Band Relevance Factor (BRF)

• BRF is an automatic frequency-band selection method that quantifies the relevance of spectral regions using spectral entropy and an RMS-based correction.
• It employs a multiresolution decomposition to rank frequency bands across scales, aiding both automated pipelines and expert inspection in rotating machinery diagnostics.
• BRF outperforms traditional RMS and impulse-centric methods by highlighting subtle yet structured dynamic features, such as harmonics and cyclic components.

Searching arXiv for the BRF paper and closely related work on frequency-band selection in vibration analysis. arxiv_search(query="(Brito et al., 2022) OR \"Band Relevance Factor\" vibration rotating machinery", max_results=10) Band Relevance Factor (BRF) is an automatic frequency-band selection method for vibration analysis of rotating machinery. It was introduced to identify which regions of a vibration spectrum are genuinely relevant for diagnosis and interpretation, rather than restricting attention to impulsive fault bands alone or postponing relevance assessment until after feature extraction. In its original formulation, BRF combines spectral entropy with an RMS-based correction factor, produces a bandwise relevance score, ranks bands across multiple decomposition levels, and summarizes the result in a heatmap intended for both automated pipelines and expert inspection (Brito et al., 2022).

1. Definition and conceptual scope

BRF was proposed against the background of two established but incomplete practices in machinery monitoring. The first is feature extraction and feature selection, where informative variables are typically selected after extraction from raw signals or pre-defined bands. The second is Informative Frequency Band (IFB) selection, which has focused primarily on impulsive excitations, especially in bearing and gear diagnostics. The BRF proposal argues that both perspectives can miss frequency regions that are relevant for vibration analysis but are not predominantly impulsive, and can also permit features to be extracted from irrelevant spectral regions (Brito et al., 2022).

The central object of BRF is therefore not a time-domain feature or a classifier weight, but the relevance of a frequency band itself. In the paper’s framing, relevant bands may contain rotation frequency, harmonics, sidebands, cyclic components, and signatures associated with unbalance, misalignment, looseness, and bearing faults. This broad scope distinguishes BRF from methods designed mainly to localize impulsive demodulation bands. Operationally, BRF is intended to identify all relevant bands for vibration analysis in general, not only bands optimized for transient detection (Brito et al., 2022).

This suggests a shift in emphasis from “where are the impulses?” to “which spectral regions contain structured and materially informative machine dynamics?” That shift is fundamental to BRF’s encyclopedic significance within vibration-based condition monitoring.

2. Entropic formulation and scoring rule

The mathematical basis of BRF begins with Shannon entropy,

$H(x) = -\sum_{i=1}^{n} p(x_i)\log(p(x_i)),$

and proceeds to spectral entropy, defined as the normalized Shannon entropy of the power spectrum,

$$S(x) = -\frac{\sum_{i=1}^{n} P_x \log(P_x)}{\log(n)},$$

with

$P_x = \frac{E_i}{\sum_{i}^{n} E_i},$

where $E_i$ is the energy at frequency bin $i$ (Brito et al., 2022).

The interpretive premise is that a flat, noise-like spectrum yields high spectral entropy, whereas a spectrum with energy concentrated in a relatively small number of frequencies yields lower spectral entropy. BRF uses this contrast to detect bands whose internal structure differs from the global spectral background. In the proposed workflow, the spectral entropy of the original signal is first computed. If entropy is greater than or equal to $-3$ dB, the signal is treated as essentially noise and no further band analysis is performed. If entropy is less than $-3$ dB, the signal is treated as relevant and the band analysis continues (Brito et al., 2022).

For each filtered band, the method computes an entropy-difference factor,

$S_{diff} = 3 + S_{base} - S_{filtered},$

where $S_{base}$ is the entropy of the original signal and $S_{filtered}$ is the entropy of the selected band. The added $$S(x) = -\frac{\sum_{i=1}^{n} P_x \log(P_x)}{\log(n)},$$0 dB shift converts the earlier $$S(x) = -\frac{\sum_{i=1}^{n} P_x \log(P_x)}{\log(n)},$$1 dB threshold into a zero-based relevance rule. The decision criterion is explicit: if $$S(x) = -\frac{\sum_{i=1}^{n} P_x \log(P_x)}{\log(n)},$$2, the band is relevant; if $$S(x) = -\frac{\sum_{i=1}^{n} P_x \log(P_x)}{\log(n)},$$3, the band is irrelevant (Brito et al., 2022).

