In some applications where the input signal is a burst signal, or the amplitude changes dramatically, the rise/fall time of the leaky integrator needs to be evaluated. The rise time can be derived from its step response; the fall time can be derived from its response with the initial conditions. The following formulas show how to calculate the rise and fall times, respectively:
- h[n] = 1 - αn+1
- h[n] = αn+1
where:
α = 1 - 1/2λ
Considering when the output of leaky integrator achieves the 90% of final value for rise time calculation, which is around 1 dB lower than final value.
For the fall time, 1% (-40 dB) is used as which will illustrate a full-scale signal fall down to -40 dBFS, this makes the fall time is around double of rising time.
The following table shows rise/fall time for reference based on different time constants:
| Time Constant | Rise Time (0 to 90%) | Fall Time (1 to 1%) | Unit |
|---|---|---|---|
| 2^0 | 1 | 1 | T4/T8 |
| 2^2 | 7 | 15 | T4/T8 |
| 2^4 | 34 | 70 | T4/T8 |
| 2^8 | 587 | 1175 | T4/T8 |
| 2^12 | 9429 | 18859 | T4/T8 |
| 2^14 | 37723 | 75447 | T4/T8 |
| 2^16 | 150900 | 301801 | T4/T8 |