What is the Laplace Transform?

Definition of Laplace Transforms

If f(t) is a function of time that is defined for all values of ‘t’, then Laplace transform of 𝑓(𝑡) denoted by ℒ{𝑓(𝑡)} is defined as ℒ{𝑓(𝑡)}=∫𝑒−𝑠𝑡𝑓(𝑡)𝑑𝑡∞0

given that the integral exists.

Where ‘s’ is a real or complex number and ′ℒ′ is the Laplace transformation operator. Since ℒ{𝑓(𝑡)} is a function of ‘s’ this can be written as F(s).

i.e., ℒ{𝑓(𝑡)}=𝐹(𝑠) which can also be written as 𝑓(𝑡)=ℒ−1{𝐹(𝑠)}, then 𝑓(𝑡) is called as “Inverse Laplace Transform” of F(s).

Table of Laplace Transforms

Significance of Laplace Transforms in Continuous-Time Signals and Systems

Laplace transforms are significant in the analysis of continuous-time signals and systems in various aspects like frequency domain analysis, damping and frequency characterization of continuous-time signals, transfer function characterization of continuous-time LTI systems, stability, and steady-state analysis in control systems, eigenfunction of LTI systems, etc.

Here we will discuss the importance of the Laplace transforms in certain aspects of continuous timeous-time signals and systems.

Transfer Function:

The transfer function of a system is the ratio of the Laplace transform of the output to the Laplace transform of the input of that system. The location of poles and zeros of the transfer function is crucial in determining the dynamic characteristics of the system. The transfer function can unify the convolution integral and differential equation representation of a system.

Damping and frequency of a continuous signal:

The growth and decay of the signal (damping) and its repetitive nature (frequency) in the time domain can be determined by using the location of poles and zeros of the Laplace transform of the concerned signal.

Transient and steady states-time analysis:

The continuous-time systems can be analyzed effectively with the help of Laplace transforms. Certain characteristics such as stability, transient, and steady-state analysis can be effectively studied using Laplace transforms, hence the Laplace transform is the prominent tool in the control theory.

Fourier transform:

Since the Laplace transform requires integration over an infinite domain it is necessary to check whether it will converge or not. It is also known as the region of convergence (ROC) in the s-plane composed of damping (𝜎) and frequency (𝜔). If the ROC includes the 𝑗𝜔 axis of the s-plane, the Laplace transform of the signal coincides with the Fourier transform of the signal. So, we can find the Fourier transform of the large class of signals by using their Laplace transform. Subtly Fourier series representation of continuous-time periodic signals has a connection with the Laplace transform, this helps in reducing the computational complexity of the Fourier series by eliminating the integration.

What are the Two-Sided Laplace Transforms?

Due to the significant application of causal signals (has no existence at t<0) and causal systems (zero impulse response at t<0) the Laplace transform is typically a one-sided transform, but the two-sided Laplace transform also exists. It is the Laplace transform applied to two different signals and systems. By separating the signal into its causal and anti-causal forms we can apply the one-sided Laplace transform, but care should be taken while applying the inverse Laplace transforms to get the correct signal.

The two-sided Laplace transform of a continuous-time function f(t) is ℒ{𝑓(𝑡)}=∫𝑒−𝑠𝑡𝑓(𝑡)𝑑𝑡∞−∞ 𝑠∈𝑅𝑂𝐶

Where 𝑠=𝜎+𝑗𝜔, and ROC is the region of convergence.

Basic Properties of the Laplace Transform

Here we will discuss the basic properties of Laplace transforms such as linearity, differentiation, integration, time-shifting, and convolution integral properties of the Laplace transform. Applying these basic properties of the Laplace transforms will reduce the time of calculation.

Property of Linearity:

If F(s) and G(s) are the Laplace transforms of two signals f(t) and g(t) respectively, ‘x’ and ‘y’ are two constant values then ℒ{𝑥𝑓(𝑡)+𝑦𝑔(𝑡)}=𝑥𝐹(𝑠)+𝑦𝐺(𝑠)

Property of Differentiation:

If F(s) is the Laplace transform of the signal, then one-sided Laplace transforms its 𝑛𝑡ℎ order derivative 𝑓𝑛(𝑡) is ℒ{𝑓𝑛(𝑡)}=𝑠𝑛𝐹(𝑠)−𝑠𝑛−1𝑓(0)−𝑠𝑛−2𝑓′(0)−⋯𝑓𝑛−1(0)

If n=1, ℒ{𝑓1(𝑡)}=𝑠𝐹(𝑠)−𝑓(0)

If n=2, ℒ{𝑓2(𝑡)}=𝑠2𝐹(𝑠)−𝑠𝑓(0)−𝑓′(0)

Property of Integration:

Laplace transform of the integral of a causal signal f(t) is ∫𝑓(𝑥)𝑑𝑥𝑡0 𝑢(𝑡)=𝐹(𝑠)𝑠

Time-shifting property:

If F(s) is the Laplace transform of the signal f(t)u(t), then the Laplace transform of the time-shifted signal 𝑓(𝑡−𝑎)𝑢(𝑡−𝑎) is, ℒ{𝑓(𝑡−𝑎)𝑢(𝑡−𝑎)}=𝑒−𝑎𝑠𝐹(𝑠)

Property of Convolution Integral:

If F(s) is the Laplace transform of the causal signal f(t), and H(s) is Laplace transform of its impulse response, then the Laplace transform of the convolution integral of 𝑓(𝑡) with ℎ(𝑡) is ℒ{(𝑓∗ℎ)(𝑡)}=𝐹(𝑠)𝐻(𝑠).

These are some basic concepts involving the Laplace transforms, there are a lot of things that are to be discussed, and we may have a further emphasis in our lectures.

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