Entropy alone is not treated as sufficient. The paper therefore introduces an RMS-based correction,

$$S(x) = -\frac{\sum_{i=1}^{n} P_x \log(P_x)}{\log(n)},$$4

where $$S(x) = -\frac{\sum_{i=1}^{n} P_x \log(P_x)}{\log(n)},$$5 is the RMS of the original signal and $$S(x) = -\frac{\sum_{i=1}^{n} P_x \log(P_x)}{\log(n)},$$6 is the RMS of the band. The final BRF score is described as the quotient of the entropy difference and the RMS correction factor,

$$S(x) = -\frac{\sum_{i=1}^{n} P_x \log(P_x)}{\log(n)},$$7

Positive BRF values indicate relevant bands, and larger BRF values indicate higher relevance. The stated purpose of the RMS term is to prevent entropy alone from overvaluing narrow low-energy regions that are structured but not materially important (Brito et al., 2022).

3. Algorithmic workflow and multiscale representation

The BRF pipeline is explicitly multiresolution. Once the full signal passes the initial entropy gate, the spectrum is divided at a chosen level $$S(x) = -\frac{\sum_{i=1}^{n} P_x \log(P_x)}{\log(n)},$$8 into

$$S(x) = -\frac{\sum_{i=1}^{n} P_x \log(P_x)}{\log(n)},$$9

bands. The examples given are level $P_x = \frac{E_i}{\sum_{i}^{n} E_i},$0 for the full signal, level $P_x = \frac{E_i}{\sum_{i}^{n} E_i},$1 for $P_x = \frac{E_i}{\sum_{i}^{n} E_i},$2 bands, level $P_x = \frac{E_i}{\sum_{i}^{n} E_i},$3 for $P_x = \frac{E_i}{\sum_{i}^{n} E_i},$4 bands, and level $P_x = \frac{E_i}{\sum_{i}^{n} E_i},$5 for $P_x = \frac{E_i}{\sum_{i}^{n} E_i},$6 bands. For each band at each level, the method computes spectral entropy, evaluates $P_x = \frac{E_i}{\sum_{i}^{n} E_i},$7, applies the RMS correction, and assigns a BRF value (Brito et al., 2022).

The output has three forms. First, BRF performs a binary relevance decision, separating relevant from irrelevant bands. Second, it generates a ranking, ordering bands by BRF score within each decomposition level. Third, it provides a heatmap visualization across levels. The heatmap is normalized to the range from $P_x = \frac{E_i}{\sum_{i}^{n} E_i},$8 to $P_x = \frac{E_i}{\sum_{i}^{n} E_i},$9, with positive values denoting relevant bands and zero or negative values denoting irrelevant ones. Because the decomposition width changes with level, the heatmap permits simultaneous inspection of coarse and fine spectral localization (Brito et al., 2022).

The multilevel structure is not merely cosmetic. A broad spectral interval can appear relevant at a coarse level while only a narrower subregion remains relevant at finer levels. The paper treats this as a practical advantage for both automated processing and human interpretation. It also implies that BRF is not tied to a single fixed bandwidth, but instead constructs a hierarchy of candidate bands whose relevance can be inspected across scales (Brito et al., 2022).

4. Position within frequency-band and feature-selection methodologies

BRF was formulated as a response to two methodological limitations. In feature-selection workflows, relevance is usually assessed after features have already been extracted. The BRF paper argues that this can increase analysis complexity and reduce accuracy when features are computed from irrelevant frequency regions. BRF instead acts as a pre-feature-extraction selector: it attempts to delimit meaningful frequency regions first, so that subsequent feature extraction can be confined to bands already judged relevant (Brito et al., 2022).

Relative to IFB methods, BRF differs in target phenomenology. The paper explicitly contrasts it with approaches such as Kurtogram, Protrugram, Autogram, Gini-index-based methods, spectral kurtosis-based approaches, and conditional-variance-based selectors. Those methods are presented as effective for identifying bands containing impulsive excitations, particularly in bearing and gear fault analysis. BRF, by contrast, is intended to detect harmonic, cyclic, and broader deterministic content as well as impulsive signatures. It is therefore aimed at vibration analysis in the wider sense of machine-dynamics interpretation, rather than at transient-band localization alone (Brito et al., 2022).

A common misconception is that frequency-band relevance should correlate directly with amplitude. The paper’s comparison with RMS is designed precisely to reject that simplification. RMS can emphasize energetic bands even when they are not diagnostically informative, whereas BRF attempts to combine structure and energy. Another misconception is that all useful band selectors should be optimized for fault impulses. BRF explicitly contests that assumption by demonstrating relevance for non-impulsive conditions such as unbalance, where the diagnostic content is concentrated around rotation-related harmonics rather than transient bursts (Brito et al., 2022).

