# Design of Infinite Impulse Response (IIR) filters

Design of Infinite Impulse Response (IIR) filters are a type of digital filter most commonly used in signal processing applications.

They are characterized by their ability to provide feedback, which allows for more flexible frequency response shaping compared to Finite Impulse Response (FIR) filters.

Designing an IIR filter involves determining the filter coefficients that will achieve the desired frequency response. There are different methods and algorithms available for IIR filter design, and I’ll outline a commonly used technique called the Butterworth filter design.

## Design of Infinite Impulse Response steps are :

1. Specify the filter requirements: Determine the desired frequency response characteristics of the filter, such as the cutoff frequency, passband ripple, and stopband attenuation.
2. Choose the filter order: The filter order determines the steepness of the roll-off between the passband and stopband. Higher filter orders result in steeper roll-offs but may introduce more complexity and computational requirements. The filter order is typically chosen based on the desired trade-off between filter performance and computational resources.
3. Normalize the filter specifications: Convert the desired cutoff frequency (or frequencies) into a normalized frequency range from 0 to 1, where 1 represents the Nyquist frequency (half the sampling rate). Normalization is necessary to make the filter design independent of the sampling frequency.
4. Select the prototype filter: Butterworth filters have a maximally flat frequency response in the passband and can be a good choice for many applications. Other types of filters, such as Chebyshev or elliptic filters, offer sharper roll-offs but may introduce ripples in the passband or stopband. The prototype filter provides the initial filter coefficients.
• Determine the pole locations: The pole locations in the complex plane define the frequency response characteristics of the filter. For Butterworth filters, the poles are uniformly distributed on a circle in the left-half plane of the complex plane. The radius of the circle determines the cutoff frequency.
• Convert the pole locations to filter coefficients: Once the pole locations are determined, they can be converted into filter coefficients using techniques like bilinear transformation or matched z-transform.
• Apply any required post-processing: Depending on the specific application, you may need to perform additional operations on the filter coefficients, such as gain adjustment or truncation.
• Implement the filter: Once the filter coefficients are obtained, they can be used in a digital signal processing algorithm, such as the difference equation representation or the direct form implementation, to process the input signal.

It’s important to note that there are several software tools and libraries available that can automate the process of IIR filter design, providing various optimization algorithms and graphical interfaces for ease of use.

These tools can significantly simplify the design process and provide efficient implementations of IIR filters.