Tone Burst Technical Reference

Introduction

There are generally two types of distortion that loudspeakers produce, time and spectrum distortion. Time and frequency distortions can be directly related to each other, but not always. The ToneBurst program focuses on Energy Decay. Here are some examples:

Time Distortions

• Phase
• Delay
• Energy Decay

Frequency Distortions

• Frequency Response
• Harmonic Distortion
• Intermodulation Distortion

I initially learned about shaped tone burst testing from Sigfried Linkwitz's website 20+ years ago: https://www.linkwitzlab.com/mid_dist.htm

After studying up on burst testing I took it one step further, stepped sweep burst testing. Time analysis of speaker tone burst signals will reveal stored energy released after the input signal has stopped. This stored energy can be caused by the drivers, speaker cabinets, or room.

Here is a program I created to test tone bursts and stepped sweep tone burts. You can read about some of its functions below, it will be expanded soon. If you decide to check it out, please remember that this program is not digitally signed, so your browser will throw a warning when you try to download it. To prevent the warning I would need to pay to get it digitally signed, I don't want to pay to get it signed. The program is free to use, there's no warranty, use at your own discretion. If you decide to use it, all I ask is that you provide attribution back to this website. Suggestions are always welcome, use the contact email in the top menu.

ToneBurst.exe

ToneBurst v2.3 — Reference Manual

ToneBurst generates calibrated tone burst WAV files and analyzes their Energy-Time Curve (ETC) using the Hilbert Transform envelope method. It is designed to measure transient response, decay characteristics, and comparative differences between a Device Under Test (DUT) and a reference recording.

Creates calibrated WAV test signals. Two modes are available: Single Burst generates one tone at a fixed frequency; Sweep generates a series of bursts stepping through a frequency range at a defined octave resolution. Output files are written to the GEN directory.

Analyzes a DUT recording against its embedded reference signal. The analyzer auto-detects whether the WAV is a single burst or a sweep using the embedded DTMF markers, then dispatches to the appropriate analysis engine. Output files are written to the IMG directory.

Compares two DUT recordings of the same test signal against each other, or against a shared reference. Produces an overlay ETC graph with crosshatch difference shading and quantitative |Diff|% statistics for each recording.

Built-in image browser for SVG and PNG output files. Displays graphs directly in the application window without needing an external viewer. Auto-fits images to window width and supports keyboard navigation through a directory of files.

ToneBurst organizes files into three subdirectories relative to the application executable. They are created automatically on first run.

ToneBurst can be run from the command line for automation and scripting. Run without arguments to launch the GUI, pass a command flag to run headlessly.

# Generate a single burst at 1 kHz
ToneBurst -genburst 1000 -wav

# Generate a sweep from 100 Hz to 20 kHz at 1/3 octave
ToneBurst -gensweep -sweepstart 100 -sweepend 20000 -sweepoct 3

# Analyze a recording (auto-detects burst or sweep)
ToneBurst -analyze DUT\my_recording.wav

# Analyze burst with SVG + ETC markers
ToneBurst -analyze DUT\burst.wav -svg -etcmarkers

# Analyze sweep, all output formats, fixed Y axis 0–10%
ToneBurst -analyze DUT\sweep.wav -svgsweep -svgmini -svgdiscrete -sweepymin 0 -sweepymax 10

# Generate with noise and distortion
ToneBurst -genburst 1000 -wav -headroom 3 -noise 60

WAV Output File Structures

Single Burst & Burst Sweep signal format specification

Single Burst WAV

A single 4-cycle Blackman-windowed tone burst at one target frequency, framed by DTMF-encoded metadata and band-limited PRN sync markers.

Burst Sweep WAV

Multiple 4-cycle bursts at logarithmically-spaced frequencies, sharing the same DTMF-first framing structure as the single burst.

DTMF Type Prefix

The first DTMF digit identifies the file type, allowing the analyzer to auto-detect and route to the correct analysis path.

DTMF Band Selection

The DTMF band is chosen automatically to avoid overlap with the test frequency. Each digit is encoded as a simultaneous pair of tones — one from the Low Group (row) and one from the High Group (column).

