Hilbert Transform (Time-Reversed Envelope) — User Guide

Extract amplitude envelope using Hilbert transform, reverse it in time, and apply it backwards to create retrograde envelope effects with preserved pitch.

Author: Shai Cohen Affiliation: Department of Music, Bar-Ilan University, Israel Version: 0.1 (2025) License: MIT License Repo: https://github.com/ShaiCohen-ops/Praat-plugin_AudioTools

Contents:

What this does Quick start Hilbert Transform Theory Parameters Audio Examples Applications

What this does

This script performs time-reversed envelope processing using the Hilbert transform — a sophisticated signal processing technique that extracts the amplitude envelope of a sound (how loud it is over time) and applies it backwards. The result: sounds that seem to "grow" into their attacks rather than decay from them. Imagine drums that swell into the hit, voices that inhale into words, or music that builds backwards into each note. The pitch and harmonic content remain unchanged — only the amplitude contour is reversed. This creates dreamlike, reversed-yet-forward, impossibly organic temporal effects.

Key Features:

Hilbert Transform — Mathematically precise envelope extraction via analytic signal
Time Reversal — Envelope reversed in time, not the audio itself
Pitch Preservation — Frequencies unchanged, only amplitude affected
Envelope Shaping — Optional sharpening/smoothing with exponent control
High-Pass Filtering — Reduces low-frequency envelope dominance
Computationally Intensive — Long processing time on extended files

What is the Hilbert Transform? The Hilbert transform creates an "analytic signal" — a complex representation of audio that separates amplitude (envelope) from phase (frequency content). Think of it as extracting the "shape" of sound over time independent of its pitch. Normal envelope followers (peak detection, RMS) are crude approximations. Hilbert transform is mathematically exact: it creates a 90-degree phase-shifted version of the signal, combines it with the original to form a complex number at each sample, then calculates the magnitude. This magnitude is the instantaneous amplitude envelope — smooth, continuous, capturing subtle amplitude variations impossible to extract otherwise. Used in: modulation analysis, speech processing, seismic analysis, and experimental music (Trevor Wishart, Curtis Roads).

Technical Implementation: (1) Convert Sound to Spectrum using FFT or DFT, (2) Create Hilbert transform via 90-degree phase shift: swap real/imaginary components and negate, (3) Convert Hilbert spectrum back to time domain → hilbert_sound, (4) Calculate envelope: sqrt(original² + hilbert²) — this is the analytic signal magnitude, (5) Scale and optionally sharpen envelope with exponent, (6) High-pass filter envelope to reduce low-frequency buildup, (7) Reverse original sound in time, (8) Multiply reversed sound by forward-time envelope (applies envelope backwards), (9) Reverse result back to forward time → sound with time-reversed envelope, (10) Peak normalize. Key insight: Because we reverse the sound, apply the envelope, then reverse again, the final result has pitch/timbre moving forward but amplitude envelope moving backward.

Quick start

In Praat, select exactly one Sound object.
Run script… → Hilbert Transform.praat.
Leave defaults (fast_fourier: no, sharpening off, highpass 50Hz) for standard envelope reversal.
Click OK — script extracts envelope, reverses it, applies backwards, plays result.
Experiment with envelope sharpening (exponent 0.5-0.8) for more dramatic swells.

Quick tip: Start with default settings on short files (under 30 seconds). Try on drums for reverse-swell effect, on vocals for inhale-into-words quality. Processing creates dreamlike, "building backwards" effect. Enable envelope sharpening (apply_envelope_sharpening: yes, exponent 0.7) for more pronounced swells. Script shows "WARNING" about runtime — Hilbert transform requires two FFT operations, making it slower than simple reverb effects.

Important: LONG PROCESSING TIME — Hilbert transform requires multiple FFT calculations. Files over 1 minute can take 30+ seconds. Files over 5 minutes can take several minutes. This is normal and unavoidable (mathematical necessity of analytic signal computation). Progress not shown during FFT — Praat will appear frozen but is working. For very long files, consider processing shorter segments. Effect cannot be previewed — must complete entire process. Save work before processing long files.

Hilbert Transform Theory

🌊 Analytic Signal Fundamentals

Real signal: Standard audio — real numbers representing amplitude

Analytic signal: Complex representation — amplitude (magnitude) + phase (angle)

Hilbert transform: Creates imaginary component that pairs with real signal

Key insight: Magnitude of analytic signal = instantaneous envelope, independent of phase/frequency

The Hilbert Transform Operation

Mathematical Foundation

Standard audio signal:

s(t) = A(t) · cos(φ(t)) where: A(t) = amplitude envelope (what we want to extract) φ(t) = instantaneous phase (frequency information)

Hilbert transform creates:

H{s(t)} = A(t) · sin(φ(t)) (90-degree phase shift — cos becomes sin)

Analytic signal combines both:

z(t) = s(t) + i·H{s(t)} = A(t) · [cos(φ) + i·sin(φ)] = A(t) · e^(iφ) (Euler's formula — complex exponential form)

Envelope extraction:

envelope(t) = |z(t)| = √[s(t)² + H{s(t)}²] = A(t) (magnitude of complex number = envelope amplitude)

Why Hilbert Transform?

