IIR Filter Design

13.1 Filter Design Specifications

Filter design starts not from a transfer function but from a specification（規格）: how much the passband may ripple, how much the stopband must be attenuated, and how wide the transition band may be. The designer's job is to find a realizable, stable, causal rational system function $H(z)$ whose frequency response stays inside the allowed tolerance region with the lowest possible cost (filter order).

13.1.1 The Tolerance Scheme

An ideal lowpass response — unity gain up to a cutoff, zero beyond — is unrealizable: it is noncausal and has infinite-length impulse response with no rational $H(z)$. Practical specifications therefore relax the ideal into a tolerance scheme （容忍度規格圖） with three regions:

• Passband（通帶） $0 \le \omega \le \omega_p$: the magnitude must stay within a ripple band, $1-\delta_p \le |H(e^{j\omega})| \le 1$ (some books use $1\pm\delta_1$).
• Stopband（阻帶） $\omega_s \le \omega \le \pi$: the magnitude must be small, $|H(e^{j\omega})| \le \delta_s$.
• Transition band（過渡帶） $\omega_p < \omega < \omega_s$: a nonzero-width "don't care" region where the response rolls off; no constraint is imposed.

13.1.2 Ripple Specifications in dB

Specifications are usually quoted logarithmically. The passband ripple $R_p$ and the minimum stopband attenuation $A_s$ (both positive dB numbers) relate to the linear tolerances as:

Typical values: $R_p = 1$ dB means $1-\delta_p = 10^{-1/20} = 0.8913$; $A_s = 40$ dB means $\delta_s = 10^{-40/20} = 0.01$. A useful intermediate quantity is the discrimination ratio built from $\varepsilon_p^2 = 10^{R_p/10}-1$ (passband) and $10^{A_s/10}-1$ (stopband); it appears in every order formula in this chapter.

13.1.3 FIR vs. IIR: Choosing the Filter Type

Before designing, choose the filter class. The lecture's comparison:

13.2 Analog Filter Design

13.2.1 Analog Filter Families at a Glance

All classical analog lowpass prototypes share the squared-magnitude template $|H_c(j\Omega)|^2 = \dfrac{1}{1+V_N^2(\Omega)}$ where $V_N$ is a polynomial or rational function of order $N$ that is small in the passband and large in the stopband. The four families differ in where they spend their approximation error:

13.2.2 Butterworth Filters

The Butterworth lowpass of order $N$ with 3-dB cutoff $\Omega_c$ is defined by its squared magnitude:

Its defining property is maximal flatness（最大平坦）: the first $2N-1$ derivatives of $|H_c(j\Omega)|^2$ are zero at $\Omega = 0$. Consequences visible in Fig. 13-2:

• $|H_c(j0)| = 1$ for every order; $|H_c(j\Omega_c)| = 1/\sqrt{2}$ ($-3$ dB) for every order — all curves pass through the same point.
• The response is monotonic in both bands; high-frequency roll-off is $-20N$ dB/decade（每十倍頻 $-20N$ dB）.
• As $N \to \infty$ the response approaches the ideal brick wall with cutoff $\Omega_c$.

Poles of the Butterworth filter

Substituting $\Omega = s/j$ gives $H_c(s)H_c(-s) = \dfrac{1}{1+(s/j\Omega_c)^{2N}}$, whose $2N$ poles are equally spaced on a circle of radius $\Omega_c$:

The poles come in pairs $\pm s_k$, none on the $j\Omega$ axis, separated by angle $\pi/N$. To build the stable, causal $H_c(s)$ we assign the $N$ left-half-plane poles to $H_c(s)$ and the mirror images to $H_c(-s)$.

Choosing the order from a spec

Requiring $|H_c(j\Omega_p)|^2 \ge (1-\delta_p)^2 = 10^{-R_p/10}$ and $|H_c(j\Omega_s)|^2 \le \delta_s^2 = 10^{-A_s/10}$ and solving the two equations of $(\Omega/\Omega_c)^{2N}$ gives:

With the integer $N$, $\Omega_c$ can be placed anywhere between the value that meets the passband exactly, $\Omega_c = \Omega_p\,(10^{R_p/10}-1)^{-1/2N}$, and the value that meets the stopband exactly, $\Omega_c = \Omega_s\,(10^{A_s/10}-1)^{-1/2N}$. The slack exists because $N$ was rounded up.

