# Analytical Calculation of the Impulse Response of an Arbitrary Filter

This is a rather general question. Suppose I have an arbitrary filter, $F$, defined in the Laplace domain; $$F=k\frac{\prod_i (s+z_i)}{\prod_j(s+p_j)}.$$ How do I calculate the impulse response of this filter analytically? As a specific simple example, consider the first order low pass filter $$F_0=\frac{1}{s+p}.$$ To be clear, what I want to get from this transformation is a function which describes the delta function response from $t=0$ to $t\rightarrow\infty$ where the delta function is applied at $t=0$.

• Lookup partial fraction expansion and inverse laplace transform tables. – porten Jan 12 '14 at 23:08
• @porten That makes sense, rather obvious in retrospect. Perhaps you could expand your comment into a full answer for posterity? – Chris Mueller Jan 13 '14 at 16:18

Your expression is the Laplace transform of the exponential function $e^{-pt}$. The Laplace transform of the dirac unit is $1$. So the inverse Laplace transform is exactly $e^{-pt}$. Maybe the fact that this function is noncausal adds a little complexity here.