8. IIR Filter Design

Introduction and Context

Infinite impulse response (IIR) filters are among the most important structures in digital signal processing. Unlike finite impulse response (FIR) filters, whose output depends only on present and past input samples, an IIR filter also uses previous output samples. This feedback gives IIR filters a potentially infinite-duration impulse response and allows them to achieve sharp frequency selectivity with relatively low order.

The practical design of an IIR filter is commonly based on an analogue prototype. Classical analogue filters such as Butterworth, Chebyshev and elliptic filters possess well-understood magnitude characteristics and stable pole configurations. Their transfer functions can be transformed into digital filters using methods such as impulse invariance and the bilinear transformation.

IIR design therefore involves more than selecting a response type. The designer must specify passband and stopband requirements, determine the minimum order, map analogue frequencies to digital frequencies, verify stability, select a suitable realization, and consider finite-word-length effects. These decisions influence computational cost, phase distortion, numerical robustness and system performance.

Learning Objectives

After completing this chapter, the reader should be able to:

1. Explain the structure and characteristics of IIR filters.

2. Distinguish Butterworth, Chebyshev Type I, Chebyshev Type II and elliptic responses.

3. Derive digital IIR filters using impulse invariance and bilinear transformation.

4. Analyse stability through pole locations.

5. Explain frequency warping and prewarping.

6. Select an appropriate prototype and transformation for an application.

7. Identify numerical, implementation and finite-precision problems.

8. Evaluate a practical IIR design using frequency-response and pole-zero analysis.

The IIR Filter as a Feedback System

A causal digital IIR filter is represented by the difference equation

y[n] = -Σ(k=1 to N) a_k y[n-k] + Σ(k=0 to M) b_k x[n-k].

Its transfer function is

H(z) = Y(z)/X(z) = (b_0 + b_1z^-1 + ... + b_Mz^-M)/(1 + a_1z^-1 + ... + a_Nz^-N).

The denominator introduces feedback. If the denominator is absent, the filter becomes an FIR filter. The poles of H(z) are the roots of the denominator, while the zeros are the roots of the numerator. For a causal rational digital filter to be bounded-input bounded-output stable, every pole must lie strictly inside the unit circle:

|p_k| < 1.

This condition is central in both design and implementation. A theoretically stable filter may become unstable in practice if coefficient quantization moves a pole outside the unit circle. For this reason, high-order filters are normally implemented as cascades of first- and second-order sections rather than as one high-order polynomial.

IIR filters are attractive because a relatively small order can satisfy demanding magnitude specifications. Their principal disadvantages are nonlinear phase, sensitivity to coefficient quantization, possible internal overflow, and a more complicated design process than that of many FIR filters.

Frequency Specifications and Filter Performance

A low-pass specification is usually expressed through four quantities:

• passband edge frequency, ω_p;
• stopband edge frequency, ω_s;
• maximum passband loss or ripple, A_p;
• minimum stopband attenuation, A_s.

For a normalized digital frequency Ω, the passband requirement may be written as

|H(e^jΩ)| ≥ 10^(-A_p/20),

and the stopband requirement as

|H(e^jΩ)| ≤ 10^(-A_s/20).

The transition band lies between the passband and stopband edges. A narrow transition band generally requires a higher-order filter. In practical systems, the order is a major design variable because it affects computational load, memory, latency, numerical sensitivity and power consumption.

The design problem is therefore a constrained approximation problem: obtain an acceptable approximation to an ideal frequency response while meeting attenuation, stability and implementation constraints.

Classical Analogue Prototypes

Butterworth Response

The Butterworth filter has a maximally flat magnitude response at zero frequency. For a normalized low-pass prototype,

|H_a(jΩ)|^2 = 1/[1 + Ω^(2N)],

where N is the order. The response is monotonic in both the passband and stopband. It has no ripple, but its transition is less sharp than that of Chebyshev or elliptic filters of the same order.

For passband edge Ω_p, stopband edge Ω_s, passband attenuation A_p and stopband attenuation A_s, the minimum order satisfies

N ≥ log10[(10^(A_s/10)-1)/(10^(A_p/10)-1)] / [2 log10(Ω_s/Ω_p)].

