EP37

EP37: Your EQ Moves Poles — Biquad Filters

双二阶H(z), 双线性变换, 极零点迁移

▶ 4:28 Signal Processing

前置知识

EP28 Your Microphone Is a Damped Oscillator

后续拓展

EP39 How Spotify Knows You Are Too Loud — The LUFS Algorithm

Overview

Turn any knob on your DAW’s parametric EQ, and somewhere inside the plugin, five floating-point numbers change. Those five numbers — $b_0, b_1, b_2, a_1, a_2$ — are the coefficients of a biquad filter, and they completely determine two things: where the poles and zeros sit in the complex $z$ -plane, and therefore what the frequency response looks like. The insight of this episode is that these two descriptions — “five coefficients” and “pole-zero constellation” — are the same object in two different languages.

A biquad is the smallest non-trivial building block of digital signal processing: a second-order IIR (infinite impulse response) filter whose transfer function is a ratio of degree-2 polynomials in $z^{-1}$ . Every EQ band in a professional plug-in is one biquad. A five-band parametric EQ chains five biquads in series. The microphone from EP28 is, after discretisation, also a biquad — its diaphragm resonance becomes a conjugate pole pair sitting just inside the unit circle.

Building a digital biquad from an analog prototype (Butterworth, Chebyshev, etc.) requires mapping the continuous $s$ -plane to the discrete $z$ -plane. The standard industrial method is the bilinear transform: replace $s$ by the rational function $\tfrac{2}{T}\tfrac{1-z^{-1}}{1+z^{-1}}$ . This map sends the imaginary axis bijectively onto the unit circle, preserving stability. Its one cost is a non-linear compression of the frequency axis — frequency warping — which must be pre-compensated during design.

The Audio EQ Cookbook (Bristow-Johnson, 1994/2016) collects closed-form formulas that absorb the bilinear algebra and expose only the three musically natural parameters: center frequency $f_0$ , gain $G$ (in dB), and bandwidth $Q$ . Raising $Q$ narrows the peak and pushes the poles toward the unit circle; raising $G$ expands the pole magnitude slightly outward. As $Q \to \infty$ the poles touch the unit circle and the filter becomes marginally unstable.

This episode also introduces dynamic range compression, which operates not in the frequency domain but in the amplitude domain. A compressor is a time-varying gain $g(t)$ whose value is estimated by a first-order IIR envelope follower. The attack and release time constants correspond directly to the coefficient $\alpha = e^{-1/(f_s \tau)}$ of that one-pole filter.

中文: “在复平面上，有两个点在缓慢移动。你转动均衡器的旋钮，它们就漂移——离单位圆越近，某个频率就被放大越多。这两个点叫做极点。今天我们来看清楚：你的EQ插件，本质上是在移动复平面上的极点。”

Prerequisites

话筒是阻尼振子（EP28） — the second-order ODE for a microphone diaphragm; quality factor $Q$ ; poles of $H(s)$ in the left half-plane as the stability criterion. EP37 is the discrete-time continuation of that story: the same poles, now inside the unit circle.

Familiarity with the $z$ -transform and the Laplace transform is assumed at the level introduced in earlier episodes. No matrix algebra is required; EP39 extends biquad chains to reverb via convolution matrices.

Definitions

Definition 37.1 (Biquad Transfer Function)

A biquad (second-order IIR) filter is defined by five real coefficients $b_0, b_1, b_2, a_1, a_2 \in \mathbb{R}$ . Its transfer function in the $z$ -domain is

$H(z) = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}}$

Multiplying numerator and denominator by $z^2$ gives the equivalent polynomial form:

$H(z) = \frac{b_0 z^2 + b_1 z + b_2}{z^2 + a_1 z + a_2}$

The zeros of $H(z)$ are the roots of the numerator polynomial; the poles are the roots of the denominator polynomial. The difference equation (time-domain implementation) is

$y[n] = b_0\,x[n] + b_1\,x[n-1] + b_2\,x[n-2] \;-\; a_1\,y[n-1] - a_2\,y[n-2]$

The system requires two past input samples and two past output samples — four numbers of state.

