Octave-Band Analysis: The Mathematical and Engineering Rationale

In industrial testing, research, and quality validation, data acquisition devices (DAQs / audio interfaces / measurement microphone front-ends) are the “front door” of the entire system. As technology and applications become more specialized, a wide variety of DAQ devices has emerged: High-precision front-ends designed specifically for acoustics and vibration General-purpose dynamic signal acquisition modules Common USB sound cards and measurement microphones Hardware is not the bottleneck anymore. The real challenge is: How do you connect, configure, and manage devices from different brands and protocols in one software platform? OpenTest is built around this pain point. With an open, multi-protocol hardware access architecture, it turns acquisition from “isolated devices” into a unified platform, enabling cross-brand, multi-device data acquisition and analysis. Multi-Protocol Hardware Access: Reducing Vendor Lock-In OpenTest supports several mainstream connection methods. You can choose the appropriate protocol based on your hardware type and driver environment (actual compatibility depends on software version and device drivers): openDAQ – For open DAQ integration. Used to connect open hardware such as CRYSOUND SonoDAQ and manage channels and acquisition parameters in a unified way ASIO / WASAPI / MME / Core Audio – Mainstream audio interfaces on Windows and macOS, supporting professional audio interfaces and USB measurement microphones such as RME, Echo, miniDSP, etc. Other proprietary protocols – Can be added according to project requirements This means you no longer need to be locked into a single hardware brand or a single piece of software. Existing devices can be brought smoothly under one platform for centralized management. Multi-Device Collaboration: One Project, Many Acquisition Tasks Complex tests often require multiple signal sources to be acquired together, for example: Dynamic signals such as microphones and accelerometers Operating parameters such as speed, temperature, pressure, torque Auxiliary audio paths for monitoring and playback With OpenTest’s multi-protocol architecture, you can manage multiple devices within the same project. For NVH and structural testing, this kind of cross-device collaboration significantly reduces repetitive work like: Recording in multiple software tools → exporting → manual time alignment → re-analysis Getting Started: Connecting Devices Quickly Connect your data acquisition device to the PC running OpenTest USB connection, or Network connection (ensure the device and PC are on the same subnet) In the Hardware Setup panel, click the “+” icon in the upper-right corner. OpenTest will automatically scan for connected devices Check the devices you want to use and click Confirm to add them to the active device list Switch to the Channel Setup list, click the “+” icon in the upper-right corner, select the channels required for the current project (channels from different devices can be combined), and click Confirm to add them to the project Select the channels; OpenTest will automatically start real-time monitoring and analysis. You can then switch to different measurement modules according to your test needs Presets + Fine Tuning: Easy to Start, Easy to Standardize To help teams enter the testing state quickly, OpenTest supports a “presets + adjustments” configuration approach: Turn commonly used hardware parameters and acquisition settings into reusable templates Apply templates directly when creating a new project to avoid starting from scratch Still keep full flexibility to fine-tune settings for different operating conditions and devices For production line or regression testing, templating adds an important benefit: uniform test conditions, comparable results, and traceable processes across time and across operators. Logging and Monitoring: Designed for Long-Term Stability For long-duration, multi-device acquisition, the worst case is discovering that something dropped out halfway. OpenTest provides observability features to address this: Device and channel status monitoring – Quickly detect disconnections, overloads, and abnormal inputs Operation and error logs – Record key actions and error events to support troubleshooting and process optimization This is especially critical for continuous production testing and durability tests, significantly reducing the chance of “realizing halfway through that nothing was actually recorded.” Typical Application Scenarios Acoustics and vibration R&D – Use the same platform to connect front-end DAQs and audio interfaces, quickly complete acquisition, analysis, and report generation Automotive NVH / structural testing – Acquire noise, vibration, and operating parameters together, minimizing cross-software alignment work Production line automated testing – Template-based configuration + monitoring/logging + automated reporting to improve consistency and traceability OpenTest’s goal is not to make you replace all your hardware, but to bring your existing hardware together on one platform so that data acquisition becomes more efficient, more controllable, and much easier to standardize. Visit www.opentest.com to learn more about OpenTest features and hardware options, or contact the CRYSOUND team for demos and application support.

