Recursive filters (IIR)

The transfer function of a second-order IIR filter is obtained

Recursive filters (IIR)

In the digital field, the expression that describes the behavior of a circuit in which a part of the output is reported at the input according to a coefficient α must respect the principle of causality, and therefore the output at instant n can be a function of itself only limited to the previous moments; for example, its simplest form is provided by the relation:

[1] y(n)=\alpha x(n)+(1-\alpha )y(n-1)

Translating into words the [1], we can say that the output of the filter is equal at any moment to a fraction α of the input at that instant, added to the output in the previous instant multiplied by the complement of α.

This mechanism is called recursive, by analogy to what happens in the computer field, since the output at instant n is a function of the output at instant n-1 which in turn is a function of the output at instant n-2, and so on.

From [1] we see that the α factor has the role of a time constant because it tells us how much the output in the current state depends on the output in the previous state and therefore we know an estimate of the speed with which the output itself varies.

In fact, for α = 1 the output depends only on the entrance at that moment (zero time constant), while for α = 0 the output is always the same as that in the previous instant and therefore there is no evolution, whatever the input (infinite time constant).

Rarely, however, will you use a filter with such a slow response as that of an RC-type analogic filter, so digital filters make sense from the second order onwards.

The description of a second-order recursive filter is provided by the expression:

[2] y(n)=a_{0}x(n)+a_{1}y(n-1)+a_{2}y(n-2)

In [2], the output in instant n is a function of the input at the same instant and the output in the previous two moments. But the output in the two previous moments was a function of the input in those moments and of the outputs in the two moments even earlier. Taking the speech to the limit, it can be concluded that the output at a certain moment is somehow a function of the outputs in all the previous moments.

Reasoning in this way, we can say that the filter has an infinite impulse response, hence the name of IIR (Infinite Impulse Response).

Discrete transfer function of an IIR filter

The transfer function of a second-order IIR filter is obtained, similarly to what is seen for FIR filters, by applying the z transform to the finite differences equation and then looking for the ratio between output and input in the z domain.

Applying the z transform to [2], we get:

Y(z)=a_{0}X(z)+a_{1}Y(z)z^{-1}+a_{2}Y(z)z^{-2}

from which the transfer function will result:

G(z)=\frac{Y(z)}{X(z)}=\frac{a_{0}}{1-a_{1}z^{-1}-a_{2}z^{-2}}

Conclusion

In a future article we will generalize the transfer function for IIR filters and compare them with FIR filters.

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