The order is rounded upward to the next integer. The Butterworth response is often preferred when a smooth magnitude characteristic is more important than the narrowest transition band.

Chebyshev Type I Response

Chebyshev Type I filters allow equiripple variation in the passband and provide a sharper transition than Butterworth filters of equal order. Their normalized response is

|H_a(jΩ)|^2 = 1/[1 + ε^2 C_N^2(Ω)],

where ε determines the passband ripple and C_N(Ω) is the Chebyshev polynomial of order N. The passband ripple is related to A_p by

ε = sqrt(10^(A_p/10)-1).

The stopband is monotonic. This response is suitable when a specified amount of passband ripple is acceptable in exchange for improved selectivity.

Chebyshev Type II Response

Chebyshev Type II filters, also called inverse Chebyshev filters, have a monotonic passband and equiripple stopband. They contain finite-frequency zeros, which create enhanced stopband attenuation at selected frequencies. They are useful when passband flatness is required but some stopband ripple can be tolerated.

Elliptic Response

Elliptic filters have equiripple behaviour in both the passband and stopband. They provide the narrowest transition band for a specified order and attenuation requirement. Their poles and zeros are distributed to optimize the approximation over both bands.

The price is greater sensitivity to parameter variation, more complicated pole-zero patterns and ripple in both bands. Elliptic filters are appropriate when order or transition width is the dominant constraint.

Comparative Interpretation

The four classical responses can be viewed as different compromises:

• Butterworth: no ripple and smooth response, but comparatively gradual transition.
• Chebyshev Type I: passband ripple and sharper transition.
• Chebyshev Type II: stopband ripple and a flat passband.
• Elliptic: ripple in both bands and the sharpest transition for a given order.

No prototype is universally superior. The choice depends on allowable ripple, attenuation, phase requirements, implementation resources and sensitivity to coefficient errors.

From an Analogue Prototype to a Digital Filter

Analogue prototype design begins with a continuous-time transfer function H_a(s). A frequency transformation and a mapping from the s-plane to the z-plane then produce H(z). A valid mapping should preserve stability: the left half of the s-plane should map into the inside of the unit circle.

Two classical approaches are impulse invariance and the bilinear transformation. They have different frequency-domain consequences and should not be treated as interchangeable.

Impulse Invariance

Impulse invariance samples the impulse response of an analogue filter. If h_a(t) is the analogue impulse response and the sampling period is T, the digital impulse response is commonly defined by

h[n] = T h_a(nT), n ≥ 0.

For an analogue transfer function expressed in partial fractions,

H_a(s) = Σ r_k/(s-p_k),

impulse invariance gives

H(z) = Σ (T r_k)/(1 - e^(p_kT)z^-1).

Thus, each analogue pole p_k maps to a digital pole z_k = e^(p_kT). If Re(p_k)<0, then |z_k|<1, so stability is preserved.

The frequency response of the resulting digital filter is related to repeated shifted versions of the analogue response:

H(e^jω) = Σ H_a(j(ω/T - 2πk/T)),

apart from scaling conventions. This expression shows the principal weakness of impulse invariance: spectral aliasing. The sampled impulse response creates periodic replicas of the analogue spectrum. If the analogue prototype has substantial energy above half the sampling frequency, these replicas overlap the baseband and distort the digital response.

Impulse invariance is most appropriate for low-pass filters whose analogue response is sufficiently attenuated before the analogue Nyquist frequency. It is generally less suitable for high-pass and bandpass filters unless aliasing is carefully examined.

Bilinear Transformation

The bilinear transformation maps the analogue s-plane to the digital z-plane according to

s = (2/T)(1-z^-1)/(1+z^-1),

or equivalently,

z = (1+sT/2)/(1-sT/2).

Substitution of this expression for s into H_a(s) produces the digital transfer function H(z).

The mapping has three important properties. First, the entire left half of the s-plane maps inside the unit circle. Second, the jΩ axis maps onto the unit circle. Third, the mapping is one-to-one, so there is no spectral aliasing. The major disadvantage is nonlinear frequency mapping, called frequency warping.

Setting z=e^jω gives

s = j(2/T) tan(ω/2).