Traditional envelope methods (crude):

Peak detection: Only captures maxima, misses subtle variations
RMS (Root Mean Square): Averages over windows, loses temporal precision
Absolute value: Doubles frequency, creates artifacts
Low-pass filtering rectified signal: Phase distortion, imprecise

Hilbert transform advantages:

Mathematically exact: True instantaneous envelope, no approximation
Smooth: Continuous envelope, no jagged edges
Phase-independent: Extracts amplitude regardless of waveform phase
Frequency-agnostic: Works equally well on bass, mids, treble
Preserves transients: Captures fast amplitude changes accurately

Frequency-Domain Implementation

The Hilbert transform is most efficiently computed in frequency domain:

Step 1: FFT — Convert time-domain sound to frequency-domain spectrum

Original spectrum: Real part: cosine components Imaginary part: sine components

Step 2: 90-degree phase shift — Swap and negate components

Hilbert spectrum formula: if row=1 then Spectrum[2,col] else -Spectrum[1,col] Translation: New real part = original imaginary part New imaginary part = -(original real part) This rotates each frequency by 90 degrees in complex plane

Step 3: Inverse FFT — Convert Hilbert spectrum back to time domain

Result: hilbert_sound Time-domain representation of 90-degree phase-shifted signal Has same frequencies as original, different phase relationship

Step 4: Envelope calculation — Compute magnitude of analytic signal

envelope = sqrt(original² + hilbert²) For each sample: Take original sample value Take hilbert sample value Square both, add, take square root Result: instantaneous amplitude at that moment

Time Reversal Process

Once envelope extracted, script applies it backwards:

Visual Representation

Original sound with natural envelope:

Time → Attack Sustain Decay /‾‾‾‾‾‾‾‾‾‾‾\___ Drum hit: sharp attack, quick decay

Extracted envelope (via Hilbert):

Envelope: /‾‾‾‾‾‾‾‾‾‾‾\___ Smooth curve representing amplitude over time

Reversed envelope:

Reversed: ___/‾‾‾‾‾‾‾‾‾‾‾\ Now builds from silence to peak, then quick cutoff

Applied to original sound (not reversed):

Result: Pitch/timbre same direction (forward) Amplitude contour opposite direction (backward) Sound "swells" into its attack, then cuts off Dreamlike, impossible-in-nature quality

Envelope Sharpening

The script offers optional envelope shaping via exponentiation:

Formula: envelope_shaped = envelope^exponent

Exponent	Effect	Use Case
1.0	No change (original envelope)	Natural envelope contour
0.8	Slight sharpening (default when enabled)	Moderate enhancement of swells
0.5	Strong sharpening (square root)	Dramatic swells, gates-like effect
0.3	Extreme sharpening	Harsh gating, abrupt transitions
2.0	Smoothing (squaring)	Gentler swells, reduced dynamics
3.0	Heavy smoothing (cubing)	Very gradual swells, compressed feel

Mathematical Effect

Sharpening (exponent < 1):

Values near 0 become even smaller → deeper valleys
Values near 1 barely affected → peaks remain prominent
Result: Increased contrast between loud and soft
Creates gate-like, "all or nothing" amplitude contour

Smoothing (exponent > 1):

Values near 0 become smaller → softens attacks
Values near 1 barely affected → peaks remain
Result: Decreased contrast, compressed dynamics
Creates gentle, flowing amplitude changes

Example with exponent 0.5 (square root):

Original envelope value: 0.25 → sqrt(0.25) = 0.5 (doubled) Original envelope value: 0.5 → sqrt(0.5) = 0.707 (increased) Original envelope value: 0.75 → sqrt(0.75) = 0.866 (slightly increased) Original envelope value: 1.0 → sqrt(1.0) = 1.0 (unchanged) Effect: Quiet parts become louder relative to peaks Creates more uniform amplitude distribution Swells feel more sudden, dramatic

High-Pass Filtering

The envelope is high-pass filtered to reduce low-frequency dominance:

Why necessary:

Hilbert-derived envelopes often have DC offset and low-frequency drift
Bass-heavy material creates envelope with strong sub-bass component
Applying unfiltered envelope can make result muddy, boomy
Low-frequency envelope modulation creates unnatural rumble

High-pass filter (default 50 Hz):

Removes sub-bass and DC from envelope itself
Envelope still affects all frequencies of original sound
Prevents low-frequency envelope components from dominating
Creates cleaner, more natural-sounding reversed envelope

Smoothing parameter (default 10 Hz):

Controls filter rolloff steepness
Lower smoothing = sharper cutoff (more aggressive filtering)
Higher smoothing = gentler rolloff (more transparent)

Adjusting Filter Parameters

Material	Cutoff (Hz)	Smoothing (Hz)	Reason
Bass-heavy (EDM, hip-hop)	80-100	15-20	More aggressive filtering to prevent boom
Balanced mix	50	10	Default — works for most material
Sparse, acoustic	30-40	5-10	Gentler filtering preserves envelope subtlety
High-frequency (cymbals)	50-70	10	Standard filtering adequate

