Local & band phase

These measurements reveal stereo opposition that the whole-track polarity measurement can dilute: a short opposed passage, or opposed treble beneath stronger aligned bass. They describe decoded stereo audio and help locate passages to audition in mono. They do not establish an encoding history, an absolute polarity, or an unintended defect.

Results in the app#

Both columns contain only a signed correlation coefficient, rounded to three decimals:

Both sort numerically from −1 to +1 and are searchable by the displayed value and column name. Missing measurements sort last in either direction. A dash means no usable measurement, while 0.000 is a measured coefficient rounded to display precision.

For either result:

• +1 means the channels are proportional with matching polarity in the measured range. Their levels need not match.
• 0 means no zero-lag correlation. This includes a quarter-cycle phase difference on a pure tone; it does not establish that the channels are unrelated.
• −1 means the channels are proportional with opposite polarity in the measured range. Unequal gains can still prevent total cancellation when summed to mono.

An opposed window is defined here as a coefficient at or below −0.5. This is a project reporting threshold for substantially negative correlation, not an industry standard, an audibility threshold or a polarity-inversion verdict. No new authenticity flag or automatic defect label is produced. The existing whole-track polarity indication keeps its own method and threshold.

Time resolution and frequency ranges#

The meter accepts exactly two channels at sample rates from 8 to 768 kHz, including stereo PCM decoded from DSD when decoding is enabled. Mono and multichannel files are not reduced to an arbitrary channel pair.

The transform size is N = next_power_of_two(ceil(sample_rate / 5)). Thus each window is at least 200 ms and less than 400 ms long. The hop is N / 2, giving 50% overlap. At 48 kHz, this is 341.333 ms, with a 170.667 ms hop. At 44.1 kHz, the window is 371.519 ms. The actual window and hop are saved in the report.

Upper limits are capped at Nyquist (sample_rate / 2). Lower edges are inclusive, upper edges exclusive. DC and the Nyquist bin are excluded. Bands entirely above Nyquist have no reading. FFT bins belong to the range containing their center frequency; the bin spacing is sample_rate / N.

Only complete windows are examined. There is no zero padding of the trailing partial window. A file shorter than one window has no result. Window starts are measured from the first decoded frame and include silent introductions. They identify the analysis window, not the exact beginning of a phase change. The minimum's window end is its start plus window_secs; when minima tie at stored precision, the first such window is retained.

Calculation#

For each channel independently, calculate a periodic Hann window and its weighted mean:

w[n] = 0.5 − 0.5 cos(2πn/N),  n = 0 … N−1
μ = Σ(w[n] × x[n]) / Σw[n]
X[k] = DFT(w[n] × (x[n] − μ))

The periodic Hann convention follows the MathWorks window documentation. Its weighting reduces leakage at window boundaries. Removing the mean prevents a constant channel bias from dominating the phase reading; DC offset remains separately available.

The implementation sums deviations from the smallest-magnitude sample before calculating the mean. This is algebraically the same formula, but a constant float offset then subtracts exactly. Choosing a small anchor also prevents an extreme outlier from swallowing ordinary samples in every subtraction. Summing a very large offset directly can lose enough precision to manufacture audible-band energy after the FFT. Removing an offset cannot recover audio already lost when a large bias was quantized to float samples.

Let L[k] and R[k] be the two complex spectra. For each frequency range, sum the positive-frequency bins in that range:

q = 2 / (N × Σw[n]²)
E_L = q × Σ |L[k]|²
E_R = q × Σ |R[k]|²
C   = q × Σ Re(L[k] × conjugate(R[k]))
correlation = C / sqrt(E_L × E_R)

The factor q comes from Parseval's identity, the two symmetric frequency halves of a real signal, and the window's mean-square gain. It makes the energy gate comparable between transform sizes. It cancels in the correlation itself. Results are clamped to −1 through +1 to contain floating-point roundoff.

A window is eligible for a particular range only when both E_L and E_R exceed 10⁻⁶, equivalent to a band RMS above −60 dBFS per channel. Each range has its own gate. A strong bass component cannot make an otherwise silent treble band eligible. The floor is a project choice to suppress meaningless ratios in silence and very weak leakage; it is not loudness weighting.

More precisely, the gate uses normalized-sample RMS strictly greater than 0.001, with 20 log10(RMS) referenced to sample magnitude one. This defines the project's floor without relying on an RMS meter's full-scale calibration convention.

