Signals & Systems : Laplace Transform

By Vishnu Pratap Singh|Updated : February 14th, 2022

Laplace Transform

The Laplace Transform is a very important tool to analyse any electrical containing by which we can convert the Integral-Differential Equation in Algebraic by converting the given situation in Time Domain to Frequency Domain

                                                  

 
  • 04-Laplace-Transform (1) is also called bilateral or two-sided Laplace transform.
  • If x(t) is defined for t≥0, [i.e., if x(t) is causal], then 04-Laplace-Transform (2) is also called unilateral or one-sided Laplace transform.

Below we have listed the Following advantage of accepting Laplace transform:

  • Analysis of general R-L-C circuits become easier.
  • Natural and Forced response can be easily analyzed.
  • The circuit can be analyzed with impedances.
  • Analysis of stability can be done easiest way.

Statement of Laplace Transform

  • The direct Laplace transform or the Laplace integral of a function f(t) defined for 0 ≤ t < ∞ is the ordinary calculus integration problem for a given function f(t).
  • Its Laplace transform is the function, denoted F(s) = L{f}(s), defined by

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  • A causal signal x(t) is said to be of exponential order if a real, positive constant σ (where σ is the real part of s) exists such that the function, e- σt|X(t)| approaches zero as t approaches infinity.
  • For a causal signal, if lim e-σt|x(t)|=0,  for σ > σc and if lim e-σt|x(t)|=∞ for σ > σc then σc is called the abscissa of convergence, (where σc is a point on real axis in s-plane).
  • The value of s for which the integral 04-Laplace-Transform (5) converges is called Region of Convergence (ROC).
  • For a causal signal, the ROC includes all points on the s-plane to the right of abscissa of convergence.
  • For an anti-causal signal, the ROC includes all points on the s-plane to the left of the abscissa of convergence.
  • For a two-sided signal, the ROC includes all points on the s-plane in the region in between two abscissae of convergence.

Properties of the ROC

The region of convergence has the following properties

  • ROC consists of strips parallel to the jω-axis in the s-plane.
  • ROC does not contain any poles.
  • If x(t) is a finite duration signal, x(t) ≠ 0, t1 < t < t2 and is absolutely integrable, the ROC is the entire s-plane.
  • If x(t) is a right sided signal, x(t) = 0, t1 < t0, the ROC is of the form R{s} > max {R{pk}}
  • If x(t) is a left sided signal x(t) = 0, t1 > t0, the ROC is of the form R{s} > min {R{pk}}
  • If x(t) is a double-sided signal, the ROC is of the form p1 < R{s} < p2
  • If the ROC includes the jω-axis. Fourier transform exists and the system is stable.

 

Inverse Laplace Transform

  • It is the process of finding x(t) given X(s)

X(t) = L-1{X(s)}

      There are two methods to obtain the inverse Laplace transform.

  • Inversion using Complex Line Integral

04-Laplace-Transform (10)

  • Inversion of Laplace Using Standard Laplace Transform Table.

Note A: Derivatives in t → Multiplication by s.

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 B: Multiplication by t → Derivatives in s.

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Laplace Transform of Some Standard Signals

04-Laplace-Transform (13)

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04-Laplace-Transform (15)

04-Laplace-Transform (16)

 Some Standard Laplace Transform Pairs

04-Laplace-Transform (17)

04-Laplace-Transform (18)

04-Laplace-Transform (19)

04-Laplace-Transform (20)

Properties of Laplace Transform

04-Laplace-Transform (22)

04-Laplace-Transform (23)

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Key Points

  • The convolution theorem of Laplace transform says that Laplace transform of convolution of two time-domain signals is given by the product of the Laplace transform of the individual signals.
  • The zeros and poles are two critical complex frequencies at which a rational function of a takes two extreme value zero and infinity respectively.

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