LAPLACE TRANSFORM: LAPLACE TRANSFORM Presented by: Nagesh Chauhan (70) Hemanshu Dave (72) Mehul Gupta (77) Sukhdev Kushwaha (86) Sidharth Tewary (107)
THE FRENCH “NEWTON”: Pierre-Simon Laplace Developed mathematics in astronomy, physics, and statistics. Began work in calculus which led to the Laplace Transform. Focused later on celestial mechanics. One of the first scientists to suggest the existence of black holes . THE FRENCH “NEWTON”
HISTORY: Euler began looking at integrals as solutions to differential equations in the mid 1700’s : Lagrange took this a step further while working on probability density functions and looked at forms of the following equation : Finally, in 1785, Laplace began using a transformation to solve equations of finite differences which eventually lead to the current transform: HISTORY
DEFINITION: The Laplace transform of a continuous function f(t ), is denoted by function F(s), defined by: w here s is a complex number: with real numbers σ and ω . Laplace transform is a replacement of Fourier transform when the continuous function f(t), signal has infinite energy. DEFINITION
PROBABILITY FUNCTION: Probability Distribution Function of a random variable X by means of the Laplace transform is given as: PROBABILITY FUNCTION
BILATERAL LAPLACE TRANSFORM: In the Laplace transform equation, when the limits of integration are from -∞ to ∞, then it is called as “Double Sided” or “Bilateral Laplace Transform” : BILATERAL LAPLACE TRANSFORM
INVERSE LAPLACE TRANSFORM: The Inverse Laplace Transform is given by the following complex integral: where γ is a real number so that the contour path of integration is in the ROC of F(s). INVERSE LAPLACE TRANSFORM
ROC: The range of values of σ for which equation: attains some finite value is called Region of Convergence (ROC). ROC
LAPLACE TRANSFORM OF SOME IMPORTANT FUNCTIONS: LAPLACE TRANSFORM OF SOME IMPORTANT FUNCTIONS
UNIT STEP FUNCTION: UNIT STEP FUNCTION
UNIT IMPULSE FUNCTION: UNIT IMPULSE FUNCTION
EXPONENTIAL FUNCTION: EXPONENTIAL FUNCTION
SINE FUNCTION: SINE FUNCTION
COSINE FUNCTION: COSINE FUNCTION
DECAYING SINE FUNCTION: DECAYING SINE FUNCTION
DECAYING COSINE FUNCTION: DECAYING COSINE FUNCTION
RAMP FUNCTION: RAMP FUNCTION
REAL-LIFE APPLICATIONS: REAL-LIFE APPLICATIONS Semiconductor mobility. Call completion in wireless networks. Vehicle vibrations on compressed rails. Behavior of magnetic and electric fields above the atmosphere.
THANK YOU: THANK YOU