Therefore, an analogue frequency Ω and a digital frequency ω are related by

Ω = (2/T) tan(ω/2),

or

ω = 2 tan^(-1)(ΩT/2).

Equal intervals on the analogue frequency axis do not map to equal intervals on the digital axis. The distortion becomes more pronounced as ω approaches π.

Prewarping

To ensure that a critical digital frequency maps exactly to the desired analogue prototype frequency, the digital specification is prewarped. If the desired digital edge is ω_c, the corresponding analogue frequency is selected as

Ω_c = (2/T) tan(ω_c/2).

For a low-pass filter, both passband and stopband edges should normally be prewarped before calculating the analogue order. The analogue design is then performed using Ω_p and Ω_s, and the bilinear transformation maps the resulting filter back to the digital domain.

For a bandpass or bandstop filter, the digital edges are first prewarped and then transformed into the normalized low-pass prototype variable. For example, an analogue bandpass transformation may use

s_L = (s^2 + Ω_0^2)/(B s),

where B is the bandwidth and Ω_0 is the centre frequency.

A Systematic IIR Design Workflow

A reliable design procedure can be organized as follows.

First, express the requirements in a consistent digital-frequency convention. State the sampling frequency, passband and stopband edges, passband ripple and stopband attenuation.

Second, choose the transformation method. Bilinear transformation is usually preferred when alias-free mapping is important. Impulse invariance may be chosen when faithful time-domain sampling and low analogue high-frequency energy are more significant.

Third, prewarp the critical frequencies if the bilinear method is used.

Fourth, normalize the frequencies and select a prototype family. The decision should reflect ripple tolerances, transition width and implementation sensitivity.

Fifth, calculate the minimum order. Use the appropriate order equation or a trusted numerical design routine, then round the order upward.

Sixth, determine analogue poles and zeros, apply the required frequency transformation, and convert the analogue transfer function to digital form.

Seventh, verify the result. Check the frequency response at all specification boundaries, pole locations, gain normalization, phase behaviour and numerical conditioning.

Finally, implement the filter in second-order sections, select scaling factors, and test it using representative signals and worst-case numerical conditions.

Stability and Realization Issues

Analogue stability is preserved by both standard mappings when the analogue prototype is stable. Under impulse invariance, a pole p in the left half-plane becomes e^(pT), whose magnitude is less than one. Under the bilinear transformation, the left half-plane maps conformally into the unit disk.

However, mathematical stability does not guarantee robust implementation. Direct-form structures may suffer from cancellation, overflow and coefficient sensitivity, particularly for high-order filters. A recommended realization is a cascade of biquads:

H(z) = G Π_i [(b_0i + b_1i z^-1 + b_2i z^-2)/(1 + a_1i z^-1 + a_2i z^-2)].

Real poles can be implemented as first-order sections or paired with another real pole. Complex-conjugate poles and zeros produce real second-order sections.

Section ordering and scaling also matter. A section with a large internal gain can overflow even when the overall filter output remains within range. State-variable scaling, saturation analysis and adequate numerical precision are therefore part of professional IIR implementation.

Worked Design Illustration

Consider an illustrative low-pass requirement with sampling frequency 8 kHz, passband edge 1 kHz, stopband edge 1.5 kHz, passband attenuation no greater than 1 dB and stopband attenuation at least 50 dB. The design will use the bilinear transformation.

The digital angular frequencies are

ω_p = 2π(1000/8000) = π/4,

ω_s = 2π(1500/8000) = 3π/8.

Prewarping gives

Ω_p = (2/T)tan(ω_p/2),

Ω_s = (2/T)tan(ω_s/2),

where T=1/8000 s. The ratio Ω_s/Ω_p, rather than the absolute scale, determines the normalized low-pass order. The designer then evaluates the Butterworth, Chebyshev and elliptic order requirements. Typically, Butterworth gives the highest order among these alternatives, Chebyshev Type I or II gives a lower order, and elliptic gives the lowest order when both ripple specifications are accepted.