Octave-band analysis can be implemented in two fundamentally different ways: FFT binning (integrating PSD/FFT bins into 1/1- and 1/3-octave bands) and a true octave filter bank (standards-oriented bandpass filters + RMS/Leq averaging). In this post, we compare how the two methods work, where their results match, where they diverge (scaling, window ENBW, band-edge weighting, latency, transient response), and how OpenTest supports both for acoustics, NVH, and compliance measurement. For a detailed explanation of the concepts, read this → Octave-Band Analysis: The Mathematical and Engineering Rationale Octave-band filter banks (true octave / CPB filter bank) Parallel bandpass filters + energy detector + time averaging A filter-bank (true octave) analyzer typically: Design a bandpass filter H_b(z) (or H_b(s)) for each band center frequency. Run filters in parallel to obtain band signals y_b(t). Compute band mean-square/power and apply time averaging to output band levels. To be comparable across instruments, filter magnitude responses must satisfy IEC/ANSI tolerance masks (class) for the specified filter set. [1][3] IIR vs FIR: why IIR (cascaded biquads) is common in practice IIR advantages: lower order for a given roll-off, lower compute, good for real-time/embedded; stable when implemented as SOS/biquads. FIR advantages: linear phase is possible (useful when waveform shape matters); design/verification can be more straightforward. For band-level outputs, phase is usually not the primary concern, so IIR filter banks are common. Multirate processing: the “secret weapon” of CPB filter banks Low-frequency CPB bands are very narrow. Implementing them at the full sampling rate is inefficient. A common strategy is to group bands by octave and downsample for low-frequency groups: Low-pass then decimate (e.g., by 2 per octave) for lower-frequency groups. Implement the corresponding bandpass filters at the reduced sampling rate. Ensure adequate anti-aliasing before decimation. Time averaging / time weighting: band levels are statistics, not instantaneous values Band levels typically require time averaging. Common options include block RMS, exponential averaging, or Leq (energy-equivalent level). In sound level meter contexts, IEC 61672-1 defines Fast/Slow time weightings (Fast ~125 ms, Slow ~1 s). [5][6] Engineering implication: different time constants produce different readings, so time weighting must be stated in reports. How to validate that a filter bank behaves “like the standard” Sine sweep: verify passband behavior and adjacent-band isolation; observe time delay effects. Pink/white noise: verify average band levels and variance/stabilization time; check effective bandwidth behavior. Impulse/step: examine ringing and time response (critical for transient use). Cross-check against a known compliant reference instrument/implementation. From band definitions to compliant digital filters: an end-to-end workflow (conceptual) Choose the band system: base-10/base-2, the fraction 1/b (commonly b=3), generate exact fm and f1/f2. Choose performance target: which standard edition and which class/mask tolerance? Choose filter structure: IIR SOS for real-time; FIR or forward-backward filtering if phase/zero-phase is required. Design each bandpass: map f1/f2 into the digital domain correctly (e.g., pre-warp for bilinear transform). Implement multirate if needed: decimate for low-frequency groups with sufficient anti-alias filtering. Verify: magnitude response vs mask; noise tests for effective bandwidth; sweep/impulse tests for time response. Calibrate and report: units and reference quantities, averaging/time weighting, method details. Time response explained: group delay, ringing, and averaging all shape readings A band-level analyzer is a time-domain system (filter → energy detector → smoother), so readings are governed by multiple time scales: Filter group delay: how late events appear in each band. Filter ringing/decay: how long a short pulse “rings” within a band. Energy averaging/time weighting: the time resolution vs fluctuation of the output level. Thus, for transients (impacts, start/stop events, sweeps), different compliant implementations can yield different peak levels and time tracks—consistent with ANSI’s caution. [3] Rule of thumb: for steady-state contributions, use longer averaging for stability; for transient localization, shorten averaging but accept higher variability and lock down algorithm details. Common real-time pitfalls Forgetting anti-aliasing in the decimation chain: low-frequency bands become contaminated by aliasing. Numerical instability of high-Q