Properties of Laplace Transforms(Bi-lateral)

1.Linearity Property:-

x_{1}(t)\leftrightarrow X_{1}(S) \ \ \ ROC : R_{1}

x_{2}(t)\leftrightarrow X_{2}(S) \ \ \ ROC : R_{2}

a\ x_{1}(t)+b\ x_{2}(t)\leftrightarrow \ \ ?.

we know that  Laplace Transform of a signal  x(t)  is  X(S) = \int_{-\infty }^{\infty }\ x(t) \ e^{-St} \ dt .

L\left \{a\ x_{1}(t)+b\ x_{2}(t)\right \} = \int_{-\infty }^{\infty }\ \left \{ a\ x_{1}(t)+b\ x_{2}(t) \right \} \ e^{-St}\ dt.

L\left \{a\ x_{1}(t)+b\ x_{2}(t)\right \} = \int_{-\infty }^{\infty }\ \left \{ a\ x_{1}(t)\ e^{-St}\ dt+b\ x_{2}(t) \ e^{-St}\ dt\right \}

L\left \{a\ x_{1}(t)+b\ x_{2}(t)\right \} = \ \ a \int_{-\infty }^{\infty }\ x_{1}(t)\ e^{-St}\ dt+b \int_{-\infty }^{\infty }\ x_{2}(t) \ e^{-St}\ dt.

L\left \{a\ x_{1}(t)+b\ x_{2}(t)\right \} = \ \ a X_{1}(S) +b X_{2}(S) .          ROC: R_{1} \cap R_{2}.

2.Time-shifting Property:-

x(t)\leftrightarrow X(S) \ \ \ ROC : R

x(t-t_{o})\leftrightarrow \ ?.

we know that  X(S) = \int_{-\infty }^{\infty }\ x(t) \ e^{-St} \ dt .

L\left \{ x(t-t_{o}) \right \} = \int_{-\infty }^{\infty }\ x(t-t_{o}) \ e^{-St}\ dt.    Lett-t_{o}=\lambda \Rightarrow dt= d\lambda

t \ limits \ : \ -\infty \ to \ \infty, \ \ \ \lambda \ \ limits \ : \ \infty \ to \ -\infty

L\left \{ x(t-t_{o}) \right \} = \int_{\lambda =-\infty }^{\infty }\ x(\lambda ) \ e^{-S(\lambda +t_{o})}\ d\lambda .

L\left \{ x(t-t_{o}) \right \} =e^{-St_{o}} \int_{\lambda =-\infty }^{\infty }\ x(\lambda ) \ e^{-S\lambda }\ d\lambda.

x(t-t_{o})\leftrightarrow \ e^{-St_{o}}\ X(S) \ , \ \ ROC:R.

from the above equation x(t-t_{o})  forms a Laplace Transform pair with e^{-St_{o}} \ X(S).

3.Frequency-shifting Property:-

x(t)\leftrightarrow X(S) \ \ \ ROC : R

?\ \leftrightarrow \ X(S-S_{o}).

we know that  X(S) = \int_{-\infty }^{\infty }\ x(t) \ e^{-St} \ dt .

L\left \{ e^{S_{o}t}\ x(t) \right \} = \int_{t=-\infty }^{\infty }\ e^{S_{o}t}\ x(t) \ e^{-St}\ dt.

L\left \{ e^{S_{o}t}\ x(t) \right \} = \int_{t=-\infty }^{\infty }\ \ x(t) \ e^{-(S-S_{o})t}\ dt .

e^{S_{o}t}x(t)\leftrightarrow \ X(S-S_{o}) \ , \ \ ROC:R.

from the above equation e^{S_{o}t}x(t)  forms a Laplace Transform pair with X(S-S_{o}).

4. Differentiation in time-domain:-

x(t)\leftrightarrow X(S) \ \ \ ROC : R

\frac{dx(t)}{dt}\leftrightarrow \ ?.

we know that  inverse Laplace Transform  x(t) =\frac{1}{2\pi \ j} \int_{\sigma - j\infty }^{\sigma +j\infty }\ X(S) \ e^{St} \ dS .

\frac{dx(t)}{dt} =\frac{1}{2\pi \ j} \int_{\sigma - j\infty }^{\sigma +j\infty }\ X(S) \ \frac{d(e^{St})}{dt} \ dS.

\frac{dx(t)}{dt} =\frac{1}{2\pi \ j} \int_{\sigma - j\infty }^{\sigma +j\infty }\ X(S) \ S \ e^{St} \ dS .

\frac{dx(t)}{dt} =\frac{1}{2\pi \ j} \int_{\sigma - j\infty }^{\sigma +j\infty } (\ S\ X(S)) \ e^{St} \ dS.

\frac{dx(t)}{dt}\leftrightarrow \ S\ X(S).

from the above equation \frac{dx(t)}{dt}  forms a Laplace Transform pair with S\ X(S)

Similarly  \frac{d^{n}x(t)}{dt^{n}}\leftrightarrow \ S^{n}\ X(S).

