Table of Contents
Designing an Infinite Impulse Response (IIR) filter that meets specific performance criteria requires a systematic approach. This guide walks you through the essential steps to derive the transfer function of an IIR filter based on given specifications, making the process accessible for students and educators alike.
Understanding Filter Specifications
Before deriving the transfer function, clearly define the filter specifications. Common parameters include:
- Cutoff frequency (fc): The frequency where the filter begins to attenuate signals.
- Passband ripple: Allowed variation in the passband amplitude.
- Stopband attenuation: The minimum attenuation required in the stopband.
- Filter type: Low-pass, high-pass, band-pass, or band-stop.
Choosing the Filter Prototype
Based on specifications, select an appropriate filter prototype. Common choices include Butterworth, Chebyshev, Elliptic, and Bessel filters. Each offers different characteristics in terms of ripple and roll-off steepness.
Deriving the Analog Transfer Function
Start with the prototype’s analog transfer function, typically expressed as:
H(s) = \frac{K}{D(s)}
where D(s) is a polynomial in s representing the poles of the filter. For example, a Butterworth filter of order n has poles evenly spaced on a circle in the left half of the s-plane.
Applying Frequency Transformations
Transform the analog prototype to match the desired cutoff frequency using the substitution:
s → \(\frac{\omega_c}{s}\)
where \(\omega_c\) is the cutoff angular frequency. This step adjusts the pole locations accordingly.
Discretizing the Filter
To implement the filter digitally, convert the analog transfer function to a discrete form using methods like bilinear transformation:
s = \frac{2}{T} \frac{1 – z^{-1}}{1 + z^{-1}}
where T is the sampling period. This process yields the digital transfer function H(z).
Expressing the Transfer Function
The final transfer function is typically expressed as a ratio of polynomials:
H(z) = \frac{Y(z)}{X(z)} = \frac{b_0 + b_1 z^{-1} + \dots + b_n z^{-n}}{1 + a_1 z^{-1} + \dots + a_m z^{-m}}
Conclusion
Deriving an IIR filter transfer function from specifications involves selecting a prototype, transforming it to the desired frequency, and discretizing it for digital implementation. Mastery of these steps enables precise control over filter characteristics, essential for various signal processing applications.