Definition 37.2 (Stability of a Biquad)

A biquad is BIBO stable (bounded-input bounded-output stable) if and only if both poles lie strictly inside the unit circle:

$|z_{\text{pole}}| < 1$

For a real-coefficient biquad the poles are either both real, or form a complex-conjugate pair $z_{1,2} = r\,e^{\pm i\theta}$ with $r = \sqrt{a_2}$ and $\theta = \arccos\!\bigl(-a_1/(2\sqrt{a_2})\bigr)$ .

The pole radius $r < 1$ corresponds to the analog stability condition $\operatorname{Re}(s_{\text{pole}}) < 0$ from

EP28

; the bilinear transform maps left half-plane to interior of unit disk.

Definition 37.3 (Bilinear Transform)

The bilinear transform is the substitution

$s \;\longleftarrow\; \frac{2}{T} \cdot \frac{1 - z^{-1}}{1 + z^{-1}}$

where $T = 1/f_s$ is the sampling period and $f_s$ is the sample rate. Solving for $z$ in terms of $s$ :

$z = \frac{1 + sT/2}{1 - sT/2}$

This is a Mobius transformation mapping the $s$ -plane to the $z$ -plane. It maps the imaginary axis ( $s = i\Omega$ ) to the unit circle ( $|z|=1$ ) and the open left half-plane ( $\operatorname{Re}(s) < 0$ ) to the open unit disk ( $|z| < 1$ ).

Definition 37.4 (Frequency Warping)

Under the bilinear transform, the digital frequency $\omega \in [0, \pi]$ (in radians per sample) and the analog frequency $\Omega$ (in radians per second) are related by the warping equation:

$\Omega = \frac{2}{T}\tan\!\left(\frac{\omega}{2}\right)$

or equivalently $\omega = 2\arctan(\Omega T/2)$ . This is a bijection from $[0,\pi]$ to $[0, \infty)$ , but it is non-linear: equal intervals in $\omega$ do not correspond to equal intervals in $\Omega$ .

To design a digital filter with a specified characteristic frequency $f_0$ (in Hz), one must pre-warp the target to the analog prototype frequency

$\Omega_0 = \frac{2}{T}\tan\!\left(\frac{\pi f_0}{f_s}\right)$

and then design the analog prototype at $\Omega_0$ before applying the bilinear substitution.

Definition 37.5 (Quality Factor Q and Pole Radius)

For a conjugate biquad pole pair at $z_{1,2} = r\,e^{\pm i\theta}$ , the pole angle $\theta \approx 2\pi f_0 / f_s$ encodes the center frequency and the pole radius $r$ encodes the bandwidth. The quality factor is approximately

$Q \approx \frac{\sin\theta}{2(1-r)}$

for $r$ close to 1. As $Q \to \infty$ , $r \to 1$ : the poles approach the unit circle and the peak height grows without bound. At $r = 1$ (poles on the unit circle) the filter is marginally unstable (undamped oscillator).

Definition 37.6 (Envelope Follower (Compressor))

A first-order IIR envelope follower estimates the instantaneous amplitude of a signal $x[n]$ by the recursion

$e[n] = \alpha\,e[n-1] + (1-\alpha)\,|x[n]|$

where the smoothing coefficient is

$\alpha = e^{-1/(f_s \cdot \tau)}$

and $\tau > 0$ is the time constant in seconds. For attack and release, separate coefficients $\alpha_A$ and $\alpha_R$ (with $\tau_A \ll \tau_R$ ) are applied depending on whether the signal is rising or falling.

The gain reduction applied by a compressor with threshold $T_{\text{dB}}$ and ratio $R:1$ is (in dB):

$G_{\text{red}}[n] = \frac{1 - 1/R}{1}\,\max\!\bigl(e_{\text{dB}}[n] - T_{\text{dB}},\;0\bigr)$

The compressor is therefore a time-varying gain $g[n] = 10^{-G_{\text{red}}[n]/20}$ multiplied sample-by-sample.

Main Theorems

Theorem 37.1 (Biquad Transfer Function and Stability)

Let $H(z)$ be a biquad with real coefficients as in Definition 37.1. Then:

$H(z)$ is BIBO stable if and only if both roots of $p(z) = z^2 + a_1 z + a_2$ satisfy $|z| < 1$ .
The Schur-Cohn stability conditions for $p(z)$ are equivalent to: $|a_2| < 1, \quad |a_1| < 1 + a_2$
The frequency response of a stable biquad is obtained by evaluating $H$ on the unit circle: $H(e^{i\omega}) = \frac{b_0 e^{2i\omega} + b_1 e^{i\omega} + b_2}{e^{2i\omega} + a_1 e^{i\omega} + a_2}$

and its magnitude is determined entirely by the distances from $e^{i\omega}$ to the poles and zeros in the $z$ -plane.

