Parametric Equalizers: Design & Optimization

• Parametric Equalizers (PEQs) are digital filter architectures composed of cascaded second-order IIR sections that allow continuous adjustment of center frequency, gain, and quality factor.
• They leverage differentiable parameterizations and gradient-based optimization to enable precise spectral shaping and efficient system correction in diverse audio applications.
• PEQs are implemented in both single and multiband systems to achieve low computational complexity and real-time performance, outperforming traditional FIR designs.

A parametric equalizer (PEQ) is a digital filter architecture composed of cascaded second-order Infinite Impulse Response (IIR) sections—commonly termed "biquads"—with each section allowing direct and continuous control over three fundamental parameters: center frequency ($f_0$), gain ($G$, in dB), and quality factor ($Q$) or bandwidth. This structure provides precise and independent manipulation of arbitrary frequency bands, in contrast to fixed-bandwidth graphic equalizers. PEQs are versatile tools in audio signal processing, facilitating tasks ranging from spectral shaping and system correction to frequency-dependent reverberation control. Recent research demonstrates highly efficient, differentiable PEQ designs that enable gradient-based optimization of their parameters, enabling state-of-the-art performance in perceptual filter fitting and computationally constrained applications (Ibnyahya et al., 25 Nov 2025, Pepe et al., 2021).

1. Mathematical Structure of Parametric Equalizers

A PEQ implements a cascade of second-order digital filters (biquads). Each biquad's transfer function in the $z$-domain is:

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

The filter coefficients $\{b_0, b_1, b_2, a_1, a_2\}$ depend on:

• Center frequency $f_0$ (Hz)
• Gain $G$ (dB), with $A = 10^{G/40}$
• Quality factor $Q$
• Sample rate $f_s$

The normalized radian frequency $\omega_0 = 2\pi f_0 / f_s$, and the intermediate $\alpha = \sin\omega_0 / (2Q)$.

For the peaking ("bell") filter:

$$\begin{aligned} & b_0 = 1 + \alpha A \ & b_1 = -2\cos \omega_0 \ & b_2 = 1 - \alpha A \ & a_0 = 1 + \alpha / A \ & a_1 = -2\cos \omega_0 \ & a_2 = 1 - \alpha / A \end{aligned}$$

After normalization by $a_0$:

$$\begin{aligned} & \hat{b}_0 = \frac{1 + \alpha A}{1 + \alpha/A} \ & \hat{b}_1 = \frac{-2\cos \omega_0}{1 + \alpha/A} \ & \hat{b}_2 = \frac{1 - \alpha A}{1 + \alpha/A} \ & \hat{a}_1 = \frac{-2\cos \omega_0}{1 + \alpha/A} \ & \hat{a}_2 = \frac{1 - \alpha/A}{1 + \alpha/A} \end{aligned}$$

For low-shelf and high-shelf bands, the coefficients further depend on $\sqrt{A}$, and a bilinear-transform warp is applied, as detailed in [(Ibnyahya et al., 25 Nov 2025), Eqns. 5–6].

2. Parameter Influence on Frequency Response

PEQ parameters have distinct, localizable spectral effects:

• Gain ($G$): Sets the amplitude at $f_0$, i.e. $|H(f_0)|_\mathrm{dB} = G$. For a bell filter, $A=10^{G/40}$ defines the peak/attenuation magnitude in linear terms.
• Quality Factor ($Q$): Controls the 3-dB bandwidth, $\Delta f \approx f_0/Q$. High $Q$ yields narrow, sharply resonant peaks; low $Q$ gives broad spectral shaping.
• Center Frequency ($f_0$): Chooses the filter’s center action.

The magnitude response for a bell filter, as given by [(Ibnyahya et al., 25 Nov 2025), Eqn. 7], is:

$$|H_B(f)| = \frac{\sqrt{ (1 - (f/f_0)^2 )^2 + (A f / f_0 Q)^2 }}{\sqrt{ (1 - (f/f_0)^2 )^2 + (f / f_0 A / Q)^2 }}$$

This quantifies the spectral locus and sharpness of influence for a given band. Sweeping $G$ modifies the height/depth at $f_0$, while $Q$ determines the width and steepness of the transition.