Digit → Row / Column Mapping

Each digit 0–9 selects one frequency from the Low Group (row index) and one from the High Group (column index). The two tones are transmitted simultaneously for 50 ms.

PRN Sync Signal

The sync signal is a sum-of-sinusoids — a deterministic, reproducible waveform constructed by adding together a large number of sine waves simultaneously. Here's each stage in detail.

Sweep Generation (-gensweep)

Frequency grid — 1/12-octave spacing starting at 100 Hz, each successive frequency is multiplied by 2^(1/12) ≈ 1.0595, the same interval as a semitone in musical pitch. This continues up to 20 kHz, producing approximately 96 frequencies covering the full audio range. Using logarithmic spacing means the frequencies are perceptually evenly spread — as many components cover 100–200 Hz as cover 10–20 kHz — which gives the signal a noise-like texture across the whole band rather than clustering energy at the low end.
The sum-of-sinusoids controls exactly which frequencies are present and at what amplitude, you can shape the spectrum to match the measurement context and guarantee that the generator and detector always use bit-for-bit identical reference waveforms regardless of platform or runtime. The cross-correlation between the recorded signal and the known reference then has a sharp, unambiguous peak at the correct sync position even if the recording is bandlimited or frequency-coloured by the device under test.
Phase randomisation — All those sine waves starting at zero phase would produce a huge coherent spike at t=0. To prevent that, each sinusoid is given a pseudo-random starting phase. The phases are generated by the classic glibc Linear Congruential Generator. The output of each step is mapped to a phase in the range 0–2π. All that's needed is that the phases are sufficiently spread out to break the coherent spike, and that the exact same phases are reproduced every time the code runs, so the generator and detector always agree on the reference waveform.
Pink Noise amplitude weighting 1/√f — Each sinusoid is scaled by √(refFreq / f), where refFreq is the lowest frequency (100 Hz). This means the 100 Hz component has full amplitude (scale = 1.0), the 200 Hz component has scale ≈ 0.707, the 1 kHz component has scale ≈ 0.316, and the 20 kHz component has scale ≈ 0.071. This is a pink weighting — amplitude falls at 3 dB per octave, which makes the power spectrum flat on a per-octave basis. Without this, the higher frequencies would dominate (white noise) the signal energy and cross-correlation would be biased toward low-frequency content in the recording.
High-pass filtering - The full-bandwidth signal is then filtered with a 2nd-order Butterworth high-pass at half the lowest burst frequency (capped at 2 kHz). This ensures the PRN doesn't overload a driver, especially small mids and tweeters. For a 1 kHz burst the PRN is high-passed at 500 Hz; for a 100 Hz sweep start it would be high-passed at 50 Hz but the 2 kHz cap applies in practice for any higher frequency tests

DUT Analysis (-analyze)

• The analysis pipeline has six distinct step that apply identically whether processing a single burst or a sweep. The sweep simply iterates steps 4–6 once per frequency.


Step 1 Reading the WAV File — The WAV is read into memory and scanned in 10 ms energy windows. The first window that exceeds 10× the RMS noise floor (or 50,000 counts absolute minimum) is flagged as the DTMF start. A Goertzel filter then reads just the first DTMF digit across all three frequency bands (LOW/MID/HIGH) simultaneously and picks the band with highest total power. Digit 1 → burst and Digit 2 → sweep. This determines which analysis path runs next.
Step 2 DTMF Parameter Decoding — The full digit string is decoded using the Goertzel algorithm at the known symbol timing (50 ms tone, 25 ms gap per digit). Every possible band is tried and the one producing the highest total energy across all digits wins. Burst (1FFFFF): 6 digits → frequency extracted as a 5-digit integer. If this fails, FFT is used as fallback. Sweep (2SSSSSEEEEEOOIIII): 17 digits → start frequency, end frequency, octave resolution, and burst interval in ms are all extracted. These values completely describe the test — no external parameters are needed.


