recognized as a valid value since it will not be two or three. It is clear that during baseline measurements, this number will always be valid, i.e. either two or three, and that was indeed observed during the experiments. It should be noted that a device can either output a two-discontinuity pulse or a three- discontinuity pulse. The two will never mix together and still be a valid signal. In other words, if a two discontinuity pulse is detected within a three-discontinuity pulse signal, the signal will be marked as effected by the exposure.

If the measurement is a baseline measurement, the output signal from Equation (2) is used to determine the mean of the pulse width and inter-pulse time of a 15 second measurement. The exponential decay is found by corresponding the output signal from Equation (2) to the original signal (right column of Figure 4). The samples from the original signal are then used to find the best fit exponential function using least squares method for

each pulse in the 15 second signal. The mean value of the exponential decay of all the pulses within the 15 second interval is found and stored with the pulse width and inter-pulse time. These values are stored for a 15 second sample of the signal.

If it is not a baseline measurement and N_{Discontinuity} is valid, then after the properties of each pulse are extracted from the signal, they are compared to the stored baseline measurement. In other words, the pulse width, exponential decay, and inter-pulse time that was extracted from the signal, are compared to the baseline measurements. If these parameters are larger than one standard deviation, which was chosen based on empirical measurements, from the stored baseline measurements, then the signal is marked as an affected signal.

The success of this method depends on extracting the baseline measurements without any field exposure accurately. In addition, the parameters must converge within a 15 second window so as to be able to make the appropriate comparisons.

The output of the recursive function Equation (2), and the corresponding features extracted from the original pulses are shown in Figure 4. Figure 5 shows the convergence of the parameters well under 15 pulses for a device operating at 1 Hz. The other devices tested showed similar results and are listed in Table 1.

It can be seen that there is a rather large difference in scale on the y-axis between Figures 5(a)-(c). This is due to the fact that the sampling time is on the order of 6 μs which is on the order of the pulse width―450 μs. The pulse rate was three orders of magnitude larger at 1 s and microsecond deviations do not show as large an effect.

Data from a security system simulator designed at the FDA [7] [8] for EMC tests on therapeutic devices were used to test the algorithm. Table 2 lists a summary of devices that were tested in the presence of electromagnetic radiation at various modulations and carrier frequencies. The output signal from total 14 sample medical devices was tested using the U3 algorithm. The medical device output signals used in the study were recorded for 15 seconds period with 150 kHz sampling rate. The table lists the type of device, namely a pacemaker, neurostimulator, ICD, or TENS, the rate the device would send output pulses, and the number of those devices tested at that rate. An operator observed the output signal of the medical device, which is shown in the “Observed Effects” column of Table 2 and compared that to automated labeling presented by the algorithm, shown in the “Detected Effects” column of Table 2. Data from the previous studies [7] showed EMI effects could either inhibit pulses or add extraneous noise pulses to the signal. Therefore, these were the two primary effects that were used as test data for various medical devices. Some of the effects were simulated by manipulating baseline data to see if the algorithm could detect the effect accordingly.

^*Device with single output (single pulse) per delivery of therapy; ^**device with three output (three pulses) per delivery of therapy.

^*Indicates simulated data was used; ^**indicates false positives detected by algorithm.

The results show that the algorithm performs as well as an observer in identifying effected signals. However, at times, the output of the U3 produced a “long peak,” which is a wide valley rather than a sharp peak at one of the discontinuities, and thus the peak detector could not correctly identify those peaks. These were false positives that were reported where the algorithm detected an effect when there was no effect. In these cases, the peak detector used could not identify some of the peaks from the U3 algorithm appropriately and thus N_{Discontinuity} was not a valid number and the signal was reported to be effected. The algorithm never failed to detect an effect, though it did report a few false positives. However, for the purpose of the device validation testing, this is considered acceptable due to the fact that if an effect within the signal is detected, the test needs to be checked for its repeatability.