3. Differentiable Parameterizations and Optimization

Analytic parameterization of PEQ coefficients ensures full differentiability. All functions used—$G\mapsto A$, $\alpha=\sin\omega_0/(2Q)$, bilinear warping, polynomial ratios—are composed of elementary differentiable operations. This allows gradient-based supervised optimization of PEQ parameters in neural or direct spectral-fit frameworks (Ibnyahya et al., 25 Nov 2025, Pepe et al., 2021). Implementation in autodiff libraries (e.g., PyTorch) yields end-to-end trainability with loss functions measuring spectral deviation from a target (e.g., reverberation time, measured magnitude response):

$$\mathrm{MSE} = \frac{1}{N_p}\sum_{p}\left[\frac{-60 m_k}{T_{60}(f_p)f_s} - 20 \log_{10}|H_{PEQ}(f_p)|\right]^2$$

(Ibnyahya et al., 25 Nov 2025) adopts this framework for fitting frequency-dependent reverberation decay, while (Pepe et al., 2021) adapts a similar approach for general room/cabin equalization.

4. PEQ Arrangement in Complex Systems

PEQs are typically implemented as a chain of $N$ biquads, configuring the first and last as low/high shelves and intermediate sections as bell filters (Ibnyahya et al., 25 Nov 2025). In multi-band designs, parameters $(f_{0,i}, Q_i)$ for each band $i$ are globally shared, while band gains $G$ may be scaled per processing context, such as feedback delay lines of differing lengths in Feedback Delay Networks (FDNs). To achieve a target decay profile $T_{60}(f)$ in FDNs, the per-delay gain scaling is set via:

$$G_k(f) = \frac{\gamma(f)\,m_k}{f_s}, \ \text{where} \ \gamma(f) = -60 / T_{60}(f)$$

This strategy allows all delay lines to use a shared filter structure, reducing both parameter count and computational complexity (Ibnyahya et al., 25 Nov 2025). In audio equalization, (Pepe et al., 2021) uses a learned bank of such PEQs, with denormalized parameter mappings for each band and source.

5. Deep Learning–Driven PEQ Design: BiasNet

(Pepe et al., 2021) introduces the "BiasNet" architecture for automatic parameter selection:

• The network consists of a virtual input (trainable bias vector), $L$ sine-activated fully connected layers, and an output layer producing normalized PEQ parameters for all bands and sources: $p_{f_c}, p_Q, p_V$.
• Output parameter denormalization translates network outputs to physical filter parameters according to defined band limits and ranges.
• The loss function $L = \gamma_1 L_1 + \gamma_2 L_2$ combines a spectral magnitude distance $L_1$—quantifying fit at various microphones/sources—and a regularization term $L_2$ for multi-source level consistency.
• Training uses measured room/cabin impulse responses, Adam optimizer, and iterative gradient descent in an offline phase. The resultant PEQ parameters achieve near-minimal third-octave mean square error (MSE) and low runtime complexity.

6. Computational and Practical Considerations

PEQs synthesized as IIR SOS offer significant runtime advantages over long FIR designs:

Scenario	MSE (BiasNet)	Runtime (BiasNet)	FIR – FD runtime
Room SISO	$1.32 \times 10^{-5}$	198 ops/sample (22 bands)	$16\,383$ ops/sample (FD8192)
Room MIMO	$1.18 \times 10^{-5}$	$1,584$ ops/sample (22 bands, 8×2)	$131\,064$ ops/sample
Car MIMO	$5.74 \times 10^{-3}$	261 ops/sample (21–29 bands)	$16\,383$ ops/sample

The IIR PEQ approach is orders of magnitude more efficient than FIR, with minimal spectral degradation (Pepe et al., 2021). This enables real-time operation on low-power DSP hardware, while still meeting stringent spectral flatness criteria. The differentiable design in (Ibnyahya et al., 25 Nov 2025) is also fully compatible with gradient-based frameworks and maintains efficiency across a scalable number of bands and delay lengths.

7. Limitations and Prospective Directions

Current PEQ approaches with differentiable optimization operate in a static, linear regime—parameters are learned offline given fixed room impulse responses or decay profiles. There is no online, adaptive update mechanism in (Pepe et al., 2021), and equalization goals are currently restricted to third-octave bands and linear phase. Future research directions include extending these methods to time-varying acoustic scenes, introducing psychoacoustic loss functions, or real-time adaptive filter training (Pepe et al., 2021).

PEQ designs that enable full parameter sharing and gain scaling across system subcomponents, as in feedback delay networks, further reduce the complexity of optimization and implementation while still supporting high-fidelity spectral shaping (Ibnyahya et al., 25 Nov 2025).