Low-pass Filter - Discrete-time Realization

Discrete-time Realization

For another method of conversion from continuous- to discrete-time, see Bilinear transform.

The effect of a low-pass filter can be simulated on a computer by analyzing its behavior in the time domain, and then discretizing the model.

From the circuit diagram to the right, according to Kirchoff's Laws and the definition of capacitance:

(V)

(Q)

(I)

where is the charge stored in the capacitor at time . Substituting equation Q into equation I gives, which can be substituted into equation V so that:

This equation can be discretized. For simplicity, assume that samples of the input and output are taken at evenly-spaced points in time separated by time. Let the samples of be represented by the sequence, and let be represented by the sequence which correspond to the same points in time. Making these substitutions:

And rearranging terms gives the recurrence relation

That is, this discrete-time implementation of a simple RC low-pass filter is the exponentially-weighted moving average

By definition, the smoothing factor . The expression for yields the equivalent time constant in terms of the sampling period and smoothing factor :

If, then the time constant is equal to the sampling period. If, then is significantly larger than the sampling interval, and .