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Detection of Action Potentials with the ContinuousWavelet TransformZoran Nenadic, D.Sc.February 6, 20081 IntroductionThe problem of detecting transient signals in a noisy environment has been studied fordecades. In the Statistical Signal Detection Theory the presence of a useful signal in abackground noise is normally cast as a hypothesis testing, where under the null hypothe-sis no signal is present. If the statistics of signal to be detected is not perfectly known,usually no uniformly most powerful (UMP) test exists, and the performance of a detectordepends on signal representation [1]. In general, a signal representation can be model basedand expansion based. When no appropriate model for the signal can be found, one usuallyresorts to a “canonical set” of basis function where the signal is projected, giving rise toexpansion coefficients. We can think of these coefficients as of signal representation in anew coordinate system. Depending on the signal representation the detection problem canbe formulated in various domains such as time domain, frequency domain, time-frequencydomain, etc. In time-frequency domain, a signal is projected onto a basis of waveformsthat are localized (subject to Heisenberg uncertainty principle) in both time and frequency,yielding a two-dimensional signal representation Tx(w,t) of a one-dimensional signal x(t).An example of this representation is a windowed Fourier Transform introduced by Gabor.A breakthrough in the theory of ...