Continuous-Time Signal Transforms

This page documents continuous-time signal processing functions in Giac.jl.

Laplace Transform

The Laplace transform is a mathematical tool for analyzing continuous-time signals and systems. GIAC provides two commands for computing Laplace transforms, available via Giac.Commands:

laplace(expr, t, s) - Computes the unilateral Laplace transform
ilaplace(expr, s, t) - Computes the inverse Laplace transform

Mathematical Definition

The unilateral Laplace transform of a continuous function f(t) is defined as:

\[F(s) = \int_0^{\infty} f(t) \cdot e^{-st} \, dt\]

Basic Usage

using Giac
using Giac.Commands: laplace, ilaplace

# Declare symbolic variables
@giac_var t s a

# Laplace transform of exponential decay exp(-a*t)
F = laplace(exp(-a*t), t, s)
# Returns 1/(a+s) which is equivalent to 1/(s+a)

# Laplace transform of unit step (constant 1)
F_step = laplace(1, t, s)
# Returns 1/s

# Laplace transform of ramp function t
F_ramp = laplace(t, t, s)
# Returns 1/s^2

# Laplace transform of t^2
F_t2 = laplace(t^2, t, s)
# Returns 2/s^3

Inverse Laplace Transform

using Giac
using Giac.Commands: ilaplace

@giac_var t s a

# Inverse Laplace transform of 1/s → unit step
f_step = ilaplace(1/s, s, t)
# Returns 1

# Inverse Laplace transform of 1/(s+a) → exponential
f_exp = ilaplace(1/(s+a), s, t)
# Returns exp(-a*t)

# Inverse Laplace transform of 1/s^2 → ramp
f_ramp = ilaplace(1/s^2, s, t)
# Returns t

Round-Trip Verification

The Laplace transform and inverse Laplace transform are mathematical inverses:

using Giac
using Giac.Commands: laplace, ilaplace, simplify

@giac_var t s a

# Verify: ilaplace(laplace(exp(-a*t))) = exp(-a*t)
original = exp(-a*t)
transformed = laplace(original, t, s)
recovered = ilaplace(transformed, s, t)
simplified = simplify(recovered)
# Result: exp(-a*t)

Common Laplace Transform Pairs

Time Domain f(t)	S-Domain F(s)	Region of Convergence
`1` (unit step)	`1/s`	Re(s) > 0
`t` (ramp)	`1/s²`	Re(s) > 0
`t^n`	`n!/s^(n+1)`	Re(s) > 0
`exp(-a*t)`	`1/(s+a)`	Re(s) > -Re(a)
`sin(w*t)`	`w/(s²+w²)`	Re(s) > 0
`cos(w*t)`	`s/(s²+w²)`	Re(s) > 0
`t·exp(-a*t)`	`1/(s+a)²`	Re(s) > -Re(a)

Using invoke_cmd

You can also use invoke_cmd directly:

using Giac

@giac_var t s a

# Using invoke_cmd
invoke_cmd(:laplace, exp(-a*t), t, s)
invoke_cmd(:ilaplace, 1/(s+a), s, t)

Notes

Unilateral Transform: GIAC uses the unilateral Laplace transform (integration from t=0 to infinity), which is standard for causal systems.
Variable Declaration: Always declare t and s as symbolic variables using @giac_var before using them in transforms.
Simplification: Use simplify from Giac.Commands to reduce results to canonical form.