Transfer Function Deep Dive: System Alignment for Sound Engineers

What the Transfer Function Actually Measures

The transfer function describes the input-output relationship of a system. In audio measurement, the "system" includes the amplifier, loudspeaker, air path, room reflections, and microphone. The measurement answers: "For every frequency I put in, what comes out?"

Unlike single-channel RTA (which shows only what the microphone hears), the transfer function compares input and output to reveal the system's true behavior — separating system response from excitation signal characteristics.

The H1 Estimator

SonaVyx uses the H1 estimator, defined as:

H1(f) = Gxy(f) / Gxx(f)

Where:

• Gxy(f) = cross-spectral density between input x and output y
• Gxx(f) = auto-spectral density of input x

H1 minimizes the effect of noise at the output (microphone noise, room noise). This makes it the correct estimator for acoustic measurement where the output is contaminated by ambient noise. The alternative H2 estimator (Gyy/Gyx) minimizes input noise — useful when the source signal itself is noisy, which is rare in controlled measurement.

Welch's Method for Spectral Estimation

Gxy and Gxx are computed using Welch's method: the input and output signals are divided into overlapping segments (50% overlap in SonaVyx), each segment is windowed (Blackman-Harris, 4-term), FFTs are computed, and the spectral products are averaged across all segments.

More averages = more stable estimate, but slower response. For live alignment, 4-8 averages provide a good balance. For documentation, 16-32 averages give a smoother, more reproducible curve.

Magnitude Response

The magnitude of H(f) in dB shows the frequency response: |H(f)|_dB = 20 log₁₀(|H(f)|). A perfectly flat system would show 0 dB at all frequencies. In practice:

• ±3 dB variation across 100 Hz - 10 kHz is excellent for a live sound system
• ±6 dB is typical for an un-tuned system in a room
• Narrow dips (nulls) exceeding -10 dB indicate comb filtering from reflections
• Broad humps indicate boundary coupling or room mode reinforcement

The low-mid buildup between 200-500 Hz visible in the magnitude response is the most common problem in live sound venues.

Coherence: The Confidence Metric

Coherence γ²(f) is the most underused and most valuable part of the transfer function measurement:

γ²(f) = |Gxy(f)|² / (Gxx(f) · Gyy(f))

It ranges from 0 to 1:

• γ² > 0.9: Excellent — the system is linear and the measurement dominates over noise
• γ² = 0.6 - 0.9: Moderate — some noise or reflections present but measurement is still useful
• γ² = 0.3 - 0.6: Poor — reflections, noise, or non-linearity significantly corrupt the measurement at this frequency
• γ² < 0.3: Unreliable — do not make EQ decisions based on magnitude data at these frequencies

What Causes Low Coherence

1. Room reflections: Energy arriving from multiple paths with different delays creates interference that varies with each measurement block. This is the most common cause.
2. Background noise: Noise at the microphone is uncorrelated with the measurement signal, reducing coherence.
3. Non-linearity: Distortion, clipping, or limiter activity creates output harmonics not present in the input.
4. Time variance: Movement (wind, people, HVAC) during the measurement changes the system between blocks.

The Critical Rule: EQ Only Where Coherence Is High

If coherence is below 0.5 at a frequency, the magnitude reading at that frequency is unreliable. Applying EQ based on unreliable data makes things worse. The practical threshold: only apply corrective EQ where γ² > 0.6. Where coherence is low, fix the cause (reduce noise, manage reflections, improve signal level) rather than EQing blindly.

Phase: Alignment and Crossover

The phase of H(f) shows timing relationships between frequency components. For system alignment, phase is critical at crossover frequencies where two loudspeaker components (e.g., main + sub, or HF + LF driver) overlap.

SonaVyx displays unwrapped phase (removing ±180° discontinuities) and computes group delay τ_g = -dφ/dω, which represents the time delay experienced by each frequency component.

Phase at Crossover

At the crossover frequency between two components, the phase responses should match (0° relative difference) for maximum summation. A 180° offset causes complete cancellation — a polarity inversion. Intermediate offsets cause partial cancellation:

• 0° phase difference → +6 dB summation
• 90° phase difference → +3 dB summation
• 120° phase difference → 0 dB (no benefit)
• 180° phase difference → complete cancellation (null)

The subwoofer alignment guide covers this in detail for the critical sub/main crossover region.

The Delay Finder

The impulse response (IFFT of the transfer function) reveals the propagation delay as the position of the first significant peak. SonaVyx's delay finder locates this peak automatically and reports the delay in milliseconds and equivalent distance (delay × 343 m/s at 20°C).

For delay tower alignment, this is the definitive measurement: set the electronic delay to match the acoustic propagation delay so that all speakers arrive at the listener simultaneously.

Practical Alignment Workflow

The Tune PA workflow in SonaVyx implements this four-step process:

1. Verify Polarity: Check each speaker component individually. Cross-correlation peak should be positive. A negative peak indicates inverted polarity — swap + and - at the amplifier.
2. Set Delay: Measure the IR for each speaker. The delay finder shows the propagation time. Set electronic delay on the later-arriving speaker to match the earlier one at the listener position.
3. Match Levels: With broadband pink noise, adjust amplifier gains so each component contributes equal level at the listening position within ±1 dB.
4. Apply EQ: With all components running together, examine the combined transfer function. Apply broad corrective EQ only where coherence exceeds 0.6. Use the AI diagnostic for automated EQ recommendations.

After alignment, use before/after comparison to verify improvement. A well-aligned system typically shows 3-6 dB improvement in coherence-weighted flatness.

SonaVyx Implementation

The transfer function engine is implemented in Rust WASM with 54 dedicated unit tests covering H1 estimation, cross-spectral density, auto-spectral density, coherence computation, phase unwrapping, group delay, Welch's method with 50% and 75% overlap, Blackman-Harris windowing, delay finding, and octave smoothing from 1/1 to 1/24. The problem detector uses the same TF data to identify feedback, comb filtering, and polarity issues automatically.