Proof.

(Part 1.) The output $Y(z) = H(z) X(z)$ . For an bounded input, the output is bounded if and only if the partial-fraction expansion of $H(z)$ has only terms $A/(z - z_k)$ with $|z_k| < 1$ , so that the corresponding time-domain sequences $A z_k^n u[n]$ decay. Poles with $|z_k| \geq 1$ produce non-decaying or growing terms, violating BIBO stability.

(Part 2.) Write $p(z) = z^2 + a_1 z + a_2$ . The Jury stability test (equivalently, Schur-Cohn criterion) for a degree-2 polynomial states: both roots inside the unit disk if and only if $p(1) > 0, \quad p(-1) > 0, \quad |a_2| < 1$

Computing: $p(1) = 1 + a_1 + a_2 > 0$ and $p(-1) = 1 - a_1 + a_2 > 0$ . These two inequalities together say $|a_1| < 1 + a_2$ , and together with $|a_2| < 1$ they form the stated conditions.

(Part 3.) The unit circle is the set $z = e^{i\omega}$ for $\omega \in [0, 2\pi)$ . Substituting into $H(z)$ gives the frequency response directly. Writing the numerator as $b_0(z - z_1)(z-z_2)$ and denominator as $(z - p_1)(z-p_2)$ , the magnitude is

$|H(e^{i\omega})| = |b_0| \cdot \frac{|e^{i\omega} - z_1|\,|e^{i\omega} - z_2|}{|e^{i\omega} - p_1|\,|e^{i\omega} - p_2|}$

Each factor is the Euclidean distance from the evaluation point $e^{i\omega}$ on the unit circle to the corresponding pole or zero in $\mathbb{C}$ . $\square$

Theorem 37.2 (Bilinear Transform: Stability Preservation)

Let $H_a(s)$ be a stable analog filter with all poles in the open left half-plane: $\operatorname{Re}(s_k) < 0$ for all poles $s_k$ . Define the digital filter obtained by the bilinear substitution of Definition 37.3:

$H(z) = H_a\!\left(\frac{2}{T}\cdot\frac{1-z^{-1}}{1+z^{-1}}\right)$

Then $H(z)$ is BIBO stable: all its poles lie strictly inside the unit circle.

Furthermore, the bilinear transform is bijective: the imaginary axis $\{s : \operatorname{Re}(s)=0\}$ maps exactly onto the unit circle $\{z : |z|=1\}$ , and no aliasing of poles occurs (unlike the impulse-invariance method).

Proof.

Write $s = \sigma + i\Omega$ . Under the bilinear map $z = (1 + sT/2)/(1 - sT/2)$ , compute:

$|z|^2 = \frac{|1 + sT/2|^2}{|1 - sT/2|^2} = \frac{(1 + \sigma T/2)^2 + (\Omega T/2)^2}{(1 - \sigma T/2)^2 + (\Omega T/2)^2}$

If $\sigma < 0$ (left half-plane), then $(1 + \sigma T/2) < (1 - \sigma T/2)$ , so the numerator $<$ denominator, giving $|z| < 1$ . If $\sigma = 0$ then numerator = denominator so $|z| = 1$ . If $\sigma > 0$ then $|z| > 1$ .

Thus the bilinear map is an order-preserving bijection between the three regions $\{\operatorname{Re}(s) \lessgtr 0\}$ and $\{|z| \lessgtr 1\}$ . Since each analog pole at $\operatorname{Re}(s_k) < 0$ maps to a digital pole with $|z_k| < 1$ , stability is preserved.

