This article covers key properties of the Laplace transform in signal and systems analysis, including linearity, time-shifting, time-scaling, convolution, and differentiation. It also discusses the initial- and final-value theorems, which help compute system behaviors and values directly from the Laplace transform without requiring the inverse transform.
The Laplace transform fulfills a number of properties that are quite valuable in various applications. In particular, by using these properties, it is possible to derive many new transform pairs from a basic set of pairs. In this tutorial, we state most fundamental properties of the Laplace transform.
- Linearity
If
$x(t)\leftrightarrow X(s)\text{ and v}(t)\leftrightarrow V(s)$
Then for any real or complex numbers a, b,
$ax(t)+bv(t)\leftrightarrow aX(s)+bV(s)$
This property says that the Laplace transform is a linear operation.
-
Right Shift in Time
If
$x(t)\leftrightarrow X(s)$
Then for any positive real number c,
$x(t-c)\leftrightarrow {{e}^{-cs}}X(s)$
The function x(t-c) is a c-second right shift of x(t). we can see that a c-second right shift in the time domain corresponds to multiplication by e-cs in the Laplace transform domain.
-
Time Scaling
If
$x(t)\leftrightarrow X(s)$
For any positive real number a,
$x(at)\leftrightarrow \frac{1}{a}X(\frac{s}{a})$
The function x(at) is a time-scaled version of the given function x(t). We can see that time scaling corresponds to scaling by the factor 1/a in the Laplace transform domain (plus multiplication of the transform by 1/a).
-
Multiplication by a power of t
If
$x(t)\leftrightarrow X(s)$
Then for any positive integer n,
${{t}^{n}}x(t)\leftrightarrow {{(-1)}^{n}}\frac{{{d}^{n}}X(s)}{d{{s}^{n}}}$
-
Multiplication by an exponential
If
$x(t)\leftrightarrow X(s)$
Then for any real or complex number a,
${{e}^{at}}x(t)\leftrightarrow X(s-a)$
By this property, multiplication by an exponential function in the time domain corresponds to a shift of the s variable in the Laplace transform domain.
-
Multiplication by sinwt or coswt
If
$x(t)\leftrightarrow X(s)$
Then for any real number ω,
$x(t)sin\omega t\leftrightarrow \frac{j}{2}[X(s+j\omega )-X(s-j\omega )]$
$x(t)cos\omega t\leftrightarrow \frac{j}{2}[X(s+j\omega )+X(s-j\omega )]$
-
Convolution
Given two functions x(t) and v(t) with x(t) and v(t) equal to zero for t<0, we define the convolution by
$x(t)*v(t)\leftrightarrow \int\limits_{0}^{\infty }{x(\tau )}v(t-\tau )d\tau \text{ t}\ge \text{0}$
Now, letting X(s) denote the Laplace transform of x(t) and v(s) denote the Laplace transform of v(t). we have the transform pair
$x(t)*v(t)\leftrightarrow X(s)V(s)$
By this property, convolution in the time domain corresponds to a product in the Laplace transform domain.
-
Integration
Given a function x(t) with x(t)=0 for all t<0, we define the integral of x(t) to be the function
$\int\limits_{0}^{\infty }{x(t)dt}\leftrightarrow \frac{1}{s}X(s)$
By this property, the Laplace transform of the integral of x(t) is equal to X(s) divided by s.
-
Differentiation in the time domain
If
$x(t)\leftrightarrow X(s)$
Then
$\overset{.}{\mathop{x}}\,(t)\leftrightarrow sX(s)-x(0)$
-
Initial-value theorem
Given a signal x(t) with transform X(s), we have
\[x(0)\leftrightarrow \underset{s\to \infty }{\mathop{\lim }}\,\text{ }sX(s)\]
\[\overset{.}{\mathop{x}}\,(0)\leftrightarrow \underset{s\to \infty }{\mathop{\lim }}\,\text{ }\!\![\!\!\text{ }{{s}^{2}}X(s)-sx(0)]\]
The initial-value theorem is useful since it allows for computation of the initial values of a function x(t) and its derivatives directly from the Laplace transform X(s) of x(t).Hence, if we know X(s) but not x(t), it is possible to compute these initial values without having to compute the inverse Laplace transform of x(t).
-
Final-value theorem
Given the signal x(t) with transform X(s), suppose that x(t) has a limit t→∞. Then the final-value theorem states that
\[\underset{t\to \infty }{\mathop{\lim }}\,x(t)\leftrightarrow \underset{s\to 0}{\mathop{\lim }}\,\text{ }sX(s)\]
- You May Also Read: Laplace Transform Introduction with Examples
Laplace Transform Properties in Signal and Systems Key Takeaways
The properties of the Laplace transform discussed in this article are crucial for simplifying the analysis and solution of complex systems in engineering and physics. These properties enable efficient transformations between time and frequency domains, making it easier to analyze system behaviors, solve differential equations, and compute initial and final conditions without the need for inverse transforms. This makes the Laplace transform an invaluable tool in signal processing, control systems, and many other applications.