Parseval's Theorem for Laplace Transform

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 −

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

Inverse Laplace Transform

The inverse Laplace transform is the method for obtaining the time domain function from its Laplace transform and mathematically, it is defined as −

$$\mathrm{L^{-1}[X(s)]\:=\:x(t)\:=\:\frac{1}{2\pi j}\int_{\sigma\:-\:j\infty}^{\sigma\:+\:j\infty} X(s) e^{st} \: ds}$$

Parseval's Theorem for Laplace Transform

Statement - The Parseval's theorem or Parseval's relation for Laplace transform states that if,

$$\mathrm{x_1(t)\:\overset{LT}\longleftrightarrow\:X_1(s) \quad \text{and} \quad x_2(t)\:\overset{LT}\longleftrightarrow\: X_2(s)}$$

Where, $\mathrm{x_1(t)}$ and $\mathrm{x_2(t)}$ are complex functions. Then,

$$\mathrm{\int_{-\infty}^{\infty}\: x_1(t) x_2^*(t) \: dt\:\overset{LT}\longleftrightarrow\:\frac{1}{2\pi j} \int_{\sigma \:+\: j\infty}^{\sigma \:-\: j\infty}\: X_1(s) X_2^*(-s^*) \: ds}$$

Proof

From the definition of the inverse Laplace transform, we have,

$$\mathrm{x_1(t) \:=\: \frac{1}{2\pi j} \int_{\sigma \:+\: j\infty}^{\sigma \:-\: j\infty}\: X_1(s) e^{st} \: ds}$$

Taking LHS of the Parseval's theorem, we get,

$$\mathrm{\text{LHS} \:=\: \int_{-\infty}^{\infty}\: x_1(t) x_2^*(t) \: dt \:=\: \int_{-\infty}^{\infty}\: \left[ \frac{1}{2\pi j} \int_{\sigma \:+\: j\infty}^{\sigma \:-\: j\infty}\: X_1(s) e^{st} \: ds \right] x_2^*(t) \: dt}$$

By rearranging the order of integration in RHS of the above equation, we get,

$$\mathrm{\int_{-\infty}^{\infty}\: x_1(t) x_2^*(t) \: dt \:=\: \frac{1}{2\pi j} \int_{\sigma \:+\: j\infty}^{\sigma \:-\: j\infty}\: X_1(s) \left[ \int_{-\infty}^{\infty}\: x_2^*(t) e^{st} \: dt \right]\: ds}$$

$$\mathrm{\Rightarrow\: \int_{-\infty}^{\infty}\: x_1(t) x_2^*(t) \: dt \:=\: \frac{1}{2\pi j} \int_{\sigma \:+\: j\infty}^{\sigma \:-\: j\infty}\: X_1(s) \left[ \int_{-\infty}^{\infty}\: x_2(t) e^{-(-s^*) t} \: dt \right]^* \:ds}$$

$$\mathrm{\Rightarrow\: \int_{-\infty}^{\infty}\: x_1(t) x_2^*(t) \: dt \:=\: \frac{1}{2\pi j} \int_{\sigma \:+\: j\infty}^{\sigma \:-\: j\infty}\: X_1(s) \left[ X_2(-s^*) \right]^*\: ds}$$

$$\mathrm{\therefore\: \int_{-\infty}^{\infty}\: x_1(t) x_2^*(t) \: dt \:=\: \frac{1}{2\pi j} \int_{\sigma \:+\: j\infty}^{\sigma \:-\: j\infty}\: X_1(s) X_2(-s^*) \: ds \:=\: \text{RHS}}$$