Hilbert Audio Processor — User Guide

Analytic-signal processing for quadrature transformation, single-sideband frequency translation, envelope extraction, and constant phase rotation, with independent odd/even channel routing.

Author: Shai Cohen Affiliation: Department of Music, Bar-Ilan University, Israel Version: 1.5 (2026) License: MIT License Repo: Praat AudioTools

Contents:

What this does Analytic signal Operations Channel routing Presets Controls Antialiasing Output behavior Visualization

What this does

Hilbert Audio Processor builds a quadrature representation of the selected Sound and uses it for four related kinds of spectral/time-domain processing: a Hilbert transform, constant-Hz frequency translation, analytic-envelope extraction, and constant phase rotation. Each real channel is processed independently, then reconstructed into the original channel layout.

The practical idea: the original waveform supplies the real component x(t), while a Hilbert-transformed version supplies a quadrature component h(t). Together they provide a two-dimensional analytic representation from which the script can derive a shifted signal, an amplitude envelope, or a rotated phase projection.

The processor is offline. It preserves the source sample rate, sample count, start time, duration, and channel count. Mono uses the Left controls. In multichannel Sounds, odd-numbered channels use the Left controls and even-numbered channels use the Right controls.

Analytic signal and Hilbert transform

For each mono channel, the script computes an exact-length FFT with To Spectrum: "no". The quadrature spectrum is formed by multiplying positive-frequency bins by −j, then converting the spectrum back to a Sound.

Analytic pair: real part = x(t) quadrature part = h(t) = Hilbert{x(t)} Analytic signal: a(t) = x(t) + j h(t) Envelope: A(t) = sqrt(x(t)^2 + h(t)^2)

DC is removed from the Hilbert component. For an even sample count, the Nyquist bin is also removed because it has no unique quadrature partner. For an odd sample count, the final FFT bin is a normal positive-frequency bin and is retained.

Exact-length FFT: v1.5 does not zero-pad the analysis transform. This keeps the reconstructed channel at the original sample count and avoids adding a separate padded time region around the source.

Five operations

1. Hilbert transform (−90 deg)

Outputs the quadrature component itself. Positive-frequency components are rotated by −90°. This is the operation formerly labelled “No shift”; v1.5 names the operation according to what it actually returns.

2. Shift up

y(t) = x(t) cos(2π Δf t) - h(t) sin(2π Δf t)

Translates the spectrum upward by a constant frequency offset Δf in Hz.

3. Shift down

y(t) = x(t) cos(2π Δf t) + h(t) sin(2π Δf t)

Translates the spectrum downward by a constant frequency offset.

Frequency translation is not conventional pitch shifting. Every spectral component moves by the same number of hertz, so harmonic ratios generally change. A 100 Hz partial and a 200 Hz partial shifted upward by 50 Hz become 150 Hz and 250 Hz, not a transposed 1:2 pair.

4. Envelope only

A(t) = sqrt(x(t)^2 + h(t)^2)

Outputs the instantaneous magnitude of the analytic pair. An optional one-pole low-pass can smooth this envelope.

5. Phase rotate

y(t) = x(t) cos(θ) - h(t) sin(θ)

Applies one constant phase rotation to the positive-frequency spectrum. In v1.5 this is computed directly in the frequency domain. DC, and Nyquist when present, are scaled by cos(θ) because their Hilbert component is zero.

Dry/wet and phase rotation: mixing a fixed phase-rotated copy with the dry signal changes overall phase and gain, but it does not create a frequency-dependent phaser or flanger response by itself. The phase angle is constant across frequency.

Channel routing

Input	Controls used	Output
Mono	Left / odd controls	Mono
Stereo	Channel 1 = Left, Channel 2 = Right	Stereo
More than 2 channels	Odd channels = Left, even channels = Right	Same number of channels

Every channel is extracted, processed independently, and recombined in its original order. The Right controls are ignored for a mono source.

