Detection of Transient Events in the Presence of Background Noise Wilfried Grange,* ,†,‡ Philippe Haas, ‡,§ Andreas Wild, ‡,§ Michael Andreas Lieb, § Michel Calame, § Martin Hegner, † and Bert Hecht* ,| Centre for Research on AdaptiVe Nanostructures and NanodeVices, Trinity College Dublin, College Green, Dublin 2, Ireland, National Center of Competence for Research in Nanoscale Science Institute of Physics, UniVersity of Basel, Klingelbergstrasse 82, CH-4056 Basel Switzerland, and Nano-Optics and Biophotonics Group, Wilhem Conrad Röntgen Research Center for Complex Materials Systems (RCCM), Department of Experimental Physics 5, UniVersity of Würzburg, Germany ReceiVed: March 5, 2008; ReVised Manuscript ReceiVed: March 5, 2008 We describe a method to detect and count transient burstlike signals in the presence of a significant stationary noise. To discriminate a transient signal from the background noise, an optimum threshold is determined using an iterative algorithm that yields the probability distribution of the background noise. Knowledge of the probability distribution of the noise then allows the determination of the number of transient events with a quantifiable error (wrong-positives). We apply the method, which does not rely on the choice of free parameters, to the detection and counting of transient single-molecule fluorescence events in the presence of a strong background noise. The method will be of importance in various ultra sensing applications. Introduction The discrimination of rare transient events (bursts) above a strong stationary background noise with a high level of confidence is a problem of broad importance in various sensing applications ranging from ultrasensitive optical detection 1–3 in biological assays 4,5 or medical diagnostics 6,7 to electromagnetic sensors 8,9 or defense applications. 7 In general, transient signals are considered detectable either if (i) their amplitude is many standard deviations above the mean value of the noise’s probability distribution and has a narrow distribution or if (ii) the waveform (the duration of the transient event) is clearly distinct from the noise’s characteristic fluctuations in time. Here, we propose a method, which is applicable if the signal bursts are neither large in amplitude nor easily distinguishable from the characteristic fluctuations of the noise. The method is based on a fast converging iterative algorithm, which determines an optimum threshold for the detection and counting of bursts. It provides a user-definable quantitative measure for the probability of false positive events due to the background noise peaks. The reliability of the method is demonstrated by Monte Carlo simulations of the burst detection process. To highlight the method’s potential, we detect and count single-molecule fluo- rescence bursts recorded in the presence of a significant stationary background noise. Results We consider a data set describing a time series of counts per time interval containing rare transient events (bursts) above a significant background noise with a Poissonian distribution. Apart from being sufficiently rare, no further assumptions are made with respect to the amplitude and shape distribution of the transient events superimposed to the background noise. Note that the results presented here are applicable for any type of background distribution as long as its shape is known. Figures 1a and b show as an example a data set representing a time trace of single-molecule fluorescence bursts as well as the respective histogram H(n). Here, n is the number of counts per 100 µs time interval (bin). Fluorescence bursts of various amplitudes are observed above the background noise. Conse- quently H(n) shows a clearly distinguishable main Poissonian noise peak and a tail that accounts for the fluorescence bursts. 3 Signal bursts cannot be fully separated from the background noise since both distributions apparently overlap. To optimally discriminate signal bursts from similar events due to background noise, a threshold must be determined above which a fluctuation is counted as a signal burst. The threshold must on one hand be low enough to miss as few as possible true signal bursts, and on the other hand, it must be high enough to minimize the probability of counting a strong fluctuation of the noise as a signal burst. Wrongly assigned bursts contribute to false positive events, whichsin view of applications e.g. in medical diagnosticss must be avoided or at least kept to a quantifiable error. To determine such an optimum threshold, the probability distribution of the background has to be recovered. Considering the normalized Poissonian distribution P(n), we have P(n) ) e -µ µ n n ! (1) where µ is the mean, and σ ) µ the standard deviation. Assuming that P(n) can be recovered with some degree of accuracy, we may consider the probability distribution of the background alone. This then enables us to determine a threshold for the burst amplitude, , by demanding that the absolute number of time intervals K for which the number of counts n exceeds the threshold  is smaller than a tolerable small number, say R. K() is determined as * Corresponding authors. e-mail: hecht@physik.uni-wuerzburg.de (B.H.); wilfried.grange@tcd.ie (W.G.). † Trinity College Dublin. ‡ These authors equally contributed. § University of Basel. | University of Wu ¨rzburg. J. Phys. Chem. B 2008, 112, 7140–7144 7140 10.1021/jp7114862 CCC: $40.75  2008 American Chemical Society Published on Web 05/14/2008