Filtering exponentials convolved with a system response function.

Abstract

<p>(<b>a</b>) Exponential on the interval [βˆ’1, 1] with Poisson noise added. Amplitude, , time constant . (<b>b</b>) Legendre spectrum of x as resulting from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0090500#pone.0090500.e283" target="_blank">eq. 5</a>. (<b>c</b>) Mean of (continuous) and inverse fLT (dashed, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0090500#pone.0090500.e306" target="_blank">eq. 6</a>) of through of the spectrum shown in <b>b</b>. (<b>d</b>) Legendre spectrum of the mean , largely lacking higher noise components. (<b>e</b>) Noisy curve is the convolution of with and . was chosen such that the curve overlaps with for large . (<b>f</b>) Legendre spectrum (gray bars) of convoluted noisy exponential shown in <b>e</b> (continuous curve). The lowpass-filtered inverse transform is shown in <b>e</b> (continuous curve) and approximates the convoluted noisy exponential. In addition, <b>f</b> shows the Legendre spectrum of , obtained through <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0090500#pone.0090500.e321" target="_blank">eq. 9</a>. The lowpass-filtered inverse transform of this spectrum is shown as the red dashed curve in <b>e</b> and approximates the original non-convoluted exponential, from which the noisy convoluted curve was generated.</p

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