For each range, the report retains:

• Correlation calculated from the sum of eligible-window energies. It is not an arithmetic average of individual coefficients.
• Minimum local correlation and the start of the corresponding window.
• Fraction of eligible windows with a coefficient at or below −0.5.
• Eligible window count, alongside the total complete-window count for coverage.

The opposed fraction counts overlapping eligible windows, not samples or seconds. It must not be presented as the percentage of the track affected. Silence and one-sided content do not contribute to the denominator. At most five summaries are retained, so memory use and report size stay bounded with track duration.

Interpretation limits#

• This is zero-lag correlation, not a phase angle in degrees and not magnitude-squared coherence. An averaged coherence estimate and the cross-spectrum angle answer different questions; see the MathWorks discussion of cross spectrum and coherence. FlacCompagnon uses the real cross spectrum here to quantify agreement or opposition within each range.
• This is not interaural coherence: Whitmer, Seeber and Akeroyd's experiments assess peak cross-correlation at the ears and show listener dependence. File L/R zero-lag correlation omits the head, room and crosstalk; that study does not calibrate our thresholds.
• Broadband and band results are energy-weighted. Multiple components can cancel each other's correlation inside the same band. This is a four-band summary, not a phase-versus-frequency curve.
• Intentional stereo widening, ambience, delays and microphone spacing can produce negative correlation. Listening and knowledge of the source are required before changing the audio.
• A brief event shorter than a window can be diluted. Hann weighting also reduces the influence of samples near each window edge. The method does not promise sample-accurate event detection.
• Windowing reduces spectral leakage without eliminating it. A tone close to a band boundary can contribute to both neighbouring bands; the edges are analysis bins, not ideal physical crossover filters. Subsonic and ultrasonic content can also leak into the measured range.
• Low-frequency estimates contain relatively few cycles. An isolated transient or a very short excerpt is less representative than sustained programme material.
• The absolute energy gate makes coverage depend on gain. Raising a very quiet file can make previously unavailable ranges measurable. A missing band is not proof of perfect mono compatibility.
• Global polarity uses whole-stream samples, while this measurement removes DC, limits the frequency range, windows the audio and gates each range. Their aggregate coefficients need not be equal.
• Invalid frames or non-finite samples withhold the entire measurement. Failed/skipped decoding and older saved reports also have no reading. Float samples above full scale are measured without clipping them first.

For a mono average M = (L + R) / 2, the measured energies obey E_M = (E_L + E_R + 2C) / 4. Negative correlation reduces this sum relative to zero cross energy, but the coefficient alone does not quantify the loss. For example, R = −0.1L has correlation −1 while its mono average is 0.45L; it does not vanish. Complete cancellation requires equal opposing channels. Conversely, quadrature tones have coefficient zero while remaining deterministically related. Audition the actual mono sum before deciding that an opposed passage needs correction.

This listening requirement agrees with the recording practitioner's discussion in Sengpiel's correlation-meter tutorial. A firsthand sampled-piano discussion also illustrates how a negative meter reading can accompany a mono sum its mixer finds acceptable. These are interpretation examples, not numerical ground truth or threshold calibration.

Design choices and cost#

Zero-lag signed correlation is retained because the question here is opposition in the mono sum. Maximizing over time lag could turn a delayed opposed tone into a positive match, hiding that evidence. Magnitude-squared coherence would describe linear relatedness instead: a single spectral segment gives a value of one at nonzero bins, and a useful estimate needs averaging across segments, as the official mscohere documentation explains. Adding such an estimate would be a different measurement with a different time-resolution tradeoff.

The four fixed bands remain summaries rather than a perceptual filter bank. On an exactly bin-centred 200 Hz tone at 32768 Hz (N = 8192, 4 Hz bins), the periodic Hann places one sixth of its energy in the lower neighbouring bin and five sixths in the centre and upper neighbour. Both adjacent bands can therefore have valid readings at a boundary. An excluded 20 kHz or Nyquist centre bin can likewise have a windowed neighbour inside a measured range. Sharper physical crossovers or auditory bands would change the method and its calibration; synthetic boundary tests make the present convention explicit.

