Inverse Laplace transform
Inverse Laplace Transform of Symbolic Expression
Compute the inverse Laplace transform of
1/s^2. By default, the inverse transform is in terms of
syms s F = 1/s^2; f = ilaplace(F)
Default Independent Variable and Transformation Variable
Compute the inverse Laplace transform of
1/(s-a)^2. By default, the independent and transformation variables are
syms a s F = 1/(s-a)^2; f = ilaplace(F)
Specify the transformation variable as
x. If you specify only one variable, that variable is the transformation variable. The independent variable is still
syms x f = ilaplace(F,x)
Specify both the independent and transformation variables as
x in the second and third arguments, respectively.
f = ilaplace(F,a,x)
Inverse Laplace Transforms Involving Dirac and Heaviside Functions
Compute the following inverse Laplace transforms that involve the Dirac and Heaviside functions.
syms s t f1 = ilaplace(1,s,t)
F = exp(-2*s)/(s^2+1); f2 = ilaplace(F,s,t)
Inverse Laplace Transform of Array Inputs
Find the inverse Laplace transform of the matrix
M. Specify the independent and transformation variables for each matrix entry by using matrices of the same size. When the arguments are nonscalars,
ilaplace acts on them element-wise.
syms a b c d w x y z M = [exp(x) 1; sin(y) 1i*z]; vars = [w x; y z]; transVars = [a b; c d]; f = ilaplace(M,vars,transVars)
ilaplace is called with both scalar and nonscalar arguments, then it expands the scalars to match the nonscalars by using scalar expansion. Nonscalar arguments must be the same size.
syms w x y z a b c d f = ilaplace(x,vars,transVars)
Inverse Laplace Transform of Symbolic Function
Compute the Inverse Laplace transform of symbolic functions. When the first argument contains symbolic functions, then the second argument must be a scalar.
syms F1(x) F2(x) a b F1(x) = exp(x); F2(x) = x; f = ilaplace([F1 F2],x,[a b])
If Inverse Laplace Transform Cannot Be Found
ilaplace cannot compute the inverse transform, then it returns an unevaluated call to
syms F(s) t F(s) = exp(s); f(t) = ilaplace(F,s,t)
Return the original expression by using
F(s) = laplace(f,t,s)
F — Input
symbolic expression | symbolic function | symbolic vector | symbolic matrix
Input, specified as a symbolic expression, function, vector, or matrix.
var — Independent variable
s (default) | symbolic variable | symbolic expression | symbolic vector | symbolic matrix
Independent variable, specified as a symbolic variable, expression,
vector, or matrix. This variable is often called the "complex frequency
variable." If you do not specify the variable, then
F does not contain
ilaplace uses the function
symvar to determine the independent
transVar — Transformation variable
t (default) |
x | symbolic variable | symbolic expression | symbolic vector | symbolic matrix
Transformation variable, specified as a symbolic variable, expression,
vector, or matrix. It is often called the "time variable" or "space
variable." By default,
t is the independent
Inverse Laplace Transform
The inverse Laplace transform f = f(t) of F = F(s) is:
Here, c is a suitable complex number.
If any argument is an array, then
ilaplaceacts element-wise on all elements of the array.
If the first argument contains a symbolic function, then the second argument must be a scalar.
To compute the direct Laplace transform, use
For a signal f(t), computing the Laplace transform (
laplace) and then the inverse Laplace transform (
ilaplace) of the result may not return the original signal for t < 0. This is because the definition of
laplaceuses the unilateral transform. This definition assumes that the signal f(t) is only defined for all real numbers t ≥ 0. Therefore, the inverse result does not make sense for t < 0 and may not match the original signal for negative t. One way to correct the problem is to multiply the result of
ilaplaceby a Heaviside step function. For example, both of these code blocks:
syms t; laplace(sin(t))
syms t; laplace(sin(t)*heaviside(t))
1/(s^2 + 1). However, the inverse Laplace transform
syms s; ilaplace(1/(s^2 + 1))
Introduced before R2006a