Bijectivity follows from the fact that the Mobius transformation $z \mapsto (1+sT/2)/(1-sT/2)$ is an invertible analytic map on $\hat{\mathbb{C}}$ (Riemann sphere); no two distinct $s$ -values map to the same $z$ -value. $\square$

Theorem 37.3 (Frequency Warping Correction)

Let $H_a(s)$ be an analog prototype with a critical frequency at $\Omega_0$ (e.g., the $-3$ dB point of a lowpass, or the center of a bandpass). After the bilinear substitution with period $T$ , the digital filter $H(z)$ has its critical feature at the digital frequency

$\omega_0 = 2\arctan\!\left(\frac{\Omega_0 T}{2}\right)$

To obtain a digital filter whose critical feature falls exactly at a desired digital frequency $\omega_d = 2\pi f_d / f_s$ , one must design the analog prototype at the pre-warped analog frequency:

$\Omega_d = \frac{2}{T}\tan\!\left(\frac{\omega_d}{2}\right) = \frac{2}{T}\tan\!\left(\frac{\pi f_d}{f_s}\right)$

The error in critical frequency if warping correction is omitted grows as $f_d / f_s$ increases: for $f_d = 0.1 f_s$ the error is under 1%, but for $f_d = 0.4 f_s$ it exceeds 25%.

Proof.

On the unit circle $z = e^{i\omega}$ , the bilinear substitution becomes:

$s = \frac{2}{T}\cdot\frac{1 - e^{-i\omega}}{1 + e^{-i\omega}}$

Multiply numerator and denominator by $e^{i\omega/2}$ :

$s = \frac{2}{T}\cdot\frac{e^{i\omega/2} - e^{-i\omega/2}}{e^{i\omega/2} + e^{-i\omega/2}} = \frac{2}{T}\cdot\frac{2i\sin(\omega/2)}{2\cos(\omega/2)} = \frac{2i}{T}\tan\!\left(\frac{\omega}{2}\right)$

So the purely imaginary value $s = i\Omega$ corresponds to $\Omega = (2/T)\tan(\omega/2)$ , i.e. $\omega = 2\arctan(\Omega T/2)$ . This is a bijective but non-linear correspondence.

Suppose the analog prototype has a feature (zero, pole, or gain crossing) at $\Omega_0$ . After bilinear substitution, that feature migrates to the digital frequency $\omega_0 = 2\arctan(\Omega_0 T/2)$ , not the intended $\omega_d$ (unless $\Omega_0 = (2/T)\tan(\omega_d/2)$ was chosen by pre-warping). Solving for $\Omega_0 = \Omega_d$ recovers the stated pre-warp formula. $\square$

Theorem 37.4 (Magnitude Response from Pole-Zero Distances)

Let $H(z)$ have poles $p_1, p_2$ and zeros $z_1, z_2$ (all complex, possibly conjugate pairs). At angular frequency $\omega$ , let

$d_{z_k}(\omega) = |e^{i\omega} - z_k|, \quad d_{p_k}(\omega) = |e^{i\omega} - p_k|$

denote the Euclidean distance from the unit-circle point $e^{i\omega}$ to zero $z_k$ and pole $p_k$ , respectively. Then

$|H(e^{i\omega})| = |b_0| \cdot \frac{d_{z_1}(\omega)\,d_{z_2}(\omega)}{d_{p_1}(\omega)\,d_{p_2}(\omega)}$

Corollary. As a pole $p_k$ approaches the unit circle ( $|p_k| \to 1$ ) from inside, the distance $d_{p_k}(\omega_k) \to 0$ at the angle $\omega_k = \arg(p_k)$ , and therefore $|H(e^{i\omega_k})| \to \infty$ : the gain peak height is inversely proportional to the distance from the nearest pole to the unit circle.

Proof.

Factor the numerator and denominator of $H(z) = b_0(z-z_1)(z-z_2)/[(z-p_1)(z-p_2)]$ . Evaluate at $z = e^{i\omega}$ :

$H(e^{i\omega}) = b_0 \cdot \frac{(e^{i\omega} - z_1)(e^{i\omega}-z_2)}{(e^{i\omega}-p_1)(e^{i\omega}-p_2)}$

Taking absolute values and using $|AB| = |A||B|$ and $|A/B| = |A|/|B|$ :

$|H(e^{i\omega})| = |b_0|\cdot\frac{|e^{i\omega}-z_1|\,|e^{i\omega}-z_2|}{|e^{i\omega}-p_1|\,|e^{i\omega}-p_2|}$

Each factor $|e^{i\omega} - c|$ is precisely the Euclidean distance $d_c(\omega)$ between two points in $\mathbb{C} \cong \mathbb{R}^2$ . This is the stated formula.