low-frequency IIR sections: use SOS/biquads and sufficient precision. Averaging in dB: always average in energy/mean-square, then convert to dB. Assuming band energies must sum exactly to total energy: standard filters are not necessarily power-complementary; verify using standard-consistent criteria instead. Octave-Band Filter Bank Analysis in OpenTest OpenTest supports octave-band analysis using a filter-bank approach:1) Connect the device, such as SonoDAQ Pro2) Select the channels and adjust the parameter settings. For an external microphone, enable IEPE and switch to acoustic signal measurement.3) In the Octave-Band Analysis section under Measurement Mode, choose the IEC 61260-1 algorithm. It supports real-time analysis, linear averaging, exponential averaging, and peak hold.4) After configuring the parameters, click the Test button to start the measurement.5) A single recording can be analyzed simultaneously in 1/1-octave, 1/3-octave, 1/6-octave, 1/12-octave, 1/24-octave, and 1/24-octave bands. Figure 1: Octave-Band Filter Bank Analysis in OpenTest FFT binning and FFT synthesis FFT binning: convert a narrowband spectrum into CPB band integrals Estimate spectrum (single FFT, Welch PSD, or STFT). Integrate/sum within each octave/fractional-octave band to obtain band power. This is common in software/offline work because a single FFT provides high-resolution spectrum that can be re-binned into any band system (1/1, 1/3, 1/12, …). Key challenge #1: FFT scaling and window corrections After an FFT, scaling depends on your definitions: 1/N normalization, amplitude vs power vs PSD, one-sided vs two-sided spectrum, and windowing. For noise measurements, ENBW is crucial; ignoring it can introduce systematic offsets. [7] A practical PSD normalization (periodogram form) # convert to one-sided PSD: multiply by 2 except DC (and Nyquist if present) This yields PSD in units of (input unit)²/Hz and supports energy consistency checks by integrating PSD over frequency. Two quick self-checks for scaling White noise check: generate noise with known variance σ²; integrate one-sided PSD over 0..fs/2 and recover ≈σ² (accounting for the ×2 rule). Pure tone check: generate a sine with amplitude A (RMS=A/√2); integrating spectral energy should recover ≈A²/2 (subject to leakage and window choice). If both checks pass, your FFT scaling is likely correct; then partial-bin weighting and octave binning become meaningful. Key challenge #2: band edges rarely align to bins → partial-bin weighting Hard include/exclude decisions at band edges cause step-like errors, especially at low frequency where bands are narrow. Use overlap-based weighting (Section 4.2.4) for the boundary bins. Does zero-padding solve edge misalignment? (common misconception) Zero-padding interpolates the displayed spectrum but does not improve true frequency resolution (which is set by the original window length). It can reduce visual stair-stepping but cannot turn 1–2-bin low-frequency bands into reliable band-level estimates. Fundamental fixes are longer windows or multirate processing/filter banks. Key challenge #3: time–frequency trade-off (window length sets low-frequency accuracy and delay) FFT resolution is Δf = fs/N. Low-frequency 1/3-octave bands can be only a few Hz wide, so achieving enough bins per band requires very large N, increasing latency and smoothing transients. Root cause: 1/3 octave is constant-Q, but STFT uses constant-Δf bins In CPB, band width scales with frequency (Δf_band ∝ f, constant-Q). In STFT, bin spacing is constant (Δf_bin constant). Therefore low-frequency CPB needs extremely fine Δf_bin (long windows), while high frequency is over-resolved. Solution routes: long-window STFT vs multirate STFT vs CQT/wavelets Long-window STFT: simplest, but high latency and transient smearing. Multirate STFT: downsample low-frequency content and FFT at lower fs, similar in spirit to multirate filter banks. Constant-Q transform (CQT) / wavelets: naturally logarithmic resolution, but matching IEC/ANSI masks requires extra calibration/validation. [4] For compliance measurements, standards-oriented filter banks are preferred; for research/feature extraction, CQT/wavelets can be attractive. FFT synthesis: constructing per-band filtering in the frequency domain FFT synthesis pushes the FFT approach closer to a filter bank: Define a frequency-domain weight W_b[k] per band (brick-wall or smooth/mask-like). Compute Y_b[k] = X[k]·W_b[k] and IFFT to get y_b[n]. Compute band RMS/averages from y_b[n]. It can easily implement zero-phase (non-causal) filtering. For strict IEC/ANSI matching, W_b and normalization must be carefully