Step 3 PRN Sync and Clock Drift Correction — A reference PRN signal is generated in software using the same function as the genenerate tab. This is cross-correlated against the recording to locate the start PRN precisely. Two-pass correlation: First pass steps every 100 samples across a ±200 ms search window around the expected position for speed. Second pass does a sample-by-sample refinement ±10 samples around the coarse peak. The sub-sample position is resolved by fitting a parabola to the three highest correlation values — sub-sample function returns a fractional sample offset so the PRN is located to better than one sample precision. The End PRN is then found by the same process, searching the second half of the file. With both PRN positions known sub-sampled precisely:
measuredSpan = endPRN_precise − startPRN_precise
expectedSpan = known from DTMF (PRN + silence + bursts + silence + noise floor)
driftRatio = measuredSpan / expectedSpan
driftPPM = (driftRatio − 1) × 1,000,000
If drift exceeds 0.5 PPM the entire recording is resampled by driftRatio using windowed sinc interpolation (DSPHelper.Resample), and all sample offsets are recalculated. If drift exceeds 1000 PPM (implausible — different clocks or wrong file) it is discarded. This makes burst timing precise even when the playback and recording devices have independent clocks.
Step 4 Noise Floor Measurement — If requested, the 1-second silence window immediately before the end PRN is used. The peak absolute sample value in that window is expressed relative to 0 dBFS — this is the noise floor in dBFS. For each individual burst analysis it is then expressed relative to the burst's own peak, giving a burst-relative noise floor in dB. If -setnoisefloor is active, this value replaces the manually set threshold for area integration.
Step 5 Per Burst Extraction and ETC Computation — For each burst slot (sweep: once per frequency; single burst: once.) The interval window is sliced from the drift-corrected recording and the burst is extracted. The first 500 samples (Burst Lead-In) are silent pre-burst padding, deliberately included because the Hilbert transform needs silence before the burst onset to avoid edge effects. Window length is 500 + WindowPeriods × samplesPerCycle — defaulting to 16 periods, extended to ensure the Windowed/Discrete Hilbert Transform (DHT) has full support at every sample.
Peak location: A frequency-adaptive RMS window, 4-cycle width, is slid across the interval to find the burst peak position. Silent intervals (below 0.1% of full scale) are skipped.
Reference generation: A mathematically perfect 4-cycle Blackman-windowed burst at the nominal frequency is synthesised and placed at sample offset 500 within an array of the same length as the DUT window. This is the ideal reference — no room acoustics, no recording chain, just the pure waveform.
Discrete Hilbert Transform envelope: Both DUT and reference windows are passed to the envelope. The DHT is a windowed FIR approximation of the ideal 2/(πk) impulse response, Blackman-windowed to suppress sidelobes, with a radius of at least 8 periods of the burst frequency. The input is zero-padded by half-width on both sides so the convolution is accurate at all positions. At each sample, the imaginary part (Hilbert output) and real part (original signal minus DC) form a complex pair; the magnitude is √(real² + imag²). This gives the instantaneous amplitude envelope. The result is converted to dB relative to peak and clamped to a −60 dB floor.
Step 6 |Diff| % Computation — Both envelopes are converted back to linear amplitude for integration. For each consecutive sample pair where at least one envelope is above the threshold, the absolute difference is computed and the area under it is integrated using the trapezoidal rule.

diffArea += 0.5 × (|DUT[i] − REF[i]| + |DUT[i+1] − REF[i+1]|) × dt

The DUT area alone is computed the same way (against zero). The |Diff| % metric is: |Diff| % = (diffArea / refArea) × 100

This expresses how much the DUT's energy-time behaviour deviates from the ideal reference, as a fraction of the reference's own energy above the threshold. A perfect transducer scores 0 %. A device with strong resonant ringing scores higher — there is no upper bound to how badly a device can behave, but the value is capped at 200 % in the code to prevent chart scaling problems.

Sweep Burst Interval

Each burst in the sweep occupies a fixed-length interval. The first 500 samples are guaranteed silent, providing a clean pre-burst padding zone for the Hilbert kernel in the analyzer.

interval = max(250 ms, 25 × period of lowest frequency)

Example: for a 100 Hz–20 kHz sweep, interval = max(250, 25 / 100 × 1000) = 250 ms = 24,000 samples at 96 kHz.