For the corollary: if $p_1 = r\,e^{i\theta}$ with $r \to 1$ , then at $\omega = \theta$ ,

$d_{p_1}(\theta) = |e^{i\theta} - r\,e^{i\theta}| = |1 - r| \to 0$

The denominator vanishes, so $|H(e^{i\theta})| \to \infty$ . $\square$

Theorem 37.5 (Serial Biquad Product Theorem)

Let $H_1(z), \ldots, H_N(z)$ be biquads in a serial cascade (output of each feeds input of the next). Then:

The combined transfer function is the product: $H_{\text{total}}(z) = \prod_{k=1}^{N} H_k(z)$
The combined system is BIBO stable if and only if every individual biquad is stable (no pole of any $H_k$ lies on or outside the unit circle).
The combined pole-zero set is the union of the individual pole and zero sets (counted with multiplicity), and the combined magnitude response is the product of the individual magnitude responses: $|H_{\text{total}}(e^{i\omega})| = \prod_{k=1}^{N} |H_k(e^{i\omega})|$

In decibels: $20\log_{10}|H_{\text{total}}| = \sum_k 20\log_{10}|H_k|$ .

Proof.

(Part 1.) In the cascade, the output of stage $k$ is $Y_k(z) = H_k(z) Y_{k-1}(z)$ , with $Y_0(z) = X(z)$ . By induction, $Y_N(z) = \bigl(\prod_k H_k(z)\bigr) X(z)$ .

(Part 2.) The poles of $H_{\text{total}}$ are the union of the poles of each factor (assuming no pole-zero cancellation). A product of rational functions has $|z_k| < 1$ for all poles if and only if each factor satisfies this. Pole-zero cancellations that would place a cancelled pole exactly on the unit circle are excluded by the assumption that each $H_k$ is individually stable.

(Part 3.) Taking absolute values of the product: $|H_{\text{total}}(e^{i\omega})| = \prod_k |H_k(e^{i\omega})|$ , since absolute value is multiplicative. Converting to dB uses the identity $\log|AB| = \log|A| + \log|B|$ . $\square$

Numerical Examples

Example 1: Peaking EQ at 1 kHz, $Q=2$ , $G=+6$ dB at $f_s=44100$ Hz.

From the Bristow-Johnson Cookbook (peaking EQ section), set $A = 10^{G/40} = 10^{6/40} \approx 1.4125$ and

\omega_0 = 2\pi f_0 / f_s = 2\pi \cdot 1000 / 44100 \approx 0.14247 \text{ rad/sample}

\alpha = \sin(\omega_0)/(2Q) = \sin(0.14247)/2 \approx 0.14197/2 \approx 0.03557

The five coefficients are:

$b_0 = 1 + \alpha A \approx 1.05025, \quad b_1 = -2\cos\omega_0 \approx -1.97984, \quad b_2 = 1 - \alpha A \approx 0.94975$ $a_0 = 1 + \alpha/A \approx 1.02514, \quad a_1 = -2\cos\omega_0 \approx -1.97984, \quad a_2 = 1 - \alpha/A \approx 0.97486$

(Dividing through by $a_0$ normalises to the standard form.) The pole radius is $r = \sqrt{a_2/a_0} \approx \sqrt{0.95092} \approx 0.9752$ , well inside the unit circle. Gain at $f_0 = 1$ kHz: exactly $+6$ dB by construction.

Example 2: Effect of increasing Q from 1 to 8.

Keep $f_0 = 1$ kHz, $G = +6$ dB, $f_s = 44100$ Hz. The parameter $\alpha = \sin\omega_0 / (2Q)$ controls the bandwidth:

$Q$	$\alpha$	Pole radius $r$	Distance to unit circle $1-r$	Approx. bandwidth
1	0.0712	0.9294	0.0706	~1000 Hz
2	0.0356	0.9650	0.0350	~500 Hz
4	0.0178	0.9824	0.0176	~250 Hz
8	0.0089	0.9912	0.0088	~125 Hz

Doubling $Q$ approximately halves the pole-to-unit-circle distance, halves the bandwidth, and narrows the EQ peak. As the table shows, at $Q = 8$ the poles are only $0.88\%$ of the unit-circle radius away from instability.

Example 3: Compressor envelope follower time constants.

