Laplace Transform & Continuous Time LTI Systems

Laplace Transform is the generalization of continuous – time Fourier Transform . The Laplace Transform of a general signal x(t) is defined as ,

$\displaystyle X(s) = \int_{-\infty}^{\infty} x(t) e^{- s t} dt $ ….(i)

It can be inferred that X(s) is a function of the independent variables corresponding to the complex variable in the exponent of e-st .
The complex variable is s = σ + j ω , where σ and  ω are real and imaginary parts respectively .

The transform relationship between x(t) and X(s) is denoted by ,

$\displaystyle x(t) \leftrightarrow X(s) $

When s = j ω , equation (i) becomes ,

$X(j \omega) = \int_{-\infty}^{\infty} x(t) e^{- j \omega t} dt $

which corresponds to the Fourier transform of x(t) , i.e.

$\displaystyle X(s)|_{s=j\omega} = L [x , t]$

Thus it can be seen that the Laplace Transform also bears a Straight forward relationship to the Fourier Transform when complex variable s is not purely imaginary . To see this relationship , let s = σ + j ω , so that

$\displaystyle X(\sigma + j \omega) = \int_{-\infty}^{\infty} x(t) e^{- j \omega t} e^{-\sigma t} dt $

$\displaystyle = \int_{-\infty}^{\infty} [ x(t) e^{-\sigma t} ] e^{- j \omega t} dt $

In particular , it can be noted that just as the Fourier transform does not converge for all signals , the Laplace Transform may converge for some values of Re(s) and not for others .
In general , the range of values of s for which the integral in equation (i) converges is referred to as the Region of Convergence (ROC)

The Region of Convergence For Laplace Transforms

Property (i) : The ROC of X(s) consists of strips parallel to the j ω – axis in the s-plane :

Since the ROC of X(s) consists of those values of s = σ + j ω for which Fourier transform of x(t)e-σ t converges , i.e. x(t)e-σ t should be absolutely integrable of those values of s ;

$\displaystyle \int_{-\infty}^{\infty} | x(t) | e^{-\sigma t} dt < \infty $

Property (ii) : For rational Laplace transforms , the ROC does not contain any poles .

Since X(s) is infinite at a pole and therefore Laplace transform of x(t) does not converge at a pole , and thus the ROC can’t contain values of s that one pole .

Property (iii) : If x(t) is of finite duration and is absolutely integrable , then the ROC is the entire s-place .

Suppose that x(t) is absolutely integrable , so that

$\displaystyle \int_{T_1}^{T_2}| x(t) | dt < \infty $

For s = σ + j ω to be in the ROC , it requires that x(t)e-σ t be absolutely integrable i.e.

$\displaystyle \int_{T_1}^{T_2}| x(t) | e^{-\sigma t}dt < \infty $ Above equation verifies that s is in the ROC when Re(s) = σ = 0 For σ > 0 , the maximum value of e-σ t over the interval on which x(t) is nonzero is e-σ t , and thus we can write ,

$\displaystyle \int_{T_1}^{T_2}| x(t) | e^{-\sigma t}dt < e^{-\sigma T_1}\int_{T_1}^{T_2}| x(t) | dt $

Since RHS of the equation is bounded , so is the LHS , hence S-plane for Re(s) > 0 must also be in the ROC with similar arguments , if σ < 0 , then

$\displaystyle \int_{T_1}^{T_2}| x(t) | e^{-\sigma t}dt < e^{-\sigma T_2}\int_{T_1}^{T_2}| x(t) | dt $

and again x(t) e-σ t is absolutely integrable and hence ROC includes the entire s-plane .

Property (iv) : If x(t) is right sided , and if the line Re(s) = σ0 is in the ROC , then all values of s for which Re(s) > σ0 will also be in the ROC .

Property (v) : If x(t) is left sided and if the line Re(s) = σ0 is in the ROC , then all values of s for which Re(s) < σ0 will also be in the ROC

Property (vi) : If x(t) is two sided and if the line Re(s) = σ0 is in the ROC will consists of a strip in the s plane that includes the line Re(s) = σ0

Property (vii) : If these Laplace transform X(s) of x(t) is rational , then its ROC is bounded by poles or extended to infinity . In addition , no poles of X(s) are contained in the ROC .

Property (viii) : If the Laplace transform X(s) of x(t) is rational , then if x(t) is right sided , the ROC is the Region in the s plane to the right of the right most pole . If x(t) is left sided , the ROC is the region in the s plane to the left of the left most pole .