Example of a recursive filter

A simple example of a recursive digital filter is given by

y_n = x_n + y_n-1

In other words, this filter determines the current output (y_n) by adding the current input (x_n) to the previous output (y_n-1).

Thus:

y₀ = x₀ + y_-1
y₁ = x₁ + y₀
y₂ = x₂ + y₁
y₃ = x₃ + y₂
... etc

Note that y_-1 (like x_-1) is undefined, and is usually taken to be zero.

Let us consider the effect of this filter in more detail. If in each of the above expressions we substitute for y_n-1 the value given by the previous expression, we get the following:

y₀ = x₀ + y_-1 = x₀
y₁ = x₁ + y₀ = x₁ + x₀
y₂ = x₂ + y₁ = x₂ + x₁ + x₀
y₃ = x₃ + y₂ = x₃ + x₂ + x₁ + x₀
... etc

Thus we can see that y_n, the output at t = nh, is equal to the sum of the current input x_n and all the previous inputs. This filter therefore sums or integrates the input values, and so has a similar effect to an analog integrator circuit.

This example demonstrates an important and useful feature of recursive filters: the economy with which the output values are calculated, as compared with the equivalent non-recursive filter. In this example, each output is determined simply by adding two numbers together.

For instance, to calculate the output at time t = 10h, the recursive filter uses the expression

y₁₀ = x₁₀ + y₉

To achieve the same effect with a non-recursive filter (i.e. without using previous output values stored in memory) would entail using the expression

y₁₀ = x₁₀ + x₉ + x₈ + x₇ + x₆ + x₅ + x₄ + x₃ + x₂ + x₁ + x₀

This would necessitate many more addition operations, as well as the storage of many more values in memory.