ilaplace

Inverse Laplace transform

Syntax

f = ilaplace(F)

f = ilaplace(F,transVar)

f = ilaplace(F,var,transVar)

Description

f = ilaplace(F) returns the Inverse Laplace Transform of F. By default, the independent variable is s and the transformation variable is t. If F does not contain s, ilaplace uses the function symvar.

f = ilaplace(F,transVar) uses the transformation variable transVar instead of t.

f = ilaplace(F,var,transVar) uses the independent variable var and the transformation variable transVar instead of s and t, respectively.

Examples

Inverse Laplace Transform of Symbolic Expression

Compute the inverse Laplace transform of 1/s^2. By default, the inverse transform is in terms of t.

syms s
F = 1/s^2;
f = ilaplace(F)

f = $$t$$

Default Independent Variable and Transformation Variable

Compute the inverse Laplace transform of 1/(s-a)^2. By default, the independent and transformation variables are s and t, respectively.

syms a s
F = 1/(s-a)^2;
f = ilaplace(F)

f = $$t\hspace{0.17em}{\mathrm{e}}^{a\hspace{0.17em}t}$$

Specify the transformation variable as x. If you specify only one variable, that variable is the transformation variable. The independent variable is still s.

syms x
f = ilaplace(F,x)

f = $$x\hspace{0.17em}{\mathrm{e}}^{a\hspace{0.17em}x}$$

Specify both the independent and transformation variables as a and x in the second and third arguments, respectively.

f = ilaplace(F,a,x)

f = $$x\hspace{0.17em}{\mathrm{e}}^{s\hspace{0.17em}x}$$

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)

f1 = $$\delta \text{<a href="matlab:doc symbolic/dirac">dirac</a>}\left(t\right)$$

F = exp(-2*s)/(s^2+1);
f2 = ilaplace(F,s,t)

f2 = $$\mathrm{heaviside}\left(t-2\right)\hspace{0.17em}\sin \left(t-2\right)$$

Inverse Laplace Transform as Convolution

Create two functions $\mathit{f}\left(\mathit{t}\right)=\mathrm{heaviside}\left(\mathit{t}\right)$ and $\mathit{g}\left(\mathit{t}\right)=\exp \left(-\mathit{t}\right)$. Find the Laplace transforms of the two functions by using laplace. Because the Laplace transform is defined as a unilateral or one-sided transform, it only applies to the signals in the region $\mathit{t}\ge 0$.

syms t positive
f(t) = heaviside(t);
g(t) = exp(-t);
F = laplace(f);
G = laplace(g);

Find the inverse Laplace transform of the product of the Laplace transforms of the two functions.

h = ilaplace(F*G)

h = $$1-{\mathrm{e}}^{-t}$$

According to the convolution theorem for causal signals, the inverse Laplace transform of this product is equal to the convolution of the two functions, which is the integral ${\int }_0^{\mathit{t}}\mathit{f}\left(\tau \right)\mathit{g}\left(\mathit{t}-\tau \right)\mathit{d}\tau$ with $\mathit{t}\ge \mathrm{0}$. Find this integral.

syms tau
conv_fg = int(f(tau)*g(t-tau),tau,0,t)

conv_fg = $$1-{\mathrm{e}}^{-t}$$

Show that the inverse Laplace transform of the product of the Laplace transforms is equal to the convolution, where h is equal to conv_fg.

isAlways(h == conv_fg)

ans = logical
   1

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)

f = 
$$\left(\begin{array}{cc}{\mathrm{e}}^x\hspace{0.17em}\delta \text{<a href="matlab:doc symbolic/dirac">dirac</a>}\left(a\right)& \delta \text{<a href="matlab:doc symbolic/dirac">dirac</a>}\left(b\right)\\ \mathrm{ilaplace}\left(\sin \left(y\right),y,c\right)& {\delta \text{<a href="matlab:doc symbolic/dirac">dirac</a>}}^{\prime }\left(d\right)\hspace{0.17em}\mathrm{i}\end{array}\right)$$

If 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)

f = 
$$\left(\begin{array}{cc}x\hspace{0.17em}\delta \text{<a href="matlab:doc symbolic/dirac">dirac</a>}\left(a\right)& {\delta \text{<a href="matlab:doc symbolic/dirac">dirac</a>}}^{\prime }\left(b\right)\\ x\hspace{0.17em}\delta \text{<a href="matlab:doc symbolic/dirac">dirac</a>}\left(c\right)& x\hspace{0.17em}\delta \text{<a href="matlab:doc symbolic/dirac">dirac</a>}\left(d\right)\end{array}\right)$$

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])

f = $$\left(\begin{array}{cc}\mathrm{ilaplace}\left({\mathrm{e}}^x,x,a\right)& {\delta \text{<a href="matlab:doc symbolic/dirac">dirac</a>}}^{\prime }\left(b\right)\end{array}\right)$$

If Inverse Laplace Transform Cannot Be Found

If ilaplace cannot compute the inverse transform, then it returns an unevaluated call to ilaplace.

syms F(s) t
F(s) = exp(s);
f(t) = ilaplace(F,s,t)

f(t) = $$\mathrm{ilaplace}\left({\mathrm{e}}^s,s,t\right)$$

Return the original expression by using laplace.

F(s) = laplace(f,t,s)

F(s) = $${\mathrm{e}}^s$$

Input Arguments

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 ilaplace uses s. If F does not contain s, then ilaplace uses the function symvar to determine the independent variable.

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, ilaplace uses t. If t is the independent variable of F, then ilaplace uses x.

More About

Inverse Laplace Transform

The inverse Laplace transform of F(s) is the signal f(t) such that laplace(f(t),t,s) is F(s). The inverse Laplace transform ilaplace(F(s),s,t) may only match the original signal f(t) for t ≥ 0.

Tips

• If any argument is an array, then ilaplace acts 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 laplace.

• 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 laplace uses the unilateral transform. This definition assumes that the signal f(t) is only defined for all real numbers t ≥ 0. Therefore, the inverse result is not unique for t < 0 and it may not match the original signal for negative t. One way to retrieve the original signal is to multiply the result of ilaplace by a Heaviside step function. For example, both of these code blocks:

  syms t;
  laplace(sin(t))

  and

  syms t;
  laplace(sin(t)*heaviside(t))

  return 1/(s^2 + 1). However, the inverse Laplace transform

  syms s;
  ilaplace(1/(s^2 + 1))

  returns sin(t), not sin(t)*heaviside(t).

Version History

Introduced before R2006a