Laplace Transform – Time Reversal, Conjugation, and Conjugate Symmetry Properties

Laplace Transform

The Laplace transform is a mathematical tool which is used to convert the differential equation in time domain into the algebraic equations in the frequency domain or s-domain.

Mathematically, if $\mathrm{\mathit{x\left ( t \right )}}$ is a time domain function, then its Laplace transform is defined as −

$$\mathrm{\mathit{L\left [ x\left ( t \right ) \right ]\mathrm{\, =\,}X\left ( s \right )\mathrm{\, =\,}\int_{-\infty }^{\infty }x\left ( t \right )e^{-st}\:dt }}$$

Time Reversal Property of Laplace Transform

Statement – The time reversal property of Laplace transform states that if a signal is reversed about the vertical axis at origin in the time domain then its Laplace transform is also reversed about the vertical axis in the s-domain. Therefore, if

$$\mathrm{\mathit{x\left ( t \right )\overset{LT}{\leftrightarrow}X\left ( s \right )}}$$

Then,

$$\mathrm{\mathit{x\left ( -t \right )\overset{LT}{\leftrightarrow}X\left ( -s \right )}}$$

Proof

By the definition of Laplace transform, we can write,

$$\mathrm{\mathit{L\left [ x\left ( t \right ) \right ]\mathrm{\, =\,}X\left ( s \right )\mathrm{\, =\,}\int_{-\infty }^{\infty }x\left ( t \right )e^{-st}\:dt }}$$

Now, by substituting $\mathrm{\mathit{t\mathrm{\, =\,}\left ( -t \right )}}$, we have,

$$\mathrm{\mathit{L\left [ x\left ( -t \right ) \right ]\mathrm{\, =\,}\int_{-\infty }^{\infty }x\left ( -t \right )e^{-st}\:dt }}$$

Let $\mathrm{\mathit{\left ( -t \right )\mathrm{\, =\,}u}}$ in RHS of the above equation, then $\mathrm{\mathit{dt\mathrm{\, =\,}du}}$,

$$\mathrm{\mathit{\therefore L\left [ x\left ( -t \right ) \right ]\mathrm{\, =\,}\int_{-\infty }^{\infty }x\left ( u \right )e^{su}\:du }}$$

$$\mathrm{\mathit{\Rightarrow L\left [ x\left ( -t \right ) \right ]\mathrm{\, =\,}\int_{-\infty }^{\infty }x\left ( u \right )e^{-\left ( -s \right )u}\:du\mathrm{\, =\,}X\left ( -s \right ) }}$$

$$\mathrm{\mathit{\therefore x\left ( -t \right )\overset{LT}{\leftrightarrow}X\left ( -s \right ) }}$$

Thus, it proves the time reversal property of the Laplace transform.

Conjugation Property of Laplace Transform

Statement – The conjugation property of the Laplace transform states that for a complex function $\mathrm{\mathit{x\left ( t \right )}}$ if

$$\mathrm{\mathit{x\left ( t \right )\overset{LT}{\leftrightarrow}X\left ( s \right ) }}$$

Then,

$$\mathrm{\mathit{x^{\ast }\left ( t \right )\overset{LT}{\leftrightarrow}X^{\ast }\left ( s^{\ast } \right ) }}$$

Proof

By the definition of Laplace transform, we have,

$$\mathrm{\mathit{L\left [ x^{\ast }\left ( t \right ) \right ]\mathrm{\, =\,}\int_{-\infty }^{\infty }x^{\ast }\left ( t \right )e^{-st}\:dt}}$$

$$\mathrm{\mathit{\Rightarrow L\left [ x^{\ast }\left ( t \right ) \right ]\mathrm{\, =\,}\left [ \int_{-\infty }^{\infty }x\left ( t \right )e^{-\left ( s^{\ast } \right )t}\:dt \right ]^{\ast } \mathrm{\, =\,}\left [ X\left ( s^{\ast } \right ) \right ]^{\ast }}}$$

$$\mathrm{\mathit{\Rightarrow L\left [ x^{\ast }\left ( t \right ) \right ]\mathrm{\, =\,}X^{\ast }\left ( s^{\ast } \right )}}$$

Or it may be represented as,

$$\mathrm{\mathit{ x^{\ast }\left ( t \right )\overset{LT}{\leftrightarrow}X^{\ast }\left ( s^{\ast } \right )}}$$

Conjugate Symmetry Property of Laplace Transform

Statement – The conjugate symmetry property of Laplace transform states that if,

$$\mathrm{\mathit{ x\left ( t \right )\overset{LT}{\leftrightarrow}X\left ( s \right )}}$$

Then, by the conjugation property, we get,

$\mathrm{\mathit{ x^{\ast }\left ( t \right )\overset{LT}{\leftrightarrow}X^{\ast }\left ( s^{\ast } \right );}}$ for complex $\mathrm{\mathit{x\left ( t \right )}}$

And if $\mathrm{\mathit{x\left ( t \right )}}$ is real function, then according to the conjugate symmetry property, we have,

$$\mathrm{\mathit{X\left ( s \right )\mathrm{\, =\,}X^{\ast }\left ( s^{\ast } \right )}}$$

Proof

By the definition of the Laplace transform, we get,

$$\mathrm{\mathit{X\left ( s^{\ast } \right )\mathrm{\, =\,}\int_{-\infty }^{\infty }x\left ( t \right )e^{-\left ( s^{\ast } \right )t}\:dt }}$$

By taking conjugation on both sides of the above equation, we have,

$$\mathrm{\mathit{X^{\ast }\left ( s^{\ast } \right )\mathrm{\, =\,}\left [ \int_{-\infty }^{\infty }x\left ( t \right )e^{-\left ( s^{\ast } \right )t}\:dt \right ]^{\ast }\mathrm{\, =\,}\int_{-\infty }^{\infty }x\left ( t \right )e^{-\left ( s^{\ast } \right )^{\ast }t}\:dt }}$$

$\mathrm{\mathit{\Rightarrow X^{\ast }\left ( s^{\ast } \right )\mathrm{\, =\,}\int_{-\infty }^{\infty }x\left ( t \right )e^{-st}\:dt\mathrm{\, =\,}X\left ( s \right ); }}$    Where, $\mathrm{\mathit{x\left ( t \right )}}$ is real

Therefore, according to the conjugate symmetry property of the Laplace transform,

$$\mathrm{\mathit{X\left ( s \right )\mathrm{\, =\,}X^{\ast }\left ( s^{\ast } \right )}}$$