Burst interval structure — 500-sample silent lead-in guarantees Hilbert padding

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Analysis Windowing

Both burst and sweep analyzers extract a time-domain window around each burst for ETC (Energy Time Curve) analysis via Hilbert transform. The window must include enough silent padding before the burst for the Hilbert kernel to operate without edge truncation.

Single Burst Window

The single burst analyzer locates the burst via PRN sync correlation, then extracts a window centered on the burst peak. The window size is based on the number of analysis periods (default 16) plus extra room for filter delay and ringing tail.

Single burst — window centered on burst peak, symmetric padding

Sweep Burst Window

The sweep analyzer extracts each burst's fixed-length interval starting from the interval boundary. The 500-sample silent lead-in provides guaranteed Hilbert padding regardless of frequency. The window starts at sample 0 of the interval.

Sweep burst — window starts at interval boundary, 500-sample guaranteed lead-in. Reference burst placed at identical position (sample 500) for consistent alignment.

Hilbert kernel size: M = max(255, samplesPerPeriod × 8). At 16 kHz / 96 kHz SR, M = 255 and minPadding = 265. The 500-sample lead-in exceeds this at all frequencies.

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ETC Analysis & Oversampling

The ETC (Energy Time Curve) is computed via Hilbert transform at the native 96 kHz sample rate for maximum waveform fidelity. The resulting dB envelope is then oversampled to 768 kHz (8×) for sub-sample alignment precision and smooth SVG rendering.

Processing Pipeline

Input

int[] samples

96 kHz native

→

Hilbert Transform

ComputeEnvelope()

96 kHz, kernel M

→

Output

dB envelope

96 kHz native

→

Oversample 8×

OversampleEnvelope()

768 kHz

→

Align

left −3 dB edge

sub-sample shift

→

Diff Area

|DUT − REF|

768 kHz precision

Why Oversample the Envelope, Not the Waveform?

The Hilbert transform produces the smoothest, most accurate envelope when operating on the original integer samples at the native sample rate. Upsampling the waveform before the Hilbert introduces sinc interpolation artifacts (ripple near transitions) that the kernel picks up as spurious envelope features. By contrast, the dB envelope is a smooth, slowly-varying curve that oversamples cleanly with simple interpolation.

Stage	Sample Rate	Purpose
Hilbert transform	96 kHz	Analytic signal envelope from original waveform — no interpolation artifacts
Envelope oversample	768 kHz (8×)	Sub-sample precision for −3 dB alignment and area integration
Fractional shift	768 kHz	Windowed sinc interpolation aligns DUT left −3 dB crossing to REF
|Diff| area	768 kHz	Trapezoidal integration of |DUT − REF| above threshold, reported as ms² and %

Alignment Method

DUT and REF envelopes are aligned by their left (rising) −3 dB crossing point. This edge is unaffected by tail ringing or distortion, giving a clean onset alignment. The shift is computed at 768 kHz resolution (1.3 µs precision) and applied via windowed sinc interpolation on the oversampled envelope.

ETC envelope alignment — left −3 dB crossing used for onset registration. Difference area (gold) computed above threshold between DUT and REF.

ETC Output Layout



Grid layout of per-frequency ETC mini-graphs with |Diff| % values

Tone Burst Bandwidth

The bandwidth of a tone burst is determined by its duration, not its frequency. For a 4-cycle burst the duration is T = 4 / f and the approximate −3 dB bandwidth is BW ≈ f / 4 — roughly 25% of the center frequency (±12.5%).

The Blackman window reduces spectral leakage (sidelobes ≈ −58 dB down) but slightly widens the main lobe, so the actual −3 dB bandwidth is closer to f / 3.5 or about 29% of center frequency.