Presets

Preset	Left / odd	Right / even	Parameters set
Custom	User choice	User choice	Uses the values entered in the form.
L: Shift Up (+50) / R: Shift Down (-50)	Shift up 50 Hz	Shift down 50 Hz	Opposing constant-Hz translations.
L: Envelope Only / R: Hilbert Phase	Envelope only	Hilbert transform (−90 deg)	Contrasting analytic-signal outputs across channel groups.
Stereo Shift Up (+200Hz)	Shift up 200 Hz	Shift up 200 Hz	Same upward translation on both routing groups.
Stereo Hilbert Phase	Hilbert transform	Hilbert transform	Quadrature output on all channels.

Preset values override the relevant operation and shift controls. Global controls such as Dry/Wet, Antialiasing, Safety Peak, Envelope Low-pass, playback, and visualization remain available.

Controls

Control	Default	Behavior
Operation L	Shift up	Operation for mono and all odd-numbered channels.
Shift amount L	500 Hz	Clamped to 0…Nyquist.
Phase angle L	90°	Used only by Phase rotate.
Operation R	Shift down	Operation for all even-numbered channels.
Shift amount R	500 Hz	Clamped to 0…Nyquist.
Phase angle R	90°	Used only by Phase rotate.
Antialiasing	On	Pre-bandlimits the source before frequency translation.
Dry/Wet	100%	Linear blend after all channels are reconstructed; clamped to 0…100%.
Safety peak	0.99	Attenuation-only output ceiling. 0 disables it.
Envelope low-pass	0 Hz	Optional one-pole smoothing of Envelope-only output; 0 disables smoothing.
Play output	On	Plays the final Sound after processing.
Draw visualization	On	Draws waveform, spectrum, and summary panels.

Exact no-op paths

Dry/Wet = 0% copies the selected Sound sample-for-sample and skips processing and safety attenuation.
Shift amount = 0 Hz returns an unchanged channel for Shift up/down.
Phase angle = 0° returns an unchanged channel for Phase rotate.

Antialiasing for frequency translation

Constant-Hz translation can push energy beyond a valid spectral boundary. When Antialiasing is enabled, v1.5 pre-filters the exact-length spectrum before the quadrature modulation.

Operation	Pre-bandlimit	Purpose
Shift up by Δf	Keep frequencies up to approximately `Nyquist − Δf`	Prevents translated energy from crossing Nyquist.
Shift down by Δf	Keep frequencies from approximately `Δf` upward	Prevents translated positive-frequency energy from crossing below 0 Hz.

The script uses Praat's compiled Hann-band filter with a transition width of approximately one millionth of an FFT bin. At sampled FFT bins this behaves effectively like the intended discrete 0/1 mask while avoiding the much slower interpreted spectral Formula used in earlier versions.

Output behavior

Sample rate: preserved.
Sample count: preserved.
Duration: preserved.
Start time: preserved.
Channel count: preserved.
Processing phase: internally referenced to local Sound time, so a non-zero xmin does not change the modulation phase.
Normalization: none. Safety Peak only reduces gain when the result exceeds the ceiling.

Dry/Wet

output = Wet × processed + Dry × original Wet = Dry_wet_percent / 100 Dry = 1 - Wet

The blend is linear and is applied only after all processed channels have been reconstructed.

Envelope smoothing

When Envelope low-pass is above 0 Hz, the analytic magnitude is smoothed by a first-order one-pole low-pass:

alpha = exp(-2π fc / fs) y[n] = (1-alpha) x[n] + alpha y[n-1]

The parameter therefore represents an actual cutoff frequency in hertz, not an arbitrary smoothing amount.

Visualization

When enabled, the script draws the standardized AudioTools figure:

Input waveform — original Sound.
Output waveform — final processed Sound after dry/wet and safety.
Input spectrum — display spectrum up to min(8 kHz, Nyquist).
Output spectrum — same display range for direct comparison.
Summary — odd/even operations, shifts, phase angles, Wet percentage, antialias state, envelope low-pass, safety ceiling, duration, sample rate, and channel count.

For multichannel Sounds, the two spectrum panels use a mono display conversion only for visualization. The actual audio processing remains independent per channel and preserves the original multichannel structure.