Suppose the selected prototype is a Chebyshev Type I filter. The passband ripple determines ε through ε=sqrt(10^(A_p/10)-1). The minimum integer order is obtained from the Chebyshev order relation using the prewarped frequency ratio. The poles of the resulting normalized prototype are scaled by Ω_p. Substitution of s=(2/T)(1-z^-1)/(1+z^-1) then yields the digital transfer function. Finally, the response is evaluated at 1 kHz and 1.5 kHz, and the poles are checked to ensure that they lie inside the unit circle.

The important lesson is not a particular numerical order but the sequence: digital specifications, prewarping, prototype selection, order calculation, analogue pole generation, bilinear mapping and digital verification.

Applied Case Study: Anti-Aliasing Decimation Filter

Background

Consider an industrial vibration-monitoring unit that samples an accelerometer at 48 kHz but later reduces the data rate to 12 kHz for storage and analysis. Before decimation by four, frequencies above the new Nyquist frequency of 6 kHz must be sufficiently attenuated.

Situation and Problem

The useful vibration band extends to 4.5 kHz. The filter must have a nearly flat passband up to 4.5 kHz and at least 80 dB attenuation from 6 kHz onward. Computational resources are limited because the filter operates continuously on an embedded processor.

Analysis

A Butterworth response would provide a smooth passband but may require a relatively high order because the transition band from 4.5 to 6 kHz is narrow. A Chebyshev Type I filter could reduce order, but passband ripple may interfere with amplitude-based vibration diagnostics. An elliptic filter would reduce order further, but ripple in both bands and greater coefficient sensitivity could complicate calibration.

The bilinear transformation is selected because the design must avoid impulse-invariance aliasing. The passband and stopband edges are prewarped using the 48 kHz sampling period. A Chebyshev Type II or a carefully constrained elliptic design may be suitable if the system tolerates the specified stopband behaviour. If measurement amplitude accuracy is paramount, a higher-order Butterworth or a low-ripple alternative may be safer.

Approach and Solution

The engineering team calculates the order for each candidate prototype, then compares multiplication count, worst-case internal gain, passband deviation and coefficient sensitivity. The selected filter is implemented as scaled second-order sections. Frequency-response measurements are made using swept sine signals, while broadband noise is used to estimate stopband attenuation. Pole locations are recomputed using quantized coefficients rather than ideal coefficients.

Outcome

The final design satisfies the passband and stopband requirements while operating within the processor budget. Testing shows that the quantized cascade remains stable and that internal states do not overflow under maximum expected vibration levels.

Learning Points

The minimum theoretical order is not the only decision criterion. A filter with one fewer section may be less reliable if its poles are poorly conditioned or its internal gain is excessive. In decimation systems, stopband attenuation must be evaluated specifically at frequencies that alias into the retained band. Prototype selection is therefore an application decision involving signal integrity, computation and implementation risk.

Applications and Professional Practice

IIR filters are used in audio equalization, biomedical instrumentation, industrial sensing, telecommunications channel selection, control systems, instrumentation amplifiers and embedded sensor interfaces. Low-order IIR filters are especially useful when memory, processor time or latency is limited.

In audio, nonlinear phase may be acceptable in some equalizers but undesirable in precision crossover networks. In biomedical systems, passband ripple can affect the amplitude of physiological components. In control applications, phase distortion and delay can alter stability margins, so the filter must be designed together with the feedback controller. In communications, a sharp transition may be valuable, but coefficient quantization and dynamic-range behaviour must be tested under modulation and noise.

Modern design software automates order calculation, pole-zero generation and coefficient conversion. Nevertheless, software output must not be accepted without checking frequency conventions, normalization, sampling rate, section scaling and stability. Automation improves productivity; it does not replace engineering interpretation.

Common Errors, Limitations and Risk Controls

A frequent error is confusing ordinary frequency in hertz with digital angular frequency in radians per sample. Another is applying the bilinear transformation without prewarping critical edges. This produces a response whose actual digital cutoff differs from the intended value.

Impulse invariance is sometimes selected without considering spectral replicas. It can produce unacceptable aliasing even when the analogue prototype itself is well designed. Conversely, bilinear transformation avoids aliasing but changes frequency spacing through warping.