designed and validated. Making FFT synthesis stream-like: OLA, dual windows, and amplitude normalization To output continuous time signals per band, use overlap-add (OLA): frame, window, FFT, apply W_b, IFFT, synthesis window, and OLA. Choose analysis/synthesis windows to satisfy COLA (constant overlap-add) conditions (e.g., Hann with 50% overlap) to avoid periodic level modulation. If the goal is to match standard filters, how should W_b be chosen? W_b[k] depends on what you want to match: Match brick-wall integration: W_b is hard 0/1 within [f1,f2]. Match IEC/ANSI filter behavior: |W_b(f)| approximates the standard mask and effective bandwidth (matches ∫|W_b|²). Match energy complementarity for reconstruction: design Σ_b |W_b(f)|² ≈ 1 (Section 7.6). You typically cannot satisfy all three perfectly at once; define your priority (compliance vs decomposition/reconstruction) up front. Energy-conserving frequency-domain filter banks: why Σ|W_b|² matters If you want band energies to sum to total energy (within numerical error), a common design aims for approximate power complementarity: IEC/ANSI masks do not necessarily enforce strict complementarity, so don’t assume exact additivity in compliance contexts. Welch/averaging strategies: how to make FFT band levels stable Use Welch averaging (segment, window, overlap, average power spectra). Average in the power domain (|X|² or PSD), then convert to dB. For non-stationary signals, consider STFT to obtain time–band matrices. Report window type, overlap, averaging count, and ENBW/CG treatment. FFT-Binning Analysis in OpenTest OpenTest supports octave-band analysis based on FFT binning:1) Connect the device, such asSonoDAQ Pro2) Select the channels and adjust the parameter settings. For an external microphone, enable IEPE and switch to acoustic signal measurement.3) In the Octave-Band Analysis section under Measurement Mode, choose the FFT-based algorithm.4) A single recording can be analyzed simultaneously in 1/1-octave, 1/3-octave, 1/6-octave, 1/12-octave, and 1/24-octave bands. Figure 2: FFT-Binning Octave-Band Analysis in OpenTest Filter-bank vs FFT/FFT synthesis: differences, equivalence conditions, and trade-offs A comparison table DimensionFilter-bank (True Octave / CPB)FFT binning / FFT synthesisStandards complianceEasier to match IEC/ANSI magnitude masks; mainstream for hardware instruments. [1][3]Hard binning behaves like band integration; matching masks requires extra weighting or standard-compliant digital filters.Real-time / latencyCausal real-time possible; latency set by filter order and averaging.Block processing adds at least one window length of delay; low-frequency resolution often forces longer windows.Transient responseContinuous output but affected by group delay/ringing; different compliant implementations may differ. [3]Set by STFT windowing; transients are smeared by windows and sensitive to window type/length.Leakage & correctionsControlled via filter design; leakage can be managed.Strongly depends on window and ENBW/scaling; edge-bin misalignment needs partial weighting. [7]InterpretabilityRMS after bandpass filtering—aligned with sound level meters and analyzers.Spectrum estimation + binning—more statistical; interpretation depends on window/averaging settings.ComputationMany filters in parallel; multirate can reduce cost.One FFT can serve all bands; efficient for offline/batch.Phase & reconstructionIIR is typically nonlinear phase (fine for levels).Frequency weights can be zero-phase; reconstruction needs attention to complementarity and transitions. When do both methods give (almost) the same answers? Band-averaged results typically agree closely when: You compare averaged band levels (not transient peak tracks). The signal is approximately stationary and the observation time is long enough. FFT resolution is fine enough that each band contains enough bins (especially at the lowest band). FFT scaling is correct (one-sided handling, Δf, window U, ENBW/CG where needed). Partial-bin weighting is used at band edges. Why differences grow for transients and short events Differences are driven by mismatched time scales: filter banks have band-dependent group delay and ringing but continuous output; STFT uses a fixed window that sets both frequency resolution and time smoothing. If event duration is comparable to the window length or filter impulse response, results depend strongly on implementation details. Error budget: where mismatches usually come from (and how to locate them quickly) Wrong averaging/combination in dB: must average and sum in the energy domain. Inconsistent FFT scaling: 1/N