A typical drum bus compressor uses $\tau_A = 5$ ms (attack) and $\tau_R = 100$ ms (release) at $f_s = 44100$ Hz:

$\alpha_A = e^{-1/(44100 \times 0.005)} = e^{-1/220.5} \approx e^{-0.004535} \approx 0.99548$ $\alpha_R = e^{-1/(44100 \times 0.100)} = e^{-1/4410} \approx e^{-0.000227} \approx 0.99977$

The attack filter has a pole at $z = \alpha_A \approx 0.99548$ and the release filter has a pole at $z = \alpha_R \approx 0.99977$ — both on the positive real axis just inside the unit circle. The closer the pole to $z=1$ , the more sluggish (longer time constant) the envelope follower.

Example 4: Signal chain pole count.

A typical vocal recording path for EP28 contains:

Microphone diaphragm (1 biquad, EP28): 2 poles, 2 zeros
High-pass filter (1 biquad): 2 poles, 2 zeros
5-band parametric EQ (5 biquads): 10 poles, 10 zeros
Compressor envelope follower (1 first-order IIR): 1 pole

Total: 15 poles, 14 zeros shaping the final audio. The constellation of these 15 points in the $z$ -plane, by Theorem 37.4 and 37.5, is the entire mathematical content of “what this recording sounds like.”

Musical Connection

音乐联系

Every EQ gesture is a pole migration.

When a mastering engineer draws a narrow boost at 3 kHz on a parametric EQ, the intuitive description is “adding presence.” The mathematical description is: two complex-conjugate poles move outward by a fraction of a percent, closer to the unit circle at angle $\theta = 2\pi \times 3000/f_s$ . Theorem 37.4 says the gain at that frequency is inversely proportional to the remaining distance to the unit circle — so the boost height is literally the reciprocal of a length in the complex plane.

The Q knob controls pole orbit radius. Musicians describe high-Q boosts as “nasal” or “ringing” — because a pole very near the unit circle has a long impulse response tail (the filter rings at its own natural frequency, just like the underdamped microphone diaphragm from

EP28

). Low-Q settings produce broad shelving shapes, corresponding to poles comfortably inside the circle with short impulse responses.

Dynamic range compression adds a time dimension: the gain $g[n]$ modulates slowly at the envelope follower’s time constant $\tau$ . Fast attack (small $\tau_A$ ) captures transients immediately, reducing the “punch” of drums; slow attack lets transients through, preserving their spike in the time-domain waveform. The trade-off is fundamental and cannot be evaded: it is the discrete-time version of the time-frequency uncertainty principle — a filter with a short impulse response (fast response to level changes) must have a broad frequency response (less selective). The envelope follower’s pole at $z = \alpha$ close to 1 has a 3 dB bandwidth of only $f_s(1-\alpha)/(2\pi)$ Hz — extremely narrow — which is why the gain changes feel smooth, not jittery.

The signal chain of Theorem 37.5 makes explicit what every recording engineer knows intuitively: the sounds of different studios, different mic chains, and different mastering processors are distinguishable because their pole-zero constellations differ. “Analog warmth” (saturation harmonics aside) is a particular distribution of poles near the real axis at low frequencies. “Air” is poles near $\theta = \pi/3$ with small $1-r$ . The vocabulary of timbral description is a folk taxonomy of pole-zero constellations.

EP39

extends this further: a reverb is a feedback delay network whose transfer function is a high-order rational function with many poles distributed across the unit disk — in the limit, approaching a dense cloud that produces the perception of diffuse space.