Other errors include rounding the order downward, ignoring the actual stopband boundary, implementing a high-order denominator in direct form, failing to test quantized coefficients, and assuming that a stable transfer function is safe under arbitrary scaling.

Professional practice requires reproducible specifications, documented design assumptions, independent verification, version-controlled coefficient files and tests using both ideal and finite-precision arithmetic. In safety-related or medical applications, design evidence should be traceable to system requirements and relevant regulatory procedures.

Emerging Design Directions

Contemporary IIR design increasingly combines classical approximation theory with numerical optimization. Designers may optimize pole-zero locations under constraints on magnitude, phase, group delay, power consumption and coefficient word length. Automated tools can search for robust second-order-section orderings and scaling factors.

Adaptive and time-varying IIR filters are used when signal statistics change, although adaptation introduces additional stability questions. Fixed-point implementation remains important in low-power devices, while floating-point processors simplify dynamic-range management. Machine-learning methods may assist in selecting filter structures or approximating design objectives, but the resulting filters must still satisfy explicit stability and frequency-response constraints.

Best-Practice Design Checklist

Before approving an IIR filter, the engineer should verify that:

1. The sampling frequency and frequency units are unambiguous.

2. Passband, stopband, ripple and attenuation requirements are explicit.

3. The prototype family is justified by application needs.

4. Prewarping has been applied where required.

5. The calculated order has been rounded upward.

6. The final digital response meets specifications at all critical edges.

7. Every pole lies strictly inside the unit circle.

8. The filter is implemented as robust first- or second-order sections.

9. Section gains and internal states have been checked.

10. Quantized coefficients have been tested under worst-case signals.

Chapter Summary

IIR filters use feedback to achieve sharp frequency selectivity with relatively low order. Their design commonly begins with Butterworth, Chebyshev or elliptic analogue prototypes. Butterworth filters are maximally flat; Chebyshev filters exchange ripple for selectivity; and elliptic filters achieve the narrowest transition through ripple in both bands.

Impulse invariance samples an analogue impulse response and preserves stability, but it introduces spectral aliasing. The bilinear transformation maps the stable analogue half-plane into the stable digital unit disk and avoids aliasing, but it produces nonlinear frequency warping. Prewarping compensates for this distortion at specified critical frequencies.

A complete design includes specification interpretation, prototype and transformation selection, order calculation, pole-zero mapping, stability verification and robust implementation. Practical engineering also requires attention to coefficient quantization, section scaling, internal overflow and application-specific phase or ripple constraints.

Key Terms

IIR filter: A digital filter whose output depends on previous outputs as well as present and past inputs.

Pole: A value of z or s at which a transfer function denominator becomes zero.

Zero: A value of z or s at which a transfer function numerator becomes zero.

BIBO stability: Stability condition in which every bounded input produces a bounded output.

Butterworth filter: A maximally flat magnitude-response filter with monotonic passband and stopband.

Chebyshev Type I filter: A filter with equiripple passband and monotonic stopband.

Chebyshev Type II filter: A filter with monotonic passband and equiripple stopband.

Elliptic filter: A filter with equiripple passband and stopband and highly selective transition.

Analogue prototype: A continuous-time reference filter used as the basis for digital design.

Impulse invariance: A transformation that samples the analogue impulse response to obtain a digital impulse response.

Bilinear transformation: A one-to-one mapping between the analogue s-plane and digital z-plane.

Frequency warping: Nonlinear correspondence between analogue and digital frequencies under bilinear transformation.

Prewarping: Adjusting analogue design frequencies so that bilinear mapping places digital critical frequencies correctly.

Spectral aliasing: Overlap of replicated spectra caused by sampling or periodic spectral mapping.

Passband ripple: Permitted variation in magnitude within the passband.

Stopband attenuation: Reduction in magnitude required in the stopband, usually expressed in decibels.

Second-order section: A first- or second-order filter section used for numerically robust implementation.

Finite-word-length effect: Performance change caused by quantized coefficients and finite numerical precision.

Prototype order: The number of poles, or effective degree, required to satisfy a filter specification.

