What is an IIR Filter?

An Infinite Impulse Response (IIR) filter is a fundamental digital signal processing tool where the output depends on current and previous input values, as well as previous output values. This recursive nature allows IIR filters to achieve a specific frequency response with a much lower computational cost (lower order) compared to Finite Impulse Response (FIR) filters.

How to Use This Calculator

This tool specifically calculates a Butterworth Lowpass Filter using the Bilinear Transform method. To use it, simply enter your Sample Rate (how many samples per second your system processes) and your desired Cutoff Frequency. The calculator provides the recursive coefficients (a and b) and the step-by-step mathematical derivation.

Key Steps in IIR Design

• Pre-warping: Digital frequencies don't map linearly to analog frequencies. We adjust the analog frequency (omega) to account for the Bilinear Transform warping.
• S-Plane Prototype: We design a stable analog filter in the s-domain.
• Bilinear Transformation: We substitute s with a function of z to move from the continuous domain to the discrete domain.
• Normalization: The final equation is normalized so that the leading feedback coefficient (a0) is 1.

Frequently Asked Questions

Why choose Butterworth? Butterworth filters are known for having a maximally flat frequency response in the passband, meaning there is no ripple in the frequencies you want to keep.

What are 'a' and 'b' coefficients? In the difference equation, 'b' coefficients are applied to inputs (feed-forward), and 'a' coefficients are applied to previous outputs (feedback).