Processing Steps Breakdown

Complete signal flow:

Original Sound — Input audio
→ Spectrum — FFT to frequency domain
→ Hilbert Spectrum — 90-degree phase shift
→ Hilbert Sound — Inverse FFT to time domain
→ Envelope — sqrt(original² + hilbert²)
→ Scaled Envelope — Peak normalization
→ Shaped Envelope — Optional exponentiation
→ Filtered Envelope — High-pass filter
→ Reversed Original — Time-reverse input sound
→ Reversed with Envelope — Multiply reversed sound by forward envelope
→ Final Sound — Reverse back to forward time
→ Normalized Output — Final peak scaling

Historical Context: The Hilbert transform was developed by David Hilbert in early 1900s for mathematical analysis. Applied to signal processing in 1940s-50s (radar, communications). Gabor (1946) used analytic signal concept for time-frequency analysis. In music: Trevor Wishart's "Audible Design" (1994) explored envelope manipulations. Curtis Roads' "Microsound" (2001) detailed granular techniques using Hilbert envelopes. Modern applications: speech processing (extracting formants), medical imaging (MRI), seismic analysis, and experimental electronic music. This script makes mathematically sophisticated envelope extraction accessible in Praat for creative audio work — what required custom C/Fortran code in 1980s now runs as simple script.

Parameters

Core Settings

Parameter	Type	Default	Description
fast_fourier	boolean	no	Use FFT (fast) vs DFT (precise but slow)
apply_envelope_sharpening	boolean	no	Enable envelope shaping via exponentiation
sharpening_exponent	positive	0.8	Exponent for envelope shaping (<1 sharper, >1 smoother)
highpass_cutoff	positive	50	High-pass filter cutoff frequency (Hz)
highpass_smoothing	positive	10	Filter smoothing/rolloff (Hz)
scale_peak	positive	0.99	Peak normalization level
play_after_processing	boolean	yes	Auto-play result when complete
show_info_report	boolean	yes	Display processing info in info window
keep_intermediate_objects	boolean	no	Keep intermediate objects for inspection

Parameter Details

fast_fourier

Options: no (DFT, default) or yes (FFT)

Why default is "no" (DFT):

Hilbert transform requires precise phase relationships
DFT provides mathematically exact results
FFT approximations can introduce subtle envelope artifacts
For envelope extraction, precision more important than speed

When to use FFT (yes):

Very long files (>2 minutes) where DFT prohibitively slow
When speed critical and minor artifacts acceptable
Batch processing many files

Performance comparison @ 1 minute 44.1kHz:

DFT: ~45-60 seconds processing time
FFT: ~15-20 seconds processing time
Quality difference: subtle (most users won't notice)

apply_envelope_sharpening

Options: no (off, default) or yes (enabled)

Disable (no): Natural envelope contour preserved

Enable (yes): Envelope shaped by exponentiation

When to enable:

Want more dramatic, sudden swells
Original envelope too gradual
Creating gated or rhythmic effects
Enhancing percussive envelope definition

When to disable:

Want natural, subtle envelope reversal
Working with classical or acoustic material
Envelope already has good definition

sharpening_exponent

Range: 0.1-10.0 (practical: 0.3-3.0)

Default: 0.8 (moderate sharpening)

Common values:

0.3-0.5: Extreme sharpening — gate-like, harsh transitions
0.6-0.8: Moderate sharpening — enhanced dynamics, clear swells
0.9-1.1: Minimal effect — nearly original envelope
1.5-2.0: Smoothing — compressed, gentle swells
2.5-3.0: Heavy smoothing — very gradual changes

Effect inactive if apply_envelope_sharpening disabled.

highpass_cutoff

Range: 10-200 Hz (practical: 30-100 Hz)

Default: 50 Hz

Purpose: Remove low-frequency components from envelope itself

Adjust lower (30-40 Hz):

Preserve more envelope low-frequency content
Gentler filtering, more transparent
Risk: possible muddiness or boom

Adjust higher (70-100 Hz):

More aggressive low-frequency removal
Cleaner, tighter envelope application
Risk: might lose subtle envelope qualities

highpass_smoothing

Range: 1-50 Hz (practical: 5-20 Hz)

Default: 10 Hz

Lower values (5 Hz): Sharp filter cutoff, more aggressive

Higher values (20 Hz): Gentle rolloff, more transparent

Recommendation: Keep default 10 Hz unless specific need

scale_peak

Range: 0.1-1.0 (practical: 0.9-0.99)

Default: 0.99

Purpose: Final output normalization

0.99: Maximum loudness with tiny headroom

0.95: Moderate safety margin

0.9: Conservative, guaranteed no clipping

play_after_processing

Options: yes (default) or no

Enable: Immediate audition after long processing wait

Disable: No playback — useful for batch work

show_info_report

Options: yes (default) or no

Enable: Display processing summary in Praat info window

Disable: Silent processing

Info report contains: Original filename, output name, duration, confirmation of processing complete

keep_intermediate_objects

Options: