Laplace Transform of Damped Sine and Cosine Functions

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

$$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-st}}\:\mathit{dt}\:\:\:\:\:\:...(1)}$$

Equation (1) gives the bilateral Laplace transform of the function $\mathit{x}\mathrm{\left(\mathit{t}\right)}$. But for the causal signals, the unilateral Laplace transform is applied, which is defined as,

$$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{\mathrm{0}}^{\infty}\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-st}}\:\mathit{dt}\:\:\:\:\:\:...(2)}$$

Laplace Transform of Damped Sine Function

The Damped Sine Function is given by,

$$\mathrm{\mathit{x}\mathrm{\left(\mathit{t}\right)}\:\mathrm{=}\:\mathit{e^{-at}}\:\mathrm{sin}\:\mathit{\omega t\:\mathit{u}\mathrm{\left( \mathit{t}\right)}}\:\mathrm{=}\:\mathit{e^{-at}}\mathrm{\left( \frac{\mathit{e^{j\omega t}-e^{-j\omega t}}}{2\mathit{j}} \right )}\mathit{u}\mathrm{\left(\mathit{t}\right )}}$$

Now, from the definition of the Laplace transform, we get,

$$\mathrm{\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\mathit{L}\mathrm{\left[\mathit{e^{-at}}\:\mathrm{sin}\:\mathit{\omega t\:\mathit{u}\mathrm{\left( \mathit{t}\right)}}\right]}\:\mathrm{=}\:\mathit{L}\mathrm{\left[\mathit{e^{-at}}\mathrm{\left( \frac{\mathit{e^{j\omega t}-e^{-j\omega t}}}{2\mathit{j}} \right )}\mathit{u}\mathrm{\left(\mathit{t}\right )} \right ]}}$$

$$\mathrm{\Rightarrow \mathit{L}\mathrm{\left[\:\mathit{e^{-at}}\:\mathrm{sin}\:\mathit{\omega t\:u\mathrm{\left(\mathit{t}\right)}}\right]}\:\mathrm{=}\frac{1}{2\mathit{j}}\mathit{L}\mathrm{\left[\mathrm{\left(\mathit{e^{-at}e^{j\omega t} -e^{-at}e^{-j\omega t}}\right)}\mathit{u}\mathrm{\left(\mathit{t}\right )}\right]}}$$

$$\mathrm{\Rightarrow \mathit{L}\mathrm{\left[\:\mathit{e^{-at}}\:\mathrm{sin}\:\mathit{\omega t\:u\mathrm{\left(\mathit{t}\right)}}\right]}\:\mathrm{=}\frac{1}{2\mathit{j}}\mathit{L}\mathrm{\left[\mathrm{\left(\mathit{e}^{-\mathrm{\left(\mathit{a-j\omega}\right)}\mathit{t}}- \mathit{e}^{-\mathrm{\left(\mathit{a\mathrm{+}j\omega}\right)}\mathit{t}} \right )}\mathit{u}\mathrm{\left(\mathit{t}\right )}\right]}}$$

$$\mathrm{\Rightarrow \mathit{L}\mathrm{\left[\:\mathit{e^{-at}}\:\mathrm{sin}\:\mathit{\omega t\:u\mathrm{\left(\mathit{t}\right)}}\right]}\:\mathrm{=}\frac{1}{2\mathit{j}}\mathrm{\left\{\mathit{L}\mathrm{\left [ \mathit{e}^{-\mathrm{\left(\mathit{a-j\omega}\right)}\mathit{t}}\mathit{u}\mathrm{\left ( \mathit{t}\right )}\right]}- \mathit{L}\mathrm{\left [ \mathit{e}^{-\mathrm{\left(\mathit{a+j\omega}\right)}\mathit{t}}\mathit{u}\mathrm{\left ( \mathit{t}\right )}\right]}\right\}}}$$

$$\mathrm{\Rightarrow \mathit{L}\mathrm{\left[\:\mathit{e^{-at}}\:\mathrm{sin}\:\mathit{\omega t\:u\mathrm{\left(\mathit{t}\right)}}\right]}\:\mathrm{=}\frac{1}{2\mathit{j}}\mathrm{\left[\frac{1}{\mathit{s}+\mathrm{\left(\mathit{a-j\omega}\right)}}-\frac{1}{\mathit{s}+\mathrm{\left(\mathit{a\mathrm{+}j\omega}\right)}}\right]}}$$

$$\mathrm{\Rightarrow \mathit{L}\mathrm{\left[\:\mathit{e^{-at}}\:\mathrm{sin}\:\mathit{\omega t\:\mathit{u}\mathrm{\left(\mathit{t}\right)}}\right]}\:\mathrm{=}\frac{1}{2\mathit{j}}\mathrm{\left[\frac{1}{\mathrm{\left ( \mathit{s}\mathrm{+}\mathit{a}\right)}-\mathit{j\omega }}-\frac{1}{\mathrm{\left ( \mathit{s}\mathrm{+}\mathit{a}\right)}\mathrm{+}\mathit{j\omega }} \right ]}\:\mathrm{=}\:\mathrm{\left[\frac{\mathit{\omega}}{\mathrm{\left(\mathit{s\mathrm{+}a}\right)^{\mathrm{2}}\mathrm{+}\mathit{\omega}^{\mathrm{2}}}}\right]}}$$

