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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.

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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:

1. The transfer function (H(z)) should be a rational function of z, and the coefficients of z should be real.
2. The poles (values, where the denominator turns 0 / output is infinite) should lie inside the unit circle of the z-plane.
3. The number of zeros should be less than or equal to the number of poles.
4. 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.
5. The left half of the s-plane should map into the inside of the unit circle of the z-plane.

• 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.