IIR Structures

Once an IIR filter transfer function H(z) is designed, it must be implemented as a concrete computational structure. The choice of structure determines how additions, multiplications, and memory elements (delay registers) are arranged, which affects numerical precision, stability, and hardware cost.

The four main IIR filter structures are Direct Form I, Direct Form II, Cascade (series), and Parallel. Each realizes the same transfer function H(z) but differs in the number of delay elements, sensitivity to coefficient quantization, and implementation efficiency.

Core Concept Explanation

A general IIR transfer function of order N is H(z) = B(z)/A(z) = (sum b_k * z^-k) / (1 - sum a_k * z^-k). Direct Form I implements this literally: the input signal is fed through the numerator polynomial (a transversal FIR section), and the result is fed through the denominator polynomial (a recursive IIR section). This requires 2N delay elements because the two sections maintain separate delay lines.

Direct Form II, also called the canonical form, swaps the order of the two sections and recognizes that they can share a single delay line. This halves the memory requirement to N delay elements for an order-N filter. The intermediate signal w[n] satisfies w[n] = x[n] + sum(a_k * w[n-k]), and the output is y[n] = sum(b_k * w[n-k]).

Cascade form factorizes H(z) into a product of second-order sections (biquads): H(z) = H1(z) * H2(z) * ... Each biquad handles one complex conjugate pole pair and one zero pair. This is the most numerically robust structure because coefficient quantization affects only one biquad at a time, keeping pole sensitivity low.

Parallel form decomposes H(z) using partial fractions: H(z) = H1(z) + H2(z) + ... Each section is computed independently and the outputs are summed. This can reduce latency in hardware and is also numerically better than monolithic Direct Form I.

Mathematical Expression

Direct Form II difference equations:

w[n] = x[n] + a1*w[n-1] + a2*w[n-2] + ... + aN*w[n-N] (recursive part) y[n] = b0*w[n] + b1*w[n-1] + b2*w[n-2] + ... + bN*w[n-N] (FIR part)

Cascade decomposition:

H(z) = product over k of [ (b0k + b1k*z^-1 + b2k*z^-2) / (1 - a1k*z^-1 - a2k*z^-2) ]

Practical Understanding

In fixed-point DSP hardware, coefficient quantization is a major concern. In monolithic Direct Form I or II, all poles of H(z) are encoded in a single high-order polynomial denominator. A small change in any coefficient can move all poles significantly. In cascade form, each biquad has only two poles, and quantization error affects only those two poles.

This is why industry DSP implementations almost universally use cascade form for IIR filters. MATLAB's default IIR filter output and most real-time audio processors use biquad cascades. Parallel form is chosen specifically when low-latency parallel computation is the primary goal.

Given:
H(z) = (1 + 0.5z^-1) / ((1 - 0.5z^-1)(1 - 0.25z^-1))
Implement as cascade of two 1st-order sections

Why this formula applies:
Cascade factorization separates poles for numerical stability.

Formula:
H(z) = H1(z) * H2(z)

Substitution:
H1(z) = (1 + 0.5z^-1) / (1 - 0.5z^-1)  [assigns numerator to first section]
H2(z) = 1 / (1 - 0.25z^-1)  [pole-only second section]

Calculation:
Section 1: w1[n] = x[n] + 0.5*w1[n-1], y1[n] = w1[n] + 0.5*w1[n-1] -- wait, rewrite:
H1 difference: w1[n] = x[n] + 0.5*w1[n-1], y1[n] = w1[n] + 0.5*w1[n-1]
Section 2: y[n] = y1[n] + 0.25*y[n-1]

Final Answer with units:
Cascade requires 2 delay registers total.
Direct Form I for same system would need 4 registers.
Cascade halves memory needs and improves coefficient sensitivity.

Exam Tip: Direct Form II uses N delays (minimum). Direct Form I uses 2N delays. Cascade form is most numerically stable for fixed-point implementation. GATE often asks which structure uses minimum delay elements — answer is Direct Form II (also called canonical form).

Mechanism Summary

• Direct Form I: 2N delays, FIR section followed by IIR section. Most straightforward but memory-intensive.
• Direct Form II: N delays (canonical). IIR and FIR share one delay line. Minimum memory.
• Cascade: product of biquads. Best numerical stability. Preferred in fixed-point hardware.
• Parallel: sum of partial fraction sections. Low latency, concurrent computation.
• Cascade pairs each pole with a nearby zero to minimize sensitivity to quantization errors.
• All four structures realize the same H(z) but differ in delay count, sensitivity, and implementation cost.

Quick Revision

• Direct Form I: 2N delays. Direct Form II (canonical): N delays.
• Cascade form: product of H_k(z) biquads. Best numerical stability for fixed-point.
• Parallel form: sum of H_k(z) from partial fractions. Allows parallel computation.
• Direct Form II difference equations: w[n] = x[n] + sum(a_k*w[n-k]), y[n] = sum(b_k*w[n-k]).
• Trap: Direct Form I is NOT canonical. Direct Form II is the canonical (minimum delay) form.
• Trap: Cascade requires factorization of H(z) into biquads; Parallel requires partial fraction expansion.
• All structures implement the same transfer function. Structure choice affects hardware cost and precision only.