Frequency	Duration	Bandwidth (−3 dB)
100 Hz	40 ms	~25 Hz
1 kHz	4 ms	~250 Hz
10 kHz	0.4 ms	~2.5 kHz
20 kHz	0.2 ms	~5 kHz

This is why tone bursts are useful for transient / time-domain analysis — they are short enough to measure time-domain behavior but have enough bandwidth to excite the system meaningfully. The trade-off is that you cannot isolate a single frequency the way you would with a long steady-state sine wave.

For speaker measurements, this bandwidth is actually desirable since it shows how the speaker responds to transient signals across a range of frequencies simultaneously, which is more representative of real audio content (music, speech) than pure tones.

4-Cycle Blackman-Windowed Reference Burst



4-cycle Blackman-windowed tone burst showing amplitude modulation envelope (dashed orange)

Optimal Octave Spacing

For adjacent burst spectra to "just touch" at their −3 dB points, we need the upper −3 dB edge of one burst to meet the lower −3 dB edge of the next.

Given

Bandwidth ≈ 29% of center frequency (±14.5%). Upper −3 dB edge: f × 1.145. Lower −3 dB edge: f × 0.855.

Derivation

f₂ × 0.855 = f₁ × 1.145
f₂ / f₁ = 1.145 / 0.855 = 1.339

2^1/n = 1.339 → 1/n = log₂(1.339) = 0.421 → n ≈ 2.37

Setting	Step Ratio	Coverage
-sweepoct 2 (½ octave)	1.414	Slight gaps between bursts
Optimal ≈ 1/2.4 octave	1.339	Spectra just touch
-sweepoct 3 (⅓ octave)	1.260	Slight overlap (~6%)

Since -sweepoct takes integers, use -sweepoct 3 or higher for continuous frequency coverage without gaps.

Parabolic Interpolation for Sub-Sample PRN Detection

Test Signal Structure



WAV file structure showing Start PRN and End PRN positions used for drift measurement

The Problem: Integer Sample Resolution

Cross-correlation finds where the PRN best matches, but only at integer sample positions. The true peak of the correlation function almost always falls between samples. A 1-sample error in a 770,400-sample span produces a 1.3 PPM apparent drift.



Cross-correlation values at integer sample positions; the true peak lies between samples

The Solution: Fit a Parabola

A parabola closely approximates the correlation peak. We use the three samples surrounding the integer maximum — y0, y1 (the peak), and y2 — to calculate where the true peak lies.



Parabola fitted through three consecutive correlation values to find the sub-sample peak

The Math

For a parabola passing through points at x = −1, 0, +1 the vertex (peak) occurs at:

δ = (y0 − y2) / (2 × (y0 − 2·y1 + y2))

Real Example (End PRN Detection)

y0 (lag−1) :  527,410,999
y1 (lag peak): 529,429,464   ← highest correlation
y2 (lag+1) :  527,361,859
y0 − y2          = 49,140
y0 − 2·y1 + y2  = −4,086,070
δ = 49,140 / (2 × −4,086,070) = −0.006013
Integer position  : 866,400
Sub-sample offset : −0.006013
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
Precise position  : 866,399.993987
              



Correlation values at three consecutive sample positions around the End PRN peak

Precision Improvement

Without Interpolation

Start PRN: 96000
End PRN: 866400
Span: 770,400 samples
Error: ~1 sample
Drift: ~1.3 PPM

With Parabolic Interpolation

Start PRN: 96000.00513
End PRN: 866399.993987
Span: 770,399.988857
Error: 0.011 samples
Drift: ~0.014 PPM

Detection Method	Resolution	Min Detectable Drift
Integer only	1 sample	~1.3 PPM
Parabolic interpolation	~0.01 sample	~0.013 PPM

~90× Better Precision

Typical consumer audio drift is 20–100+ PPM; professional gear is 1–20 PPM. With parabolic interpolation we can accurately measure even professional-grade clock accuracy.

|Diff| % Metric

The |Diff| % metric compares the Energy Time Curve (ETC) of the Device Under Test (DUT) to an ideal 4-cycle Blackman-windowed reference burst. It measures how much the burst "spreads" or "rings" compared to the ideal.

