High-pass Filter - Discrete-time Realization

Discrete-time Realization

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

Discrete-time high-pass filters can also be designed. Discrete-time filter design is beyond the scope of this article; however, a simple example comes from the conversion of the continuous-time high-pass filter above to a discrete-time realization. That is, the continuous-time behavior can be discretized.

From the circuit in Figure 1 above, according to Kirchoff's Laws and the definition of capacitance:

\begin{cases} V_{\text{out}}(t) = I(t)\, R &\text{(V)}\\ Q_c(t) = C \, \left( V_{\text{in}}(t) - V_{\text{out}}(t) \right) &\text{(Q)}\\ I(t) = \frac{\operatorname{d} Q_c}{\operatorname{d} t} &\text{(I)} \end{cases}

where is the charge stored in the capacitor at time . Substituting Equation (Q) into Equation (I) and then Equation (I) into Equation (V) gives:

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 continuous-time RC high-pass filter is

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

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

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