Questions: Parallel Filter Realization Structures

Question 1

A 6th-order IIR filter is implemented in parallel form on a fixed-point DSP. A quantization error is introduced in the coefficients of one second-order section. The effect on the filter's pole locations is:

• A. All 6 poles shift simultaneously, since the overall transfer function is the sum of all sections
• B. Only the poles of the affected section shift; the other sections' poles remain unchanged
• C. The poles of adjacent sections shift due to output correlation at the summing node
• D. All poles shift equally since the sections share the same input signal

Answer: (B)

This is the key advantage of parallel realization. In direct-form implementation, all N poles are jointly determined by N global coefficients — a quantization error in any one coefficient perturbs all the poles. In parallel form, each section has its own local coefficients governing only its own poles. A quantization error in section 3 shifts only section 3's pole pair; sections 1, 2, 4, 5, and 6 are completely unaffected. This locality makes parallel realization far more robust in fixed-point environments, especially for high-order filters with poles near the unit circle.

Question 2

In a parallel filter implementation, careful scaling of each section is required primarily because:

• A. Sections with higher Q factors have slower group delay, causing phase cancellation at the summing node
• B. All sections receive the same full-amplitude input, so large gain differences between sections can cause overflow in high-gain sections or noise dominance in low-gain sections
• C. The summing node introduces nonlinear distortion if input signals are not phase-aligned
• D. Sections with real poles have different gain profiles than sections with complex conjugate poles

Answer: (B)

In cascade realization, each section's input is the output of the previous section, allowing section-by-section sequential scaling. In parallel form, all sections receive the same original input at full amplitude simultaneously. If one section has peak gain of 1000 and another has peak gain of 0.001, the high-gain section overflows while the low-gain section's output is buried in quantization noise. Each section must be individually normalized to use the full dynamic range available, with a compensating scale factor at the summing node.

Question 3

In parallel filter realization, the sections are 'independent' in the sense that each section receives primarily the output of the previous section as its input.

• T. True
• F. False

Answer: False

This describes cascade (series) realization, not parallel. In parallel form, all sections share the same common input signal. The independence of parallel sections refers to their pole sensitivity — coefficient errors in one section do not perturb other sections' poles — not to signal routing. In cascade form, sections are chained: section 2's input is section 1's output. In parallel form, every section simultaneously processes the same original input.

Question 4

Parallel realization is particularly beneficial for high-order IIR filters implemented in fixed-point arithmetic because quantization errors affect only the local poles within each section.

• T. True
• F. False

Answer: True

For high-order direct-form filters, poles near the unit circle are highly sensitive to coefficient quantization — small perturbations can push poles outside the unit circle, causing instability. Parallel form partitions the poles into small groups (1 or 2 per section), each governed by its own local coefficients. This locality prevents quantization errors from propagating across poles and makes the implementation numerically robust even when poles are near-marginal.

Question 5

Explain how partial fraction decomposition leads to the parallel realization structure, and why this structure improves coefficient sensitivity compared to direct-form realization.

Model answer: Partial fraction decomposition rewrites H(z) = N(z)/D(z) as a sum of simpler rational terms, each corresponding to one pole (or complex-conjugate pole pair): H(z) = H₁(z) + H₂(z) + ... + H_k(z). Each term is a low-order (1st or 2nd order) filter that can be realized independently. Since addition distributes over LTI systems, feeding the same input to all sections and summing their outputs yields the original H(z). The sensitivity improvement comes from localization: in direct form, the full-order denominator polynomial D(z) collectively determines all poles, so any coefficient error perturbs all of them. In parallel form, each section's coefficients control only that section's pole(s), confining quantization errors to their local section.

The connection to partial fractions from calculus is direct: the same algebraic decomposition that simplifies integration also simplifies filter implementation by breaking a complex rational function into manageable pieces.