Step 1 — Record the Signals

The reference is the ideal 4-cycle Blackman burst. The DUT signal is recorded through the speaker and room, and may show additional ringing after the main burst.



Reference signal (top) vs. DUT signal (bottom) showing post-burst ringing

Step 2 — Compute ETC (Envelope via Hilbert Transform)

The Hilbert Transform extracts the amplitude envelope, which is then converted to dB scale (0 dB = peak).



ETC envelopes in dB scale; the −40 dB threshold (orange dashed) defines the integration boundary

Step 3 — Calculate Area Above −40 dB Threshold

We integrate the linear envelope (not dB) above the −40 dB threshold. The shaded regions below represent the areas being compared.



Shaded areas above the −40 dB threshold for Reference (top) and DUT (bottom)

Step 4 — Calculate |Diff| %

|Diff| % = |DUT Area − Reference Area| / Reference Area × 100%

= |19.5 − 17.0| / 17.0 × 100% = 2.5 / 17.0 × 100% = 14.7%

Visual Summary



Side-by-side comparison: ideal burst vs. burst with ringing tail

Interpretation Guide

|Diff| %	Rating	Interpretation
0–5%	Excellent	Minimal ringing or resonance
5–15%	Good	Some coloration
15–30%	Fair	Noticeable resonance
30–50%	Poor	Significant ringing
>50%	Bad	Severe resonance issues

What the Metric Captures

Speaker resonance (cone continues moving after signal stops), room reflections (sound bouncing back to microphone), cabinet vibrations (enclosure ringing), and port turbulence (in ported speakers).

Noise Floor Measurement

The noise floor is measured from a dedicated 500 ms silence window located just before the End PRN marker.

Measurement Window Location



The 500 ms noise measurement window is located between the last silence gap and the End PRN

Step 1 — Locate the Window

noiseFloorSilence    = SampleRate × 0.500    # 48,000 samples at 96 kHz
noiseFloorEndSample  = endPrnPosition
noiseFloorStartSample = endPrnPosition − noiseFloorSilence
              

Step 2 — Calculate RMS

RMS (Root Mean Square) measures the average power of the signal in the noise window.

RMS = √( (s₀² + s₁² + s₂² + … + sₙ²) / N )



Visualisation of noise samples in the measurement window (exaggerated amplitude)

Step 3 — Convert to dBFS

Convert the RMS value to decibels relative to full scale (maximum possible digital value).

dBFS = 20 × log₁₀( RMS / Full Scale Max )

For 24-bit audio: Full Scale Max = 8,388,607 (2²³ − 1)

Example:  RMS = 3,340
dBFS = 20 × log₁₀( 3,340 / 8,388,607 )
     = 20 × log₁₀( 0.000398 )
     = 20 × (−3.4)
     = −68 dBFS
              

Step 4 — Per-Burst Relative Noise Floor

For each burst's mini ETC graph, the noise floor is calculated relative to that burst's peak amplitude, giving the signal-to-noise ratio for that specific frequency.



Per-burst ETC graph with the noise floor shown as an orange dashed line relative to the burst peak

Relative Noise Floor = 20 × log₁₀( Noise RMS / Burst Peak Amplitude )

Example: 20 × log₁₀( 3,340 / 6,700,000 ) = 20 × log₁₀( 0.000499 ) = −66 dB

Two Different Noise Floor Values

Absolute (Console Output)

"Noise Floor: −58.1 dBFS (measured from silence window)"

Measured relative to full scale (0 dBFS). Same value for the entire file. Tells you the actual noise level.

Relative (Per-Burst Graph)

Orange dashed line at −XX dB on each mini ETC graph.

Measured relative to each burst's peak amplitude. Different per burst (lower freq = higher amplitude). Shows SNR for that burst.

Why Measure in the Silence Window?

Captures system noise without signal interference. Located after all bursts so no burst energy leaks in. 500 ms provides enough samples for a stable RMS measurement. Positioned before the End PRN so the PRN doesn't corrupt the measurement.

Noise sources captured: ADC quantization noise, preamp/mic noise, environmental noise (room, HVAC, etc.), and electromagnetic interference.