Limits and Open Questions

Fixed-point coefficient quantisation. On embedded hardware (guitar pedals, hardware EQ units) the biquad coefficients are stored as fixed-point integers. For high-Q filters near the unit circle, tiny rounding errors in $a_1, a_2$ can displace poles outside the unit circle, causing instability. The standard remedy is to use second-order sections (Direct Form II Transposed) and to limit $Q$ based on word length, but no universal rule covers all parameter ranges. This is an active concern in embedded DSP design.
Non-minimum phase zeros. The Bristow-Johnson peaking EQ formulas place zeros symmetrically with respect to the unit circle (if a zero is at radius $r$ , its reflection at $1/r$ is also a zero). Minimum-phase designs place all zeros inside the unit circle, giving the fastest impulse response; but some filter types (linear-phase FIR, all-pass sections) deliberately use zeros outside the unit disk. The interaction between minimum-phase and non-minimum-phase sections in a serial chain is not always intuitive.
Compressor parameter interactions. The standard compressor model treats attack and release as independent first-order IIR filters. In practice, the knee shape, look-ahead (delay compensation), and program-dependent release create higher-order dynamics that a single-pole envelope follower does not capture. A rigorous mathematical model of compressor transient behaviour remains an open problem in audio engineering.
Loudness and nonlinear perception. The magnitude response $|H(e^{i\omega})|$ is a linear, time-invariant quantity. Human loudness perception is neither linear nor time-invariant: the equal-loudness contours (Fletcher-Munson curves) are level-dependent, and the auditory system has its own compressive nonlinearity (outer hair cell electromotility). Matching a linear biquad EQ to a perceptual loudness target requires additional psychoacoustic modelling not captured by Theorem 37.4.
Multi-rate and oversampled biquads. Many modern plug-ins run biquads at 2× or 4× oversampled rates to reduce frequency warping near Nyquist and to allow the nonlinear saturation components (operating on the upsampled signal) to alias-free back down. The interplay between the bilinear transform pre-warping (Theorem 37.3) and the oversampling ratio introduces design dependencies that are not yet standardised in the literature.

Conjecture (Optimal Pole Placement for Perceptual Transparency)

There exists a measurable function

\Phi: \mathbb{C}^N \to \mathbb{R}_{\geq 0}

that maps a pole-zero constellation of size

N

to a perceptual distance from the flat (identity) response, such that two EQ settings with the same

\Phi

-value are indistinguishable by trained listeners in a double-blind ABX test. The conjecture is that

\Phi

is not simply

\int |H(e^{i\omega}) - 1|^2\,d\omega

(the

L^2

error on the unit circle), but involves frequency-dependent weighting by the auditory filter bank bandwidth (ERB scale). Falsification criterion: find two pole-zero constellations

A, B

with

\Phi_{\text{ERB}}(A) = \Phi_{\text{ERB}}(B)

but with ABX discrimination rate significantly above chance in a listener panel of

\geq 20

trained engineers.

Academic References

Oppenheim, A. V., & Schafer, R. W. (2009). Discrete-Time Signal Processing (3rd ed.). Prentice Hall. Ch. 5–7 (IIR filter design, bilinear transform, stability of second-order sections).
Proakis, J. G., & Manolakis, D. G. (2006). Digital Signal Processing: Principles, Algorithms, and Applications (4th ed.). Prentice Hall. Ch. 8 (design of digital filters from analog prototypes; Schur-Cohn stability).
Bristow-Johnson, R. (2016). Cookbook formulae for audio equalizer biquad filter coefficients. Available at: https://webaudio.github.io/Audio-EQ-Cookbook/audio-eq-cookbook.html (Engineering reference; not peer-reviewed).
Zölzer, U. (2011). DAFX: Digital Audio Effects (2nd ed.). Wiley. Ch. 2 (parametric EQ biquad structures), Ch. 4 (dynamic range processors and envelope followers).
Regalia, P. A., Mitra, S. K., & Vaidyanathan, P. P. (1988). The digital all-pass filter: A versatile signal processing building block. Proceedings of the IEEE, 76(1), 19–37. (Pole-zero geometry and allpass decomposition of biquads.)
Moorer, J. A. (1979). About this reverberation business. Computer Music Journal, 3(2), 13–28. (Early formulation of cascaded biquad chains for artificial reverberation; connects to EP39.)
Smith, J. O. (2007). Introduction to Digital Filters with Audio Applications. W3K Publishing. Available at: https://ccrma.stanford.edu/~jos/filters/ (Comprehensive coverage of pole-zero analysis, frequency response geometry, and biquad design.)
Bohn, D. A. (2008). Bandpass filter design. Rane Note 147. Rane Corporation. (Engineering treatment of Q factor, bandwidth, and biquad coefficient sensitivity in audio applications.)
Kuo, S. M., & Lee, B. H. (2001). Real-Time Digital Signal Processing. Wiley. Ch. 6 (fixed-point implementation of biquad filters, coefficient quantisation effects, and stability margins).
Välimäki, V., & Reiss, J. D. (2016). All about audio equalization: Solutions and frontiers. Applied Sciences, 6(5), 129. doi:10.3390/app6050129. (Survey of perceptual EQ design, auditory weighting of frequency response error, and open problems in transparent equalisation.)