13.2.3 Butterworth Design Example

13.2.4 Chebyshev Type I Filters

The Chebyshev Type I filter trades passband flatness for a sharper knee: it is equiripple in the passband（通帶等漣波） and monotonic in the stopband.

$T_N$ is the order-$N$ Chebyshev polynomial: it oscillates between $\pm1$ for $|x|\le 1$ (hence equal ripple between $1$ and $1/\sqrt{1+\varepsilon^2}$ in the passband) and grows like $\cosh$ — much faster than $x^N$ — outside, giving rapid attenuation. Properties:

• Passband ripple is set by $\varepsilon$: $\;\varepsilon = \sqrt{10^{R_p/10}-1}$.
• $|H_c(j0)| = 1$ for odd $N$, $\;1/\sqrt{1+\varepsilon^2}$ for even $N$ (count the ripple extrema).
• Poles lie on an ellipse in the $s$-plane (minor axis along $j\Omega$).
• Required order: $$N \;\ge\; \frac{\cosh^{-1}\!\sqrt{\dfrac{10^{A_s/10}-1}{10^{R_p/10}-1}}}{\cosh^{-1}(\Omega_s/\Omega_p)}$$ — typically about half the Butterworth order for the same spec.

13.2.5 Chebyshev Type II Filters

Chebyshev Type II (inverse Chebyshev) flips the trade: monotonic, maximally-flat-like passband and equiripple stopband（阻帶等漣波）, achieved by placing transmission zeros on the $j\Omega$ axis in the stopband:

It needs the same order as Type I for a given spec, but has no passband ripple — useful when the passband carries a signal whose amplitude must not be distorted. Cost: a rational (not polynomial) transfer function with finite zeros, and the response no longer decays monotonically to zero at high frequency (it returns to the ripple ceiling $\delta_s$).

13.2.6 Elliptic (Cauer) Filters

Elliptic filters are equiripple in both bands, replacing $T_N$ with the Jacobi elliptic rational function $R_N$:

By the Chebyshev approximation theorem, distributing the error equally over both bands is optimal: for given $(\delta_p,\delta_s,\Omega_p,\Omega_s)$ the elliptic filter achieves the spec with the minimum possible order（最小階數） of any rational filter. The price is the most nonlinear phase and strong group-delay peaking near the band edge.

13.2.7 Analog Frequency Transformations

All formulas above are for a lowpass prototype. Other shapes come from algebraic substitutions on $s$ (here the prototype has cutoff 1 rad/s):

In digital IIR design the same idea can be applied either in the analog domain (before the $s\to z$ map) or directly in the $z$-domain with all-pass substitutions; MATLAB's butter(N, Wn, 'high') etc. handle this automatically (§13.4).

13.3 Digital Filters from Analog Prototypes

13.3.1 What a Good s→z Mapping Must Do

We have a good $H_c(s)$; we need $H(z)$. Any useful transformation from the $s$-plane to the $z$-plane must satisfy two requirements:

1. Frequency axis maps to the unit circle: the essential frequency-selective behavior on the $j\Omega$ axis must appear on $z = e^{j\omega}$ (possibly with a distorted frequency axis).
2. Stability is preserved: the left half-plane $\operatorname{Re}\{s\}<0$ must map inside the unit circle $|z|<1$, so a stable, causal $H_c(s)$ yields a stable, causal $H(z)$.

Two classical mappings satisfy both: impulse invariance and the bilinear transformation.

13.3.2 Impulse Invariance

The impulse invariance method（脈衝響應不變法） defines the digital filter by sampling the analog impulse response:

This is exactly the sampling theorem applied to $h_c(t)$ (the factor $T$ normalizes the gain so that $H(e^{j0}) \approx H_c(j0)$). Two consequences:

• Linear frequency mapping: within the baseband, $\;\omega = \Omega T$. Analog band edges translate as $\omega_p = \Omega_p T$, $\omega_s = \Omega_s T$ — no distortion of the frequency axis.
• Aliasing（混疊）: the shifted copies $H_c(j(\omega-2\pi k)/T)$ overlap unless $H_c$ is bandlimited — and no realizable filter is. The digital response is the analog one plus aliased tails, which raises the stopband floor (Fig. 13-5).

