Offset Exponentials

Use two one-pole TIIRs, to make an offset exponential:

$$h(n) = \left\{\begin{array}{ll} a e^{c n} + b, & n=0,1,2,\ldots,N-1 \\ [5pt] 0, & \hbox{otherwise} \\ \end{array} \right.$$

• The constant portion $b$ requires only one multiply
  (by $b$ ) since the pole for this TIIR filter is at $z=1$
• Resets for pure integrators are needed less often than for growing exponentials
• Using a cascade of digital integrators, any polynomial impulse response is possible
• A cubic-spline impulse response requires four integrators