There are two forward FFTs per complete window, with O(N log N) work, and no per-window allocation. The spectral scan stops at 20 kHz or Nyquist, whichever comes first; at 768 kHz it visits about 6827 bins instead of all 131072 positive bins. The two spectra, frame buffer and Hann coefficients use 48N bytes before the FFT plan and scratch: 768 KiB at 48 kHz, 12 MiB at 768 kHz. Summaries remain fixed-size. Decimating before the FFT could reduce the high-rate cost, but would require a specified anti-alias filter and renewed band/gate verification; this audit does not substitute unverified resampling.

Saved reports#

JSON adds an optional local_phase object with window_secs, hop_secs, analyzed_windows, broadband, and four bands. Each band contains its effective low_hz, high_hz, and an optional summary. Summaries contain correlation, minimum_correlation, minimum_start_secs, opposed_fraction and eligible_windows. A band with high_hz <= low_hz is above Nyquist and has no summary.

CSV adds phase_window_s, phase_hop_s, and phase_analyzed_windows, then five numeric fields for each range:

The suffixes are correlation, minimum, minimum_start_s, opposed_fraction, and eligible_windows. The frequency labels are nominal; use the row's sample_rate to determine Nyquist. Fractions remain in 0–1 and timestamps in seconds. Missing values are blank, measured zeros remain zero. File fingerprints remain the final two columns. Existing JSON reports load with local_phase: null and require reanalysis to obtain these values.

Automated verification#

cargo test -p flaccompagnon-core local_phase
node --test tests/analysis-cells.test.mjs tests/search.test.mjs

Unit tests live in core/tests/unit/analysis/local_phase.rs; the independent expected values come from sine phase identities, Hann's analytical three-bin powers and sums of orthogonal tone energies. They cover 0°, 60°, 90° and 180°, unequal gains, opposite treble under aligned bass, a short opposed passage with a known position, large constant DC rejection, ordinary audio after an extreme zero-weight first sample, Parseval normalization in all four bands, bin boundaries, the exclusive 20 kHz edge, energy-weighted aggregation, the RMS gate, invalid input, incomplete windows, supported rates and Nyquist limits. core/tests/local_phase.rs passes an independently generated 24-bit WAV through the complete decoder and verifies the reported timing, bands, CSV and JSON, including loading an older report. Frontend tests cover signed sorting, values, missing readings, tooltips and search.

Manual fixtures with FFmpeg or Audacity#

Run these from a folder where you want to keep the test audio. -n prevents overwriting an existing file. All three signals stay comfortably below clipping.

1. Brief opposition hidden by the whole-track average#

ffmpeg -n -f lavfi -i 'aevalsrc=0.2*sin(2*PI*1000*t)|if(between(t\,2\,3)\,-1\,1)*0.2*sin(2*PI*1000*t):s=48000:d=6' -c:a pcm_s24le phase-local.wav

Expect Local phase ≈ −1.000, with its minimum window contained within 2–3 s and a nonzero opposed-window share. The midrange band is opposed there too. The whole-track correlation remains positive, approximately +0.667. In Audacity, generate a stereo tone, split the channels and invert only one channel between 2 and 3 s for an equivalent test. Listen to a mono mix to hear the cancellation in that passage.

2. Opposed treble under aligned bass#

ffmpeg -n -f lavfi -i 'aevalsrc=0.2*sin(2*PI*100*t)+0.1*sin(2*PI*9000*t)|0.2*sin(2*PI*100*t)-0.1*sin(2*PI*9000*t):s=48000:d=5' -c:a pcm_s24le phase-bands.wav

Expect Local phase ≈ +0.600 and Band phase ≈ −1.000. On hover, bass is about +1, treble about −1 with 100% opposed windows, and the two unused middle bands should be unavailable. The analytical broadband value is (0.2² − 0.1²) / (0.2² + 0.1²) = 0.6. On summing the channels, the 9 kHz component cancels while the bass remains.

3. Quadrature is a measured zero#

ffmpeg -n -f lavfi -i 'aevalsrc=0.2*sin(2*PI*1000*t)|0.2*cos(2*PI*1000*t):s=48000:d=5' -c:a pcm_s24le phase-quadrature.wav

Expect both columns near 0.000 and an opposed-window share of 0%. The midrange band has a measurement; the other bands are unavailable. Changing the cosine to the same sine as the left channel produces +1.000, and negating that sine produces −1.000. These checks distinguish real zero correlation from missing data.