IIR Filter Design

8.1  From analog prototype to digital filter

Designing IIR coefficients directly is difficult because feedback makes the error depend nonlinearly on the coefficients, and an unconstrained search can create unstable poles. A standard alternative is to design a well-understood analog prototype first and then transform it into a digital filter.

The analog prototype is a rational function of \(s\), while the required digital filter is a rational function of \(z^{-1}\). A mapping \(s\mapsto f(z)\) connects the two.

The two main mappings have different trade-offs. Impulse invariance preserves samples of the analog impulse response but can alias the frequency response. The bilinear transform avoids aliasing but warps the frequency axis. Before applying either method, identify the specification, choose an analog prototype family and form the required band shape.

8.1.1  What the specification says

A filter specification gives allowable ranges rather than one exact curve. It defines the permitted passband variation, maximum stopband level and the two band edges. The transition band between the edges is left unconstrained, giving the design enough freedom to be realizable.

The original requirements are stated in digital frequency \(\omega\), because the final filter is digital. The corresponding analog frequencies \(\Omega\) must be derived according to the chosen transformation. They are intermediate design values, not the original specification.

8.1.2  Which analog prototype

Every design in this lecture begins from one of the four classical prototypes of Lecture 5. Each is defined by its magnitude-squared response, and for the Butterworth family that response is as flat as a rational function of the given order can be made at \(\Omega=0\):

Here \(N\) is the order, \(\Omega_c\) is the cutoff and \(\epsilon\) controls the passband tolerance. The denominator polynomial \(B(s)\) distinguishes the prototype families and determines their left-half-plane poles. The first design decisions are therefore the family and the required order.

8.1.3  Getting the band shape you want

Prototype tables normally use a normalized low-pass filter with unit cutoff. Standard substitutions in \(s\) convert it into a low-pass, high-pass, band-pass or band-stop response. The transformation relocates the band edges while preserving the prototype's ripple levels.

8.2  Impulse invariance

The most literal reading of "carry the analog filter into discrete time" is to keep its impulse response and simply read it at the sampling instants. That is the whole idea of impulse invariance, and everything else in this section is a consequence of it.

Impulse invariance samples the analog impulse response: \(h[n]=T h_a(nT)\). The interval \(T\) sets the analog-to-digital frequency scaling; in the normalized examples it is chosen conveniently. As in Lecture 3, sampling in time creates repeated spectral copies and therefore a risk of aliasing.

If the analog response were exactly zero above \(\pi/T\), the spectral copies would not overlap inside \(|\omega|\le\pi\). The digital response would then be a frequency-scaled version of the analog response:

Real analog filters are not perfectly band-limited, so neighbouring copies can overlap the baseband. The mapping remains linear, but high-frequency energy folds into lower digital frequencies. A low-pass prototype may decay enough for this error to be small. A high-pass or band-stop response retains substantial high-frequency energy, so impulse invariance is usually unsuitable.

To obtain a digital transfer function, first write \(H_a(s)\) in partial fractions. Each analog pole produces an exponential term in \(h_a(t)\). Sampling that exponential produces a geometric sequence with a simple \(z\)-transform:

A stable analog pole has \(\Re(s_k)<0\). Its digital image satisfies \(|z_k|=e^{\Re(s_k)T}<1\), so it lies inside the unit circle. Impulse invariance therefore preserves stability.

8.2.1  Three things Eq. (8.6) does not say

The formula maps poles, not zeros. Digital zeros appear after the partial-fraction terms are recombined over a common denominator and depend on both poles and residues. Do not map analog zeros using \(z=e^{sT}\); impulse invariance is not a complete point-by-point mapping of the two planes.

Changing the normalized value of \(T\) does not remove aliasing. The digital specification is fixed in \(\omega\), so changing \(T\) rescales both the analog band edges and the sampling boundary together. To reduce fold-back, use a higher-order prototype with faster high-frequency decay or design extra stopband margin.

Other time-domain mappings are possible. Step invariance, for example, preserves samples of the analog step response. A waveform-invariance method is useful when transient properties such as rise time, overshoot or ringing matter more than an exact frequency-axis mapping.

MATLAB corner for Example 8.1

This script designs Example 8.1's \(N=6\) analog Butterworth prototype, applies impulse invariance via impinvar(), and checks the resulting digital filter against the discrete-time specification.

Goal: confirm the impulse-invariance design meets the passband/stopband targets, and show its (nonlinear, but nearly constant in-band) group delay.