conventions, one-sided vs two-sided, Δf, window normalization U. Missing window corrections: ENBW for noise; coherent gain/leakage for tones. Using nominal frequencies to compute edges instead of exact definitions. No partial-bin weighting at band boundaries (especially harmful at low frequency). Multirate/anti-alias issues in filter banks. Different averaging time constants/windows between methods. True method differences: brick-wall binning vs standard filter skirts/roll-off imply systematic offsets. A strong debugging approach: first match total mean-square using white noise (scaling/ENBW/partial-bin), then validate band centers and adjacent-band isolation using swept sines or tones. Engineering checklist: make 1/3-octave analysis correct, stable, and reproducible Choose a method: compliance → filter bank; offline statistics → FFT binning For regulations/type testing/instrument comparability: prefer IEC/ANSI-compliant filter banks and report standard edition and class. [1][3] For offline processing, large datasets, or flexible band definitions: FFT binning can be efficient, but scaling and boundary weighting must be rigorous. If you need per-band time-domain signals (modulation, envelope, etc.): consider FFT synthesis or explicit filter banks. Selecting FFT parameters from the lowest band (example) Example: fs=48 kHz, lowest band of interest is 20 Hz (1/3 octave). Its bandwidth is only a few Hz. If you want at least M=10 bins per band, you may need Δf_bin ≤ bandwidth/10, implying a very large N (e.g., ~100k points; 2^17=131072). This illustrates why real-time compliance often favors filter banks. Typical mistakes that prevent results from matching Summing magnitude |X| instead of power |X|² or PSD. Averaging in dB instead of in linear power/mean-square. Ignoring ENBW/window scaling for noise. [7] Computing band edges from nominal frequencies. Not stating time weighting/averaging conventions (Fast/Slow/Leq). [5][6] Recommended validation flow (regardless of implementation) Tone-at-center test (or sweep): verify that energy peaks in the correct band and adjacent-band rejection behaves as expected. White/pink noise: verify expected spectral shape in band levels and assess stability/averaging time. Cross-implementation comparison: compare your implementation with a known reference on identical signals; isolate scaling vs definition vs filter-skirt differences. Record and freeze parameters (band definition, windowing, averaging) in the test report. Reproducibility checklist: include these in reports so others can recompute your levels Band definition: base-10 or base-2? b in 1/b? exact vs nominal used for computation? reference frequency fr? Implementation: standard filter bank (IIR/FIR, multirate) vs FFT binning/synthesis; software/library versions. Sampling/preprocessing: fs, detrending/DC removal, anti-alias filtering, resampling. Time averaging: Leq / block RMS / exponential; time constants, block size, overlap, averaging frames; Fast/Slow context if relevant. FFT details (if used): window type, N, hop, zero-padding, PSD normalization, one-sided handling, ENBW/CG, partial-bin weighting. Calibration/units: input units and reference quantities (e.g., 20 µPa), sensor calibration factors and dates. Output definition: RMS vs peak vs band power; 10log vs 20log conventions; any band aggregation steps. If you remember one line: document “band definition + time averaging + FFT scaling/window treatment (if any)”. Most disputes disappear. Quick formulas and numeric example (ready for code/report) Base-10 one-third-octave constants G = 10^(3/10) ≈ 1.995262 r = 10^(1/10) ≈ 1.258925 # adjacent center-frequency ratio k = 10^(1/20) ≈ 1.122018 # edge multiplier about center f1 = fm / k f2 = fm * k Example: the 1 kHz one-third-octave band fm = 1000 Hz f1 = 1000 / 1.122018 ≈ 891.25 Hz f2 = 1000 * 1.122018 ≈ 1122.02 Hz Δf ≈ 230.77 Hz Q ≈ 4.33 OpenTest integrates both methods. Download and get started now -> or fill out the form below ↓ to schedule a live demo. Explore more features and application stories at www.opentest.com. References [1] IEC 61260-1:2014 PDF sample (iTeh): https://cdn.standards.iteh.ai/samples/13383/3c4ae3e762b540cc8111744cb8f0ae8e/IEC-61260-1-2014.pdf [3] ANSI S1.11-2004 preview PDF (ASA/ANSI): https://webstore.ansi.org/preview-pages/ASA/preview_ANSI%2BS1.11-2004.pdf [4] HEAD acoustics Application Note: FFT - 1/n-Octave Analysis - Wavelet (filter bank description): https://cdn.head-acoustics.com/fileadmin/data/global/Application-Notes/SVP/FFT-nthOctave-Wavelet_e.pdf [5] IEC 61672-1:2013 (IEC page): https://webstore.iec.ch/en/publication/5708 [6] NTi Audio Know-how: Fast/Slow time weighting (IEC 61672-1 context): https://www.nti-audio.com/en/support/know-how/fast-slow-impulse-time-weighting-what-do-they-mean [7] MathWorks: ENBW definition example: https://www.mathworks.com/help/signal/ref/enbw.html

