MECH7029 Chap.2 Frequency Bands, Weighting and Level Addition
Frequency Bands, Weighting and Level Addition
Spectra decide what can be controlled
This chapter organises the audible range into octave and one-third-octave ratios, then explains how a time signal becomes a spectrum. It applies A-weighting band by band and adds independent levels through linear energy.
Tone extraction from background and coherent-wave addition are kept separate because they require different operations even when every input is reported in decibels.
Band labels describe ratios, not equal widths
An octave doubles frequency, so its centre is the geometric mean of its edges. One-third-octave analysis subdivides the same ratio logarithmically.
The widths therefore grow with centre frequency, unlike a narrowband spectrum with equal hertz spacing. A pure tone creates concentrated content, an impulse spreads energy broadly, and an irregular record can produce a continuous spectrum. Analysis resolution determines which features remain distinguishable.
A-weighting is applied to individual frequency content before overall energetic addition.
Low-frequency bands receive large negative corrections because hearing sensitivity is lower there under the weighting model. The weighted total cannot be recovered reliably by attaching one generic correction to an already combined unweighted level.
Choose the linear domain from the relationship
Independent bands or incoherent sources combine through pressure-squared contributions.
Coherent waves instead require phase-aware pressure addition before squaring. Background correction subtracts linear energy only when total and background measurements represent compatible conditions; if their levels are close, small uncertainty makes the extracted source unstable. An independent total must exceed its largest contributor but remain below the equal-source upper bound set by contributor count.
What this chapter covers
- 01
Octave and one-third-octave bands
- 02
Geometric centre frequency
- 03
Tone, impulse and broadband spectra
- 04
A-weighting by frequency band
- 05
Energetic addition of independent levels
- 06
Background subtraction in linear units
Worked example · free
Combine three octave-band levels
- 3Convert every band to a linear energy-like term.
- 2Sum the terms and return to decibels.
- 2Bound the result using the loudest band.
Key terms
- Octave Band
- A band whose upper frequency is twice its lower frequency.
- Centre Frequency
- The geometric midpoint that divides a ratio-defined band evenly.
- Spectral Density
- Acoustic energy distributed per unit frequency.
- A-weighting
- A frequency correction representing the ear's unequal sensitivity.
- Broadband Noise
- Noise carrying energy across a continuous frequency range.
- Coherent Addition
- Combination of same-frequency amplitudes with their phase relation retained.
Frequency Bands, Weighting and Level Addition FAQ
Why is the centre of an octave not the arithmetic midpoint?
An octave is defined by frequency ratios, so the centre must have the same ratio to each edge. The geometric mean supplies that property, while an arithmetic midpoint divides only the numerical interval and gives unequal ratios.
How do I remove measured background from a total level?
Convert the total and background levels to linear energy-like quantities, subtract the background term, and convert the positive remainder back to decibels. Subtracting the dB labels directly reports only a level difference, not source level.
When must phase be included in sound addition?
Phase matters when waves at the same frequency remain coherent, because instantaneous pressures can reinforce or cancel. Independent bands or unrelated sources are normally combined through time-mean squared pressure after their cross terms average away.
Exam move
Start by drawing three paired sketches from memory: a sinusoid and its tone spectrum, a short pulse and its broad spectrum, and an irregular signal with broadband content. Label whether the frequency bins are narrowband, octave or one-third octave, and state how their widths differ. Then practise centre-frequency and edge calculations through ratios rather than memorised lists.
For level work, use a fixed four-line layout: identify independence or coherence, convert each relevant level to a linear quantity, perform the addition or subtraction once, and convert the result back. Before calculating, bracket an independent total between the largest component and the equal-source upper bound. A result outside that interval is wrong regardless of calculator precision.
Add A-weighting by writing each band correction beside its unweighted level, forming corrected band levels, and only then adding them energetically. Run a separate background problem in which total and background levels approach one another; explain why subtraction becomes sensitive and why a source-only result may be unreliable.
Finish with one phase example to keep coherent pressure addition separate from independent energy addition. In the error ledger, record whether a mistake came from band geometry, weighting order, source relationship, background compatibility or premature rounding. Rework the physical classification before the arithmetic when the same error returns.
Finish by choosing a plausible control from the spectrum and stating which band would show success; this turns frequency analysis from a calculation exercise into evidence for an engineering decision.