Review Questions

Short Answer Questions

1. Why can an IIR filter achieve a sharp transition with lower order than many FIR filters?

2. State the pole-location condition for BIBO stability of a causal digital IIR filter.

3. What type of ripple is associated with a Chebyshev Type I filter?

4. Why is prewarping needed in bilinear-transformation design?

5. What is the principal spectral disadvantage of impulse invariance?

Long Answer Questions

1. Explain the characteristics and design trade-offs of Butterworth, Chebyshev Type I, Chebyshev Type II and elliptic filters.

2. Derive the frequency relationship produced by the bilinear transformation and explain frequency warping.

3. Describe a complete workflow for designing a digital low-pass IIR filter from an analogue prototype.

4. Compare impulse invariance and bilinear transformation with respect to stability, aliasing and frequency mapping.

5. Discuss why second-order-section implementation is generally preferred for high-order IIR filters.

Analytical / Application Questions

1. An application permits no passband ripple but requires a narrow transition band. Which prototype families would you compare, and what trade-offs would guide the final choice?

2. A bilinear-designed filter meets its analogue prototype specification but misses the desired digital stopband edge. Identify the likely design error and propose a correction.

3. An implemented filter is theoretically stable, but fixed-point testing shows oscillation. Analyse the likely causes and recommend a verification and implementation strategy.

22. Multiple Choice Questions with Answers

Question 1

Which characteristic most directly distinguishes an IIR filter from an FIR filter?

A. It uses feedback from previous output samples.

B. It always has linear phase.

C. It cannot contain poles.

D. It has a finite-duration impulse response.

Correct Answer: A. It uses feedback from previous output samples.

Question 2

For a causal digital IIR filter, which pole configuration guarantees BIBO stability?

A. All poles lie on the unit circle.

B. All poles lie outside the unit circle.

C. All poles lie strictly inside the unit circle.

D. At least one pole lies at z = 1.

Correct Answer: C. All poles lie strictly inside the unit circle.

Question 3

Which prototype has a maximally flat magnitude response and no passband ripple?

A. Elliptic

B. Butterworth

C. Chebyshev Type I

D. Chebyshev Type II

Correct Answer: B. Butterworth

Question 4

What is the main disadvantage of impulse invariance?

A. It cannot preserve stability.

B. It produces frequency warping but no aliasing.

C. It introduces spectral aliasing through replicated analogue spectra.

D. It maps stable poles outside the unit circle.

Correct Answer: C. It introduces spectral aliasing through replicated analogue spectra.

Question 5

Under the bilinear transformation, the analogue frequency corresponding to digital frequency ω is

A. Ω = (T/2)tan(ω/2)

B. Ω = (2/T)tan(ω/2)

C. Ω = (2/T)sin(ω)

D. Ω = ωT/2

Correct Answer: B. Ω = (2/T)tan(ω/2)

Question 6

Why are critical frequencies prewarped before bilinear-transformation design?

A. To introduce controlled aliasing.

B. To make the analogue prototype unstable.

C. To compensate for nonlinear analogue-to-digital frequency mapping.

D. To remove all passband ripple.

Correct Answer: C. To compensate for nonlinear analogue-to-digital frequency mapping.

Question 7

A designer wants the narrowest transition band for a specified order and accepts ripple in both bands. Which prototype is most appropriate?

A. Butterworth

B. Chebyshev Type II

C. Elliptic

D. FIR moving-average

Correct Answer: C. Elliptic

Question 8

Which implementation is generally more numerically robust for a high-order IIR filter?

A. One direct-form polynomial section

B. A cascade of scaled second-order sections

C. An unscaled single delay element

D. A filter with all coefficients rounded to zero

Correct Answer: B. A cascade of scaled second-order sections

Question 9

If an analogue stable pole is p and impulse invariance is used, its digital pole is

A. p/T

B. 1 + pT

C. e^(pT)

D. e^(-p/T)

Correct Answer: C. e^(pT)

Question 10

In the decimation case study, why must the stopband begin before the new Nyquist frequency?

A. To increase passband ripple.

B. To prevent unwanted components from aliasing into the retained band.

C. To make the analogue prototype noncausal.

D. To eliminate the need for coefficient quantization.

Correct Answer: B. To prevent unwanted components from aliasing into the retained band.