Implementation via partial fractions

In practice nobody samples $h_c(t)$ literally; the pole mapping is done algebraically. Expand the (simple-pole) prototype and map each term:

Each analog pole $s_k$ becomes a digital pole at $z_k = e^{s_k T}$. Since $\operatorname{Re}\{s_k\}<0 \Rightarrow |z_k| = e^{\operatorname{Re}\{s_k\}T} < 1$, stability is preserved. Note that zeros do not map this way — only poles.

13.3.3 The Bilinear Transformation

The bilinear transformation（雙線性轉換） avoids aliasing entirely by using an algebraic, one-to-one map between the $s$-plane and the $z$-plane:

(It derives from the trapezoidal-rule discretization of the integrator $1/s$.) Mapping properties — check them by substituting $s = \sigma + j\Omega$:

• $s = j\Omega$ (entire imaginary axis) $\to$ $|z| = 1$ (unit circle, traversed once): one-to-one, hence no aliasing.
• $\operatorname{Re}\{s\} < 0 \to |z| < 1$: stability preserved.
• $s=0 \to z=1$; $\;\Omega\to\pm\infty \to z=-1$ (i.e. $\omega = \pm\pi$): the infinite analog axis is compressed into one revolution of the unit circle.
• Order is preserved: an $N$-th order $H_c(s)$ gives an $N$-th order $H(z)$; each analog zero at $s=\infty$ becomes a digital zero at $z=-1$.

13.3.4 Frequency Warping and Prewarping

Setting $z = e^{j\omega}$ in the bilinear map gives $s = \frac{2}{T}\,\frac{1-e^{-j\omega}}{1+e^{-j\omega}} = j\frac{2}{T}\tan(\omega/2)$, i.e. the frequency axes are related nonlinearly:

For small $\omega$, $\Omega \approx \omega/T$ (the map is locally linear), but band edges at higher frequencies are warped. The fix is prewarping（預翹曲）: translate every critical digital frequency into the analog domain using the exact warping formula before designing the prototype:

Then after the bilinear map the digital filter hits $\omega_p,\omega_s$ exactly. Note $T$ cancels out of the final $H(z)$ — the spec is prewarped by $T$ and warped back by $1/T$ — so one conventionally sets $T=1$ (or $T=2$) for convenience.

13.3.5 Complete Design Procedure

For the impulse-invariance route, replace steps 2 and 5 by the linear translation $\Omega_{p,s} = \omega_{p,s}/T$ and the pole-by-pole map of Eq. (13.3.2) — and over-design the stopband slightly to absorb aliasing.

13.4 Design in MATLAB and Other Filter Shapes

MATLAB (Signal Processing Toolbox) wraps this whole chapter in two calls per family: an order-selection function and a design function. Digital frequencies are normalized so that $1.0$ means the Nyquist frequency $\pi$ rad/sample:

Other shapes ("Other Filter Shapes", slide 31): pass 'high', 'stop', or a two-element Wn = [w1 w2] for bandpass — MATLAB applies the frequency transformations of §13.2.7 to the lowpass prototype internally, then the bilinear transform with built-in prewarping:

% lowpass meeting wp=0.4pi, ws=0.6pi, Rp=1 dB, As=40 dB
[N, Wn] = buttord(0.4, 0.6, 1, 40);     % -> N = 9, Wn = 0.4230
[b, a]  = butter(N, Wn);                % digital LPF via bilinear (prewarped)

[b, a] = butter(6, 0.7, 'high');        % highpass, cutoff 0.7*pi
[b, a] = ellip(4, 0.5, 50, [0.25 0.45]);% bandpass (order doubles to 8)
[b, a] = cheby2(5, 40, [0.3 0.5], 'stop'); % bandstop
freqz(b, a, 2048)                       % always verify against the mask

13.5 Chapter Summary

Exercises

wp = 0.3;  ws = 0.4;          % band edges / pi
Rp = 0.5;  As = 50;           % dB

[Nb, ~ ] = buttord (wp, ws, Rp, As);   % Nb = 20
[Nc, ~ ] = cheb1ord(wp, ws, Rp, As);   % Nc = 9
[Ne, Wn] = ellipord(wp, ws, Rp, As);   % Ne = 6

[b, a] = ellip(Ne, Rp, As, Wn);        % minimum-order realization
[H, w] = freqz(b, a, 2048);
plot(w/pi, 20*log10(abs(H))); grid on
xlabel('\omega/\pi'); ylabel('|H| (dB)'); axis([0 1 -100 5])
hold on                                 % spec mask
plot([0 wp],[-Rp -Rp],'r--', [ws 1],[-As -As],'r--')