The region of convergence (ROC) of Laplace transform of the damped sine function is $\mathit{Re}\mathrm{\left(\mathit{s}\right)}$ > -a as shown in Figure-1. Hence, the Laplace transform of the damped sine function along with its ROC is given by,

$$\mathrm{\mathit{e}^{-\mathit{at}}\:\mathrm{sin}\:\mathit{\omega}t\:\mathit{u}\mathrm{\left(\mathit{t}\right)}\overset{\mathit{LT}}{\leftrightarrow}\mathrm{\left[\frac{\mathit{\omega}}{\mathrm{\left(\mathit{s\mathrm{+}a}\right)^{\mathrm{2}}\mathrm{+}\mathit{\omega}^{\mathrm{2}}}}\right]};\:\mathrm{ROC}\to \mathrm{Re}\mathrm{\left(\mathit{s}\right)}>\:-\mathit{a}}$$

Laplace Transform of Damped Cosine Function

The damped cosine function is given by,

$$\mathrm{\mathit{x}\mathrm{\left(\mathit{t}\right)}\:\mathrm{=}\:\mathit{e^{-at}}\:\mathrm{cos}\:\mathit{\omega t\:\mathit{u}\mathrm{\left( \mathit{t}\right)}}\:\mathrm{=}\:\mathit{e^{-at}}\mathrm{\left( \frac{\mathit{e^{j\omega t}\mathrm{+}e^{-j\omega t}}}{2} \right )}\mathit{u}\mathrm{\left(\mathit{t}\right )}}$$

By the definition of the Laplace transform, we have,

$$\mathrm{\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\mathit{L}\mathrm{\left[\mathit{e^{-at}}\:\mathrm{cos}\:\mathit{\omega t\:\mathit{u}\mathrm{\left( \mathit{t}\right)}}\right]}\:\mathrm{=}\:\mathit{L}\mathrm{\left[\mathit{e^{-at}}\mathrm{\left( \frac{\mathit{e^{j\omega t}\mathrm{+}e^{-j\omega t}}}{2} \right )}\mathit{u}\mathrm{\left(\mathit{t}\right)}\right]}}$$

$$\mathrm{\Rightarrow \mathit{L}\mathrm{\left[\:\mathit{e^{-at}}\:\mathrm{cos}\:\mathit{\omega t\:u\mathrm{\left(\mathit{t}\right)}}\right]}\:\mathrm{=}\frac{1}{2}\mathrm{\left\{\mathit{L}\mathrm{\left[\mathit{e^{-at}e^{j\omega t}}\:\mathit{u}\mathrm{\left(\mathit{t}\right )} \right]}\mathrm{+} \mathit{L}\mathrm{\left[\mathit{e^{-at}e^{-j\omega t}}\:\mathit{u}\mathrm{\left(\mathit{t}\right )} \right]}\right\}}}$$

$$\mathrm{\Rightarrow\mathit{L}\mathrm{\left[\:\mathit{e^{-at}}\:\mathrm{cos}\:\mathit{\omega t\:u\mathrm{\left(\mathit{t}\right)}}\right]}\:\mathrm{=}\frac{1}{2}\mathrm{\left\{\mathit{L}\mathrm{\left[\mathit{e^{-\mathrm{\left(\mathit{a-j\omega}\right)\mathit{t}}}\:\mathit{u}\mathrm{\left ( \mathit{t}\right)}}\right ]}\mathrm{+}\mathit{L}\mathrm{\left[\mathit{e^{-\mathrm{\left ( \mathit{a+j\omega} \right )\mathit{t}}}\:\mathit{u}\mathrm{\left ( \mathit{t}\right)}}\right]}\right\}}}$$

$$\mathrm{\Rightarrow \mathit{L}\mathrm{\left[\:\mathit{e^{-at}}\:\mathrm{cos}\:\mathit{\omega t\:u\mathrm{\left(\mathit{t}\right)}}\right]}\:\mathrm{=}\frac{1}{2}\mathrm{\left[\frac{1}{\mathit{s}+\mathrm{\left(\mathit{a-j\omega}\right)}}\mathrm{+}\frac{1}{\mathit{s}+\mathrm{\left(\mathit{a\mathrm{+}j\omega}\right)}}\right]}}$$

$$\mathrm{\Rightarrow \mathit{L}\mathrm{\left[\:\mathit{e^{-at}}\:\mathrm{cos}\:\mathit{\omega t\:\mathit{u}\mathrm{\left(\mathit{t}\right)}}\right]}\:\mathrm{=}\frac{1}{2}\mathrm{\left[\frac{1}{\mathrm{\left ( \mathit{s}\mathrm{+}\mathit{a}\right)}-\mathit{j\omega }}\mathrm{+}\frac{1}{\mathrm{\left ( \mathit{s}\mathrm{+}\mathit{a}\right)}\mathrm{+}\mathit{j\omega }} \right ]}\:\mathrm{=}\:\mathrm{\left[\frac{\mathit{s+a}}{\mathrm{\left(\mathit{s\mathrm{+}a}\right)^{\mathrm{2}}\mathrm{+}\mathit{\omega}^{\mathrm{2}}}}\right]}}$$

The ROC of the Laplace transform of the damped cosine function is also $\mathit{Re}\mathrm{\left(\mathit{s}\right)}$ > -a as shown in Figure-1. Hence, the Laplace transform of the damped cosine function along with its ROC is given by,

$$\mathrm{\mathit{e}^{-\mathit{at}}\:\mathrm{cos}\:\mathit{\omega}t\:u\mathrm{\left(\mathit{t}\right)}\overset{\mathit{LT}}{\leftrightarrow}\mathrm{\left[\frac{\mathit{s+a}}{\mathrm{\left(\mathit{s\mathrm{+}a}\right)^{\mathrm{2}}\mathrm{+}\mathit{\omega}^{\mathrm{2}}}}\right]};\:\mathrm{ROC}\to \mathrm{Re}\mathrm{\left(\mathit{s}\right)}>\:-\mathit{a}}$$