<P> where Q c (t) (\ displaystyle Q_ (c) (t)) is the charge stored in the capacitor at time t (\ displaystyle \ scriptstyle t). Substituting equation Q into equation I gives i (t) = C d ⁡ v out d ⁡ t (\ displaystyle \ scriptstyle i (t) \; = \; C (\ frac (\ operatorname (d) v_ (\ text (out))) (\ operatorname (d) t))), which can be substituted into equation V so that: </P> <Dl> <Dd> v in (t) − v out (t) = R C d ⁡ v out d ⁡ t (\ displaystyle v_ (\ text (in)) (t) - v_ (\ text (out)) (t) = RC (\ frac (\ operatorname (d) v_ (\ text (out))) (\ operatorname (d) t))) </Dd> </Dl> <Dd> v in (t) − v out (t) = R C d ⁡ v out d ⁡ t (\ displaystyle v_ (\ text (in)) (t) - v_ (\ text (out)) (t) = RC (\ frac (\ operatorname (d) v_ (\ text (out))) (\ operatorname (d) t))) </Dd> <P> 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 Δ T (\ displaystyle \ scriptstyle \ Delta _ (T)) time . Let the samples of v in (\ displaystyle \ scriptstyle v_ (\ text (in))) be represented by the sequence (x 1, x 2,..., x n) (\ displaystyle \ scriptstyle (x_ (1), \, x_ (2), \, \ ldots, \, x_ (n))), and let v out (\ displaystyle \ scriptstyle v_ (\ text (out))) be represented by the sequence (y 1, y 2,..., y n) (\ displaystyle \ scriptstyle (y_ (1), \, y_ (2), \, \ ldots, \, y_ (n))), which correspond to the same points in time . Making these substitutions: </P>

Low pass filter and high pass filter circuit diagram