In audio and vibration testing, engineering teams often find themselves jumping between multiple software tools and data acquisition systems from different vendors. Interfaces vary, workflows are fragmented, and new engineers can spend a significant amount of time just learning the tools before they can focus on the engineering problem itself. OpenTest, developed by CRYSOUND, is a next-generation acoustic and NVH testing platform designed for engineers, researchers, and manufacturers. Built around the principles of an open ecosystem, AI-driven intelligence, and high compatibility, it allows users to complete the entire workflow—from acquisition to reporting—within a single software environment. OpenTest supports three operating modes: Measure, Analysis, and Sequence, covering both laboratory validation and repetitive production testing. Core capabilities include real-time monitoring and analysis, FFT and octave analysis, sweep analysis, sound power testing, sound level meter functions, and sound quality analysis. The platform also provides standard test reports and dedicated sound power reports that comply with international standards. On the hardware side, OpenTest connects to a wide range of multi-brand DAQ devices via mainstream audio protocols such as openDAQ, ASIO, and WASAPI, as well as optional proprietary drivers such as NI-DAQmx, enabling unified management of CRYSOUND SonoDAQ, RME, NI, and other devices within a single platform. On the software side, its modular plugin architecture exposes interfaces for Python, MATLAB, LabVIEW, C++ and more, making it easy for teams to package in-house algorithms and domain applications as plugins and deploy them within the same environment. From Acquisition to Report: A Three-Step Quick-Start Workflow 1. Installation and Basic Connectivity – Let the Signals In Download the latest installer from the official website www.opentest.com and complete the installation. Connect your DAQ device to the PC; for your first trial, you can simply use the built-in PC sound card to run a quick test. In the OpenTest setup section, scan for available devices and select the devices and channels you want to use. Once added to the project, your basic connectivity is complete. 2. Run Basic Tests with Real-Time Analysis – See It First, Then Optimize In the channel management view, select the input/output channels you want to use and configure key parameters such as sensitivity, sampling rate, and gain. The system automatically activates the Monitor panel, where you can view real-time waveforms, FFT spectra, and key metrics such as RMS level and THD at a glance. When needed, you can enable the built-in signal generator to output excitation signals and use the recording function for long-duration acquisition, preserving data for later comparison and analysis. 3. Perform In-Depth Analysis and Reporting in the Measure Module – Turning Data into Decisions Switch to the Measure module to access advanced applications such as FFT analysis, octave analysis, sweep analysis, sound power testing, sound level meter, and sound quality—providing everything you need for deeper investigation. Use the data set functionality to review and overlay historical records, so you can compare different samples, operating conditions, or tuning strategies side by side. Waveforms and analysis results can be exported at any time. With the reporting function, you can generate test reports with a single click, closing the loop from test execution to final deliverables. Who Is OpenTest For? New acoustic and vibration test engineers who want to establish a complete workflow quickly using a single toolchain. Laboratories and corporate teams that need to manage multi-brand hardware and consolidate everything into one unified software platform. Project teams in automotive NVH, consumer electronics, and industrial diagnostics that require high channel counts, automation, and AI-enhanced analysis capabilities. Wherever you are on your testing infrastructure journey, OpenTest lets you start with a free entry-level edition and adopt an open, intelligent, and scalable ecosystem with a low barrier to entry. Visit www.opentest.com to explore detailed features, supported hardware, and licensing and plan options, and book a demo to see how OpenTest and CRYSOUND can help you build an efficient, open, and future-ready acoustic and vibration testing platform.