5.Differentiation in S-domain:-

x(t)\leftrightarrow X(S) \ \ \ ROC : R

?\leftrightarrow \frac{dX(S)}{dS}.

we know that  X(S) = \int_{-\infty }^{\infty }\ x(t) \ e^{-St} \ dt .

\frac{dX(S)}{dS} = \int_{-\infty }^{\infty }\ x(t) \ \frac{de^{-St} }{dS}\ dt.

\frac{dX(S)}{dS} = \int_{-\infty }^{\infty }\ x(t) \ e^{-St} \ (-t)\ dt .

\frac{dX(S)}{dS} = \int_{-\infty }^{\infty }\ (-t \ x(t)) \ e^{-St} \ dt.

\frac{dX(S)}{dS}\leftrightarrow \ -t\ x(t) \ \ \ ROC:R.

from the above equation \frac{dX(S)}{dS}  forms a Laplace Transform pair with -t\ x(t).

6. Time-reversal property:-

x(t)\leftrightarrow X(S) \ \ \ ROC : R

x(-t)\leftrightarrow \ \ ?.

we know that  X(S) = \int_{-\infty }^{\infty }\ x(t) \ e^{-St} \ dt .

L\left \{ x(-t) \right \} = \int_{-\infty }^{\infty }\ x(-t) \ e^{-St} \ dt.          Let  -t=\ \lambda ,      -dt=\ d\lambdat \ limits \ : \ -\infty \ to \ \infty, \ \ \ \lambda \ \ limits \ : \ \infty \ to \ -\infty.

L\left \{ x(-t) \right \} = \int_{\lambda =\infty }^{-\infty }\ x(\lambda ) \ e^{S\lambda } \ (-d\lambda ) .

L\left \{ x(-t) \right \} = \int_{\lambda =-\infty }^{\infty }\ x(\lambda ) \ e^{-(-S) \lambda } \ d\lambda.

x(-t)\leftrightarrow \ X(-S).

from the above equation x(-t)  forms a Laplace Transform pair with X(-S).

7. Time-Scaling property:-

x(t)\leftrightarrow X(S) \ \ \ ROC : R

x(at)\leftrightarrow \ \ ?.

we know that  X(S) = \int_{-\infty }^{\infty }\ x(t) \ e^{-St} \ dt .

L\left \{ x(at) \right \} = \int_{-\infty }^{\infty }\ x(at) \ e^{-St} \ dt.          Let  at=\ \lambda ,      dt=\ \frac{d\lambda}{a}t \ limits \ : \ -\infty \ to \ \infty, \ \ \ \lambda \ \ limits \ : \ -\infty \ to \ \infty.

L\left \{ x(at) \right \} = \frac{1}{a}\int_{\lambda =-\infty }^{\infty }\ x(\lambda ) \ e^{(\frac{-S}{a})\lambda } \ d\lambda .

x(at)\leftrightarrow \frac{1}{a} \ X(\frac{S}{a}), \ \ \ if \ a>0.

x(-at)\leftrightarrow \frac{1}{a} \ X(\frac{-S}{a}), \ \ \ if \ a<0 \ and \ (a\neq -1).

 

8. Convolution in Time-domain:-

x_{1}(t)\leftrightarrow X_{1}(S) \ \ \ ROC : R_{1}

x_{2}(t)\leftrightarrow X_{2}(S) \ \ \ ROC : R_{2}

x_{1}(t) * x_{2}(t)\leftrightarrow \ \ ?.

we know that  X(S) = \int_{t=-\infty }^{\infty }\ x(t) \ e^{-St} \ dt .

L\left \{ x_{1}(t) * x_{2}(t) \right \} = \int_{t=-\infty }^{\infty }\ \left \{ x_{1}(t) * x_{2}(t) \right \}\ e^{-St} \ dt.

L\left \{ x_{1}(t) * x_{2}(t) \right \} = \int_{t=-\infty }^{\infty }\ \int_{\tau =-\infty }^{\infty }\ x_{1}(\tau )\left \{ x_{2}(t-\tau ) \right \}\ e^{-St} \ dt \ d\tau.

L\left \{ x_{1}(t) * x_{2}(t) \right \} = \int_{\tau =-\infty }^{\infty }\ x_{1}(\tau )\left \{ \int_{t=-\infty }^{\infty } x_{2}(t-\tau )\ e^{-St} \ dt \right \} \ d\tau.

L\left \{ x_{1}(t) * x_{2}(t) \right \} = \int_{\tau =-\infty }^{\infty }\ x_{1}(\tau )\left \{ e^{-S\tau } X_{2}(S) \right \} \ d\tau.

L\left \{ x_{1}(t) * x_{2}(t) \right \} = \left \{ \int_{\tau =-\infty }^{\infty }\ x_{1}(\tau ) e^{-S\tau } \ d\tau \right \} \ X_{2}(S).

L\left \{ x_{1}(t) * x_{2}(t) \right \} = \ X_{1}(S) \ X_{2}(S)

x_{1}(t) * x_{2}(t)\leftrightarrow \ X_{1}(S) \ X_{2}(S), ROC : R_{1} \cap R_{2}.

 

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Author: vikramarka

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.