Figure 2. Filtering exponentials convolved with a system response function.
(a) Exponential on the interval [−1, 1] with Poisson noise added. Amplitude, 
, time constant 
. (b) Legendre spectrum of x as resulting from eq. 5. (c) Mean 
of 
(continuous) and inverse fLT (dashed, eq. 6) of 
through 
of the spectrum shown in b. (d) Legendre spectrum of the mean 
, largely lacking higher noise components. (e) Noisy curve is the convolution of 
with 
and 
. 
was chosen such that the curve overlaps with 
for large 
. (f) Legendre spectrum (gray bars) of convoluted noisy exponential shown in e (continuous curve). The lowpass-filtered inverse transform is shown in e (continuous curve) and approximates the convoluted noisy exponential. In addition, f shows the Legendre spectrum of 
, obtained through eq. 9. The lowpass-filtered inverse transform of this spectrum is shown as the red dashed curve in e and approximates the original non-convoluted exponential, from which the noisy convoluted curve was generated.