IIR Basics

An IIR filter (Infinite Impulse Response filter) is a digital filter whose output depends not only on the current and past input samples but also on past output samples. This feedback mechanism makes the impulse response of an IIR filter theoretically infinite in duration, which distinguishes it fundamentally from FIR filters. IIR filters are a core topic in GATE DSP and in analog-to-digital filter conversion methods.

Core Concept Explanation

The defining characteristic of an IIR filter is the presence of feedback. The general difference equation of an IIR filter is: y[n] = sum from k=0 to M of b_k·x[n-k] minus sum from k=1 to N of a_k·y[n-k]. The b_k coefficients define the feedforward (numerator) part, and the a_k coefficients define the feedback (denominator) part. Because past output values feed back into the computation, any initial excitation of the filter continues to influence all future outputs, producing an infinitely long impulse response.

In the z-domain, the transfer function of an IIR filter is: H(z) = B(z)/A(z) = [sum of b_k·z^(-k)] / [1 + sum of a_k·z^(-k)]. The poles of H(z) arise from the denominator polynomial A(z). Unlike FIR filters which have only zeros (no poles except at the origin), IIR filters have poles, and the location of these poles determines both the frequency selectivity and the stability of the filter.

The connection to analog filter design is the primary reason IIR filters are practical. Analog filters such as Butterworth, Chebyshev, and Elliptic have well-established pole-zero designs that achieve sharp frequency selectivity with low order. IIR digital filter design methods convert these analog prototype filters to equivalent digital filters, inheriting the efficiency of the analog design.

Stability of IIR Filters

An IIR filter is BIBO stable (Bounded Input Bounded Output stable) if and only if all poles of H(z) lie strictly inside the unit circle in the z-plane, meaning |p_k| < 1 for all poles p_k. If any pole lies on or outside the unit circle, the filter is unstable and its output will grow without bound for a bounded input. This stability condition is a fundamental difference from FIR filters, which have no poles (except at the origin) and are always stable.

Practical fixed-point implementations of IIR filters are also susceptible to limit cycle oscillations, which are small self-sustained oscillations that occur due to quantization of the feedback coefficients. This is a direct consequence of the feedback structure and does not occur in FIR filters. For this reason, IIR filters in fixed-point systems require careful coefficient quantization and overflow management.

Mathematical Expression

The transfer function H(z) = B(z)/A(z) can be rewritten in terms of poles and zeros as: H(z) = G · product of (1 - z_k·z^(-1)) / product of (1 - p_k·z^(-1)), where z_k are the zeros, p_k are the poles, and G is a gain constant. The frequency response is obtained by evaluating H(z) on the unit circle: H(e^jω) = H(z)|_{z=e^jω}. The magnitude response |H(e^jω)| gives the filter gain at each frequency, and the phase response angle(H(e^jω)) gives the phase shift. For IIR filters, this phase response is generally nonlinear, which means different frequencies experience different time delays through the filter.

Numerical Example

For a simple first-order IIR filter with a given pole location, verify stability and compute the magnitude response at a specific frequency by evaluating H(z) on the unit circle.

Given:
First-order IIR filter: H(z) = 1 / (1 - 0.5·z⁻¹)
Pole at z = 0.5
Find: Is it stable? What is |H(e^jω)| at ω = π/2?

Why this formula applies:
Stability check uses |pole| < 1.
Frequency response: substitute z = e^jω into H(z).

Formula:
H(e^jω) = 1 / (1 - 0.5·e^(-jω))
|H(e^jω)| = 1 / |1 - 0.5·e^(-jω)|

Substitution at ω = π/2:
e^(-jπ/2) = cos(-π/2) + j·sin(-π/2) = 0 - j·1 = -j
1 - 0.5·(-j) = 1 + 0.5j

Calculation:
|1 + 0.5j| = sqrt(1² + 0.5²) = sqrt(1 + 0.25) = sqrt(1.25) = 1.118
|H(e^jπ/2)| = 1 / 1.118 = 0.894

Final Answer:
Pole at z=0.5 → |0.5| < 1 → Filter is stable.
|H(e^jπ/2)| = 0.894 (approximately -0.97 dB gain at ω = π/2)

Exam Tip: GATE often tests IIR stability by giving H(z) and asking whether it is stable. Simply find the poles (roots of the denominator) and check if all pole magnitudes are less than 1. If the denominator is (1 - az^(-1)), the pole is at z = a, and the filter is stable only if |a| < 1. This single check is sufficient.

Why IIR Filters Are Preferred Over FIR in Certain Applications

• IIR filters achieve the same magnitude response sharpness with a much lower filter order compared to FIR filters. A 5th order Butterworth IIR filter can match what a 50th order FIR filter achieves near the cutoff frequency.
• Lower order means fewer multiplications and additions per output sample, which directly translates to lower computational load and lower power consumption in hardware implementations.
• However, IIR filters have nonlinear phase, which causes phase distortion. For applications such as EEG signal processing or audio equalization where phase coherence matters, FIR filters are preferred despite their higher order.
• The Butterworth IIR filter has maximally flat magnitude in the passband. The Chebyshev Type I has equiripple in the passband. Chebyshev Type II has equiripple in the stopband. Elliptic (Cauer) has equiripple in both bands and achieves the sharpest transition for a given order.
• Stability of IIR filters must be checked after any coefficient quantization, because rounding analog prototype poles to finite precision can move them outside the unit circle.

Quick Revision

• IIR filter difference equation: y[n] = sum b_k·x[n-k] - sum a_k·y[n-k]. Past outputs feed back into the system.
• Transfer function: H(z) = B(z)/A(z). Poles arise from denominator A(z).
• Stability condition: ALL poles must satisfy |p_k| < 1 (inside unit circle). Even one pole on or outside the unit circle makes the filter unstable.
• IIR filters have nonlinear phase. FIR filters can have exactly linear phase.
• Lower order than FIR for same sharpness: key practical advantage of IIR.
• Analog prototypes used in IIR design: Butterworth (maximally flat), Chebyshev I (passband ripple), Chebyshev II (stopband ripple), Elliptic (ripple in both, sharpest).
• Exam trap: FIR filters are always stable. IIR filters may be unstable. Do not assume stability without checking pole locations.