%% impulse_invariance_demo.m - Example 8.1's N=6 Butterworth (Eq. 8.6)
Ap = 20*log10(1/0.89125); As = 20*log10(1/0.17783);
Wp = 0.2*pi; Ws = 0.3*pi; T = 1;
 
[N, Wc] = buttord(Wp, Ws, Ap, As, 's');   % N=6, Wc=0.7032 (Step 2)
[zb, pb, kb] = butter(N, Wc, 's');
[numA, denA] = zp2tf(zb, pb, kb);         % analog H_c(s), Step 3
 
[numD, denD] = impinvar(numA, denA, 1/T);  % impulse invariance, Eq. (8.6)
 
[H,w] = freqz(numD, denD, 1024);
figure('Color','w');
subplot(1,3,1); plot(w/pi, 20*log10(abs(H)), 'LineWidth',1.3); grid on;
ylim([-40 5]); xline(0.2,'g--'); xline(0.3,'r--'); title('log-magnitude (dB)');
subplot(1,3,2); plot(w/pi, abs(H), 'LineWidth',1.3); grid on; title('magnitude');
subplot(1,3,3); grpdelay(numD, denD, 1024); title('group delay (nonlinear near transition)');

8.3  The bilinear transform

The bilinear transform is an algebraic substitution rather than a sampling operation, so it does not create spectral copies. It maps each relevant point in the \(s\)-plane to one point in the \(z\)-plane:

8.3.1  Stability is preserved exactly

Substitute \(s=\sigma+j\Omega\) into Eq. (8.7) and compare the magnitudes of the numerator and denominator. The sign of \(\sigma\) determines whether the image lies inside, on or outside the unit circle:

The complete left half-plane maps inside the unit circle, so every stable analog pole becomes a stable digital pole. Unlike impulse invariance, the mapping is one-to-one and does not fold multiple analog frequencies onto one digital frequency.

8.3.2  Frequency warping

The price of that one to one map is paid on the boundary between the two regions. Put \(z=e^{j\omega}\) into Eq. (8.7) and factor \(e^{-j\omega/2}\) out of the numerator and the denominator, which is the step that turns the algebra into a tangent:

Evaluating the inverse mapping on \(z=e^{j\omega}\) gives a purely imaginary value of \(s\), so the unit circle maps onto the \(j\Omega\)-axis. The resulting frequency relationship is nonlinear:

At low frequencies, \(\tan(\omega/2)\approx\omega/2\), so the mapping is nearly linear. As \(\omega\) approaches \(\pi\), the tangent tends to infinity. The entire unbounded analog frequency axis is therefore compressed into the finite digital interval, avoiding aliasing but producing frequency warping.

8.3.3  Prewarping

Because the mapping is nonlinear, using the naive analog edges \(\Omega=\omega/T\) would place the final digital edges incorrectly. Prewarping applies the inverse mapping first, so the analog prototype is designed at frequencies that will transform to the required digital edges:

Prewarping makes each specified band edge land exactly at its target. Frequencies between the edges are still warped, but this does not violate an ordinary tolerance mask because the transition band is unconstrained.

8.3.4  Where the poles and zeros land

The bilinear transform maps both poles and zeros. If an analog filter has fewer finite zeros than poles, its remaining zeros are regarded as lying at \(s=\infty\). These map to \(z=-1\), so a bilinear low-pass has zeros at the Nyquist frequency.

The substitution can change the overall gain, so normalize the final filter at a suitable reference frequency: DC for a low-pass, \(\omega=\pi\) for a high-pass, and the centre frequency for a band-pass.

MATLAB corner for Example 8.2

The script designs the filter in two ways: with MATLAB's bilinear() command and by applying the prewarped substitution by hand. Comparing the two responses checks the algebra and shows why prewarping is needed to place the digital cutoff at the requested frequency.

Goal: verify \(|H(e^{j\pi/4})|=1/\sqrt2\) exactly, and show the full magnitude/phase response of the resulting one-pole digital filter.

