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

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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 x(t) is a time domain function, then its Laplace transform is defined as −

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

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

$$\mathrm{x(t)\:\overset{LT}\longleftrightarrow\:X(s)}$$

Then,

$$\mathrm{x(-t)\:\overset{LT}\longleftrightarrow\:X(-s)}$$

Proof

By the definition of Laplace transform, we can write,

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

Now, by substituting t = (−t), we have,

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

Let (−t) = u in RHS of the above equation, then dt = du,

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

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

$$\mathrm{\therefore\:x(-t)\:\overset{LT}\longleftrightarrow\:X(-s)}$$

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 x(t) if

$$\mathrm{x(t)\:\overset{LT}\longleftrightarrow\:X(s)}$$

Then,

$$\mathrm{x^*(t)\:\overset{LT}\longleftrightarrow\:X^*(s^*)}$$

Proof

By the definition of Laplace transform, we have,

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

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

$$\mathrm{\Rightarrow\:L[x^*(t)]\:=\:X^*(s^*)}$$

Or it may be represented as,

$$\mathrm{x^*(t)\:\overset{LT}\longleftrightarrow\:X^*(s^*)}$$

Conjugate Symmetry Property of Laplace Transform

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

$$\mathrm{x(t)\:\overset{LT}\longleftrightarrow\:X(s)}$$

Then, by the conjugation property, we get,

$$\mathrm{x^*(t)\:\overset{LT}\longleftrightarrow\:X^*(s^*);\:\:\text{for complex }\: x(t)}$$

And if x(t) is real function, then according to the conjugate symmetry property, we have,

$$\mathrm{X(s)\:=\:X^*(s^*)}$$

Proof

By the definition of the Laplace transform, we get,

$$\mathrm{X(s^*) \:=\: \int_{-\infty}^{\infty}\:x(t)e^{s^*t}\:dt}$$

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

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

$$\mathrm{\Rightarrow X^*(s^*) = \int_{-\infty}^{\infty} x(t) e^{-st} \, dt = X(s)}$$

Where, x(t) is real

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

$$\mathrm{X(s)\:=\:X^*(s^*)}$$