Editing Recurrence relation (section)

=== Digital signal processing ===
In [[digital signal processing]], recurrence relations can model feedback in a system, where outputs at one time become inputs for future time. They thus arise in [[infinite impulse response]] (IIR) [[digital filter]]s.

For example, the equation for a "feedforward" IIR [[comb filter]] of delay <math>T</math> is:

:<math>y_t = (1 - \alpha) x_t + \alpha y_{t - T},</math>

where <math>x_t</math> is the input at time <math>t</math>, <math>y_t</math> is the output at time <math>t</math>, and <math>\alpha</math> controls how much of the delayed signal is fed back into the output.  From this we can see that

:<math>y_t = (1 - \alpha) x_t + \alpha ((1-\alpha) x_{t-T} + \alpha y_{t - 2T})</math>
:<math>y_t = (1 - \alpha) x_t + (\alpha-\alpha^2) x_{t-T}  + \alpha^2 y_{t - 2T}</math>

etc.