What is an Infinite Impulse Response Filter (IIR)?

The infinite impulse response is a type of digital filter that is used in Digital Signal Processing applications. A filter’s job is to allow certain types of signals to pass and block the rest. The infinite impulse response filter is unique because it uses a feedback mechanism. It requires current as well as past output data. Though they are harder to design, IIR filters are computationally efficient and generally cheaper.

IIR filter bloack diagram — IIR filter block diagram

Contents

Methods to design an Infinite Impulse Response filter

Direct method – Designing analog filters and then converting them to digital.
Approximation of derivatives method – Approximating a differential equation (continuous-time (d/dt)) by an equivalent difference equation. We have covered the approximation of derivatives method to design IIR filters here.
Impulse invariance method – simple s-plane to z-plane mapping. The impulse invariance method to design IIR filters is covered here.
Bilinear Transformation method (BLT) – An improvement over the impulse invariance method. We have covered the Bilinear transform method to design IIR filters here.
Approximation of standard analog filters – Butterworth, Chebyshev, and Elliptic analog filter designs can be approximated as IIR filters. These analog filters have an infinite response too.
Spectral transformation – We can create different types of digital IIR filters using a digital IIR low pass filter via the spectral transformation method.

Transfer function of an IIR filter

$H(z)\quad =\quad \sum _{ n=0 }^{ \infty }{ h(n){ z }^{ -n }\quad =\quad \frac { \sum _{ k=0 }^{ M }{ { b }_{ k }{ z }^{ -k } } }{ 1+\sum _{ k=1 }^{ N }{ { a }_{ k }{ z }^{ -k } } } }$

What are the conditions to design a stable Infinite Impulse Response (IIR) filter?

To design a stable and causal IIR filter, the following requirements are necessary:

The transfer function (H(z)) should be a rational function of z, and the coefficients of z should be real.
The poles (values, where the denominator turns 0 / output is infinite) should lie inside the unit circle of the z-plane.
The number of zeros should be less than or equal to the number of poles.
For the effective conversion from an analog filter to a digital one, the imaginary axis of the s-plane should map into the unit circle of the z-plane. This establishes a direct relationship between the analog frequency and digital frequency in two domains.
The left half of the s-plane should map into the inside of the unit circle of the z-plane.

Advantages of an IIR filter

IIR filters are more versatile.
They are computationally easier to implement.
They are cheaper too.
The infinite response of the IIR filter is a cool feature when you are looking for amplification of signals. Not so much when you wish to attenuate them, though.

Disadvantages of an IIR filter

Practically realizable digital IIR filters do not have a linear phase response. The linear phase is a property where the phase response of a filter is a linear function of the frequency.
IIR filters cannot implement integer math, which is an easy to use mathematical representation because it’s cheap, fast, and easy to implement.
IIR filters have a higher chance of being unstable.
Additionally, they also have a higher chance of being affected by quantization operations like truncation and rounding. This is due to the feedback mechanism that introduces poles in the transfer function. On the contrary, FIR filter transfer functions do not have poles.

This information should suffice into what the core aspect of an IIR filter is. We will dive into the technical depth of designing IIR filters in this digital signal processing course. You can also check out the difference between IIR and FIR filters here.