Four part :    The Laplace Transform and its applications                                                             

 

 

The study of systems involves the solution of differential equations or evaluation of the superposition integral .

 These techniques can result in tedious mathematical operations using classical methods however these solutions

can be achieved faster using the Laplace Transform which changes differential equations in algebraic manipulations.

 

Let’ us define The Laplace Transform :

 

                             

 

With 

 

Conditions for the Existence of a Laplace Transform of f(t) = F(s)
1)	is piecewise continuous on .
2)	is of exponential order as . That is, there exist real constants , , and such that for all .
Note that conditions 1 and 2 are sufficient, but not necessary, for to exist.

 

 

The parameter s is assumed to be positive and large enough to ensure that the integral converges ,

we see that s is complex in such case the real part of s must be positive and large enough to ensure convergence.

 

 

1)  The first Translation theorem

           If   L{f(t)}=F(s)    then  

 

 

2)      Multiplying by t and tⁿ

If L{f(t)}=F(s)    then     

 

 

3)      Dividing by t

 

      If L{f(t)}=F(s)    then     provided

Table of standard transform

 

Standard transform
a

 

 

4) Linearity Property

If a and b are constants while f(t) and g(t) are functions of t then

{a f(t) + b g(t)} = a{f(t)} + b{g(t)}

 

5) Inverse Transform

 

If   L{f(t)}=F(s)    then  

 

Table of inverse transforms

 

Table of inverse transforms
	a

 

 

5)      Laplace transform of derivatives

 

 

 

Where x₀ = value of x at t=0 and x₁ = value of

     

 

 

7) Laplace Transforms of Integrals

7.1.    If   alt="$\{g(t)\}$" v:shapes="_x0000_i1086"> , then alt=MATH v:shapes="_x0000_i1087">

7.2.    For the general integral

,

Where       is the value of the integral when .

 

 

8) Laplace of unit step function of the form 

 

   

 

9) Laplace of     (The second shift theorem )

 

   

 

 

10) If f(t) is a periodic function of period T we have :

 

      

 

 

 

11) Dirac Delta (unit impulse ) Laplace

 

      

 

 

 

 

 

Go to problems for the four  part

 

Go home

Go to first part : Principles of Signal and Systems Modeling Concepts

Go to second part : System Modeling and Analysis in the time domain

Go to third part : Fourier series