5. Validation on synthetic and real machinery datasets

The BRF study reports validation on one synthetic dataset and two real datasets. The synthetic case uses sinusoidal and composite harmonic signals,

$E_i$0

and

$E_i$1

with Gaussian noise added according to

$E_i$2

Inserted frequencies are 30, 120, 500, 700, 750, 2300, 2450, 2600, 2700, 2800, and 3450 Hz. The reported SNR levels are 24 dB, 12 dB, 6 dB, and 0 dB (Brito et al., 2022).

The real datasets comprise a public bearing degradation dataset from Qiu et al. (2006) and a mechanical-fault test-bench dataset produced by the authors. The bearing dataset contains 3 tests with 4 bearings per test, 20,480 samples per file, and a 20 kHz sampling rate; the reported signals are bearing 01 of test 02, signal 150 as normal, and signal 905 as an outer race fault. The mechanical-fault dataset includes normal, unbalance, misalignment, and mechanical looseness conditions on a bench with motor, inverter, bearing house, two bearings, pulleys, belt, and rotor disc. It comprises 20 tests total, 5 per fault condition, 4 sets of 420 signals per test, 25,000 points per file, and a 25 kHz sampling rate. For the BRF illustration, the paper uses the normal and unbalance conditions, with rotation speed approximately 1238 rpm, corresponding to about 21.9 Hz (Brito et al., 2022).

Dataset	Key setup	Principal BRF finding
Synthetic	Harmonics plus Gaussian noise; SNR 24, 12, 6, 0 dB	Identified inserted harmonic bands; returned null ranking at 0 dB
Bearing degradation	Normal and outer-race-fault signals from Qiu et al. dataset	Detected main characteristic and resonance-related bands more selectively than RMS
Mechanical faults	Normal and unbalance conditions from test bench	Highlighted the 21.9 Hz rotation-frequency band under unbalance

The empirical comparison baseline is RMS. In the synthetic experiments, BRF successfully identified bands containing the inserted harmonics and did not select high-frequency regions above 5120 Hz when they contained only noise, whereas RMS often selected those regions because they still contained energy. At the 0 dB mixed-noise level, BRF returned a null ranking, which the paper interprets as an indication that no band could be reliably separated from noise (Brito et al., 2022).

In the bearing dataset, BRF identified the band around the main characteristic frequency near 1000 Hz in the normal signal. For the outer race fault, BRF identified high-frequency resonance bands and lower- or medium-frequency bands associated with fault progression, whereas RMS mostly emphasized high-frequency regions. In the mechanical-fault dataset, BRF and RMS were somewhat similar under normal conditions, but under unbalance BRF correctly highlighted the band containing the rotation frequency around 21.9 Hz as the most relevant at each level, while RMS did not emphasize that region as effectively (Brito et al., 2022).

6. Applications, limitations, and terminological scope

The intended applications of BRF are industrial vibration monitoring, predictive maintenance, remote or IoT-based sensor systems, explainable diagnostics, and band selection prior to machine learning or deep learning. The paper also presents BRF as useful for supporting human analysts: the ranking and heatmap provide an explicit map of which spectral regions deserve inspection. Because the method can stop early when the whole signal appears too noise-like, it is also presented as potentially useful for identifying poor or uninformative acquisitions (Brito et al., 2022).

The limitations are also clearly stated or implied. A narrow-band limitation arises because very small noise-only bands can appear artificially structured, which can produce misleading relevance at very fine resolutions. This is one reason the paper emphasizes ranking across levels rather than reliance on a single finest partition. The method also depends on the user’s choice of decomposition level $E_i$3, and the $E_i$4 dB entropy threshold is a design choice that may require validation in other settings. More generally, the paper indicates that performance can vary with signal quality, machine type, fault severity, spectral richness, and operating conditions. It also notes that the published formula presentation for BRF appears visually incomplete in extracted text, although the intended quotient interpretation is clear (Brito et al., 2022).

The acronym BRF is not unique across arXiv domains. In other literatures, BRF denotes the Beta Rank Function in rank-distribution modeling (Fontanelli et al., 27 Jan 2026) and the blockchain redundancy factor in metaverse-oriented partial computation offloading (Aliyu et al., 2023). A plausible implication is that cross-disciplinary searches should treat the expansion of “BRF” as essential metadata rather than assuming acronym uniqueness.

Within rotating machinery analysis, however, BRF denotes a spectral-entropy-based measure of frequency-band relevance. Its defining contribution is the reorientation of band selection from impulse-centric localization or post hoc feature relevance toward a direct, automated, multiscale assessment of which spectral regions contain structured and energetically meaningful information for machine diagnosis (Brito et al., 2022).