Digital Devices
Digital devices have the advantage of being versatile, and are not susceptible to unexpected output variation due to heat or noise. The discrete nature of a computer however, does not allow for all of history to be computed. Some finite range must exist. Therefore, the number of data points that can be stored in memory (N), determines the oldest data point in memory, so that the value a is never more than N samples old. The effect is that any history older than a is completely forgotten, and no longer influences the output.
A solution to this problem is the Coopmans approximation, which allows old data to be forgotten more gracefully (though still with exponential decay, rather than with the power law decay of a purely analog device).
Read more about this topic: Fractional-order Integrator
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—H.G. (Herbert George)