How to find analytic description of filtered signal I am looking for an exact analytic description of a filtered signal. I have an electronic circuit whose input is a monoexponential decay. First (1) the signal gets filtered by a simple RC-Lowpass. Then (2) a second order Bessel Low Pass Filter is applied.

The respective transfer functions are:
\begin{align}
TF_{\rm RC}(s) &amp;= \frac{R}{s+\frac{1}{R C}}\\
TF_{\rm Bessel}(s) &amp;= \frac{a_1}{a_1+b_1s+s^2}
\end{align}

The input signal is:
$$I_{\rm in}(t) = \frac{V}{R_1 + R_2} + R_1\cdot V_{\rm cmd}\cdot e^{-t\frac{R_1 + R_2}{CR_1R_2}}\frac{1}{R_2(R_1 + R_2)}$$

My idea was to calculate the output in the $s$-domain:
$$ V_{\rm out}(s) = I_{\rm in}(s)\cdot TF_{RC}(s)\cdot TF_{Bessel}(s) $$
and then apply the inverse laplace transform to obtain $V_{\rm out}$(t).
The inverse transform is what troubles me. I tried to calculate it with MATLAB, but it doesn't return an analytic function. I guess this comes from the complex poles of the bessel filter. Could you give me a hint on how to calculate the output?

Edit:
$$ TF(s) =  \frac{6.092e12}{s^3 + 5.367e04*s^2 + 1.038e09*s + 6.074e12}$$

$$ I_{in}(t) = \frac{10e-3}{500e6 + 10e6} + 500e6*10e-3*e^{-t*\frac{500e6 + 10e6}{10e-12*500e6*10e6}}\frac{1}{10e6*(500e6 + 10e6)} $$

$$ TF(s)*I_{in}(s) = \frac{4.9e-21*(1.2e24*s + 2.5e26)}{s*(s + 1.0e4)*(s^3 + 5.4e4*s^2 + 1.0e9*s + 6.1e12)} $$

PFE yields:

$$ TF(s)*I_{in}(s) = \frac{1.1e-9 - 4.8e-11i}{s + 2.2e4 - 1.1e4i} + \frac{1.1e-9 + 4.8e-11i}{s + 2.2e4 + 1.1e4i} + \frac{6.6223e-7}{s + 10200.0} + \frac{-6.6441e-7}{s + 10235.0} + \frac{1.9666e-11}{s}  $$

Inverse Laplace transform of the upper term will yield a complex result. I know that I can simplify the first two terms by bringing them to a common denominator and thereby get rid of the complex numbers. The inverse transform is however still complex.

Edit: calculation with symbolic parameters yields:
$$ H(s) = \frac{a*s+b}{c*s^2+d*s+f} $$
$$ H(t) = a*dirac(1, t) + (2*b*sin((t*(- d^2 + 4*c*f)^{1/2})/(2*c))*exp(-\frac{d*t}{2*c}))\frac{1}{(4*c*f - d^2)^{1/2}} $$

I don't quite understand where the dirac comes from, but this is basically the same result as doing the calculation with complex numbers and the taking the real part.
Assuming your algebra is correct, there is nothing wrong with complex numbers in the Laplace domain. Complex poles reflect an oscillatory signal - there is no oscillation at all if all poles are strictly real, just exponential growth or decay. The variable $s$ is itself a complex number, and the transfer function is defined over all $s$, i.e. the entire complex plane.