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Decibels & Magnitude-Response Helpers

Overview & Motivation

Audio, filter, and control-system specifications express signal levels and filter performance in decibels (dB) because the human auditory system and most engineering metrics scale logarithmically with amplitude ratio. Specifying a stop-band attenuation of 60 dB or a pass-band ripple of 0.1 dB is natural and compact; the equivalent linear ratios (1 000 : 1 and 1.01161 : 1) are not. A small set of conversion primitives — ToDecibels, FromDecibels, and two derived helpers for attenuation and ripple — centralises this conversion and eliminates scattered, error-prone inline 20·log10 expressions throughout the rest of the library.

Mathematical Theory

Magnitude Decibel Conversion

For a positive amplitude ratio $r > 0$, the equivalent level in decibels is:

$$L_{\mathrm{dB}} = 20 \log_{10}(r)$$

The factor 20 (rather than 10) reflects the voltage/pressure convention: power is proportional to the square of amplitude, so a doubling of amplitude ($r = 2$) gives a 6 dB increase, matching the $10 \log_{10}(4) = 6.02$ dB power equivalent.

Inverse Conversion

$$r = 10^{L_{\mathrm{dB}}/20}$$

This inverse is exact for all finite $L_{\mathrm{dB}}$; no guard is needed on the output side.

Zero and Negative Input Guard

$\log_{10}(0) = -\infty$; negative ratios are physically meaningless. Both cases are mapped to a finite floor value $L_{\min}$ chosen well below any engineering specification of interest:

$$L_{\mathrm{dB}} = \max!\left(20\log_{10}(r),; L_{\min}\right), \quad r > 0$$ $$L_{\mathrm{dB}} = L_{\min}, \quad r \leq 0$$

A floor of $-160,\mathrm{dB}$ corresponds to an amplitude ratio below $10^{-8}$, safely beyond the dynamic range of any practical floating-point computation in 32-bit single precision.

Derived Helpers

Stop-band attenuation between a pass-band ratio $r_p$ and a stop-band ratio $r_s$:

$$A = L_{\mathrm{dB}}(r_p) - L_{\mathrm{dB}}(r_s)$$

Pass-band ripple between the maximum and minimum in-band ratios $r_{\max}$ and $r_{\min}$:

$$\Delta = L_{\mathrm{dB}}(r_{\max}) - L_{\mathrm{dB}}(r_{\min})$$

Both are simple differences in decibel space, exploiting the logarithm identity $\log(a/b) = \log a - \log b$.

Complexity Analysis

Operation Time Space Notes
ToDecibels O(1) O(1) One log10 + one max + one mul
FromDecibels O(1) O(1) One pow
AttenuationDb O(1) O(1) Two ToDecibels + one subtraction
RippleDb O(1) O(1) Two ToDecibels + one subtraction

No state, no buffers. All operations are pure functions.

Step-by-Step Walkthrough

Converting a ratio of 10 to decibels:

  1. Input $r = 10$; guard passes ($r > 0$).
  2. Compute $20 \cdot \log_{10}(10) = 20 \cdot 1 = 20$.
  3. Apply floor: $\max(20, -160) = 20$.
  4. Output: $20,\mathrm{dB}$.

Round-trip for $r = 0.5$:

  1. ToDecibels(0.5) = $20 \cdot \log_{10}(0.5) \approx -6.0206,\mathrm{dB}$.
  2. FromDecibels(-6.0206) = $10^{-6.0206/20} \approx 0.5$.

Pitfalls & Edge Cases

  • Passing $r = 0$ produces $-\infty$ from log10; the floor guard prevents propagation into downstream computations.
  • Negative ratios indicate a programming error (signed sample values must not be passed directly as ratios without taking absolute value first); they are silently floored rather than raising an exception, consistent with the no-exception policy.
  • With fast-math enabled, the compiler may fuse or reorder floating-point operations. The log10 result is still monotone and the floor remains correct because it uses std::max, which is not reordered away.
  • FromDecibels has no floor; at very large positive dB values the result overflows to +inf in float — this is expected behaviour for out-of-range inputs.

Variants & Generalizations

  • Power decibels use $10 \log_{10}(\cdot)$ (factor 10 rather than 20). The amplitude convention used here ($\times 20$) is correct for voltage, pressure, and filter transfer-function magnitude.
  • Field-quantity vs. power-quantity disambiguation: IEEE 60268 / IEC 61672 mandate $20 \log_{10}$ for sound pressure level; the same convention applies to filter magnitude response.
  • The floor can be parameterised if a stricter or looser sentinel is required; the default of $-160,\mathrm{dB}$ is conservative for 32-bit float.

Applications

  • Filter specification: stop-band attenuation and pass-band ripple in dB are the primary acceptance criteria for IIR/FIR designs.
  • Frequency response plots: FrequencyResponse::Calculate() already returns magnitude in dB using $20 \log_{10}$; these helpers provide the same conversion for ad-hoc analysis.
  • Controller gain margin is expressed in dB; converting from a linear ratio with ToDecibels avoids duplication.
  • Audio dynamic processing (compressor thresholds, limiter ceilings) and acoustic measurement both use dB natively.

Connections to Other Algorithms

  • control_analysis::FrequencyResponse internally applies $20 \log_{10}(|H|)$ on its magnitude output vector; these helpers are the scalar equivalent exposed for library consumers.
  • Pass-band ripple computed by RippleDb feeds directly into filter-design acceptance testing alongside the step/transient-response metrics in the evaluation primitives family.

References & Further Reading

  • Proakis, J. & Manolakis, D., "Digital Signal Processing", 4th ed., Prentice Hall, 2007 — Appendix A (decibel notation).
  • Zolzer, U., "DAFX: Digital Audio Effects", 2nd ed., Wiley, 2011 — Chapter 2 (level and gain in dB).
  • IEC 61672-1:2013, "Electroacoustics — Sound level meters — Part 1: Specifications."