An 8th-order IIR filter implemented as a single monolithic difference equation is numerically unstable on a fixed-point DSP. Refactoring it as four cascaded 2nd-order biquad sections fixes the problem. Why?
BEach biquad has only 3 denominator coefficients instead of 9, so coefficient quantization errors cause much smaller pole displacements
CBiquads operate at lower sample rates, reducing the numerical error per sample
DThe cascade eliminates all feedback paths, which are the source of numerical instability
The key issue is polynomial root sensitivity: for a high-degree polynomial, a small change in one coefficient can dramatically shift roots (poles) — potentially outside the unit circle, causing instability. For a 2nd-degree polynomial with 3 denominator coefficients, root sensitivity is far lower. By factoring into biquads, each section's poles depend on only 3 coefficients rather than all 9 in the monolithic 8th-order polynomial. Option D is wrong — biquads still have feedback (they are IIR sections with poles).
Question 2 Multiple Choice
In a cascade realization, all pole-zero pairings produce mathematically identical frequency responses. Why, then, do engineers carefully choose which poles to pair with which zeros?
ADifferent pairings have different computational costs because some biquads require more multiplications
BDifferent pairings produce different intermediate signal levels, affecting overflow and round-off noise with finite-precision arithmetic
CCertain pole-zero pairings are prohibited because they produce unstable biquad sections
DThe ordering of sections affects the overall phase response even though magnitude is identical
Mathematically (in infinite precision), all pairings and orderings produce the same H(z). In practice with finite-word-length arithmetic, each section amplifies and attenuates the signal differently depending on how its poles and zeros are matched. Poor pairings can cause intermediate signals to overflow or be lost in round-off noise. The standard rule is to pair each pole pair with the zero pair closest to it in the z-plane, which tends to keep each biquad's gain near unity, preventing both overflow and noise buildup.
Question 3 True / False
Placing the highest-Q (most resonant) biquad sections first in a cascade chain maximizes dynamic range.
TTrue
FFalse
Answer: False
High-Q sections have poles near the unit circle and produce large gain near their resonant frequency. Placing them first means the input signal is amplified early in the chain, risking overflow before later sections can attenuate it. The recommended heuristic is to place high-Q sections in the MIDDLE of the chain: early sections shape the signal to a manageable level, middle sections apply the demanding resonance at a controlled amplitude, and late sections complete the shaping. This minimizes both early-stage overflow and late-stage noise amplification.
Question 4 True / False
A cascade realization and a parallel realization of the same transfer function both use 2nd-order sections, but only the cascade computes the overall response as a product of section transfer functions.
TTrue
FFalse
Answer: True
In cascade (series) realization, sections are connected in series — the output of each feeds the input of the next. The overall H(z) = H₁(z) · H₂(z) · ... · Hₖ(z) — a PRODUCT, because multiplication in the z-domain corresponds to series connection. In a parallel realization, sections operate simultaneously on the input and their outputs are summed, giving H(z) = H₁(z) + H₂(z) + ... — a SUM. Both factorizations are valid representations of the same H(z) but have different coefficient sets and numerical properties.
Question 5 Short Answer
Explain why high-order IIR filters are typically implemented as cascades of 2nd-order sections rather than as a single high-order difference equation, and what role pole-zero pairing plays.
Think about your answer, then reveal below.
Model answer: A single high-order polynomial difference equation suffers from high coefficient sensitivity: small rounding errors in the coefficients (unavoidable in fixed-point arithmetic) cause large displacements of the roots (poles and zeros), potentially destabilizing the filter. This sensitivity grows rapidly with polynomial degree. By factoring H(z) into 2nd-order biquad sections, each section's roots depend on only 3 denominator coefficients, giving far lower sensitivity. Pole-zero pairing determines the gain profile of each biquad section: pairing each pole pair with its nearest zero pair keeps each section's gain near unity, preventing intermediate signal overflow or excessive noise amplification between sections.
Biquads are the universal building block because any complex-conjugate pole pair and zero pair can be represented by one biquad's 5 real coefficients. All even-order filters factor exactly into biquads; odd-order filters add one first-order section. Standard DSP hardware is often optimized specifically for biquad computation.