%% bilinear_lpf_demo.m - Example 8.2's prewarped first-order BLT design (Eq. 8.10)
wc = pi/4;                       % target digital cutoff (rad)
Wc = tan(wc/2);                  % Step 1: prewarp, T=2 convention (fs=0.5)
numA = Wc;                       % Step 2: analog numerator
denA = [1 Wc];                   % Step 2: H_a(s) = Wc/(s+Wc)
[numD, denD] = bilinear(numA, denA, 0.5); % Step 3: bilinear, fs = 1/T = 0.5
 
[H, w] = freqz(numD, denD, 1024);
[~, idx] = min(abs(w - wc));
fprintf('|H(e^{j pi/4})| = %.4f  (target 1/sqrt(2) = %.4f)\n', abs(H(idx)), 1/sqrt(2));
 
figure('Color','w');
subplot(1,2,1);
plot(w/pi, abs(H), 'LineWidth', 3); grid on;
xline(0.25, 'r--', 'LineWidth', 2);
yline(1/sqrt(2), 'k:', 'LineWidth', 1.5);
title('magnitude');
 
subplot(1,2,2);
plot(w/pi, unwrap(angle(H))*180/pi, 'LineWidth', 3); grid on;
title('phase');

8.3.5  What warping does to phase

Prewarping corrects selected band edges, not the complete frequency-axis shape. In particular, an analog linear-phase response does not remain linear after the nonlinear mapping. Applying the transform to an ideal delay shows this directly:

The bilinear transform is well suited to standard magnitude specifications because prewarping can place their important edges exactly. Impulse invariance is useful when preserving sampled time-domain behaviour or a linear frequency mapping is more important, provided the analog response decays enough to keep aliasing small. For high-pass and band-stop analog prototypes, the bilinear transform is normally required.

8.4  Changing the band shape in the \(z\)-domain

Band transformation can also be performed after a digital low-pass filter has been created. This is especially useful with impulse invariance: first transform a decaying analog low-pass prototype, then use a digital substitution to obtain the required high-pass or band-stop response.

Write \(H_{lp}(Z)\) for the digital low-pass filter already in hand and \(H(z)\) for the filter wanted. Both are rational in the inverse variable, so it is convenient to state the substitution that way as well:

The substitution \(G(z^{-1})\) must remain rational, map the unit circle onto itself, and map the inside of the unit circle into itself. These conditions preserve implementability, frequency-response meaning and stability. They lead to an all-pass substitution built from first-order factors:

The condition \(|\alpha_k|<1\) preserves stability. A single real factor maps one digital low-pass filter into another low-pass filter with a different cutoff. Let \(Z=e^{j\theta}\) denote the prototype frequency and \(z=e^{j\omega}\) the transformed frequency:

The all-pass substitution rearranges the frequency axis without changing the prototype's magnitude values. An equiripple prototype therefore keeps the same ripple heights while its band edges move. Solving Eq. (8.15) at the desired edge determines \(\alpha\); related substitutions produce high-pass, band-pass and band-stop filters.

Here \(\theta_p\) is the prototype passband edge and the \(\omega\) values are the required output edges. Band-pass and band-stop mappings use two all-pass factors, so an order-\(N\) prototype becomes an order-\(2N\) filter and can shape two transitions.

There are two practical routes. We can change the band shape in the analog domain before converting the prototype, which is convenient when using analog design tables. Alternatively, we can convert a digital low-pass filter first and then apply a substitution in the \(z\)-domain. The second route reuses an existing digital design and is useful for high-pass or band-stop filters based on impulse invariance. In the low-pass-to-low-pass substitution, the parameter \(\alpha\) also provides a convenient way to tune the cutoff without redesigning the whole structure.

8.5  Choosing between the two methods

Both are stability-preserving, linear-algebra-simple transforms of the same analog prototype, but they disagree everywhere else:

For most standard frequency-selective filters, the bilinear transform is preferred because it avoids aliasing and places prewarped edges exactly. Impulse invariance remains useful when preserving samples of an analog system's impulse response or transient behaviour is the main objective.

?Problems with solutions

Work each one before reading the solution.

Summary and key formulas

The reference sheet: the two transforms' key equations, and the property-by-property comparison.

• Both methods start from a Lecture 5 analog prototype and transform it, because IIR filters are essentially never optimized directly (Lecture 6, Section 6.2).
• Impulse invariance samples the analog impulse response; the pole rule \(z_k=e^{s_kT}\) guarantees stability but the method can alias if the prototype is not sufficiently band-limited.
• The bilinear transform is a one-to-one algebraic map: no aliasing, exact stability preservation, but a nonlinear frequency-warping relation that must be compensated by prewarping the design edges.
• In practice, the BLT is the default for standard frequency-selective filters; impulse invariance is reserved for cases where the time-domain impulse-response shape itself must be preserved.
• Both examples in this lecture transform a Lecture 5 Butterworth prototype, and the same recipe applies unchanged to Chebyshev or elliptic prototypes.