Questions: IIR Filter Design and Realization

Question 1

A designer implements a 12th-order IIR lowpass filter in Direct Form II on a fixed-point DSP. During testing, the filter output diverges exponentially. What is the most likely cause?

• A. The bilinear transform was applied incorrectly, mapping stable analog poles to unstable z-domain positions
• B. Coefficient quantization in the high-order Direct Form II polynomial caused poles to drift outside the unit circle
• C. The sampling rate was set too low, causing the passband to alias into the stopband
• D. The Butterworth prototype specification was incorrect, selecting a filter order too high for the cutoff frequency

Answer: (B)

High-order IIR filters in Direct Form II are numerically sensitive: the coefficients of a high-degree polynomial in z must be represented with limited fixed-point precision, and small quantization errors cause significant displacement of pole locations. Poles that were designed to sit safely inside the unit circle can drift outside it, producing instability. This is not a design error in the bilinear transform — it is an implementation problem unique to high-order direct-form structures. The remedy is cascade realization as second-order biquad sections, which confines each polynomial to order 2 and dramatically reduces sensitivity.

Question 2

Why must the analog cutoff frequency be pre-warped before applying the bilinear transform when designing a digital IIR filter?

• A. Because the bilinear transform introduces aliasing that shifts the entire passband upward by a fixed amount
• B. Because the bilinear transform maps analog to digital frequencies nonlinearly, compressing high frequencies, so the digital cutoff will not match the intended frequency without correction
• C. Because the impulse invariance method requires pre-warping to prevent aliasing of the imaginary-axis poles
• D. Because pre-warping corrects for the group delay introduced by the feedback poles in the IIR structure

Answer: (B)

The bilinear transform maps the entire analog frequency axis (−∞ to +∞) onto the digital frequency range (−π to +π) through a nonlinear (tangent) compression. A specified analog cutoff frequency Ω_c does not land at the corresponding digital frequency ω_c = Ω_c · T after transformation — it gets compressed. Pre-warping adjusts the analog prototype's cutoff to Ω_c = (2/T)·tan(ω_c/2) before designing, so that after the bilinear transform is applied, the digital filter's cutoff lands exactly where intended. Without pre-warping, the passband edge will be systematically off.

Question 3

An elliptic IIR filter achieves a steeper rolloff than a Butterworth filter of the same order by allowing equiripple in both the passband and the stopband.

• T. True
• F. False

Answer: True

Butterworth filters achieve maximally flat magnitude in the passband — zero ripple — at the cost of a gradual transition from passband to stopband. Chebyshev filters trade passband (or stopband) flatness for a steeper rolloff. Elliptic (Cauer) filters allow equiripple in both the passband AND the stopband, achieving the steepest possible rolloff for a given filter order — they are optimal in the minimax sense. The tradeoff is more complex pole-zero geometry and less predictable group delay. For strict attenuation specs with tight order constraints, elliptic filters deliver the most selectivity.

Question 4

Cascade realization of an IIR filter as second-order biquad sections requires more multiply-add operations per output sample than Direct Form II, which is why Direct Form II is preferred in practice.

• T. True
• F. False

Answer: False

The computational cost of cascade realization and Direct Form II is approximately the same — both require roughly 2N multiplications and additions per sample for an N-th order filter. The reason cascade realization is preferred in fixed-point implementations is numerical stability, not computation. In cascade form, each biquad has only 5 coefficients and represents a well-conditioned second-order polynomial; quantization errors cause small, manageable pole displacement. High-order Direct Form II is numerically ill-conditioned. Computation is essentially equivalent; stability is the decisive factor.

Question 5

Explain why cascade realization of an IIR filter as a series of second-order sections is preferred over Direct Form II for high-order fixed-point implementations.

Model answer: In Direct Form II, the filter's poles are roots of a single high-degree polynomial whose coefficients must be stored in fixed-point arithmetic. Small quantization errors in high-degree polynomial coefficients cause large shifts in pole locations — sensitivity grows rapidly with order — and poles can drift outside the unit circle, destabilizing the filter. In cascade realization, H(z) is factored into second-order sections (biquads), each with only five coefficients representing one conjugate pole pair and nearby zeros. Each biquad's poles are far from the unit circle boundary in that section's coefficient space, and quantization errors cause much smaller, manageable displacements. The filter remains stable even under fixed-point limitations.

The numerical sensitivity of polynomial root locations to coefficient errors is a classical result: roots of high-degree polynomials can shift dramatically in response to tiny coefficient perturbations. By decomposing the filter into biquads, the cascade approach avoids high-degree polynomial representation entirely, confining sensitivity to small, isolated second-order polynomials. This is the key insight that makes cascade of biquads the standard for practical IIR implementation.