Laplace Transform of Damped Sine and Cosine Functions

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

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

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

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

Laplace Transform of Damped Sine Function

The Damped Sine Function is given by,

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

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

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

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

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

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

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

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

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

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

Laplace Transform of Damped Cosine Function

The damped cosine function is given by,

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

By the definition of the Laplace transform, we have,

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

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

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

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

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

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

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