Laplace Transform

Poke a system (impulse) and the Laplace Transform maps its time response into the s-plane, where pole positions decide ring vs fade.

What it is

A maths tool that turns a system's full time behaviour - both the ring and the fade - into one tidy formula.

Key facts

• Definition: F(s) = integral from 0 to infinity of f(t) * e^(-s*t) dt, where f(t)=signal in time, s=complex frequency, e=2.718, t=time in seconds.
• s = sigma + j*omega: sigma (real part) = decay/growth rate in 1/seconds, omega (imaginary part) = angular frequency in radians/second, j = sqrt(-1).
• omega = 2 * pi * f, so a frequency f in hertz maps to omega in radians/sec; pi = 3.14159.
• POLES = values of s where the system blows up (denominator = 0); ZEROS = where output = 0 (numerator = 0). Pole positions decide the whole response.
• Pole in LEFT half-plane (sigma < 0) = STABLE, response fades out. Pole in RIGHT half (sigma > 0) = UNSTABLE, response grows/oscillates forever.
• Pole exactly ON the imaginary axis (sigma = 0) = pure undamped ringing that never dies (a perfect oscillator).
• Time constant tau = 1/|sigma| seconds; after 1 tau a 1st-order response has decayed to 37% (1/e), after 5 tau to under 1% (settled).
• -3 dB = half power point = the cutoff frequency of a filter; voltage is at 70.7% (1/sqrt(2)) there.
• +6 dB = double the voltage/pressure; +10 dB = perceived 'twice as loud'; +3 dB = double the power.
• Filter slopes from pole count: 1 pole = 6 dB/octave, 2 poles = 12 dB/oct, 4 poles = 24 dB/oct (Linkwitz-Riley LR4, the live default).

How it works

1. Take the system's time behaviour f(t) - e.g. how a speaker cone moves after a pulse.
2. Multiply it by e^(-s*t) and integrate over all time; this 'weighs' it against every possible decay+frequency combo.
3. The result H(s) is the transfer function: output divided by input in the s-domain.
4. Find the POLES (denominator zeros) - their positions on the s-plane fully predict ring, fade, and stability.
5. Left-half poles = it settles; right-half = it runs away; distance from the axis sets how fast and how much it rings.
6. Engineers PLACE poles to design a filter, crossover, or EQ with the exact tone and damping they want.

Real examples

• A speaker box modelled as a 2nd-order high-pass: Laplace maths predicts how tightly the cone stops after a kick drum (the Q / damping).
• A 12 dB/octave Linkwitz-Riley crossover at 80 Hz is literally designed from poles placed on the s-plane.
• An analogue parametric EQ bell curve is a transfer function H(s) born from the Laplace Transform.
• A subwoofer high-pass (rumble filter) at 30 Hz: pole location sets how fast it rolls off and whether it overshoots.
• A compressor's attack/release smoothing is a 1st-order Laplace system with a single time constant tau.

How it helps in live sound

• Pick crossover TYPE on your DSP knowing the maths: Linkwitz-Riley 24 dB/oct (4th order) is the live-sound default because both halves sum flat at crossover (-6 dB each).
• Butterworth filters ring less but the two bands sum to +3 dB at the crossover point, causing a bump; Bessel = best phase but gentle slope.
• Higher filter ORDER = steeper slope but more phase shift and more potential ringing near cutoff; do not stack pointless steep filters.
• Set sub high-pass (rumble/HPF) ~30-40 Hz to stop wasting amp power on inaudible infrasonic energy and cone over-excursion.
• A high-Q (>1) EQ boost is an underdamped resonant peak: it RINGS in time, smearing transients; use lower Q (0.7-1.0) for tone, narrow only to kill feedback.
• 'Phase' problems at a crossover are pole/transfer-function behaviour, not a bug; flip polarity or use the matching LR alignment your processor offers.