Detection of Action Potentials with the Continuous Wavelet Transform

The problem of detecting transient signals in a noisy environment has been studied for decades. In the Statistical Signal Detection Theory the presence of a useful signal in a background noise is normally cast as a hypothesis testing, where under the null hypothesis 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 detector depends on signal representation [1]. In general, a signal representation can be model based and expansion based. When no appropriate model for the signal can be found, one usually resorts to a “canonical set” of basis function where the signal is projected, giving rise to expansion coefficients. We can think of these coefficients as of signal representation in a new coordinate system. Depending on the signal representation the detection problem can be formulated in various domains such as time domain, frequency domain, timefrequency domain, etc. In time-frequency domain, a signal is projected onto a basis of waveforms that 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 wavelets offered a powerful alternative to windowed Fourier Transform, where a onedimensional signal x(t) is represented in time-scale domain by virtue of a wavelet transform Tx(a, b).